摘要:

Cryogenic Photolysis Studies Part 2.-Inf'rared Spectrum of Nitrosomethane Monomer J. BARNES,~ STEPHENBYAUSTIN HARRYE. HALLAM,* WARING~ AND J. RONALDARMSTRONG Department of Chemistry, University College of Swansea, Singleton Park, Swansea SA2 8PP Received 8th July, 1974 The photolysis of trans-t-butyl nitrite in an argon matrix at 20 K gave rise to absorptions due to acetone, nitrosomethane and cis-t-butyl nitrite. Isomerisation was dominant at higher concentra- tions ;decomposition was dominant at lower concentrations. Vaporisation of nitrosomethane dimer and trapping of the products at 20 K yielded the spectrum of nitrosomethane monomer and the trans dimer. Ultra-violet irradiation dissociated the dimer, leaving nitrosomethane monomer. On warming to room temperature and recooling to 20 K the cis-nitrosomethane dimer was obtained, which could be dissociated by photolysis to regenerate the monomer. A vibrational assignment of the monomer is presented.The structure and properties of C-nitroso compounds have been reviewed by Gowenlock and Luttke.l Nitrosomethane is produced in the pyrolysis or photo- lysis of t-butyl nitrite and by the reaction of methyl radicals (from e.g., the photolysis of iodomethane) with nitric oxide. Early results were confused since the final product obtained is nitrosomethane dimer or, in some cases, the tautomer formaldoxime. Gowenlock and Trotman carried out a detailed investigation of the pyrolysis and photolysis products of t-butyl nitrite and established that the nitrosomethane dimer exists in two isomeric forms, cis and trans : CH3 CH3 CH3 0 \/ \/IN=N N-N / 4 d\ 0 0 0 CH3 The cis dimer was produced on warming the trapped pyrolysis products to ca.230 K or on ultra-violet irradiation of the trans dimer. The tram dimer was produced on photolysis of t-butyl nitrite or on heating the cis dimer or on dissolving the cis dimer in a solvent of low dielectric constant. Monomeric nitrosomethane could be generated by heating the dimer in the gas phase (the activation energy for dissociation was found to be ca. 90 kJ mol-I). Tautomerisation to formaldoxime occurred most readily in aqueous solution. Gowenlock et aL2* and Luttke reported infi-a-red and ultra-violet spectra of the two dimers. Luttke attempted to record the infrared spectrum of the monomer by dissociat- ing the dimer in a heated gas cell, but experienced difficulty due to rapid tautomerisa- j. present address :Department of Chemistry &Applied Chemistry, University of Salford, Salford M54WT.$present address :Atomic Energy Research Establishment, Harwell Oxfordshire OX11 ORA. 11-1 1 CRYOGENIC PHOTOLYSIS STUDIES tion. He reported the N-0 stretching mode at 1564 cm-I and the C-N stretching mode at ca. 842 cm-l. The electronic spectrum of the monomer has been reported by Dixon and Kroto following flash photolysis of t-butyl nitrite ; absorptions at 660 and 677 nm were observed. Coffey, Britt and Boggs obtained the microwaye spectrum of the monomer by vaporisation of the trans dimer at room temperature.The matrix isolation technique is a powerful method of stabilising transient species for spectroscopic stu~ly.~ Thus we embarked on an investigation of the photolysis products of t-butyl nitrite isolated in argon matrices in an attempt to obtain the complete vibrational spectrum of nitrosomethane monomer. EXPERIMENTAL The cryogenic apparatus and procedure have been described previously. O t-Butyl nitrite, obtained from Fluka Chemicals, and acetone were dried over anhydrous calcium chloride and fractionally distilled. Photolysis experiments were carried out using a Philips medium pressure mercury lamp. In the first series of experiments (photolysis of t-butyl nitrite in argon matrices) spectra were recorded on a Perkin-Elmer 225 spectrometer ; wavenumbers quoted are accurate to _+ 1 cm-l.In the second series of experiments (deposi- tion in an argon matrix of the product of the vaporisation of nitrosomethane dimer) spectra were recorded on a Perkin-Elmer 457spectrometer ;wavenumbers quoted should be regarded as accurate to +3 cm-l. RESULTS Idrared spectra of t-butyl nitrite in argon matrices at 20 K were recorded at matrix to absorber (M/A) ratios of 2000, 500 and 100. No significant changes were observed between the spectra. Spectra of t-butyl nitrite in an argon matrix at M/A 500 were recorded after different photolysis times with the medium pressure mercury lamp. A number of new absorptions appeared (fig. 1) which increased in intensity with increasing photolysis time. Identical behaviour was observed in several experi- ments under different conditions, including photolysis during deposition.Photo-Iysis of more dilute matrix samples (M/A 2000) gave rise to a different distribution of intensities of the product absorptions (table 1). Analogous results were obtained when the experiments were repeated in nitrogen matrices. Also recorded in table 1 are the more intense bands observed in the spectra of acetone trapped in argon matrices at M/A 500. There is an excellent correspondence between a number of the absorptions produced on photolysis of the t-butyl nitrite and the spectrum of acetone. The only medium or strong band in the acetone spectrum not observed in the photolysis product spectrum (2972 cm-') would be obscured by a t-butyl nitrite absorption.The small shifts observed for a number of bands, particularly the C=O stretch, may be accounted for by the presence of another photolysis product in the same or an adjacent trapping site. Coe and Doumani and Tarte l2 proposed that the photolysis of t-butyl nitrite proceeds via intramolecular rearrangement to give acetone and nitrosomethane as primary products : (CH3)3CONO+hv + (CH3)JCO+CHSNO. However McMillan, Calvert and Thomas l3 found that there is an induction period in the appearance of nitrosomethane, whereas acetone is produced immediately. Dixon and Kroto 'also observed an induction period in nitrosomethane production in their flash photolysis experiments.Thus the mechanism is apparently : (CH3)3CONO+]ZV + (CH3)3CO* +NO (CH3)3CO* + (CH3)2CO+CH3 CH3+NO + CHSNO. A. J. BARNES, H. E. HALLAM, S. WARING AND J. R. ARMSTRONG I I I-lI 1 I 1800 I600 lL00 1230 1000 wavenumber/cm-l II I000 8 00 600 400 FIG.1 .-Infrared spectra of t-bury1 nitrite in argon matrices at M/A = 500 before photolysis and after 150 min photolysis with a medium pressure mercury lamp. Absorptions which appear on photo- lysis are arrowed. (a) 1800 to 1000cm-' region. (6) 1000 to 400 cm-' region. In a matrix at 20 K it is unlikely that the fragments could escape from the cage and thus it would not be expected that appreciable quantities of methyl radicals and nitric oxide would be stabilised. No evidence was found for infra-red absorption due to either methyl or nitric oxide; consequentIy it would seem reasonable to assign the CRYOGENIC PHOTOLYSIS STUDIES absorptions, not due to acetone, observed after the photolysis of t-butyl nitrite to nitrosomethane.However, there are two sets of bands with different concentration dependences. The set which is more intense in the M/A 2000 spectrum parallels the intensity variation of the acetone absorptions and may thus be assigned to nitroso- methane. The complexity of the spectra undoubtedly causes several nitrosomethane bands to be obscured, thus an attempt was made to obtain nitrosomethane monomer trapped in an argon matrix without other interfering species. Nitrosomeihane dimer was prepared by vapour phase photolysis of t-butyl nitrite in a silica bulb with the bottom part blackened to prevent photolysis of the liquid (cf.Gowenlock and Trot- man '). The following series of experiments was performed, infrared spectra being recorded after each stage. TABLE1.-cOMPARISON OF THE INFRARED SPECTRUM OF ACETONE WITH ABSORPTIONS (Cm-') PRODUCED BY THE PHOTOLYSIS OF t-BUTYL NITRITE IN AN ARGON MATRIX (CH 3)3CONO photolysis products 3019 (W -3018 (m) CH3 asym. stretch 301 5 (w) -I --2972 (m) CH3 asym. stretch -2962 (vw)1718 (m) (s) 1722 (vs) C=O stretch 1599 (w)1597 (s) 1555 1551 1460 1435 1443 (m) CH3 asym. defs.1429 (s) 1406 (m) CH3 asym. def. 1361 (s) CH3 sym. def. 1354 (s) CH3sym. def. 1223 (s) C-C stretch1216 1091 (m) CH3rock 865 -860 736 (ms) (m) 729 (m) (m679 (w)--570 (w)531 (w) (mw) 529 (ms) C==O i.p.bend * T. Shimanouchi, Tables of Molecular Vibrational Frequencies (NSRDS-NBS 39, 1972). (I) The vapour from the heated nitrosomethane dimer was deposited on the cold window of the cryostat simultaneously with a stream of argon matrix gas (no attempt was made to estimate the matrix ratio achieved). (2) The deposited products were irradiated with a medium pressure mercury lamp for 45 min using a filter to cut out wavelengths below ca. 250 nm. A, J. BARNES, H. E. HALLAM, S. WARING AND J. R. ARMSTRONG (3) The window was allowed to warm up to 77 K (thereby evaporating off the argon matrix gas) and then recooled to 20 K.(4) The window was allowed to warm up to room temperature and then recooled to 20 K. (5) The material on the window was irradiated with the medium pressure mercury lamp for 40 min using a filter to cut out wavelengths below CIZ. 250 nm. (6) The irradiation was repeated without the filter. The absorptioiis observed (fig. 2) after stages 1, 2, 4 and 6 are recorded in table 2. The nitrosomethane dimer vaporisation products originally deposited apparently contained at least two species, since a number of the absorptions disappeared on photolysis. The species remainirrg is designated A, that removed by the photolysis is called B in table 2. Heating the window to 77 K and then recooling to 20 K had little effect on the spectrum, despite the loss of the argon matrix material : the 1297 cm-l absorption of species B reappeared only weakly and the 1410 cm-l absorption of species A split into a doublet. However, on warming to room temperature and recooling to 20 K the absorptions due to species A were replaced by an entirely new spectrum (designated species C in table 2).The effect of photolysis with the filtered mercury lamp was to reduce the intensities of the bands due to species C, while the spectrum of A reappeared. Unfiltered photolysis led to complete conversion of species C to species A. __ I, I I 1 I 3200 28001600 1400 1200 1000 800 600 400 wavenumber 1cm-I FIG.2.-Infrared spectra of (a) products of vaporisation of nitrosomethane dimer deposited in an argon matrix at 20 K; (6) deposited products after 45 min filtered mercury lamp photolysis ;(c) Droduct after warming to room temperature and recooling to 20 K :(d)product as (c) after 60 min photolysis, using an untiltered lamp.Similar results were obtained when the experiment was repeated with a different flow rate of argon gas (the rates used were in the range 10-20mmol h-l) and the experiments should perhaps be regarded as solid state, rather than matrix, photolysis in view of the slight changes resulting from the loss of the matrix in stage 3. CRYOGENIC PHOTOLYSIS STUDIES TABLE2.-ABSORPTIONS OBSERVED (Cm-') AFTER DEPOSITION OF THE PRODUCTS OF THE VAPORISATION OF NITROSOMETHANE DIMER IN AN ARGON MATRIX species species species stage stage stage stageA B C (1) (2) (4) (6)* * * 3059 (w) * 3055 (mw) *3019 (w) * * * 2991 (mw) * * 2955 (w) *2951 (mw) *2917 (w) * * * 2901 (mw) * * * 1549 (s) * 1511 (w) *1484 (w) *1443 (w) *1422 (m) 1411 (m) * * * * 1410 (s) *1401 (m) * 1388 (m)1383 (vs) * (w)*1371 (mw) * * *1348 (s) 1343 (m) * * 1311 (m) * 1304 (w) *1297 (s) * * 1162 (w) * 1142 (m) * * 1128 (w) 1104 (m)1069 (m) 1057 (m) 1028 (s) * * 967 (w) * 949 (m) * * * 916 (mw) * * * 870 (m) *749 (m) *622 (m) *619 (s) * * 574 (m) 536 (s) * *480 (m) *470 (m) * = observed.DISCUSSION From the detailed study by Gowenlock and Trotman of the isomers of nitroso-methane dimer, the most probable identification of species A, B and C is monomer, trans dimer and cis dimer respectively. Species A is produced by the photolysis of either B or C, as would be expected if A is the monomer and B and C are the dimers.A. J. BARNES, H. E. HALLAM, S. WARING AND J. R. ARMSTRONG Warming species A to room temperature leads to species C, corresponding to Gowen-lock and Trotman's observation that the cis dimer is produced from the monomer under similar conditions. The possibility of species B being formaldoxime may be rejected on two grounds : (i) the absorption maximum of formaldoxime is below 210 nm, but species B is photolysed by wavelengths above 250 nm (the absorption maxi- mum of the trans dimer is at CQ. 275 nm); (ii) the absorptions observed fur species B do not include a strong band above 1600 cm-I expected for formaldoxime.'" The (relatively) slow dissociation of species C on filtered-mercury-lamp photolysis may be explained since the absorption maximum of the cis dimer is to lower wavelength than that of the trans ciimer,2 and thus nearer to the cut-off of the filter used.The possibility of secondary photolysis may be rejected, since the different species may be readily interconverted by using the appropriate conditions. These identifications of species €3 and C may be confirmed by comparison of the absorptions observed in these experiments (table 2) with the spectra reported by Gowenlock et d The trans dimer in a KBr disc displays strong absorptions at 2p 3049, 1397, 1286, 1134 and 936 cm-'. Species €3 has bands at 1297, 1142 and 949 cm-1 corresponding to the last three, while the first two would be obscured by absorp- tions due to species A.The strong band observed for species B at 536 cm-I was out- side the range of the original study. Similarly the cis dimer in a KBr disc was reported 2, to exhibit strong absorptions at 3049,2967,1667,1471,1387, 1341,1107, 1061, 1017 and 740 cm-l. It can be seen from table 2 that species C exhibits strong absorptions corresponding to all these bands with the exception of that at 1667 cm-'. The low tenperature spectra obtained here do, as expected, show more detail than the previous (KBr disc) spectra. The weak absorption observed at 1303ern-' coincides with a characteristic band of cis nitrosoalkane dimers, attributed to the N--N stretch, which had not previously been observed for the cis dimer of nitroso- methane.Species A may thus be assigned with confidence as nitrosomethane monomer. Comparison of the absorptions produced by the photolysis of t-butyl nitrite in argon matrices with those of species A shows that the bands which are more intense at high M/A ii; ihe photolysis of t-butyl nitrite (1555, 1411, 1351, 865 and 570 cm-l) corre- spond to the most intense absorptions of nitrosomethane monomer. The absorptions which are more intense at low M/A (1598, 938, 736 crn-I and other, weaker, bands) do not correspond with the spectrum of any of species A, B or C. The possibility that this set of absorptions is due to formaldoxime may be rejected, since 1598 cm-' is too low a frequency for the C=N stretch of formaldoxime, nor do the other unassigned absorptions correspond with the spectrum of formaldoxime. The absorp- tions cannot be attributed to secondary photolysis products, since they are observed after.a short photolysis time and increase steadily in intensity with increasing photoly- sis time.Unless photolysis of t-butyl nitrite aggregates in an argon matrix can lead to hither- to unsuspected decomposition products, the only remaining possibility is that iso- merisation of the t-butyl nitrite is occurring. The nitro compound, 2-methyl-2- nitroprspane, may be ruled out as the major product since no strong absorptions corresponding to the NO2stretchin2 modes of the compound (1543 and 1346 cm-I) l6 were observed.Brown and Pirnentel l7 observed cis-trans isomerisation of methyl nitrite on photolysis in an argon matrix. Tarte has investigated the geometrical isomerism of the alkyl nitrites, and found that t-butyl nitrite is predominantly in the trans form, He gives the vapour phase N-0 stretching frequency of the trans and cis isomers as 1655 and 1610 cm-I respectively. The argon matrix frequency for the trans isomer (the only conformer present in the low temperature matrix) is 1638 cm-l, CRYOGENIC PHOTOLYSIS STUDIES thus 1598 cm-l is very close to the value expected for the cis isomer. A further verification is provided by the characteristic ON0 bending frequency, normally found l8 at ca.600 cm-l for the trans isomer and ca. 680 cm-l for the cis isomer. This mode may be assigned as 580cm-l in the spectrum of unphotolysed trans-t- butyl nitrite, and the photolysis product shows a corresponding absorption at 679 cm-l. Comparison of the principal bands of the trans-t-butyl nitrite with the bands produced by photolysis, which are not assignable to either acetone or nitrosomethane, TABLE3.-cOMPARISON OF THE PRINCIPAL BANDS (Cm-l) OF tranS-t-BUTYL MTRITE IN AN ARGON MATRIX WITH THE ABSORPTIONS ASSIGNED TO Cis-f-BUTYL NITRITE ~~UIIS(CH 3) JONO cis(CH3))30NO assignment 3002 (m) 3015 (w)2995 (s) 2962 (vw) CH3 asym. stretches 2989 (s) 2945 (m) 2914 (m) CH3 sym. stretches 2883 (m) 1638 (vs) 1598 (s) N-0 stretch 1476 (s) 1464 (m) 1460 (w) CH3 asym.defs 1395 (s) 1371 (s) 1363 (w) CM3 sym. defs1369 (s) 1268 (s) *1252 (m) 1246 (m) CH3rocks 1197 (vs) * 1180 (m) 1037 (s) 957 (m) 938 (s) 808 (vs) 916 (m) skeletal stretches 736 (ms)769 (vs) 729 (m)761 (vs) 580 (w) 679 (w) ON0 bend * these bands are more intense in the photolysed spectrum. TABLE4.-ASSIGNMENT OF THE FUNDAMENTAL VIBRATIONS OF NITROSOMETHANE MONOMER IN AN ARGON MATRIX mode CH3NO/Ar CH,NO(g) a CHjCHO b A' AN CH, asym. stretch CH3 sym. stretch N-0 stretch CH3 asym. deformation CH, sym. deformation CH3rock C-N stretch CNO bend CH3 asym. stretch CH3 asp. deformation CH3 rock torsion 2991 (mw) 2901 (mw) 1549 (s) 1410 (s) 1348 (s) 967 (w) 870 (m) 574 (mw) 2955 (w) 1410 (s) 916 (mw) 1564 842 3005 (m) 1441 (s) 919 (m) 2917 (-) -1352 (s) -I 2967 (m) 1420 (s) 867 (m) 150 (w) a ref.(6). T. Shimanouchi, Tables of Molecular Vibrational Frequencies (NSRDS-NBS 39, 1972). A. J. BARNES, PI. E. HALLAM, S. WARING AND J. R. ARMSTRONG shows a clear parallel in each spectral region (table 3). Thus, photolysis of trans-t- butyl nitrite aggregates leads to a mixture of cis and trans isomers, presumably via a process involving adjacent t-butyl nitrite molecules. The bands observed for nitrosomethane monomer may be assigned to the funda- mental modes by comparison with the isoelectronic molecule acetaldehyde (table 4). Only one band was observed in the region of the CH3 asymmetric deformations, thus these are both assigned at 1410cm-l.The C-N stretching frequency is nearly 30 cm-I higher than the gas phase value of 842 cm-I reported by LuttkeY6 but this band was observed in the gas phase only as a shoulder on the intense 888 cm-1 band of formaldoxime. The 3059 cm-l band of nitrosomethane monomer may be assigned as 2 x 1549 = 3098 cm-I, and the 1128 cm-l band as 2 x 574 = 1148 crn-'. Only the weak band at 1162 cm-I remains unassigned. Using the structural data given by Coffey et aL8 and reasonable values for the force constants associated with the methyl group, a local syminetry force field treat- ment gave the following values : KN+ 1030 N m-1 Kc-+ 380Nm-l KCNo 160 N m-l. The N-O stretching force constant may be compared with values of 1010 N m-1 reported.by Shurvell et aZ.19 for CF3N0 and 1104 N m-1 reported by Jacox and Milligan2* for HNO (the latter value corresponds to the authors' assignment By which Is supported by recent data from electronic spectra 21). CONCLUSiONS The high yield of cis-t-butyl nitrite compared with that of acetone and nitroso- methane in the photolysis of trans-t-butyl nitrite in the more concentrated matrix samples supports the mechanism of dissociation into an excited (CH3)3C0 radical and nitric oxide, rather than the intramolecular rearrangement mechanism. Either the (CH3)3C0breaks down to give acetone and methyl, which combines with the nitric oxide trapped in the same cage to give nitrosomethane, or the (CH3)3C0 attacks a neighbouring t-butyl nitrite molecule, picking off NO to give cis-or trans- t-butyl nitrite according to the relative orientation of the two molecules.The latter process, favoured by the cage effect at high concentrations of t-butyl nitrite, results in the conversion of the initially predominantly trans-t-butyl nitrite to a mixture of cis and trans isomers. This is a useful method of generating a non-equilibrium mixture of conformational isomers for spectroscopic examination. Nitrosomethane monomer may be conveniently obtained in the low temperature solid phase by trapping a mixture of monomer and dimer from vaporisation of the dimer, followed by photolytic dissociation of the dimer component. We thank I.C.I. Ltd for the award of a Research Fellowship (to A.J. B.) and S.R.C. for financial support. We are grateful to Dr. S. Suzuki for performing the force constant calculations. B. G. Gowenlock and W. Luttke, Quart. Rev., 1958,12, 321. B. G. Gowenlock and J. Trotman, J. Chenz. Soc., 1955,4190. L. Batt, B. G. Gowenlock and J. Trotman, J. Chem. Soc., 1960, 2222. L. Batt, J. K. Brown, B. G. Gowenlock and K. E. Thomas, J. Chem. Soc., 1962, 37. W. Luttke, 2.Elektrocliem., 1957, 61, 976. W. Luttke, 2.Elekfrocliem., 1957, 61, 302. CRYOGENIC PHOTOLYSIS STUDIES R. N. Dixon and H. W. Kroto, Proc. Roy. Soc. A, 1964,283,423. D. Coffey, C. 0.Britt and J. E. Boggs, J. Chem. Phys., 1968, 49, 591. Vibrational Spectroscopy of Trapped Species, ed. H. E. Hallam (Wiley, London, 1973). lo A. J. Barnes and J.D. R. Howells, J.C.S. Faraday ZI, 1973, 69, 532. l1 C. S. Coe and T. F. Doumani, J. Amer. Chem. Soc., 1948, 70, 1516. l2 P. Tarte, Bull. SOC.roy. Sci. Lizge, 1953, 22, 226. l3 G. R. McMillan, J. G. Calvert and S. S. Thomas, J. Plys. Chem., 1964, 68, 116. l4 S. Califano and W. Luttke, 2.phys. Chem., 1956, 6, 83. l5 B. G. Gowenlock, H. Spedding, J. Trotman and D. H. Whiffen, J. Chem. SOC.,1957, 3927. l6 J. F. Brown, J. Amer. Chem. Soc., 1955, 77, 6341. H. W. Brown and G. C. Pimentel, J. Chem. Phys., 1958, 29, 883. P. Tarte, J. Chem. Phys., 1952, 20, 1570. l9 H. F. Shurvell, S. C. Dass and R. D. Gordon, Canad. J. Chem., 1974,52, 3149. 2o M. E. Jacox and D. E. Milligan, J. Mol. Spectr., 1973, 48, 536. 21 P. N. Clough, B. A. Thrush, D. A. Ramsay and J. G. Stamper, Chent.Phys. Letters, 1973,23, 155. (PAPER 4/1366)

ISSN:0300-9238    

DOI:10.1039/F29767200001

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Vapour Phase Raman Spectra of the Molecules MH, (M = C, Si, Ge or Sn) and MF, (M = C, Si or Ge) Ranian Band Intensities, Bond Polarisability Derivatives and Bond Anisotropies BY ROBERT AND ROBINJ. H. CLARK*S. ARMSTROKG Christopher Ingold Laboratories, University College London, 20 Gordon Street, London WC1H OAJ Received 5th March, 1975 The vapour phase Raman spectra of the molecules MH4 (M = C, Si, Ge or Sn) and MF, (M = C,Si or Ge) have been recorded with 488.0 and/or 514.5 nm excitation at pressures of 0.5-1.0 atrn and at temperatures of ca. 295 K. The intensities of all four Raman-active fundamentals of each niole- cule have been determined relative to that of the vl(nl)band of methane as external standard, and this has permitted the calculation of Raman scattering cross sections for each band.Molecular (%;.) and bond (Zhx)polarisability derivatives have been calculated. The cCLHvalues are similar, for a gjven M, atom, t,o the cCLF values and all follow trends established previously for other ligands (X), Orcx rn clsix < ZCeX < iiinx. Bond anisotropies (~Mx)have been deduced from the Raman intensities of the v2(e)fundamental of each molecule. This method only yields the modulus of the ym value, but, when taken in conjunction with Kerr effect results, it is concluded that ym is positive in each case. Quadratic force fields for each molecule are established on the assumption of the Wolkenstein intensity theory, and mean square amplitudes of vibration are also calculated. Many studies have been made on the Raman spectrum of gaseous methane,' studies which have been concerned primarily with analyses of the vibration-rotation structure of the Raman-active bands.Studies along similar lines have only very recently appeared on the other Group IV spherical top hydrides and deuterides, SiH, and SiD4,29 GeH, and GeD4,5 and SnH,.6 Several studies of the frequency-corrected molar intensities (i.e. Raman scattering cross-sections, RSC or do/d92 values) of the v,(a,) fundamental of methane have been carried out, and, as argued el~ewhere,~ the most probable value seeins to be 3.035 x cm2 mol-l sr-l for 514.5 nm excitation. This value is the average of that of Schrotter and Bernstein,8 as corrected by Bern~tein,~ as corrected by Holzer.' ' and that of Holzer and Moser,lo Intensity studies on the Raman-active fundamentals of the other spherical top hydrides have not previously been carried out. In this work, the RSC values for all four Raman-active fundamentals of silane, germane and staiinane as well as for the three non-totally symmetric modes of methane were determined by comparison with that of the vl(al) fundamental of methane. Such studies permit the calculation of both molecular (Ej) and bond (&x) polarisability derivatives, bond anisotropy derivatives (yhx), bond anisotropies (yMx), force constants based on the Wolkenstein force field, as well as mean square amplitudes of vibration. Owing to the fact that the intensity of a Raman band is determined by the square of a polarisability derivative, it follows that the sign of the latter is not determined by Raman studies.However, by linking the present Raman results with Kerr effect measurements l2 it is possible to determine the sign of yILIx. Siniilar studies have also been carried out on the spherical top tetrafluorides CF4, 11 RAMAN SPECTRA OF MH, AND MF4 SiF, and GeF,, for which the only previous Ranian intensity results relate to measure-ments on all four bands of CF4 '' and on the vl(al) bands of CF, 13* ',and SiF4.13 The present intensity results are, however, the first to be obtained for these molecules with laser excitation. EXPERIMENTAL PREPARATION OF COMPOUNDS The methane and carbon tetrafluoride were obtained from Matheson Co.Inc. Samples of silane, germane and stannane were prepared by the methods of Norman et aZ.15and of silicon tetrafluoride and germanium tetrafluoride as described by Hoffman and Gutowsky.l INSTRUMENTAL The Raman spectra were recorded by use of a Spex 1401 spectrometer with Coherent Radiation model 52 Arf and Kr+ lasers. The scattered radiation was collected at 90" and focused by afj'0.95 lens onto the entrance slit of the 0.75 m Czerny-Turner monochsomater after having been passed through a polarisation scrambler. The gratings (Bausch and Lonib 1200 lines mid) were blazed at 500 nm. The method of detection was photon counting in conjunction with a cooled RCA C31034 (grade I) phototube and linear display. The power available at 488.0 nm and 514.5 nm was approximately 1.3 and 1.6 W respectively.Band areas were determined either with a Kent Chromalog Two integrater or by the cut-and-weigh procedure. The area measurements are accurate to +lo % unless otherwise stated. All quoted relative molar band intensities have been corrected for the spectral response of the instrument. Quoted wavenumbers of band maxima are believed to be accurate to & 0.5 cm-1 (neon calibration lines). The cell arrangement (extra-cavity) was the same as that used previ~usly.~ The cell (volume ca. 60 cm3) was filled with methane to a pressure of ca. 1 atm. The Raman signal strength was then optimised and the vl(al) band centred at 2917 cm-l scanned 6-12 times. The cell was then evacuated and the second compound allowed in, also to a pressure of ca. 1 atm.After temperature equilibrium had been re-established, the same procedure was repeated for each band. The entire procedure was then repeated using the bracketing technique, an average of five different samples of each molecule being used. Possible decomposition of the sample in the laser beam was checked by measuring the area of its vl(al) band frequently throughout a run. Only for germanium tetrafluoride and stannane was decomposition detected (stannane exploded after a period of 2-5 h in the laser beam). However, by contrast with the observations of Oskam, who used in-cavity techniques, no gradual loss of intensity of the stannane peaks was noted during the course of a run. The laser beam traversed the cell once only.In the case of carbon tetrafluoride, the intensity of 2v2(A1) (which has p = 0.0) has been added to that of the vl(al) fundamental ;this was deemed to be the most satisfactory proce- dure for dealing with the Fermi resonance problem, as has been suggested e1~ewhere.l~ The same procedure has been adopted for silicon tetrafluoride where 2v4(A1) (which also has p = 0.0) is in Fermi resonance with vl(al). This adds 4.2 % in the first case and 6.8 % in the second case to the measured intensities of the vl(al) band. RESULTS The intensities of all four fundamentals of the tetrahydrides MH, (M = C, Si, Ge or Sn) and tetrafluorides MF4 (M = C, Si or Ge) have been measured relative to that of the v,(al) band of methane as external standard.As indicated in the intro- duction, the scattering activity of the standard has been taken to be 1226 x cm4 g-' mol-' (203.5 Nx cm4 g-') which is equivalent to a Raman scattering cross section of 3.035 x cm2 mol-' sr-l for 514.5 nm excitation, to an i?i value of 2.13 A2N3 g-* and thus to an i?& value of 1.07 A2. The results are given in table 1. R. S. ARMSTRONG AND R. J. H. CLARK TABLERELATIVE MOLAR INTENSITIES AND RAMANSCATTERING CROSS SECTIONS OF RAMAN-ACTIVE BANDS OF THE MOLECULES STUDIED molecuIe FIcrn-1 a ZzMi lZiM2 da/U CH4 2916.7 1.oo 3.035 x 1533.3 0.120 0.364 514.5 nm 3019.5 1.900.626 0.0051305.9 -0.015 SiH4 2185.7 2.59 10.0~x 10-30 972.1 0.82 3.19 488.0 nm 2189.1 1.04 4.04 913.3 -0,Ol -0.04 2155.7 2.65 8.04~10-30 972.1 0.85 2.58 514.5 nm 2189.1 1 .065 -0.03 3.23 913.3 -0.01 GeH, 21 10.6 3.28 12.7~x 10-30 930.6 1.19 4.63 488.0 nm 2111.5 5.33-0.02 1.37 821.O N 0.08 10-3021 10.6 3.30 10.0~ 930.6 1.23 3.73 514.5 nm 2111.5 4.16-0.02 1.37 821.O -0.06 SnH4 1907.8 5.01 15.2x 10-j' 753.3 2.57 7.80 514.5 nm 1905.1 2.22 0.02 -0.06 6.74 681.O CF4 908.4 0.310 0.904X 434.5 0.093 0.28 514.5 nm 1283.0 0.071 0.22 63 1.2 0.131 0.40 10-30SiF4 800.8 0.344 1.04~ 264.2 0.105 0.320 514.5 nm 1029.6 0.0473 0.144 388.7 0.108 0.328 GeF4 735.0 0.611 1.855x 202.9 0.420 1.275 514.5 nm 800.1 0.165 0.501 273.1 0.273 0.829 a The quoted wavenumbers come from ref.(1) (CH,), (2) (SiH4), (4) (GeH4), (6) (SnH,), and (19) (CF4, SiF4 and GeF4).b The scattering cross sections are quoted for 295 K. At this tempera- ture the 2917 cm-l band, vl(al), of the standard, methane, has da/dQ = 3.889 x for 488.0 nm excitation and 3.035 x cm2 mol-l sr-I for 514.5 nm excitation. RAMAN SPECTRA OF MH4 AND MF4 The relative molar intensities have been converted to polarisability derivatives by use of the relationship -=-[ ]12Ml fg2 45Ei2+7yi2 I1Mz 91 45Ei2-+7y12 where 1-exp(-hcv,/kT) 1-exp(-hcv2/kT) (2) 5; and ~f are, respectively, the mean molecular polarisability derivative and the aniso- tropy derivative with respect to the normal coordinate Qj, degeneracy gj,vo is the exciting frequency, and vj is the frequency shift of the normal modej. The Raman scattering cross-section for the jth fundamental (dal/dQ) is given by the expression7 (dOjldC2) = 0.969 44 x 10-37fj(SA)j (3) where (SA)j, the scattering activity, is defined to be (SA), = gj(45E5’++yi2).(4) Bond polarisability derivatives were calculated via the relationship where mxis the mass of the X atom in a molecule of general forniula MX4. The relative scattering activities of the different fundamentals, together with the mean molecular polarisability derivatives, molecular anisotropy derivatives, MX bond lengths, and MX bond anisotropies yMx(= all -EL) are given in table 2. This last quantity has been deduced from yi by way of the relationship 17*l8 3 Y; = e. (;) 2.-FREQUENCY-CORRECTED RELATIVE MOLAR INTENSITIES OF RAMAN-ACTIVETABLE FUNDA-MENTALS OF GROUP IV TETRAHYDRIDES AND TETRAFLUORIDES, AND MOLECULAR POLARISABILITY DERIVATIVES OBTAINED THEREFROM 1W2MI PlM2) polarisability derivatives Vl(U1) vz(4 v3(f2) v4(t2) c(y9 Y; Y; y; rMxc IYMX CH, 1.00 0.046 0.664 -0.002 1.07 0.82 2.54 0.12 1.091 0.32 SiH4 1.660 0.179 0.669 -0.002 1.38 1.62 2.55 0.14 1.480 0.85 GeH4 2.08 0.246 0.825 -0.003 1.50 1.89 2.83 0.18 1.527 1.02 SnH4 2.585 0.395 1.14 -0.003 1.72 2.40 3.33 0.16 1.701 1.455 CF4 0.062 0.0070 0.021 0.0161 1.14 0.32 0.455 0.395 1.323 0.645 SiF4 0.057 0.0038 0.011 0.0069 1.11 0.235 0.32 0.26 1.55 0.56 GeF4 0.091 0.010l) 0.0274 0.010~ 1.40 0.38 0.52 0.32 1.67 0.98 a Mean values for exciting lines Arf 488.0 nm and 514.5 nm.b The units of &hxare A2,off2, y 1 and yi areA28-3 (x N*),of rm are& and of (all -a&x are A3.C TablesofInteratomic Distances and Configuration in MolecuZesand Ions (Chem. SOC. Spec. Publ., No. 1 1 and 18, The Chemical Society, London, 1958 and 1965). Survey spectra of silane, germane and stannane are given in fig. 1-3. Scale expanded spectra of silane and germane are as good as, and of stannane are superior to, those obtained by Oskam and co-workers 2-5 who employed a more powerful laser than that used for this work, and also the in-cavity technique. The spectra of R. S. ARMSTRONG AND R. 3. H. CLARK the tetrafluorides (fig. 4-6) are similar to those obtained by Clark and Rippon,lg with one important exception. The unusual contour of the v3(t2)band of germanium tetrafluoride was reinvestigated in an attempt to understand the reason for the apparent very strong Q-branch.It became clear that germanium tetrafluoride 11xo.05 111 i/crn-' FIG.1.-Vapour phase Raman spectrum of silane at 295 K. The instrumental settings were as foi-lows :scanning speed (s.s.) 20 cm-l min-l, slit widths (s.w.) 200/300/200 pm, slit height (s.h.) 10 mm, time constant (t.c.) 0.25 s, gain 5 K (inset for vl, 100 K ; 1 K = 1000count s-ll n GeH, (464torr) iicrn-' FIG.2.-Vapour phase Raman spectrum of germane at 295 K. The instrumental settings were the same as for fig. 1, except for S.S. 40 cm-I min-'. attacks silicone grease during the course of the measurements to produce some silicon tetrafluoride and that the ~3(t2)band of the former is coincident with the v,(a,) band of the latter.This is evident because in an 1-Lscan of the v3(t,) band of ger- RAMAN SPECTRA OF MH, AND MF4 inanium tetrafluoride (fig. 6) the Q branch is barely evident whereas in an Itotalscan it is very pronounced. The contour of a non-totally symmetric band should, of course, be the same for an IL as an Itotalscan. Accordingly, suitable corrections (amounting to 10 %) were made to the Itotalspectrum of germanium tetrafluoride in the vl(t2) region to allow for the contribution made by the underlying vl(nl) band of the silicon tetrafluoride impurity. Additional weak (presumably impurity) bands at 676, 648 (medium), 616, 538, 478 and 390 cm-I also occur. SnH, (461 torr) I I I I, I ,I ,!I> 4 I I I , I I I 2100 2000 woo 1800 1700 900 eoo 700 600 ilcrn-' FIG.3.-Vapour phase Raman spectrum of stannane at 295 K.The instrumental settings were S.S. 40cm-' min-l, S.W. 150/200/150 pm, s.h. 10mm, t.c. 0.4s, gain 5 K (inset for vl, 50 K). CF, (760tom) G1crn-l FIG.4.-Vapour phase Raman spectrum of carbon tetrafluoride at 295 K. The instrumental settings were S.S. 40cm-l mine', S.W. 200/300/200 pm, s.h. 10 mn, t.c. 1 s, gain 500 count s-l (inset for vl, 5K). The situation restricts the accuracy of the intensity data on germanium tetra- fluoride to & 15 %. The present intensity results for carbon tetrafluoride differ substantially from those obtained by Holzer who used mercury arc excitation. Whilst the intensities of the vl(a,)and v,(e) bands are greater than Holzer's (by ca.25 %) those of the v,(t2)and R. S. ARMSTRONG AND R. J. H. CLARK v4(t2)bands are smaller (by ca. 65 % and 10 % respectively). Although no explana-tion for this difference can be offered, it is widely recognised that the presently used techniques are superior to those used previously. I I I I I 1 1 I If I 1 1 1 1 1 It00 1000 900 800 11 400 300 200 i/crn-l FIG.5.--B;apour phase Raman spectrum of silicon tetrafluoride at 295 K. The instrumental settings were the same as for fig. 4,except for the gain being 1 K (inset for vl, 10 K). Ge F4 '(760torr ) -1 ' ' I ' ' 800 700 300 200 100 v/cm-l FIG.6.-Vapour phase Raman spectrum of germanium tetrafluoride at 295 K.The instrumental settings were the same as for fig. 5. The diagram also includes an I1 scan, which demonstrates that the Q branch of the Y3(r2) fundamental is coincident with that of the v,(al)fundamental of the silicon tetrafluoride impurity. RAMAN SPECTRA OF MH4 AND MF4 DISCUSSION POLARISABILITY FUNCTIONS The bond polarisability derivatives (ELx)deduced as described above, and listed in table 2, lie in the order z& < ZSiH < ZLeH < ainH and && < 6$iF < a&& i.e. taken in conjunction with earlier results,' abx values are shown to increase in the order E;,(1.07) < E&(1.14) < E&(2.38) < EbBr(3.O3)* and ZiiH(1.38) 2 ZiiF(l.11) < EiiC1(2.68)< E4iBr(2.91)* < EkiI(4.21)" and E&,,(1.50) % EbeF(l.40)< E&.ec1(3.57) < ELeBr(3.61)*< &,,(5.02)* and Z&( 1.72) < EknC1(3.71) (all values are in A2;values known only for cyclohexane solutions are indicated by an asterisk).The interpretation of these trends is not simple, but the features of dependence of Ejitx values on both p, the fractional covalent character of the MX bond, and on some power function of the bond length seem clear.* The present values for both ELF and EkiF (1.14 and 1.11 A2 respectively, each & 5 %) are greater than the values previously foundby Long and Thomas13 (0.94-L 5 % and 0.90+_20% A2 respectively, after conversion to the same value for the intensity of the standard as used herein). Even after making due allowance for the estimated experimental error in each of the Ehx values, a small discrepancy remains, and it is concluded that the present measurements, being based on photon counting rather than photographic techniques, are the more accurate.Moreover, the previous measurements on silicon tetrafluoride are evidently subject to considerable uncertainty owing to attack by the compound on the mirrors. Two other values for &F for carbon tetrafluoride, again based on mercury arc measurements, appear in the literature.ll* l4 One of these l4 is 1.03A2& 5 %, after conversion to the present intensity scale, and thus it is in agreement (within the experimental uncertainty) with the present results. The other" is 0.96A2 (again after conversion to the present intensity scale), but the errors in this value were not specified.The parallel and perpendicular bond polarisability derivatives, ail and respec-tively, have been deduced from the relationships 20-23 atx = +(a;]+2a;) (7) and iyf = (2/J3)[L,,(ail -ai)+L,i(ail -aL)/?-], = 3 or 4 (8) * The semi-theoretical delta function expression for ZLx reducesto Zhx = (2/3>(Xtp/Zeffa,)(~n>r3, where x and Zeff are the geometric means of the electronegativities and effective nuclear charges, respectively, of M and X, Zeffin each case being taken to be the atomic number minus the number of inner-shell electrons, a. is the Bohr radius, p is the Pauling fractional covalent character, n is the MX bond order and r is the equilibrium MX internuclear distance.R. S. ARMSTRONG AND R. J. H. CLARK where L4iare entries in the t2block of the L matrix, which relates symmetry to normal coordinates (see later). The results are given in table 3, from which it is seen that xi] > a? and the ratio ctl/ajl is in the range 0.15-0.33, i.e. similar to that found for the MC1, MBr and MI bonds of molecular tetrahalides in the vapour phase.? Also in common with earlier results, cc\l is seen to be much larger and in general more sensitive to the nature of the chemical bond than is the case for a;; ail increases in the order MH > MF < MC1 < MBr < MI. TABLE3.-PARALLEL AND PERPENDICULAR BOND POLARISABILITY DERIVATIVES (A2)OBTAINED FROM THE WOLKENSTEINFORCE FIELD molecule (334SiH4 .1; 2.47 2.75 a; 0.37 0.70 a;/.;I 0.15 0.25 GeH4 3.01 0.74 0.25 SnH4 3.47 0.84 0.24 CF4 2.31 0.51 0.22 SiF4 2.02 0.66 0.33 GeF4 2.77 0.71 0.26 CALCULATION OF w FIELDS A description of the procedures used for the calculation of symmetry force con- stants from fhdarnental frequencies together with Raman band intensities has been given elsewhere.'* 20-23 Owing to sign ambiguities in the columns of the L matrices, corresponding to 180" changes of phase of the normal coordinates, the six- teen possible L matrices are reduced to four distinct ones.2o Of these, two are un- acceptable in that the off-diagonal elements Lijare comparable with (MH,) or larger than jMF4) the diagonal elements Lii ;moreover, the calculated values for the mean square amplitudes of vibration for these two solutions are inconsistent with the electron diffraction values.Of the remaining two matrices, for only one is the same set of and a1 values obtained from eqn (8) for both i = 3 as well as i = 4, and hence this solution is regarded as the only acceptable one.* The results are to be compared with those obtained by use of fundamental frequencies and Coriolis con- stants as constraints to the force field.16 The agreement between the two sets of force constants is not good, and this suggests (as indicated elsewhere) 24 that73 2og the Wolkenstein assumptions are not completely valid. Root mean square amplitudes of vibration have also been calculated and compared with some values obtained by electron diffraction measurements.The results seem satisfactory, when the insensi- tivity of these parameters to the force field is recalled. Both the force constants as well as the root mean square amplitudes of vibration are available on request. BOND ANISOTROPIES Bond anisotropies, as obtained from Raman intensity measurements on the v,(e) fundamental of each spherical top MX4 molecule, are determined in magnitude but not in sign (cf. table 2). Values for anisotropic bond polarisabilities, all and al, may be obtained by combining the two solutions for y with values of the sum, o! 11 +2x1 (bL+2bTin the context of Kerr effect studies). The latter are determined from 9 (a,]f2aJ = --EP (9)47cN * On this basis, the only acceptable soIution to the Wolkenstein force field calculations described in ref.(7) and (21) is that labelled W2 for all tetrahalides except TiCI4 and TiBr4, for which it is that labelled W1. RAMAN SPECTRA OF MH4 AND MF4 where is the bond electronic polarisation. may be obtained from R,, the bond refraction extrapolated to infinite wavelength, or approximately from 0.95 RD,where RDis the bond refraction for the NaD line. TABLE4.-ANISOTROPIC BOND POLARISABILITIES a OBTAINED FROM MEAN POLARISABILITIES AND BOND ANISOTROPIES FOR GROUP IV TETRAHYDRLDES AND TETRAFLUORIDES bond (all +2a_~) IYMXI all alia 11 a II aA da II C-H 1.955 0.32 0.865 0.545 0.63 0.44 0.76 1.73 Si-H 3.555 0.85 1.75 0.90 0.51 0.62 1.47 2.37 Ge-H 4.05 1.02 2.03 1.01 0.50 0.67 1.69 2.52 Sn-H 5.07f 1.455 2.66 1.205 0.45 0.72 2.175 3.02 C-F 2.12 0.645 1.14 0.49 0.43 0.28 0.92 3.29 Si-F 2.495 0.56 1.205 0.645 0.53 0.455 1.02 2.24 Ge-F 2.88 0.9s 1.61 0.6, 0.39 0.30 1.29 4.3 a In units of A3.b Obtained by application of eqn (9). CR. J. W. Le Fkvre, B. J. Orr and G. L. D. Ritchie, J. Chem. Soc., 1966, 273. d Derived from EP(Si-H) or EP(si-F), K. L. Wamaswarny and H. E. Watson, Proc. Roy. Soc. A, 1936, 156, 144. e From 0.95 RD (Ge-H), J. Satge, Ann. Clzim., 1961,6, 519. f From 0.95 RD (Sn-H), P. M. Christopher and J. M. Fitzgerald, Jr., Austral. J. Clzem., 1965,18, 1709. g Derived from EP(C-F), K. L. Ramaswamy,Proc. Indian Acad. Sci. A, 1935,2,630. h From 0.95 Rn (Ge-F), Comprehensive Inorganic Chemistry, ed. J. C. Bailar, H.J. Emeleus, R. S. Nyholm and A. F. Trotman-Dickenson (Pergamon, London, 1973), vol. 2, p. 23. The two sets of anisotropic bond polarisabilities appear in table 4 for each bond studied. Separate application of the two sets of data to appropriate molecules yields molecular anisotropy values which may be compared with those obtained experi- mentally from the Kerr effect. With this approach no firm distinction could be drawn regarding the sign of yCqH owing to the small absolute value thereof (0.32A3). However for Si-H, Ge-H, C-F and Si-F bonds, for which the necessary Kerr effect data already exist, there is indicated a clear preference for taking yMxto be positive. The detailed arguments are presented el~ewhere.~~ By analogy (and in the absence of supporting Kerr effect data), it is suggested that YMX values for Ge-F and Sn-H bonds also have positive signs.Thus it is concluded that all > a1 and that both all and a1 increase in the order CCCX< asix < aGeX < aSnX. One of us (R. S. A.) thanks the Royal Society for the award of a Commonwealth Bursary. The S.R.C. and the Administrators of the University of London Central Research Fund are thanked for financial support. A. Weber, in The Raman Efect, ed. A. Anderson (Dekker, New York, 1973), vol. 2, p. 543, and references therein. H. W. Kattenberg and A. Oskam, J. Mol. Spectr., 1974, 49, 52. D. V. Willetts, W. J. Jones and A. G. Robiette, J. Mol. Spectr., 1975, 55, 200. H. W. Kattenberg, W. Gabes and A. Oskam, J.Mol. Spectr., 1972,44,425. H. W. Kattenberg, R. Elst and A. Oskam, J. Mol. Spectr., 1973, 47, 55. H. W. Kattenberg and A. Oskam, J. Mol. Spectr., 1974, 51, 377.’R. J. H. Clark and P. D. Mitchell, J.C.S. Faraday 11, 1975, 71, 515. H. W. Schrotter and H. J. Bernstein, J. MuZ. Spectr., 1964, 12, 1. H. J. Bernstein, J. Mol. Spectr., 1967, 22, 122. W. Holzer and H. Moser, J. Mol. Spectr., 1964, 13, 430. 11 W. Holzer, J, Mol. Spectr., 1968, 25, 123. l2 C. G. Le Fkvre and R. J. W. Le Fkvre, in Physical Methods of Chemistry, ed. A. Weissberger and B. Rossiter (Wiley, London, 1972), vol. I, part IIIC, chap. VI,p. 399 ;R. S. Armstrong,R.J. W. Le F&vre and K. R. Skamp, J.C.S. Dalton, to be published. R. S. ARMSTRONG AND R. J. H. CLARK l3 D.A. Long and E. L. Thomas, Trans. Faraday Soc,, 1963,1026. l4 G. W. Chantry and L. A. Woodward, quoted by K. A. Taylor and L. A. Woodward, Proc. Roy. Soc. A, 1961, 264, 558. A. D. Norman, J. R. Webster and W. L. Jolly, Inorg. Synth., 1968, 11, 170. l6 C. J. Hoffman and H. S. Gutowsky, Inorg. Synth., 1953, 4, 145. l7 W. F. Murphy, W. Holzer and H. J. Bernstein, Appl. Spectr., 1969, 23,211. G. W. Chantry, in The Raman Eflect, ed. A. Anderson (Dekker, New York, 1971), vol. 1. l9 R. J. H. Clark and D. M. Rippon, J. Mol. Spectr., 1972, 44,479. 2o R. J. H. Clark and P. D. Mitchell, J. Mol. Spectr., 1974, 51, 458. 21 D. A. Long, Proc. Roy. SGC.A, 1953,219,203. 22 D. A. Long, D. C. Milner and A. G. Thomas, Proc. Roy. SOC.A, 1956,237,197. 23 G. W. Chantry and L. A. Woodward, Trurts. Faraday SOC.,1960, 5&,1110. 24 K. A. Taylor and L. A. Woodward, Proc. Roy. SOC.A, 1961,264, 55s. 25 R. S. Arrnstrong and R. J. H. Clark, J.C.S. Dalton, to be published. (PAPER 5 /442)

ISSN:0300-9238    

DOI:10.1039/F29767200011

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Modulated Perturbation Theory for Molecular Interactions Part 1.-An Exact Second-order Calculation for the Ground State of Hi BY VALERIO * MICHELE and GIUSEPPEMAGNASCO, BATTEZZATI? FIGARI Istituto di Chimica Industriale dell’universiti, 161 32 Genova, Italy Received 21st March, 1975 A perturbation theory which includes exchange appropriate for calculations of molecular inter- actions over the whole range of internuclear separations from the united atom to the separate atoms is proposed in terms of a modified MS-MA approach. Modulation of Ho by suitable parameters is introduced to a first order to keep the electronic component of the Coulombic perturbation V small at all separations, so giving better convergence of the perturbation expansion. An exact calculation of the interaction energy for the ‘Xistate of Hd shows that the results obtained in second order are capable of accuracy over the whole range of internuclear distances.The separation of the interaction energy into Coulombic and penetration components allows a detailed analysis of the physical nature of the interaction to be made. To obtain the complete potential energy curve for atomic and molecular inter- actions by perturbation theory (p.t.) it is necessary to go beyond the first order to account for the long-range part of the potential and at the same time to take into account naclear and electron exchange which is dominant at small separations. Attempts to include such symmetry restrictions into a higher order pet. have stimulated a great deal of research.l It should be emphasized that such a perturbation expansion is neither unique when truncated in low order lm3nor can it converge rapidly at short distances because of the magnitude of the perturbation. Nonetheless both the physical insight that can be obtained into the nature of the interaction and the fact that improvement over the first approximation is dictated by the functional form of the Perturbation itself render a perturbation approach useful. The aim of this series is to present a perturbation theory, which includes exchange, suitable for giving accurate results in the calculation of molecular interactions over the whole range of internuclear separations R from the united atom (R = 0) to the separate atoms (R= co).This is achieved in ternis of a modified Murrell-Shaw- Muslier-Amos (MS-MA) p.t. 49 based on a inodulation of the unperturbed Hamil- tonian Ho to keep the electronic component of the Coulombic perturbation V small for any internuclear separation. In this way a large part of the interaction is accounted for in first order so giving a better converging expansion, especially at small separations. Although giving a first order wavefunction (w.f.) lacking the correct overall sym- metr~,~.6$ the MS-MA theory was chosen because (i) it gives, in second order, the correct asymptotic behaviour of the Coulombic energy at large separations and almost correct (98 %) asymptotic behaviour of the exchange energy, and (ii) it allows a second-order calculation including exchange to be carried out simply in terms of the first-order polarization function a10ne.~ t Laboratorio di Cibernetica e Biofisica del C.N.R., 16032 Camogli, Italy.$ This can give some trouble in calculating expectation values,l but has no effect on the energy obtained in second order by perturbation theory. 22 V. MAGNASCO, M. BATTEZZATI AND G. FIGARI The theory, which enables a separation of the energy into Coulombic and pene- tration (exchange) components to be has been applied here to an exact second-order calculation of the interaction energy for the ground state (2Z:) of the hydrogen molecular ion H;, for which accurate values for comparison are known over a wide range of internuclear separations. 9 Variational approximations to the second-order energy, the study of excited states, and the extension of the theory to cover heteronuclear diatomic molecules and many-electron atoms and molecules which can be tackled in terms of a double per- turbation expansion l3 will form the object of further papers in this series.METHOD According to the Chipman-Bowman-Hirschfelder generalized p.t. for exchange interactions,l the first few MS-MA perturbation equations are : (ffo-~oMo = 0 (14 (~0-E0)~1+~(~-~1)40 (W= 0 WO-Eo)42 +(y-Ed41 -E2A40 = 0, (1c) where Ho is the unperturbed hamiltonian (an unsymmetric hermitean operator), V = H-Ho the unsymmetric hermitean perturbation, A the antisymmetrizer (a projection operator satisfying A2 = A = At) which commutes with the total hamil- tonian (AH = HA) but does not commute with either Ho or V.The related energy corrections are Eo = mi(A$olHol#'o>, Mz = (60lA40)-~7<+old)o> = 1 (24 El = Mo2(A40pq40) (2b) E2 = Mi(4OIV-Elpl). (2c) E2 can be put in the alternative form 79 l4 where is the first-order polarization functi0n.l. *, Conventional ~.t.~* l5 for H; takes the unperturbed systems to be a ground state hydrogen atom located on nucleus A (see fig. 1) (Sa) the interaction with the other proton at B being taken as perturbation 11v = --f-l'b R' This separation of the hamiltonian is appropriate at large distances but fails in the region near the united atom (u.a.) where it cannot provide a good description of the system. In fact the perturbation series for the electronic energy 16* l7 based on (sa), (5b)and (3a) has a rather poor convergence in the u.a.limit Eo = -5, Ei = -1, E2 = -3. (6) S Atomic units (a.u.) are used throughout this paper: 1 hartree (a.u. of energy) = 27.21 eV = 2635.5 kJ mol-I ; 1 bohr (a.u. of length) = 0.529 17 x lo-* cm. MOLECULAR INTERACTIONS This behaviour can be understood in ternis of the multipolar expamion of the Hamiltonian, namely I FIG.1.-Reference coordinatesfor Hi (z-axis from A to B along R). which shows that an enhanced charge-shift is occurring on nucleus A when the proton at B approaches the hydrogen atom. As a consequence, a perturbation theory with good convergence properties at all internuclear separations R should allow in low order a close "following '' of such charge modifications which may be very large at short range.This can be done most easily by including in Ho from the very beginning the majority of the spherical part of the perturbation through the introduction of an " effective " nuclear charge co which accounts for the charge-shift occurring between the interacting partners. Accordingly, our modulated p.t. will be based on an unperturbed Ho describing a 1s electron in the field of a nuclear charge +co located on A (with eigenfunction & and eigenvalue Eo) COH, = -+v2--, qbo = (c:/n)+exp(-Cora), E~ = -12COY (84ra the perturbation due to the proton at B being Co-1v = ---1 +-. 1 pa rb The first term on the right in (8b)can be interpreted as the intra-atomic component of the perturbation V, and acts so as to screen the spherical part of the remaining interatomic potential.We expect co to change from 1 in the limit of large R (vanish-ingly small intra-atomic perturbation) to 2 in the limit of the united atom (vanishingly small electronic perturbation). Therefore we have as R -+ 0 (He+) Eo + -2, E,"+ 0, while as R + 00 (H+H+) zav = ----32,2-r: +Q(R-4)~2 2~3 V. MAGNASCO, M. BATTEZZATI AND G. FIGARI Since the first order energy (p = coR) Eo+G = 4301 = MXA4oIHel40) P is an upper bound to the true energy eigenvalue, we can use the variation theorem to determine co for each internuclear separation R, obtaining in this way the best starting point for our perturbation expansion.This yields for co 1 +e-P(3 ++Qp -p2++p3)+e-2P(3+6p++p2)+eV3q1+$p +$p2 +3p3)co = 1+e-p(2 +2p -+p2++p3)+e-2P(1 +2p ++p2-+p3-+p") (124 with the asymptotic expansions 331 5 +o(p6)small R (12b)co = 2-+p2+p3-2518p4 +18op and co = 1 +e-q( 1 +$p -+p2)+o(e-2p) large R. (1 24 The modulated first-order equation for 4: becomes (co fixed) to be solved with the condition (40fq5p>= 0. We notice that 47 may be split into a purely spherical part 6:' satisfying whose solution is and a part @" satisfying the equation which has been solved in confocal elliptic coordinates p = (ra4-rb)/R, v = (r,-rb)/R by Dalgarno and Lynn l8 (see also Coulson and Robinson 19* 20). It is lossible to show 21 that 4:' contributes to the polarization (Coulombic) energy ' not to the .I: overall second-order energy.For the gerade state of Hz A = $(1 +P),and from (3a)and (8b)we get P being the operator representing nuclear inversion through the midpoint and SOO= {&$'40). The integrals occurring in (16) can be obtained from ref. (15) simply by scaling R in p and taking account of #;'. Asymptotic expansion (see Appendix) of (16) near the united atom limit (He+) shows that the leading term is now of order R4,t as it should be for a second-order t The conventional MS-MA theory (co = 1) yields the ma. asymptotic expansion EZ = -++ RZ+o(~3). MOLECULAR INTERACTIONS theory satisfying the molecular cusp conditions at the two nuclei,16 and that the term of order R5involves a logarithmic term,17 whereas at large separations we get the correct multipolar expansion of the second-order polarization energy 22 (leading term -2.25R-4).The modulation of Ho introduced in first order to keep the elec- tronic component of the perturbation small at all separations yields in second order therefore a p.t. which is qualitatively correct over the whole range of internuclear separations. CALCULATIONS AND RESULTS Calculations for the 'C: electronic ground state of Hi have been performed on the CII 10070 digital computer of the University using double precision and especially developed numerical routines for the accurate evaluation of the exponential integral 24 The results for a wide range of internuclear separations are collected in tables 1-3.The interaction energy for Hi(2Zl)can be written AE = E-E; = AEo+E1+E,+. .. (174 where E: = -3 is the ground state energy of the isolated hydrogen atom, and AEO = -$(c$-1) (184 1.-Hi(2zi). BESTVALUES OF THE MODULATION PARAMETER AND EXACT MS-MATABLE PERTURBATION ENERGIES up TO SECOND ORDER (a.u.) R co EQ Ei EZ 1.o 1.5 2.0 2.5 3.5 4.0 5 .o 6.0 7.0 8.O 9.0 10.0 12.5 15.0 20.0 3.a 1.537 93 1.361 42 1.238 69 1.153 67 1.094 88 1.054 80 1.028 31 1.002 02 0.995 08 0.995 27 0.996 94 0.998 31 0.999 14 0.999 87 0.999 98 0.999 99 -1.182 62 -0.926 74 4.767 18 4.665 48 4.599 38 4.556 30 4.528 71 4.502 03 4.495 09 4.495 28 4.496 95 -0.498 31 4.499 14 -0.499 87 -0.499 98 4.499 99 0.741 62(+0) 0.359 62(+0) 0.180 68(+0) 0.867 22(- 1) 0.349 38(-1) 0.635 15(-2) -0.861 96(-2) -4.171 74(-1) -0.139 80(-1) 4.875 70(-2) 4.478 27(-2) -4.240 78(- 2) 4.115 08(-2) 4.158 84(-3) 4.195 54(-4) 4.245 17(-6) -0.151 75(-1) -0.188 15(-1) 4.183 79(- 1) 4.162 21(-1) 4.135 33(- 1) 4.108 68(-1) 4.847 51(-2) 4.482 22(- 2) 4.259 64(- 2) 4.138 08(-2) 4.753 63(- 3) 4.432 68(-3) 4.263 90(-3) -0.973 84(-4)-0.454 34(- 4) -0.141 85(-4) the energy due to the radial polarization of the atom by the interacting proton.When c0 # 1 a large part of the interaction can thus be accounted for in the first order of perturbation theory, the multipolar part of the exchange-polarization of the atom remaining as a small effect in second order. The best perturbation expansion occur- ring at any R is given in table 1, and the interaction energies summed to various orders are collected in table 2.The exact second-order MS-MA calculation including exchange and the modulation of the perturbation yields interaction energies which are well within 2 % of the exact values given by Peek.12 When co = 1 we recover exactly the results of Chalasinski,' but at large distances (10 bohr) the modulated results are consistently better. The improvement at short separations is on the other hand rather dramatic (table 1 and fig. 3). V. MAGNASCO, M. BATTEZZATI AND G. FIGARI At very short distances the electronic interaction energy for the gerade "C: state of Hl may be written in terms of the u.a. expansion AE, = E,-E: = AEo+E,"+E2+ . . . 17b) where Eg = -2 is now the energy for the 2Sground state of He+, and AEO = -+(~;-4) (18b) is the energy associated with the decrease of the nuclear charge of the u.a.The first order term Efaccounts for the division of the u.a. nuclear charge producing the TABLE2.-H:('E:). INTERACTION ENERGIES TO VARIOUS ORDERS AND COULOMBIC AND PENETRATION COMPONENTS OF THE INTERACTION UP TO SECOND ORDER (a.u.) R AEE AEo = AE@, AE(2) AEcb AEFD 1.o 0.482 20(- 1) -0.682 62(+0) 0.590 01(-1) 0.438 26(- 1) 0.105 84(+0) -0.620 22(-1) 1.5 -0.823 20(- 1) -0.426 74(+0) -0.671 21(-1) -0.859 36(- 1) -0.224 50(-2) -0.836 91(-1) 2.0 -0.102 63(+0) -0.267 18(+0) -0.865 05(-1) -0.104 88(+0) -0.211 18(- 1) -0.837 67(- 1) 2.5 -0.938 23(-1) -0.165 48(+0) -0.787 57(-1) -0.949 79(- 1) -0.207 29(- 1) -0.742 49(-1) 3.0 -0.775 63(- 1) -0.993 86(- 1) -0.644 48(-1) -0.779 82(-1) -0.163 66(-1) -0.616 15(-1) 3.5 -0.608 55(-1) -0.563 08(-1) -0.499 56( -1) -0.608 25(-1) -0.119 40(- 1) -0.488 84(-1) 4.0 -0.460 85(-1) -0.287 14(- 1) -0.373 34(-1) -0.458 09(- 1) -0.839 54(-2) -0.374 14(-1) 5.0 -0.244 20(- 1) -0.203 07(-1) -0.192 05(-1) -0.240 28(-1) -0.400 48(-2) -0.0022(--1) 6.0 -0.11969(-1) 0.490 02(-2) -0.908 05(- 2) -0.116 77(-1) -0.196 17(-2) -0. 71 52(-2) 7.0 -0.559 40(-2) 0.471 08(-2) -0.404 62( -2) -0.542 71(-2) -0.103 35(-2) -0.439 35(-2) 8.0 -0.257 04( -2) 0.304 85(--2) -0.173 42(-2) -0.248 78(-2) -0.590 20(-3) -0.189 76(-2) 9.0 -0.119 54(-2) 0.168 29( -2) -0.724 85(-3) -0.115 75(-3) -0.361 44(-3) -0.796 09( -3) 10.0 -0.578 73(-3) 0.852 98(-3) -0.297 89( -3) -0.561 79(-3) -0.234 10(-3) -0.327 68(-3) 12.5 -0.130 53-3) 0.128 09(-3) -0.307 58(-4) -0.128 14(-3) -0.942 98(- 4) -0.338 43(-4) 15.0 -0.489 38(-4) 0.165 16(-4) -0.303 87(--5) -0.484 73(-4) -0.451 33(-4) -0.333 96( -5) 20.0 -0.142 59(-4) 0.217 79(-6) -0.273 79(--7) -0.142 12(-4) -0.141 82(-4) -0.300 18(-7) a Referred to separate systems (E: = -0.5 a.u.) and including nuclear repulsion.IJ b AE(n) = AE, + C Ei. c Exact interaction energies as given by Peek. l2 i=l TABLE INTERACTION ENERGIES a (a.U.) NEAR THE UNITED ATOM 3.-Hz(2El). ELECTRONIC (He+) R AE0(l)b AEe(*)c CO A&Cl) d AEe(2)f AE g 0.0 0.5 0.0 2.0 0.0 0.0 0.0 0.1 0.504 53 0.016 04 1.979 93 0.021 79 0.021 72 0.021 75 0.2 0.516 47 0.056 11 1.937 40 0.071 67 0.071 11 0.071 4 0.4 0.554 90 0.172 61 1.832 73 0.201 16 0.197 75 0.199 2 0.5 0.578 65 0.237 63 1.778 88 0.268 24 0.262 74 0.265 0 0.6 0.604 10 0.302 65 1.726 19 0.333 25 0.325 52 0.328 5 0.8 0.657 57 0.426 13 1.626 93 0.453 43 0.441 50 0.445 5 1.o 0.711 63 0.536 15 1.537 93 0.559 00 0.543 82 0.548 2 1.25 0.777 04 0.653 44 1.441 87 0.671 71 0.654 01 0.658 2 1.5 0.838 31 0.750 77 1.361 42 0.766 21 0.747 39 0.751 0 1.75 0.894 74 0.831 86 1.294 43 0.845 86 0.826 93 0.829 8 2.0 0.946 22 0.900 12 1.238 69 0.913 49 0.895 11 0.897 3 a Referred to the united atom (E: = -2 a.u.), b first-order Heitler-London, C second-order MS-MA without modulation, first-order theory with modulated perturbation (this paper), fsecond- order MS-MA theory with modulated perturbation (this paper), gexact values as given by Bates et al.' diatomic molecule,' and contains a substaiitial contribution froin exchange inter- actions (table 4).The overall second-order term is small, suggesting that a first-order p.t. based on the united atom would be adequate at small separations provided ex- change and modulation of the perturbation are properly accounted for. The elec- tronic interaction energies (17b) near the u.a. resulting from our modulated pt. are MOLECULAR INTERACTIONS TABLE4.-Hz ('Z:). SEPARATION OF COULOMBIC AND PENETRATION COMPONENTS OF THE IPJJERACTION~ENERGY(REFERRED TO SEPARATE ATOMS) INTO CONTRIBUTIONS FROM DIFFERE~T ORDER (a.U.) R A,?g Eib Eib EF E,Pn 0.2 0.4 0.6 0.8 1.o 0.123 23(+0)' 0.320 54(+0)" 0.510 13(+0)" 0.676 53(+0)= 0.817 37(+0)" 0.123 40(-1) 0.261 91(-1) 0.143 99(-1) -0.169 go(--)' -0.555 63(-1)' -0.642 34(-1)-0.141 47(+0) -0.174 25(+0) -0.173 17(+0)-0.155 96(+0) -0.639 02(-1)-0.145 57(+0)-0.191 27(+0) -0.206 11(+0) -0.202 8l(+O) 0.636 72(-1) 0.138 07(+0) 0.166 52(+0) 0.161 23(+0) 0.140 78(+0) 1.o 1.5 2.0 2.5 3.0 4.0 6.0 8.0 10.0 12.5 15.0 20.0 -0.682 62( +0)-0.426 74(+0) -0.267 18( +0)-0.165 48(+0)-0.993 86(- 1) -0.287 14(- 1) 0.490 02(-2) 0.304 85( -2)0.852 98(- 3) 0.128 09(-3) 0.165 16(-4) 0.217 79(-6) 0.944 43(+0) 0.526 19(+0) 0.307 93(+0) 0.182 14(+0) 0.105 89(+0) 0.294 57(- 1) -0.488 05(-2)-0.304 37(-2)-0.852 61(-3)-0.128 08(-3)-0.165 16(-4) -0.217 79(-6) -0.155 96(+0) -0.101 70(+0)-0.618 63(-I)-0.373 92(- 1) -0.228 71(-1)-0.913 81(-2)-0.198 14(-2) -0.595 OO(-3)-0.234 47( -3)-0.943 07( -4)-0.451 33(-4) -0.141 82(-4) -0.202 81(+0) -0.166 57(+0)-0.127 25(+0)-0.954 20(- 1) -0.709 53(-1) -0.380 77(- 1) -0.173 89(-2)-0.298 25(-3)-0.307 66(-4)-0.303 88(-5)-0.273 79(-7) -0.910 01(-2) 0.140 78(+0) 0.828 87(- 1) 0.434 84(- 1)0.211 70(-1) 0.933 73(-2) 0.663 09( -3)-0.615 07(-3)-0.158 63(-3)-0.294 25(-4)-0.307 68(-5)-0.300 81(-6) -0.263 95(-8) a Referred to united atom (EZ = -2 a.u.).b Electronic contribution only. R/bohr FIG.2.-Electronic interaction energies referred to the united atom (He+) according to different perturbation theories which include exchange. (1) Exact AEe (full line) ; (2) first-order Heitler- London ; (3) second-order MS-MA without modulation ; (4) first-order u.a.theory ;l6 (5) first-order and (0)second-order MS-MA with modulated perturbation. V. MAGNASCO, M. BATTEZZATI AND G. FIGARI compared in table 3 with the exact values and with those resulting from the MS-MA theory without modulation, and are plotted against R in fig. 2 together with the first-order u.a. results of Byers-Brown and P0wer.l The percentage errors = 100 x (approximate -exact)/exact in the u.a. electronic energies resultkg from the different theories are plotted against R for the u.a. region in fig. 3. The validity of the u.a. modhikited p.t. over the whoIe range of internuclear separations is a-pparent from the figiire.At very small seperations our results seem consistently better than those recently obtained from a modified form of Van Vleck's p.t. for degenerate 26 5r -25-201J~lll~~llltllll~ll,, 0 0.5 '1.0 1.5 2 Rlbohr FIG.3.-Percentage error (see text) in the electronic energy near the u.a. (He+) according to different perturbation theories which include exchange. (l), (2) First- and second-order MS-MA with modu- lated perturbation ;(3) second-order MS-MA without modulation ;(4) first-order u.a, theory.16 Of interest in the description of the physical nature of the interaction is the separa- tion of the interaction energy into Coulombic and penetration (exchange) compo- nents, which can be done following our previous work.8-10 In terms of the permuta- tion operator P we can write : Gb= (40lmbo) (194 w = [I +~401~40~l-1~~401~~-~~~l~o~ U9b) MOLECULAR INTERACTIONS AECbconsists of the quasi-classical Coulombic interaction between nuclei and elec- tron 27*28 (" radial " polarization +Coulombic interaction between static densities +Coulombic interaction between transition densities).AEPnis a symmetry-dependent electronic contribution resulting from the physical identity of the interacting particles 10 1 2Rlbohr 3 4 FIG.4.-Hd(2Cl). Exact second-order MS-MA calculation with modulated perturbation : inter-action energy and its Coulombic and penetration components near the bond region. Top :electronic components of the interaction referred to the united atom (He+) at small internuclear separations.and consists of a first-order term (196) describing " non local " interference effects associated with the interpenetration of static charge distributions ** lo and a second- order term (206) giving the exchange part of the polarization. These definitions are completely general and are valid over the whole range of internuclear separations (only electronic contributions being considered in the u.a. region) and agree with the conventional definition of Coulombic and exchange components 79 15* 29* 30 in the limit of large R where overlap and modulation become negligible. The results of such an analysis for Hz(2Cgf)are given in table 4 and in the last two columns of table 2, and are plotted against R in fig.4. The relative importance of Coulombic and penetration components changes according to the internuclear separation. If we broadly distinguish three regions : V. MAGNASCO, M. BATTEZZATI AND G. FIGARI (a) small R,near the united atom; (b) medium range A,near the bond region; (c) large I? ; th!: calculations show that in region (a) the electronic interaction energy is dominated by the Couloinbic component mainly through its zeroth-order term AEo (right top of fig. 4) ;in region (b)the Coulombic components (including nuclear repulsion) cancel to some extent and the largesz term is the first-order penetration which appears as the main fxtor responsible for the existence of a stable chemical bond at R = 2 bohr (GECbwould account only for about 20 % of the bond energy) ; in region (c) the weak attraction is dominated by the second-order Coulombic energy whose leading term goes as and describes the ‘‘dipole ” polarization of the hydrogen atom by the proton.Note, however, that the range of the first-order penetration is much larger than is usudly believeds. 31 and the neglect of exchange effects in the van der Waals region is expected to a-f€ect seriously the calculated value of the interaction. CONCLUSION In conclusion we make the following remarks. (i) Although higher-order p. t. calculations including exchange are difficult and may be no more accurate than the corresponding variational calculations, the per- turbation approach seems preferable for the reasons already mentioned in the intro- duction and also because different physical effects are easily recognised in diEerent orders of the perturbation expansion.Low order treatments may be sufficiently accurate if a convenient unperturbed hamiltoniaii can be chosen on the basis of physical intuition and mathematical corxnience so as to give a rapidly converging series. (ii) The modulation of the perturbation in first order yields a perturbation expan- sion which shows excellent convergence at any R,this being ensured by the perturba- tion vanishing in the limits R -+ 0 and R -+ m. (iii) The second-order MS-MA theory including exchange does seem appropriate for describing long-range “ physical ”as we!l as short-range “ chemical ”interactions, including the very short-range interactions near the united atom.(iv) An exact second-order calculation attains chemical accuracy (roughly, to wit hiil 1 kcal) over the whole range of internuclear separations. (v) The separation of the interaction energy into Couloinbic and penetration com- ponents gives a physically meaningfill picture of the various factors determining the variation of the interaction with internuclear separation, a fact important in develop- ing accurate semiempirical models of molecular interactions. Points (iii) +(iv) are particularly useful because to continue the calculation through the third order of p.t. would require knowledge of the second-order polarization fum- tion +pZ 32 or the first-order exchange functions 8 clnd col defined in the generalized Chipman-Bowman-Hirschfelder theory.In view of the rather complicated expres- sion resulting for the exact second-order energy this would obviously be very difficult. This point calls also for simpler variational approximations to the second-order energy to allow for practical calculations for any system whose first-order equation (3b)cannot be solved in an exact way. APPENDIX Asymptotic expansions of the interaction energy for H2+(2Z,+)according to the second- order MS-MA theory with modulation of the perturbation. MOLECULAR INTERACTIONS (i) Small R (D = 2R) AE:') = 3D2-4D3+&D4+~05+O(D6) (AW E2 = --'034995D4+[~-$(y+1n 2D)]05 +O(D6) (AW to be compared with the exact expansion l7 AEe = +D2-$D3+&$D4+[#-$(y+1n 2D)]D5+O(D6), (A21 y = 0.57721, Euler's constant.For the leading terms of the Coulombic and penetration components near the u.a. we have : = 402 -2~3AE~ +o(~4) (A34 Eib(e) = 5D3+O(D4) (A3b) E';" = -3~2+%~3+0(04) (A34 Eib = -+D2+3D3+O(D4) (A44 EP,"= $D2-5D3+0(D4) (A4b) AEzb = ~D2-+D3+0(D4), AEP"= 0(D4). (A5) (ii) Large R (p = R) which coincides with the exact long-range multipolar expansion of the second-order golariza- tion energy.22 Leading terms of Coulombic and penetration components of the interaction energy at large separations are : AEo = -e-R(1++R-$R2)+O(e-2R) (A74 Eib = e-R(1+4R-+R2)+O(e-2R) (A7b) which shows that to first order AEo exactly cancels out Efb leaving as the dominant first- order contribution the penetration component (A7c).In second order (A66) already gives the Coulombic component, hence : EP," = -&R e-R+O(e-R), (AW giving for the asymptotic overall components up to second order : 2.25AEC~= --R4 +OW6) (A94 AEPn= -3R e-R+O(e-R). (A9b) While (A9a) is exact, (A9b) gives 98 % of the exact asymptotic value of the exchange energy (-R e-R) given by Herring.33 V. MAGNASCO, M. BATTEZZATI AND G. FIGARI We acknowledge financial support by the Italian National Research Council (C.N.R.). We thank Mrs. Graziella Bitossi for typing the manuscript. D. M. Chipman, J. D. Bowman and J. 0. Hirschfelder, J. Chem. Phys., 1973, 59, 2830, and references therein. J. 0. Hirschfelder, Chem. Phys. Letters, 1967, 1, 325. P. 0.Lowdin, Int.J. Quantum Chem., 1971, 2S, 137. J. N. Murrell and G. Shaw, J. Chem. Phys., 1967,46,1768. J. I. Musher and A. T. Amos, Phys. Rev., 1967, 164, 31. A. T. Amos and J. I. Musher, Chem. Phys. Letters, 1967, 1, 149.'G. Chalasinski and B. Jeziorski, Int. J. Quantum Chem., 1973, 7, 63. V. Magnasco, G. F. Musso and R. McWeeny, J. Chem. Phys., 1967,47,4617. V. Magnasco and G. F. MUSSO,Atti Accad. Ligure Sci.Lettere, 1970, 27, 207. lo V. Magnasco and G. F. MUSSO,J. Chem. Phys., 1974, 60, 3744. D. R. Bates, K. Ledsham and A. L. Stewart, Phil. Trans., 1953,246,215. l2 J. M. Peek, J. Chem. Phys., 1965,43,3004. l3 V. Magnasco, Chem. Phys. Letters, 1974,26, 192. l4 W. A. Sanders, J. Chem. Phys., 1969,51,491. l5 D. M. Chipman and J. 0.Hirschfelder, J.Chem. Phys., 1973,59,2838. W. Byers-Brown and J. D. Power, Proc. Roy. SOC.A, 1970,317,545. l7 W. Byers-Brown and E. Steiner, J. Chem. Phys., 1966,44, 3934. l8 A. Dalgarno and N. Lynn, Proc. Phys. Soc. A, 1957,70,223. l9 C. A. Coulson and P. D. Robinson, Proc. Phys. SOC.A, 1958,71, 815. 'O P. D. Robinson, Proc. Phys. Soc. A, 1958,71,828. 21 V. Magnasco and M. Battezzati, to be published. 22 A. Dalgarno and J. T. Lewis, Proc. Roy. Soc. A, 1955,233,70. 23 C. A. Coulson and W. E. Duncanson, Phil. Mag., 1942,33,754. 24 J. Miller and R. P. Hurst, Math. Tables Aids Comp., 1958, 12, 187. 25 S. Y. Chang, J. Chem. Phys., 1973, 59,1790. 26 T. S. Nee, R. G. Parr and S.Y. Chang, J. Chem. Phys., 1973,59,4911. 27 H. C. Longuet-Higgins, Proc. Roy. SOC.A, 1956, 235, 537. 28 R. McWeeny, Rev. Mod. Phys., 1960,32,335. 29 J. 0.Hirschfelder and W. J. Meath, Ah. Chem. Phys., 1967,12,3. 30 P. R. Certain, J. 0. Hirschfelder, W. Kolos and L. Wolniewicz,J. Chem. Phys., 1968, 49, 24. 31 F. 0. Ellison, J. Chem. Php., 1961, 34, 2100. 32 G. Chalasinski and B. Jeziorski, Int. J, Quantum Chem., 1973, 7,745. 33 C. Herring, Rev. Mod. Phys., 1962,34,631. (PAPER 5/548) Tr-2

ISSN:0300-9238    

DOI:10.1039/F29767200022

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Graph Theory of Free Radicals Validation of a Recent Assertion and Its Relation to the Pairing Theorem BY COLINL. HONEYBOURNE Physical Chemistry Laboratories, The Polytechnic, Ashley Down, Bristol 7 Received 1st May, 1975 The criteria for the occurrence of a symmetric eigenvector in odd graphs are deduced ;these con- firm the validity of a recent assertion concerning the cause of such an eigenvector. The relationship to the Coulson-Rushbrooke Pairing Theorem is noted. In a recent paper it was asserted that those odd graphs of N vertices having an unmarked unvalued starred partial graph display a spectrum containing an eigenvector such that the coefficients at all unstarred vertices are zero and those at all starred vertices have the value kG*where 8 = (N+ 1)/2.The adjective "symmetric " was chosen to describe eigenvectors of this type which proved extremely important in discussing the behaviour of the odd a-electron in free radica1s.l The purpose of this paper is to demonstrate the validity of the proposed relationship between the occur- rence of a symmetric eigenvector and a starred partial graph that is unmarked and unvalued. The terminology to be used is closely allied to that adopted by Longuet- Higgins in his paper concerning the eigenvalue problem of alternant hydrocarbons.2 Although the general form of the work presented below is similar to that used in discussions of the Coulson-Rushbrooke Pairing Theorem 2* 4* (CRT), it is em- phasised that symmetric eigenvectors can arise in graphs which do not obey the CRT, and frequently do not arise in graphs which do obey the CRT.THEORY A PROPERTY OF CERTAIN MATRICES Attention is drawn to the following property of the characteristic polynomial of matrices which can be factored into a particular blocked form. Consider a matrix of dimension N x N (where N is odd) which has a leading diagonal block A* (see fig. 1) of dimension G x G with a set of equal diagonal elements, a*, and off diagonal elements of zero. The detailed forms of the blocks B, B and A" will be considered later because they do not affect the present argument. The diagonal form and dimensions of A* dictate that the characteristic polynomial, f(~),tlways contains a factor (a* -E) regardless of the values of the matrix elements in B, B and A".Thus one eigenvalue, and the associated eigenvector, of such a matrix may be determined with particular ease. ADJACENCY MATRICES OF ODD ALTERNANT GRAPHS An alternant graph is one which may have alternate vertices distinguished by a star with no two starred vertices being adjacent to each other.2* It has become 34 C. L. HONEYBOURNE 35 customary to apply the adjective " alternant " only when all vertices are unmarked and all edges are equally valued ;such graphs obey the CRT (e.g., the alternant hydro- carbons). However, in this work we will apply the adjective &' alternant " to any framework for which the starring criterion holds whether it be unmarked and unvalued marked (0-o-o-o-o),(0-0-0-0-o), valued (o---o-o-o---o) or marked and valued (0-0--0--0-0).The adjacency matrix of a bipartite graph has non-zero matrix elements for vertices and edges and matrix elements of zero elsewhere. The adjacency matrix, M, of any odd alternant graph may be partitioned into blocks as shown in fig. 1. If A* is to have the form required to exploit the property described above, then all starred vertices must be equally marked (i.e., with a*). A" is diagonal with diagonal elements FIG.1. at,,(v = G+ 1, G+2, . . .,N) and B and B have zero matrix elements except for edges connecting adjacent starred and unstarred vertices. The CRT holds 2* when all the a;" = a* ;there are no restrictions on the magnitude of non-zero elements in B or B.6 In what follows the restriction on the matrix elements a;,, is lifted; these may be unequal to a* and to each other.Further, the restriction is imposed that all non- zero elements in B and B take the same value. Clearly, the characteristic polynomial of M has at least one solution Ej = a* if A* takes the prescribed form. THE STARRED PARTIAL GRAPH Odd alternant graphs relevant to n-electron free radicals may contain vertices of degree a, b or c1 according as that vertex is connected to 1, 2 or 3 other vertices. The starred partial graph, G* of an odd graph G is assembled from the starred vertices in G and new edges of type A, B or C1according as the omitted unstarred vertices are of degree a, b or c. In order that the adjacency matrix A4of G should contain a block A* it is sufficient that G*should be unmarked ;i.e., all starred vertices in G must be identically marked. The criterion specified for B and B indicates that G must be unvalued.Thus, .-@-.-o-e has the required form whereas 0-0-0-0-0and 0---0-~-0 ---0 do not. The connectedness of the vertices in G will determine the form of B and B,the valuing of the edges in G*, and the form of the secular equations. In what follows we will show that G must only contain unstarred vertices of degree b in order that the eigenvalue cj = a* is associated with a " symmetric " eigenvector -this then determines that if G* is unvalued as well as unmarked a symmetric eigenvector does occur. GRAPH THEORY OF FREE RADICALS LINEAR ODD ALTERNANT GRAPHS The adjacency matrices of those odd linear graphs that are unvalued and equally marked at all starred sites have the general form 0 I I 1 I 0 \ \ I \ \ I \\ I M= 0 \'\ \' \ '\ I I 1 I ! 'a* I a* ! P FIG.2.where all non-zero elements in B and B are given the value j. We find that the characteristic polynomial, J(E),of the secular determinant D of A4 is given by 6 where the dk(&)are formed by striking out the first k rows of D, the single kth column of the starred block, and the first (k-1) columns of the unstarred block ;we refer to the subdeterminant obtained from A* and B as the starred block and to that from A" and B as the unstarred block. The secular equations given below have the following terminology: p, A, cr) are running indices for starred vertices, v, IC, z are running indices for unstarred vertices and j labels the eigenvalue with the corresponding eigenfunction coefficients being labelled cy, or c;~.(a* -Ej)CTl +flc;2 =o These equations determine that, for Ej = a*, all coefficientsat unstarred vertices (the cgy)are zero and that all other coefficients have the value _+awhere the normalisation condition determines that a = /-*. The eigenvalue E~ = a* is a consequence of (i) the alternant nature of the odd graph and (ii) the identical marking at all starred vertices. C.L. HONEYBOURNE The symmetric form of the eigenvector (Lee, c:: = cy2N--Q+ is a further consequence of all uiistarred vertices being of degree b, thereby generating the relationships cTp = 3-CTL.Clearly the above criteria may be summarised in the statement that “ the starred partial graph must be unmarked and unvalued ”. NON-LINEAR ODD ALTERNANT GRAPHS The criteria for the occurrence of at least one solution of the form E~ = u* have already been discussed. A detailed examination of the various types of secular equa- tion is necessary to determine the general criterion for the solution E~ = a* to be associated with a symmetric eigenvector. The deduction of the general criterion proceeds stepwise as follows : starred site (degree a) (a” -Ej)qfl + starred site (degree b) pqv+(a” -Ej)CYP +pi”, =o (11) starred site (degree c) pcyv+(a” -Ej)CYP +pcj9,+ = 0 (111) unstarred site (degree b) flcj*,+(a&-~JciD, +acj*, =o (W unstarred site (degree c) flcx +(a:v -cj)cjOv-tSc?, +Pc& = 0 (V) unstarred site (degree a> pcyp+(a:” -.sj)cj”y =o (W (i) All odd alternant graphs have at least one secular equation of type I from the convention that the number of starred vertices is >d.Hence, at least one of the csv (say, cjo2) is always zero. (ii) This particular unstarred vertex must be of degree b or c to which secular equations of type IV or V apply respectively. Type IV propagates the required rela- tionship (cY1 = -&, say) whereas Type V destroys any such relationship except for the trivial case of all coefficients being zero. The only sequence of interest is that beginning *-o-*. . .. 1231234 123 o4 (iii) The three possible sequences are *-o-*,*-o-*-o... . and *-o-*/ . \05 All three give = -cy3, with cj”2 = 0; the second gives c’j4 = 0 as a consequence of being linear. If the latter is to be incorporated into a larger graph, then vertex 4 is of either degree b or degree c; only the first of these propagates the relationship of the general form cyP = +cj*,, because the unstarred site of degree c gives cTp+cya+ cya = 0. The third possible sequence mentioned above may either be the final sequ- ence or be incorporated into larger graphs with vertices 4 and 5 in sites of various degrees of connectedness -these will be dealt with below. (iv) If the third possible sequence is final, then it must be relabelled o-*-o/*\* GRAPH THEORY OF FREE RADICALS and the presence of a secular equation of type V prevents the occurrence of cyp = rfI.$A. *7 4/ 12 ”0 \(v) If vertices 4 and 5 are both of degree b (Le., *-o--* 1 0 5\ *6 then IC;~~ = Icy?] = lty61 = lcj*,l and cj2 = c;4 = cj”6 = 0as required. (vi) If one site is of degree n (say 4) and one of degree b we have cj2+cj4+c;~= 0 and pc73+c;4(a24-~j) = 0. If az4 # a* then c74 # 0 and if ai4 = a* then c;~= c:3 = 0 ;in both instances the symmetry relation is destroyed. The occurrence of an unstarred vertex of degree a is not possible in. a Kekul6 structure of a n-electron mono- radical. (vii) If one or both of vertices 4 or 5 are of degree c the required relationship cannot occur [see (ii) and (iii) above]. (viii) Any extension of the graph only introduces cases already considered.(ix) In all the cases where the relationship cyp = +cyA is propagated, all unstarred vertices are of degree b; whenever this relationship is not propagated, at least one vertex is not of degree 6. (x) Therefore, if an unmarked starred partial graph is unvalued, the eigenvalue E~ = a* of the related complete graph will be associated with a symmetric eigenvector. A consequence of the above deduction is that a symmetric eigenvector cannot occur in odd alternant cyclic systems. These must contain even-membered rings attached to one (or three. . .) odd-membered side chains which automatically gene- rates an unstarred vertex of degree c in either the ring (e.g., benzyl) or the side chain (e.g., iso-propenyl phenyl).A symmetric eigenvector does occur in, say, trivinyl- methyl or divinyl butadienyl methyl and allied appropriately star-marked systems obtained during discussions of the effect of the odd unpaired e1ectron.l Inspection of secular equation (IV) shows that if, in contrast to the foregoing, # /IvA,the required symmetry property cyp = & cy~is not obtained. Although the pair of edges at a given unstarred vertex of degree b must therefore have the same value, this value can differ from that assigned to any other pair of such edges. The restriction imposed on the non-zero matrix elements in B and B can be partially lifted. CONCLUSION It has been shown that, for there to be an eigenvector with zero coefficients at unstarred vertices and with coefficients of -t [(N+ 1)/2]-* (i.e., +t-*)at all starred vertices, then a graph G of N vertices must be (i) odd, (ii) alternant with respect to the starring criterion, (iii) equally marked at all starred vertices, (iv) contain unstarred vertices of degree b only, (v) have identically valued edges at a given unstarred vertex.Criteria (i)-(iv) are commensurate with the requirement that the starred partial graph G* be unmarked and unvalued as asserted previously.’ Criterion (v) does not pre- clude the requirement that G* shall be unvalued because it is the degree of an unstarred vertex in G that determines the valuing of G*.I The Coulson-Rushbrooke Pairing Theorem does not demand criteria (iv) and (v) but does demand that (vi) all vertices be equally marked [this includes (iii) but is more C.L. HONEYBOURNE restrictive]. In recent work unvalued graphs which fulfil (i)-(iv) and (vi) were classi- fied as I(CR) and those fulfilling only (i)-(iii) and (vi) were classified as II(CR). Those graphs which do not obey the pairing theorem, but which do exhibit a symmetric eigenvector in their spectrum, are obtained by identically marking I(CR) graphs at all starred vertices and marking unstarred vertices at random. Extensive valuing can also occur provided that criterion (v) is complied with. The value of j in cJ = a* has not been investigated although it has been found (and asserted) in earlier work thatj = G. In odd alternant hydrocarbons j = 6' by the CRT, and j = G in recent work on perturbed free radicals in which, in units of the standard Hiickel beta, p = 1 and either CI* = 0 and a:,, = 1 or 0, or a* = 4/(N+1) and aiv = 0.l In both categories the perturbation imposed does not alter the sym- metries of the highest doubly filled, partially filled and lowest empty orbitals from those observed in the parent hydrocarbon.A Referee is thanked for calling attention to recent relevant work on graph theory, Recent Advances in Graph Tt2eory (Academia, Prague, 1975). C. L. Honeybourne, J.C.S. Faraday II, 1975, 71, 1343, H. C. Longuet-Higgins, J. Chem. Phys., 1950, 18, 265. C. A. Coulson and G. S. Rushbrooke, Proc. Camb. Phil. SOC.,1940, 36, 193. C. A. Coulson and H. C. Longuet-Higgins, Proc. Roy. Soc. A, 1947,192,16. A. Graovac, I. Gutman, N. TrinajstiC and T. ZivkoviC, Theor. Chim. Acta, 1972, 26, 27, and references therein. L. Salem, The Molecuiar Orbital Theory of Conjugated Systems (W. A. Benjamin, New York, 1975), p. 37. (PAPER 5 1813)

ISSN:0300-9238    

DOI:10.1039/F29767200034

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Model Interpretations of the Far Infrared Absorptions in Compressed Gaseous and Liquid Bromotrifluoromethane, CBrF, BY GRAHAMJ. DAVIES Post Office Telecommunications Headquarters, Dollis Hill, London NW2 7DT AND MYRONEVANS* Physical Chemistry Laboratory, South Parks Road, Oxford OX1 342 Received 2nd May, 1975 The far infrared (2-100 cm-’) absorption of the symmetric top molecule bromotrifluoromethane (CBrF3) has been measured for the gas phase in pressure range 1.2-46.2 bar and for the liquid. Whereas the peak of the broad bands observed moves only slightly from 9.8 cm-l at 1.2 bar to 11.O cm-l in the liquid at the same temperature (295 K), there are pronounced differences in the bandshape which are reflected in considerable damping of the time functions obtained by Fourier transformation of the frequency data.A simple model of bimolecular collision induced dipolar absorption accounts fairly well for the high frequency shoulders which appear in the spectra of the compressed gas. Using this model, preliminary values of the quadrupole (Q) and octopole moment (Q) of the molecule have been obtained. A mechanism of collision disturbed torsional oscillation of the permanent dipole (p) within potential wells accounts more satisfactorily for the absorption in the liquid than the J-diffusion model, which assumes free rotation between instantaneous collisions. Several recent studies 1-4 of the far infrared (10-200 cm-l) absorption bands of small linear molecules in the compressed gas and liquid states have had the aim of gaining some insight to the liquid state by comparison with the bimolecular collision induced absorption.Present theories 5s of this latter type of infra-red absorption are restricted by the form of the intermolecular potential U(R)assumed, and since this is still usually approximated by a Lennard-Jones potential,’ the experimental absorption is adequately predicted in intensity and bandshape only for the simplest molecules such as H2 and NZ,and then only in the gas phase where interactions involving more than two molecules are rare. As expected, neglect of the electrostatic part of U(A)(e.g. the quadrupole-quadrupole interaction energy 4, in highly quad- rupolar and anisotropic molecules such as cyanogen and carbon dioxide leads to disagreement between the experimental and predicted frequency of maximum absorp- tion (Ymax) and to a fallacious temperature variation of the quadrupole moment (Q? when this is regarded as the unknown quantity linking the observation and theory.However, it is now possible to study larger but “ pseudo-spherical ” molecules in this way with the help of advances in experimental and theoretical techniques. Frost et al. O* have recently extended the treatment of bimolecular collision induced absorption to symmetric top molecules which allows this type of study to be made with molecules, other than linear, which contain a permanent dipole moment. The use of a liquid helium cooled Rollin detector with phase-modulated Michelson inter- ferometers has extended the spectroscopic range down to 2 cm-1 so that overlap with the “ microwave region ” is now routine.40 G. J. DAVIES AND M. EVANS The molecule BrCF, is well suited to this type of study since it is spheroidal with a small dipole moment (0.65 D), and, by symmetry, a quadrupole moment which is small enough not to affect U(R) adversely, and yet large enough to produce a significant absorption in comparison with that of the permanent dipole. There are at least two classes of information which may be defined. First, values of the apparent quadrupole moment (Q) and the octopole moment (Q) may be obtained from the integrated intensity of the absorption band with the equation : 03 A = 1 a(V) dV = A'p2N+(B'p2 +CQ2+Dn2)N2 where a(?) is the experimental absorption coefficient (in neper cm-') at frequencies i,N the number density (molecule cm-,) and A', B', C and D are theoretical constants. (If both Q and R are unknown, then (1) must be used in conjunction with frequency or time domain curve fitting, see evaluation section below).The above equation assumes that collisions involving more than two molecules are rare, so that the terms in N3etc. are small, that overlap and translational absorptions l2 are small, and that cross terms between the induced dipole moment and the permanent dipole are negligible. Secondly, information about the fluctuation in space with time of the molecular permanent dipole moment vector (u) (i.e., a unit vector along the permanent dipole) may be obtained following the effect of progressive compression and liquefaction of the macroscopic sample on the far infrared bandshapes.A convenient function which can be used for this purpose is where t denotes time elapsed from an arbitrary point t = 0 and the angular brackets denote ensemble averaging. f(t) is proportional l3 to the Fourier transform of a@): Ex(?)exp(2niikt) dv -a, [1-exp(-hcF/kT)] if cross-correlations (vector products of u between different molecules) are neglected and the internal field factor l4 is considered static (independent of V). " Dynamic " internal field factors have been reviewed by Brot l5 and are important only for large p in the liquid phase. Eqn (2) is valid only if the absorption due to the permanent dipole l6 is Fourier transformed. Given accurate estimates of IQl and IQl, it would be possible to subtract the induced dipolar absorption in the gas phase using eqn (1).However, in the liquid phase eqn (1) no longer applies (because of many body collisions) and the relative contributions of permanent and induced dipole to the bandshape are usually very difficult to estimate.l' [Gordon's sum rule l8 may be used l9 to estimate the contribution to the integrated intensity (A) of all rotational modes (microwave and far infrared) of the permanent dipole. In liquids such as chlorobenzene l9 the excess over this sum is of the order of 30-50 % of the total observed A.] Experimentally, CBrF, is convenient because the rotational contribution of the permanent dipole peaks at 9.8 cm-I, so that the whole band is situated in the measur- able range 2-100 cm-l. Finally, the critical properties make it straightforward to study the liquid in equilibrium with several atmospheres pressure of vapour and also to obtain high gas number densities by heating to above the critical temperature.FAR I.R. OF CBrF, EXPERIMENTAL Submillimetre wavelength measurements 2o were carried out at Dollis Hill with two interferometers evacuated to remove water vapour and modified to cover the spectral ranges (i) 2 to 31 cm-l; (ii) 20-100 cm-l. In both cases the air-cooled lamp-housing was replaced by a more efficient water-cooled unit and phase-modulation 21$22 was incorporated. The apparatus was left switched on to ensure maximum stability.Over the range 20-100 cm-l, where a quartz Golay detector was used, signal to noise ratios 21 as great as 1000 were Resolution in this region was 4 cm-l. For the range 2 to 31 cm-I a Rollin InSb, liquid helium cooled detector 24 was employed together with a 4 mm black-polyethylene filter. Signal to noise ratios as great as 10 000 were obtained, whilst the reproducibility of three consecutivi: runs was estimated to be of the order of 0.1 %. The resolution was 2 cm-l. The spectra of the compressed gas and liquid were taken in a high pressure cell having 7 mm 2 cut crystalline quartz windows, and with an adjustable path length. The spectrum of the detected power was computed for a number of interferograms for two thicknesses of the given gas or liquid sample, and the power coefficient was calculated from the ratio of the averages.Surface and internal-reflection effects were thus almost eliminated.25 The absorption of the lower density vapour was measured with a fixed path length (147.6 mni) cell consisting of a gold-plated copper light pipe with poly-(4-methylpent-l-ene) windows embedded in a stainless steel sheath. The radiation was focused onto the detector with a light cone. The nominal purity of the sample (Matheson Ltd.) of CBrFJ used was 99.0 mole % (minimum), the main contaminants being other freons and air. The specimen was distilled onto type 3-A zeolite at liquid nitrogen temperature in adjoint chambers of either cell to remove moisture as far as possible.The CBrF3 was thereafter stored in these chambers as a liquid under its own vapour pressure at room temperature (295 K). High pressures were obtained by heating the assembly above the critical temperature (340 K). The liquid absorption was observed at 295 K after distillation into the optical path. The pressure, as monitored by a Budenberg gauge, was constant to within &2 % of its value once equilibrium was obtained. N was calculated with generalised compressibility curves 26 at each pressure and its uncertainty is estimated to be N & 3.5 %. RESULTS The far infrared spectra of gaseous CBrF, at low and high pressure and of the liquid in equilibrium with the vapour at 295 K are illustrated in figs. 1, 3, 5 and 7.The experimental A values are shown in table 1, together with the relevant observed T,,,, the wavenumber corresponding to amax,the maximum in each absorption band. TABLEAM FAR INFRAREDABSORPTIONSOF CBrF3 GAS AND LIQUID pressure/ 1021N/phase temp./K bar molecule cm-3 Alneper cm-2 iimax/cm-l gas 295 1.2 0.03 0.57 9.8 gas 295 3.3 0.08 2.05 9.8 gas 357 28.4 0.77 36 10.7 gas 357 46.2 1.94 204 11.0 liquid 295 -6.22 388 11.0 In the figures, the bars denote the theoretical frequencies and relative intensities corresponding to the J + J+n (n = 1, 2, 3) absorption due to rotation of the permanent dipole (n = l), dipole induced (n = 1); quadrupole induced (n = 2); and octopole induced (n = 3) dipole absorptions. Qualitatively, it is apparent that the envelope of the pure rotational lines observed at 1.2 bar and 3.3 bar develops an extended high frequency tail at 28.4 bar, which is G.J. DAVIES AND M. EVANS more significant at 46.2 bar and relatively less so in the liquid. The overall integrated intensity A is linearly dependent on N at the Iowest pressures but increases as a higher power of N thereafter. In the liquid, AlN is relatively much smaller than in the 00 0 0 : 0 -lo i-12 . 14 i/cm-' FIG.1 .-Measured absorption at 1.2 bar and 3.3 bar, 295 K (0). The bars represent the frequencies and relative integrated intensities of some J --f J+ 1 pure rotational absorptions. I I I I I 1 I I I I 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 time/ps FIG.2.-Plot offexp(t)for the 1.2 bar experimental absorption (+) andfth(t) from eqn (6)(e).compressed gas, as for nitrous oxide and the non-dipolars carbon dioxide and ~yanogen.~The non linear dependence of A upon N suggests that induced dipolar absorption involving two or more molecules is occurring in addition to rotational FAR I.R. OF CBrF, absorption of individual permanent dipoles. In the following evaluation of the results the induced dipole is assumed to arise from the effect of a second molecule (expanded in point multipoles and modulated by its rotational motion) in inducing a 0 20 40 60 80 i/crn-' FIG.3.-Measured absorption at 28.4 bar, 357 K (0). ---,Alternative extrapolations of high frequency necessary because of low power throughput in this region. I Frequencies and relative intensities of J -+Jfl (pure dipolar and dipole-infuced dipole) absorptions; i as above for quadrupole-induced dipole (J --f J+ 2) lines; i octopole-induced (J +J+ 3) lines.I I I I I 1 I I 1 1 0 0 0.5 1.0 1.5 2.0 2.5 3 0 3.5 4 0 4 5 timelps FIG.4.-Plot offex&) (9,I) (with uncertainty bars due to high frequency extrapolation at 28.4 bar and 357 K). -, free rotor function fth(t) [eqn (6)]at 357 K ; ---,eqn (7) with IQl = 4.0x e.s.u., lsll = lox e.s.u. dipole in the first in a bimolecular collision. Interactions involving more than two molecules are neglected, which has the consequences that the apparent values of lQ1 G. J. DAVIES AND M. EVANS (the quadrupole moment) and lsZl (the octopole moment) needed to account for the experimental liquid A/N using this model [eqn (l)] will be less than the corresponding constants derived from the gas phase data.This reflects the reduced efficacy of 0 0 0 0 0 0 0 0 0 0 OO 0 0 Glcm-' FIG.5.-As for fig. 3 with data at 46.2 bar, 357 K, no extrapolation being necessary. I I 1 1 1 1 0 I.o 2.0 3.0 4.0 FIG.6.-Plot of fez&) timelps computed with two different algorithms from the experimental .(a of fig. 5 (-0-0-); at 357 K (---) ; eqn (7) with 1 Ql = 4.2x e.s.u., lSrl = 11 x e.s.u. (-1. multibody interactions in the liquid in generating induced dipoles, and the breakdown of the bimolecular model. FAR I.R.OF CBrF, \ \ \ \ '0 ooooOuoo OO oo~ocooo -1-O?1---L.-.---L -L--Io000*00oaLJ 20 40 60 80 100 i'/crn-l FIG.7.-The far infrared absorption of CBrF,(I) at 295 K. EVALUATION OF THE RESULTS GAS AT 1.2 AND 3.3 bar At these low pressures, the linear dependence of A upon N suggests that the observed bands, each peaking at 9.8 cm-', arise from rotational absorption of the permanent dipole, and that we are observing the Boltzmann envelope of the J 3J+ 1 transitions (separated by 2B =0.139 96 cm-I and thus well outside our resolving capabilities). The solution of the Schrodinger equation for the pure rotational motion of a symmetric top molecule gives the following equation for A(T) the intensity and frequency of each (unbroadened) J 3J+ 1 absorption : J (J +1)2 -K2 A(9 = J, K, J+1 -exp[ -E(J, K)/kT] (3) '('7 K= -J where 3(2Z+ 1)(412 +41+ 3) for K =0, 3(2Z+ 1)(412 +4I+ 3) for K # 0 or K =a multiple of 3, 3(2Z+ 1)(412 +41) for K # 0 or_K # a multiple of 3.h/8n2I,c; B =h/8n2ZBc;V(J) =2B(J+ 1); BJ(J+ 1) +(A0-B)K2;i J (2J+ l)S(I,K) exp(-hcE/kT). J K=-J In this equation p is the permanent dipole moment, J and K the rotational quantum numbers, I the nuclear spin quantum number of the off-axis nuclei (F), I, and IBthe moments of inertia about axes parallel and perpendicular, respectively, to the three fold symmetry axis (C-Br), and 2,the rotational partition function. Thus we have A' of equation (1) as : A' =CA(9) J G. J. DAVIES AND M.EVANS The molecular constants used in evaluating &V) are given in table 2, along with polarisability parameters to be used later. TABLE2.-MOLECULAR CONSTANTS OF CBrF3 anisotropy mean molecular of polari-Bman/cm-l polarisability sabilitytemp./K Zr A/cm-1 B/cm-1 I [eqn (211 1024a&m3 10246/cm3 295 452474 0.192 1 0.06998 21 0.5 9.790 5.3+0.3 28 0.5 357 602351 0.192 1 0.06998 0.5 10.777 5.3 0.3 0.5 The similarity of the position of the calculated P,,, at 295 K of 9.790 cm-1 and the observed of 9.8 cm-I suggests that a contour drawn through the J -+ J+ 1 delta functions would describe the observed bandshape adequately. For comparison therefore we have used the Heisenberg continuum representation of eqn (3) in the time domain with the experimental " rotational velocity " correlation function f(t)of the introduction section.This function is used because it is essentially l3 the Fourier transform of a@). It is the negative of the second derivative of the vectorial correlation function ((u(0) 9 u(t))) this being essentially the Fourier transform of a(F)/i2,which requires accurate data in the region (0.001-10) cm-l for its evaluation. Gordon 29 has shown that the dipole correlation function is given by : co (u(0) u(t)> = I(P) cos(2nVct) di (4)1 -co where I(V) = 3hn(?)a(F)/16n4F[l-exp(-hcS/kT)]. Here, n(?) is the frequency dependent refractive index, approximately unity in the gas phase, t the time and u the dipole unit vector. We have : l3 d2 (40) * ti@)> = -Tii"<@> u(t)> hence using J = 5/2B-1, we obtain from eqn (3), (4) and (5) : as the continuum representation of eqn (3).Thus : fth(l) = <ri(o) 'k(t))th/<Zi(o) li(O>>th. This is compared in fig. 2 with the real part of eqn (2), the corresponding experi- mental representation. Whereas both curves are very similar, fexp(t) shows very slight collision damping. However the functions in fig. 2 can safely be taken as the norm unaffected by induced dipolar absorption. THE COMPRESSED GAS AT 28.4 AND 46.2 bar As can be seen in fig. 3-6, the absorption bands of the compressed gas have developed high frequency shoulders. This is reflected in the time domain (fig. 4 and 6) where fexp(t) in both cases is considerably damped in comparison with &(t) for free rotation [eqn (6)].The high frequency absorption has been interpreted in terms FAR I.R. OF CBrF3 of eqn (l), whose constants A', B', C and D have recently been predicted by Frost," and are given in table 3 in terms of the rotational constants J and K, the intermolecular separation R, the intermolecular potential energy U(R),and the polarisability factors cq, = 1/3 (all +2al) and 6 = (all -aL). E, S,2, etc. are those defined for eqn (3). TABLE 3.-FROST EQUATIONS parameter JdJ) J4x3 toB' 2B(J+ 1) -1 4nP4exp[-U(R)/kT]dR C ({ 1-exp[-AcGl(J)lkT])x3hcZr o K= -.I exp(-EhClknG(J"4Crifi(J,K)+ (40/3)8'fi (J,K)K2LJ(J+2)11), where fi = (J-K+ l)(J+K+ l)/(J+ 1). 2B(2J+3) 4x3 1" ~TR-~exp[-U(R)/kT]dR 5 {[I-exp(-hc<,(J)/kT])x3h~Zr0 K= -.I exp(-13c/kT)%(J)[18 (J,K)+ (4 815)8'ff2'(JYW11, wherefi.= (J-K+ 2)(J-K+ 1)(J+ K+ 2)(J+K+ l)/[(J+1)(J+2)(2J+ 3)] ~TR-~D 6B(J+2) 2s" exp[ -U(R)/kT]dR 6 {(1 -exp[ -hci,(J)/kn)x3hcZr 0 K--J exp(-Ehc/kT)G3(J)[(24x'+8y)a: + (1 76x'/9+ 16y/3)~8~]}, 5(J+ K+ 3)(J+ K+ 2)(J+ K+ 1)(J-K+ 3)(J-K+ 2)(J-K+ 1)with x' = (J+ 2)(J+ 3)(W+2)(2J+ 3)(2J+ 5) 3 (J+ K+ 3)(J+ K+ 2)(J+ K+ 1)(J-K+ 3)(J-K+ 2)(J-K+ 1)Y= (2J+ 2)(W+ 3)(2J+ 4)(W+ 5)(W+6) Y 3(J-K+ 2)(J-K+ 1)(J+ K+ l)(J+K+ 2)W= (2J+ 2)(W+ 3)(J+ 2) Thus in using eqn (1) we again suffer an aesthetic difficulty in that it yields sets of discrete line spectra (fig. 3 and 5) whereas the experimental absorption is obviously a broad continuum at our resolution and sample pressure.There are also weaknesses inherent in the Frost theory because cross-relaxation between overlapping lines 30 of J --+ J+n transitions is not considered ; also there is the fact that an eigenstate of the interacting pair is taken as the product of the eigenstate of the isolated molecule. This is adequately correct only for a purely central intermolecular potential U(R), which can be approximated by a Lennard-Jones form. Moreover, each J -+ J+n line will be broadened in practice, so much so that J + J+2 lines have rarely 31 been individually resolved. There remain the vexed questions of translational ti and overlap absorption which have been adequately treated only for small linear molecules such as hydrogen. In the case of the relatively heavy CBrF3 molecule pure translational absorption (AJ = 0) should occur at very low fre- quencies and thus not affect A/N or the bandshape very much, although this effect is responsible for the width of the induced infrared lines (deviating, of course, from Frost's delta functions).Van Kranendonk and Kiss found the total A/N due to overlap induction and the interference effect between the quadrupolar and overlap moments in hydrogen to be about 8 % of the total. Ho, Birnbaum and Rosenberg found that neglect of overlap absorption would lead to a value of the quadrupole moment in C02 only about 1% too large. Therefore we feel justified in avoiding an explicit treatment of the overlap absorption in CBrF3 in this work. The limiting uncertainty in the evaluation of eqn (1) is, in fact, linked to the choice of the Lennard-Jones parameters used with the (assumed) radial distribution function.=We have taken values of ~/k 423 K, c = 4.4 A corrected to the same extent as those for CClF3 as was found necessary by Barnes and Sutton 32 to account for their second dielectric virial coefficient data. G. J. DAVIES AND M. EVANS Since we have two unknowns in eqn (1) (lQl and IQl), if octopole-induced absorption is considered important, an analysis of the variation of A with N is not alone sufficient to determine both IQl and IQl. To do this a comparison of bandshape is necessary, using iterative techniques until the optimum matching between experi- mental results and eqn (1) is obtained.This procedure is conveniently and sensitively carried out in the time domain by comparing FeXp(t)with the Fourier transform of eqn (1). In a way analogous to that used in deriving eqn (6) from eqn (3), it can be shown that : * Fa(?) cos(2nnl?ct)dV -co [I-~XP(-hc?/kT)] (V -2BK)2(i+2BK)2K24;(i7--2BK)(G+2BK)~4/2B+4$6~ (G -2B)(G+2B) [18a;f (V, K)+9S2f2(\), K)/i] cos(27rikt) dV +I Here, Fm E = ncn/Z,, I, = J 4nR-" exp[ -U(R)/kT]dR, 0 rK= exp[ -(Ao -B)K2hc/kT], ( = BhclkT, 5[V +6B(K+l)](V +6BK)CV+6B(K-I)] x x"' --[G-6B(K-l)](i-GBK)[V-6B(K+ l)]V 6B(c+6B)(2i-12B)(2i-6B)(2i +6B) G[i+6B(K+ l)](i+6BK)[V+6B(K- I)] x y' = [V-6B(K-l)](V-6BK)[S-6B(K+ l)] 12B(2~-12B)(2v-6B)(2G +6B)(2G+12B) 3(i-6BK)[V-6B(K+l)][V+ 6B(K- I)](? +6BK)w! = 6B(2i-12B)(2V-6B)i FAR I.R.OF CBrF3 The radial distribution integrals (In)in (7) were evaluated with the tables of Buckingham and Pople 33 and Fth(t), normalised as usual to unity at t = 0, was computed on a CDC 7600. The best matches between Fth(t) andf,,,(t) obtained are displayed in fig. 4 and 6, which were those obtained with lQl = 4.,, x e.s.u.,IQl = lox and [ Ql = 4.8x e.s.u., lln[ = 11 x e.s.u. respectively. The matches show the neglect of broadening in each J -+ J+n line, and neglect of cross-relaxation between overlapping lines in the considerable underdamping of Fth(t) as compared withf,,,(t). This, together with the uncertainty in the (Lennard- Jones) parameters of an uncertain angle-independent representation of the inter- molecular potential render the mean values of lQl = (4.4k0.4) x e.s.u.and [Szl = (11-12) x e.s.u. preliminary and tentative. Nevertheless, by symmetry, CBrF3 is not expected to have a large [Q[and our values do not contradict this. I 1 I I I I I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 time/ps FIG.8.-Plot offexp(t) from the data of fig.7 (-) ; free rotor If&)] at 295 K [.. . ..(l)]; F"(t) with Tr = 7 PS,T = 0.65PS, = 0.8PS [---(2)]. THE LIQUID AT 295 K Qualitatively, in the frequency domain (fig. 7), ij,,, has moved only slightly, from the 9.8 cm-l of the gas at 1.2 bar to 11 .O cm-l in the liquid at the same temperature. This observation, together with the fact that the high frequency tail of the absorption is less pronounced than in the compressed gas might lead one to the conclusion that the liquid phase absorption is merely a much broadened version of that of the free rotor.That free rotation is not occurring in the liquid can be seen from fig. 8, wheref,,,(t) is highly damped in comparison with the free rotorfth(t). The function fexp(t) also shows slight superimposed oscillations on the main curve, a phenomenon which has been observed 13* 34 in dense dipolar liquids, where the oscillations are much more pronounced. Neglecting thus the hypothesis of free molecular rotation in the liquid, we assume that the potential experienced by a given molecule due to its nearest neighbours does not have spherical symmetry. One therefore has a certain number of potential wells separated by Using Kauzmann's treatment of this situation one obtains, G.J. DAVIES AND M. EVANS neglecting the inertia of the molecule, an expression identical to the well known solution of Deb~e.~’ Brot 38 has recently revised Kauzmann’s treatment using multiple wells and including the inertial term. The consequences of Brot’s model are : (i) while they are resident in one of the wells, the molecules undergo torsional oscillation (libration) ; (ii) the time of jump (2,) from one well to the next or another is not negligible. With a mean time between adiabatic collisions of z and a mean of residence in the well of 2, the autocorrelation function of a unit dipole vector undergoing such motion is given 39 by : F(t) = (40) U(t>>B,ot where sin at V(t)= exp(-t/z) 1 for coo > 112 = exp(-t/z)( 1+t/z) for coo = l/z a = [co; -1/z2]*, b = [l/r2-co;]*, where wo = 2nv’,c is the proper frequency of libration and where H(t-tj) = exp[-(t+tj)/7,3 [1+ (t-tj)]/za].The second derivative [F”(t)]of F(t),whose negative gives (li(0) zi(t)}Brot, has been evaluated 34 for zr = 7.0 ps, z = 0.65 ps, z, = 0.8 ps and is displayed in fig. 8 with the experimental functionf,,,(t). Superficially, the agreement is good although F”(t)contains three arbitrary parameters (z,za and z,) which reduces its significance somewhat. Nevertheless, zr = 7 ps ought to be roughly comparable with the Debye relaxation time zD (microwave data is needed to confirm this). It is instructive to compare F(t)(with an apparent mean time between “ collisions ” of z = 0.65 ps) with the equivalent autocorrelation function derived from a model which allows of free rotation between the assumed instantaneous collisions.Such is McClung’s J-diffusion model for symmetric top m01ecules,~~ and several of these functions for CBrF, are given in fig. 9 for various mean times between collisions (z). The Brot function F(t) is plotted on the same scale on the same figure, and loses correlation much less rapidly than the McClung function with the same z of 0.65 ps. Unfortunately, since accurate data down to about 0.001 cm-1 are unavailable, the equivalent experimental function is unknown, but judging from the close similarity between the derivatives fexp(t) and F”(t), it will probably resemble Brot’s function rather than McClung’s.The bandshape andf,,,(t) are both influenced by induced absorption, mainly at higher frequencies, and which may be responsible for the short time oscillations in the time function. Gordon l8 predicts the total integrated intensity per molecule due to all rotational type motions (in the microwave and far infrared) as :($) =-2np2 Gordon 3c21A = 2.55 x lo-’’ cm FAR I.R. OF CBrF, whereas the observed value is (table 1) 6.2 x cm. This is reduced to 4.5 x cm after a correction for the (assumed) static l5 internal field.149 l7 [Such a correction does not affect the bandshape and thus not fexp(t)]. This still leaves almost 50 % of the absorption unaccounted for by permanent dipolar absorption, mostly at higher frequencies. 1.0 0.8 0.6 n.c 2=: 0.4 0.2 0.0 timelps FIG. 9.-Vectorial autocorrelation functions calculated from the extended diffusion model of McClung 40 (-).From top to bottom, the curves are for values of the mean time between collisions (T) of 0.1,0.2,0.3,0.5,1.0ps and the free rotor (7 -+ m). F(t)from eqn (8) withTr = 7 ps, T = 0.65 psand Ta = 0.8 ps (---). DISCUSSION Although there is no great change in Ymax from the fairly dilute gas to the liquid, the bandwidth and bandshape change considerably. This indicates that the molecular interaction and motion also vary considerably with progressive compression and liquefaction. The basically simple Frost representation suffers from predicting a series of delta functions at discrete frequencies corresponding to various J + J+n transitions and consequently the equivalent Heisenberg representation [eqn (7)] does not resemble the observed broad continuum absorption Vex&)]satisfactorily.However, the values of I Ql and lsZl obtained agree fairly well for different I?, indicating that three body interactions, leading to terms in N3 in eqn (l), are not significant at these molecular number densitites. Obvious improvements in the theoretical description would be an account of translational absorptions (AJ = 0) and especially translational broadening of each J +J+n line, together with cross-relaxation between overlapping lines. Working in the time domain seems to be a sensitive and convenient method of comparing predicted and experimental absorptions.In the liquid, the problem of separating the contribution of induced dipoles remains unsolved, but the model of permanent dipolar libration within potential wells generated by neighbouring molecules seems to give a fairly satisfactory repre- G. J. DAVIES AND hl. EVANS sentation of the absorption (fig. 8). The comparison of fig. 9 suggests that extended diffusion models such as that of McClung are less realistic in the liquid phase. Experimentally, microwave data on compressed gas-liquid systems are needed to evaluate the autocorrelation function (u(0) u(t)) which will be considerably less affected by induced absorption (being insensitive to data above 10 cm-l), and frequency dependent refractive index data in the whole region (0.001-100) cm-l is needed in order to evaluate the effect of the dynamic internal field in the liquid.We thank the Director of Research at the Post Office for permission to publish this work; M. W. E. thanks S.R.C. for a post-doctoral fellowship. A. I. Baise, J.C.S. Faraduy 11, 1972,68, 1904. I. Darmon, A. Gerschel and C. Brot, Chem. Phys. Letters, 1970, 7, 53. G. Birnbaum, W. Ho and A. Rosenberg, J. Clzem. Phys., 1971, 55, 1028; J. E. Harries, J. Phys. B, 1970, 3, 704. M. Evans, J.C.S. Furuday 11, 1973, 69, 763. J. H. van Kranendonk and Z. J. Kiss, Cunud.J. Phys., 1959,37, 1187. ti J. D. Poll and J. H. van Kranendonk, Cunud. J. Phys., 1961, 39, 189.H. Sutter and R. H. Cole, J. Chem. Phys., 1970,52, 132. D. R. Bosomworth and €3.P. Gush, Canud. J. Phys., 1965, 43, 751. M. Evans, J.C.S. Furaday 11, 1975, 71, 71. lo B. S. Frost, J.C.S. Faruduy 11, 1973, 69, 1142. l1 G. J. Davies and M. Evans, J.C.S. Furuduy 11, 1975, 71, 1275. l2 M. Evans, Mu!. Phys., 1975, 29, 1345. l3 A. Gerschel, I. Darmon and C. Brot, Mol. Phys., 1972, 23, 317. l4 J.-L. Greffe, J. Goulon, J. Brondeau and J.-L. Rivail, J. Chim. Phys., 1973, 70, 282. l5 C. Brot, Dielectrics and Related Molecular Processes (Chem. SOC., London, 1975), vol. 2. l6 W. G. Rothschild, J. Chem. Phys., 1968, 49,2250. l7 G. W. F. Pardoe, Thesis (Universityof Wales, 1969) ; Mansel Davies, Ann. Rep. Chem. SOC. A, 1970, 67, 67; M. W. Evans, Spectrochim.Acta, 1974, 30A, 79; I. Larkin and M. Evans, J.C.S. Faraday ZZ,1974,70,477 ; G. J. Davies, J. Chamberlain and M. Davies, J.C.S. Faradzy 11, 1973, 69, 1223 ; B. Lassier and C. Brot, J. Chim. Phys., 1968, 65, 1723. l8 R. G. Gordon, J. Chenz. Phys., 1963,38, 1724. l9 M. Davies, G. W. F. Pardoe, J. E. Chamberlain and H. A. Gebbie, Trans. Faraday Soc., 1968, 64, 847; G. W. F. Pardoe, Trans. Faruduy SOC., 1970, 66, 2699. 2o G. Vanasse and H. Sakai, Prugr. Optics, 1967, 6, 261. 21 J. Chamberlain, ZnfraredPhys., 1971, 11, 25. 2z J. Chamberlain and H. A. Gebbie, Infrared Phys., 1971, 11, 57. 23 G. J. Davies and J. Chamberlain, J. Phys. A, 1972, 5, 767. 24 P. E. Clegg and J. S. Huizinga, I.E.R.E. Conf. Infrured Techniques, Reading, 1971.25 J. Chamberlain,Infrared Phys., 1972, 12, 145. 26 0.H. Hougen, K. M. Watson and R. A. Ragatz, Chemical Process Principles Charts (Wiley,New York, 1964). 27 A. H. Sharbough, B. S. Pritchard and T. C. Madson, Plzys. Rev., 1950, 77, 3021. 28 J. A. Berm and L. Kevan, J. Phys. Chem., 1969, 73, 3560. 29 R.G. Gordon, J. Chem. Phys., 1965,43,1307. 30 G. Birnbaum, personal communication. 31 S. Weiss and R. H. Cole, J. Chem. Phys., 1967, 46, 644. 32 A. N. M. Barnes and L. E. Sutton, Trans Furuday Suc., 1971, 67, 2915. 33 A. D. Buckingham and J. A. Pople, Trans Furaduy Suc., 1955, 51, 1173. 34 M. Evans, J.C.S. Furuduy II, 1974,70, 1620. 35 J. Frenkel, Acta Physiochiin. U.S.S.R., 1935, 3, 23. 36 W. Kauzmann, Rev. Mud. Phys., 1942,14, 12. 37 P. Debye, Polar Molecules (Chem Catalog Co., 1929). 38 C. Brot, J. Physique, 1967, 28, 789. 39 B. Lassier and C. Brot, Chem. Phys. Letters, 1968, 1, 581. 40 R. E. D. McClung, J. Chem. Phys., 1969,51,3842 ; 1972,57,5478.

ISSN:0300-9238    

DOI:10.1039/F29767200040

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Derivation of Molecular Spectra from the Polarized Spectra of Monoclinic Crystals and its Application to the Electronic Spectrum of Bis(methoxyacetato)diaquacopper(n) BY MICHAELA. HITCHMAN Chemistry Department, University of Tasmania, G.P.O. Box 252C, Hobart, Tasmania 7001, Australia Received 5th May, 1975 A method of determining the molecular absorption tensor of molecules having a single symmetry axis (pointgroups Cs,Cz and Czh)and crystallizing in a monoclinic space group is described. The method is applied to the electronic spectrum of bis(methoxyacetato)diaquacopper(u) and shows that the " d-d " electronic transitions are polarized approximately midway between the bond direc- A tions and the bisectors of OCuO bond angles, probably closer to the bond bisectors, with the most intense absorption occurring when the electric vector lies within theche late ring.A possible explanation for this is considered. The method is also used to derive the g-tensor of the molecule and the non-coincidence of the principal in-plane g-axes and absorption directions is discussed. There has been considerable interest 1-5 in the electronic structure of metal com- plexes having only a single axis of symmetry (point group C,,C2or C2h). In favour- able cases the ground states of such molecules may be studied by e.p.r. spectroscopy and considerable progress has been made in the theoretical interpretation of the g-tensors of molecules of this type.1-3 The intensity of absorption of light is also a tensor quantity, and the principles involved in the deduction of the molecular absorp- tion ellipsoid from single crystal measurements are similar to those encountered in the evaluation of the molecular g-tensor.Despite the fact that an increasing number of complexes of low symmetry are being studied by polarized crystal spectr~scopy,~-~ no attempt has as yet been made to derive the absorption ellipsoid in its complete form, though this should greatly help the elucidation both of the nature of the excited states, and of the intensity inducing mechanisms operating in complexes of this kind. This paper describes a method for deducing this information. DISCUSSION The complete specification of a tensor of a molecule having one symmetry axis z requires the measurement of 4 tensor elements, these being used to derive the 3 principal values and the orientation of the in-plane axes with respect to arbitrarily chosen x and y axes.The conditions limiting the experimental determination of these tensor elements have been discussed in detail for the'derivation of molecular g-tensors from crystals having exchange-narrowed e.p.r. signal^,^ and exactly similar arguments apply to the interpretation of the optical spectra of crystals. First, the crystal sym- metry must be monoclinic or triclinic; the subsequent discussion will be limited to electric dipole transitions in monoclinic crystals. Secondly, two measurements must be made on each of two crystal planes, and at least one of these planes must 54 M.A. HITCHMAN not contain the unique crystal axis. In optical spectroscopy the measurements must be made with the electric vector parallel to the two extinction directions of each crystal face. If this condition is not satisfied, rotation of the vector occurs as the light passes through the crystal and the knowledge of the precise orientation of the vector in the crystal coordinates is lost. Normally, the absorption is measured as a function of wavelength, thus yielding a spectrum. The measurement of the wave- length dependence of the absorption is complicated by the fact that except when it lies along the unique crystal axis, the extinction direction of a crystal face can in principle vary as a function of wavelength. However, while the possibility of such changes should always be investigated experimentally, in practice, for metal com- plexes they are usually found to be too small to be detected.To define the molecular absorption ellipsoid, a molecular coordinate system must be defined. For molecules of C,,C2 or C,, symmetry, z coincides with the unique axis of the point group, while x and y can be chosen as any orthogonal directions perpendicular to z. The first step in the derivation of the molecular spectra is the calculation, for each crystal spectrum, of the projections made by the electric vector upon the molecular axes of each of the two independently oriented molecules in a monoclinic space group, and qualitative discussions of molecular spectra have often been made by correlating changes in the relative intensities of absorption peaks with changes in these projec- tions.If it is assumed that the principal absorption directions coincide with the chosen molecular axes (i.e., if the off-diagonal elements of the absorption tensor are set equal to 0) then the calculation of the molecular spectra is straightforward and this process has been carried out for several c~mplexes.~-~~ It has been suggested lo that this calculation is best performed by comparing the changes in the absorbance with polarization direction for each crystal face [see eqn (4) and (5) of the Appendix], as this then excludes errors due to the measurements of crystal thickness and mini- mizes those due to variations in optical quality between crystal faces.While this method is adequate for molecules in which all of the molecular axes are defined by symmetry, e.g., point groups DZhor C,, (and, incidentally, is also applicable to orthorhombic crystal systems) it requires extension for the lower symmetry molecules under consideration here. The additional determination of the principal directions of the absorption tensor in the xy molecular plane is accomplished by the measurement of the appropriate off-diagonal tensor element, followed by diagonalization of the tensor. In order to deduce the four molecular tensor elements, use must be made of the four parameters defining the crystal absorption ellipsoid. The method described here therefore makes use not only of the absorption magnitudes, but also of the direction of maximum and minimum absorption in the crystal face not containing the unique axis.The mathe- matical procedure by which this is done is given in the Appendix. Note that the method assumes that the extinction directions coincide with the directions of maxi- mum and minimum absorption of light. In practice, this will almost always be the case in studies of spin-allowed "d-d" spectra, as the refractive index of the crystal (upon which the extinction directions depend) will be dominated by its imaginary component, which is proportional to the intensity of light absorption. However, the extinction directions and directions of maximum and minimum absorption are not formally required to be coincident,13 and this feature should always be checked experimentally.Particular care should be exercised in this respect when the ligands contain aromatic ring systems, as the neighbouring intense internal ligand transitions could affect the extinction directions. When experimental measurements can be made on three crystal faces, all four tensor elements can be obtained without making use of this directional information (see Appendix). SPECTRA OF Cu (RIIeOAc),(H,B), APPLICATION OF THE METHOD TO THE OPTICAL SPECTRUM OF B I s(MET HOXY A cE T AT 0)DI AQUAcoPPER (I I) As a typical application of the method we consider the optical spectrum of the complex bis(methoxyacetato)diaquacopper(II), [Cu(MeOAc),(H,O),]. This centro- symmetric complex has approximate C,, symmetry (fig.1) and crystallizes in the monoclinic space group P21/nwith unit cell parameters a = 692, b = 726, c = 1010 pm, y = 96.7".14 However, in the subsequent discussion the b and c axes will be interchanged to comply with the more usual convention defining b as the unique axis and p as the monoclinic angle. To deduce the molecular spectrum we define a molecu- lar coordinate system as follows : z lies along the Cu-O(H,O) direction, y is ortho- gonal to z and the Cu--O(acetate) direction, and x is orthogonal to y and z (i.e., x is almost exactly along the Cu-O(acetate) direction (fig. 1). The spectra of the (OlO), (001) and (100) crystal faces have been reported by Bew et a1.l' The extinction I 6 FIG.1.-The inolecular geometry of bis(inethoxyacetato)diaquacopper(rr) ;the hydrogen atoms have been omitted for the sake of clarity.The molecular coordinate system x,y and z, and directions of the in-plane g axes and electronic absorption axes are illustrated. At the band maximum EX makes an angle of 27+ 7" with the x axis, while the angle made by gx is -8f4". directions in the (010) face happen to lie along the directions of the a and c* axes and were found to be coincident with the directions of maximum and minimum absorption.16 The spectra measured with the electric vector along the a, b and c axes are shown in fig. 2 [the a spectrum is conimon to the (010) and (001) faces]. The principal molecular absorbances E~,cy and cZ (given in arbitrary units) calculated by the method given in the Appendix are also shown in fig.2, while the orientation of the principal absorbance axes x and y (principal tensor parameters will be indicated by bold type) at the band maximum is illustrated in fig. 1. DISCUSSION OF THE ERRORS INHERENT IN THE RESOLUTION OF MOLECULAR SPECTRA The errors occurring in the derivation of a molecular absorption spectrum are derived basically frorn two sources. First, there is the experimental error in the M. A. HITCHMAN measurements (absorbance, angle etc.) ; this is generally easily estimated. Secondly, there is an additional uncertainty due to the transformation from crystal to molecular coordinates, followed by the averaging over the two independently oriented molecules in the unit cell and solution of the set of simultaneous equations to obtain the absorp- tion tensor elements. It is this latter source of error, which has usually been ignored in the past, which can sometimes cause large uncertainties in the molecular spectra. l7 A major contributing factor here is the relative orientation of the two independent molecules in the unit cell, as this decides the degree of independence of the set of equations which must be solved to obtain the absorption tensor elements.The true accuracy of the molecular spectra can be estimated by calculating these for the widest possible extremes of experimental data, and this was done in the case of Cu(MeOAc),-(H20)2assuming a possible error of & 0.03 in all absorbance values and & 3" for the extinction direction in the ac crystal plane.The resulting uncertainty was k0.05, I L I I 1 15 10 energy/103 cm-l FIG,2.-The crystal and molecular spectra of bis(methoxyacetato)diaquacopper(Ir). The crystal spectra are given for the electric vector parallel to the a,b and c crystal axes. See text for the defini- tion of b and c, and the method of calculation of the x, y and z spectra. 0.04 and 0.06 absorbance units in cx, E~ and cZ, respectively, at the absorbance maxi- mum (these values were somewhat lower in the less intense regions of the spectrum). The uncertainty in the angle a between the principal absorption directions and the molecular axes was & 7" at the absorbance maximum, increasing to & 10" on either side of this (9 and 16 x lo3 cm-I). As expected, the uncertainty in the molecular spectra is somewhat greater than that in the crystal spectra, though the effect is not very pronounced, indicating that in this particular case the relative orientation of the molecules does not seriously jeopardize the resolution of the molecular spectra.It is noteworthy that the z spectrum shows a slight negative absorbance in the region 15 to 18 x lo3 cin-l suggesting the existence of some experimental error not taken into account in the present treatment. SPECTRA OF Cu (M~OAC)~(H~O)~ APPLICATION OF THE METHOD TO THE DETERMINATION OF MOLECULAR g-TENSORS The method described here for the determination of the electronic absorption ellipsoid is equally applicable to the deduction of the g-tensors of molecules of C2,or similar symmetry (or indeed to other tensors, such as molecular magnetic susceptibility).The appropriate values of g2 and q need only be substituted for the absorbance values in eqn (4) and (6) of the Appendix. The principal crystal g-values g1 = 2.088 5, g2 = 2.166 2, g3 = 2.344 5 have been reported l5 for Cu(Me0Ac) with g2 lying along the b crystal axis and g1 making an angle of y = -38" with the a crystal axis. Allowing an uncertainty of f0.03 in the values of g2, and 53" in y gives principal molecular g-values of g, = 2.036 k0.015, g, = 2.348 +0.008 and g, = 2.212+0.011, with g, making an angle of a = -8k4" with the molecular x axis. These compare favourably with the values of g, = 2.028 k0.004, g, = 2.368 +0.001, g, = 2.223&0.001, a = 0.6" found by Dawson et aZ.,3 and g, = 2.026 6, g, = 2.344 7, g, = 2.224 1 obtained by Bew et a2.l with g, defined along the Cu-0 (carboxylate) direction.INTERPRETATION OF RESULTS The molecular spectrum of Cu(MeOAc),(H,O), shows three peaks at -9.5, 12.25 and -16.25 x lo3cm-l (fig. 2). Each of these is considerably more intense in x, than y or z polarization, and the peak at -16.25x lo3cm-1 is absent from the z spectrum. The angle a defining the direction of in-plane polarization of the transi- tions is 26+ 10" at 9.5 x lo3cm-l, 27+7" at 12.25 x lo3cm-1 and 16+ 10" at 16.25 x lo3 cm-l, a positive sign indicating rotation from the x axis into the chelate ring. An angle of 40" would occur if the transitions were polarized along the bond angle bisectors, while one of -5" would be expected if the transitions were polarized most nearly along the bond directions.It would seem that the electronic transitions are polarized approximately midway between these extreme situations, though, particu- larly for the more accurately defined intense transition at 12.25 x lo3cm-l, the A polarization directions more nearly correspond to the bisectors of the OCuO bond angles.For a centrosymmetric complex such as Cu(MeOAc),(H,O), the "d-d" transi-tions occur by a vibronic intensity mechanism. In the C2,point-group vibrational modes are available to allow each "d-d" transition in every polarization, so that formal selection rules cannot be used to assign the observed spectrum.Note, how- ever, that one transition is absent in one polarization, that at -16.25 x lo3cm-l in y. If the eflectiue vibronic pointgroup were D2, with x, y and z defined as pre- viously (i,e., approximately along the bond directions), then the transition dxy-+ ~Z ~Z-would be forbidden in y polarization. This lends some support to the hypothe- sis that the peak at -16.25x lo3cm-l is due to a transition between the orbitals containing these functions (both dx2-y2 and dxybelong to A, in the C,, pointgroup). What seems fairly certain is that as quite strong bonds occur along each of the direc- tions x, y and z, the lowest energy peak at -9.25 x lo3cm-l is due to a transition between the level derived from d3z2-rz, and the ground state orbital.This would suggest that the most intense peak at 12.25 x lo3cm-l is due to one or both of the transitions from the two levels derived from the dxyand dyzorbitals. It should be stressed, however, that the assignment of the two higher energy peaks is tentative. In the earlier study of the optical spectrum of Cu(MeOAc),(H,O), it was assumed that the directions of polarization of the optical transitions were coincident with the M. A. HITCHMAN principal g-axes. However, the full analysis of the electronic spectrum presented here shows that the in-plane polarization directions are in fact quite different from the directions of the g-axes (fig. 1) as the latter are quite close to the bond directions. This clearly demonstrates the value of determining the direction of in-plane polariza- tion in the analysis of the spectra of complexes of this kind, and also shows the fallibility of the assumption which has sometimes been made 6* that the polariza- 59 tion directions of the optical spectra must coincide with the principal g-axes.As has been stressed el~ewhere,~. 19* 2o the g-tensor and electronic absorption tensor are derived from quite different sources, and in fact in a low symmetry complex of this type, the difference in the orientation of these tensors can provide useful information on the electronic structure of the compound. In a comparatively ionic complex such as Cu(MeOAc),(H,O), the g-tensor is dominated by the metal parts of the molecular orbital containing the unpaired electron.The orientation of the g-tensor therefore essentially reflects the effect of the ligand field upon the d-orbitals, and the very large difference between g, and g, and the close coincidence of the g axes with the Cu-0 (carboxylate) and Cu-0 (methoxy) bond directions results from the fact that the oxygen atom of the carboxylate group produces a much greater o-perturbation than that of the methoxy group.3 As shown previously,lg 3* l9 a large difference in 6-bonding power produces significant admixture of the dX2-,,z and d3z2-rZorbitals, and this tends to dominate the in-plane g anisotropy. The polarization directions of the '' d-d " electronic transitions, however, are dominated by the comparatively small ligand components of these molecular orbitals, so that it is the electronic structure of the ligands rather than simply that of the coordination polyhedron around the copper ion which is important in deciding the intensities and polarization properties of the electronic transitions.While the complicated electronic structure of the molecule precludes a detailed interpretation of the absorption intensities in Cu(MeOAc)(H,O), with the data presently available, it is noteworthy that many planar complexes of DZhsymmetry of the general form Cu(acetylacetonate), exhibit strong in-plane polarization of the electronic transitions, the absorption being most intense when the electric vector bisects the chelate ring.g* lo*21 There is also some evidence 22 that this is the case in complexes of the type trans-bis(N-alkylsalicyla1diminato)copper(II), which have a rather similar microsymmetry to Cu(MeOAc),(H,O),.In all of these complexes, as in Cu(MeOAc),(H,O),, the most intense spectrum occurs when the electric vector is directed approximately towards the centre of the chelate ring. The polarization properties of the "d-d " transitions in these complexes have been related to the symmetry of neighbouring charge transfer states in which an electron is trans- ferred from an essentially non-bonding set of" lone pair '' ligand orbitals to the half- filled orbital centred largely on the copper lop 23 The carboxylate and methoxy- oxygen atoms in Cu(MeOAc),(H,O), also contain "lone pair " orbitals lying approximately in the plane of the chelate ring and the fact that the polarization direc- tions of the electronic transitions lie more nearly along the bisectors of the bonds than along the bonds themselves, and the relative intensities of the x and y spectra, suggest that possibly a similar intensity mechanism may be making a significant contribution in this complex also.This is interesting, as Cu(MeOAc),(H,O), differs from the acetylacetonato complexes in the fact that the low symmetry of the ligand field produces a ground state wavefunction with lobes which differ in extent along x and y, the former being much larger than the latter.3 The intensity mechanisms operating in transition metal compounds are by no means well understood and it would seem that the investigation of the polarization directions of the electronic transitions in low symmetry complexes should provide useful information in this area.For instance, it would be interesting to know the in-plane polarization direc- SPECTRA OF Cu (McOAc),(H20), tions of a planar complex of C2h symmetry in which only one of the donor atoms of each chelating ligand has a ‘‘ lone pair ” of electrons available to take part in a charge transfer transition (e.g., a trans amino acid complex). APPENDIX We consider the absorption occurring when polarized light passes through a crystal. Let the orientation of the electric vector in the crystal be dehed by three angles y, 8 and 4 (fig. 3). Here 8 and 4 define the crystal face perpendicular to the light path, and y defines b a ii c FIG.3.-Definition of the angles 7,8and 4 used to define the position of the eIectric vector E in the crystal coordinates a, b, and c.the position of the electric vector E in the face. The projections of E on the crystal coor- dinates a, b and c are given by :24 Eu = Eccos /?* (1 -E,” sin2 p* -E,”)* (14 (the sign is decided by which of the quadrants defined by the (100) and (001) crystal planes E occupies; the positive sign is taken when E is in a quadrant containing the positive a axis) : Eb = sin y sin 8 (W E, = (sin tj cos 8-cos y tan 4) cos 4/sin p*. (14 These projections are converted to a set of orthogonal crystal coordinates a’,b’, c’ by the relationships : E,. L= E, sin /?*; Ebr = Eb; E,.= Ec-Ea COS /?* (2) The projections Exl,Ex, etc. made by the electric vector upon the two independent molecules (labelled 1 and 2) in the monoclinic cell are : Eyl = .yu’ Yb’ Yc’ ; E~2 -xb’xb’ 42 = Ya’ -Yb‘Yc’ -&] (3) 2,‘Ez1 [”‘zb’ 2,’ [&I EZ, -2,’ -2b’ Z,J [E~J Here the orthogonal molecular coordinate system is defined in accordance with the point- group C,,C2or C2h(Le., z is defined by symmetry, x and y are not) and xa*, etc., are the M. A. HITCHMAN projections niade by the molecular coordinate vectors upon the orthogonal crystal co-ordinates. The absorbance of light A by the crystal is given by : where gij is the molar extinction coefficient tensor defining the light absorbance of the mole- cule (note &,,,.zyx), c is the molar concentration of the complex and t is the crystal thickness.To measure ~~ithe absorbance must be measured with E along each of the extinction direc- tions of two crystal faces. Let the measured absorbances at a fixed wavelength be A(1) and 42) for the first face and 43) and A(4) for the second face. Substitution into eqn (4), followed by division of the equation containing A(1) by that containing A(2),and that con- taining A(3) by that containing A(4), and some straightforward algebra yields two equations of the form : (41)[Exl(2)2+Ex2(2)21/42) -ado2 -Ex2(1)21&xx+ -* * +2(A(l)[EX1(2)E,l(2>+ Ex2(2)EYZ (2)1/&2) -Ex1(1PY 1(1)-Ex2 (1142(1>IEx, = 0 (54 (A(3)[ExI (4124-Ex2(4)’I/A (4) -Ex1(3)2 -E,2@)2) Ex, -I---* +2 (A(3)[Ex1(4)Ey 1(4) + Ex2 (4)Ey2 (91/A(4) -Ex1(3)EyL(3) -Ex2 (3)Ey 2 (3)} = 0 (5b) Note that by taking the ratios A(l)/A(2)and A(3)/A(4),the thicknesses of the two crystal faces are removed from the equations.Note also, that when E,, = 0 (i.e., when the complex belongs to a pointgroup such as D2hwhere the axes are defined by symmetry elements) these two equations are sufficient to allow the ratios E,, : E~~ etc. to be determined. Let the second crystal face be one which does not contain the unique crystal axis. For this face (defined by 8 = 6’ and 4 = 4) the maximum absorbance A(4) occurs when E is defined by the angle y= ij, i.e., dA/dq = 0 when = #.* Differentiation of eqn (4) yields : From eqn (3) : From eqn (2) : From eqn (1) : d(E,)/dy = cos y sin 8 (9b) d(E,)/dV = [cos + (cos ycos 6 +sin ytan +)]/sinz /?*.(94 * As noted in the text, although there is no formal requirement that the extinction directions should coincide with the directions of maximum and minimum absorption, this should almost always be the case in practice where spin-allowed “d-d” transitions are being studied. If this condition is not met, the four tensor elements can be determined if the spectra of a third crystal face can be measured. The three equations of the form given in eqn (5) can then be solved for the ratios E~~ : Eyy :EZZ :Exy. = SPECTRA OF Cu (MeOAc),(H,O), Substitution of the values ij, 8 and $ into eqn (6), via the relationships given in eqn (7)-(9), plus the two expressions of the form given in eqn (5) yields three simultaneous equa- tions which may be solved to give the ratios of the tensor elements E,, :E,, :gZz :E,,,.Dia-gonalization yields the ratios of the principal tensor elements zx:E, : E= and the angle between the principal x and y axes and the molecular x and y axes. If the thickness of one of the crystal faces is known, substitution into eqn (4) readily gives the absolute values of the prin- cipal tensor elements. To generate the molecular spectrum the process is performed at successive wavelengths, the whole process being conveniently performed by a computer program.t M. A. Hitchman, C. D. Olsen and R. L. Belford, J. Chem. Phys., 1967, 50, 1195. M. A. Hitchman, B. W. Moores and R. L. Belford, Inorg. Chem., 1969, 8,1817.K. Dawson, M. A. Hitchman, C. K. Prout and F. J. C. Rossotti, J.C.S. Dalton, 1972,1509. B. W. Moores and R. L. Belford, Electron Spin Resonance of Metal Complexes, ed. T. F. Yen (Plenum, New York, 1969), p. 17. B. J. Hathaway and P. G. Hodgson, Spectr. Acta, 1974, 30A, 1465. C. W. Reimann, G. F. Kokosyka and H. C. Allen, J. Res. Nut. Bur. Stand. A, 1966,70, 1.’R. J. Dudley, B. J. Hathaway and P. G. Hodgson, J.C.S. Dalton, 1972, 882. J. Ferguson, J. Chem. Phys., 1961,35, 1612. R. L. Belford and J. W. Carmichael, Jr., J. Chem. Phys., 1967, 46,4515. lo M. A. Hitchman and R. L. Belford, Inorg. Chem., 1971, 10,984. l1 P. L. Meredith and R. A. Palmer, Inorg. Chem., 1971,10, 1049. l2 S. G. Lipson and H. Lipson, Optical Physics (Cambridge University Press, 1969), pp.308-311. l3 J. A. Mandarino, Amer. Miner., 1959, 44, 65. l4 C. K. Prout, R. A. Armstrong, J. R. Carruthers, J. G. Forrest, P. Murray-Rust and F. J. C. Rossotti, J. Chem. Soc. A, 1968, 2791. l5 M. J. Bew, D. E. Billing, R. J. Dudley and B. J. Hathaway, J. Chem. SOC.A, 1970, 2640. l6 B. J. Hathaway, personal communication. l7 T. S. Piper and R. L. Belford, Mol. Phys., 1962, 5, 1969. l8 D. E. Billing, R. Dudley, B. J. Hathaway, P. Nicholls and I. M. Procter, J. Chem. SOC.A, 1969, 265 ; B. J. Hathaway, M. J. Bew and D. E. Billing, J. Chem. SOC.A, 1970,1090. l9 M. A. Hitchman, J. Chem. SOC.A, 1970,4. 2o R. J. Dudley, B. J. Hathaway and P. G. Hodgson, J.C.S. Dalton, 1972, 882; R. J. Dudley,B. J. Hathaway, P. G. Hodgson, P. C. Power and D. J. Loose, J.C.S. Dalton, 1974, 1005; B. J. Hathaway and P. G. Hodgson, Spectrochirn. Acta A, 1973, 1465. 21 F. A. Cotton and J. J. Wise, Inorg. Chem., 1967, 6, 917; B. J. Hathaway, D. E. Billing and R. J. Dudley, J. Chem. Soc. A, 1970, 1420. 22 J. Ferguson, J. Chem. Phys., 1961, 35, 1612. 23 M. A. Hitchman, Znorg. Chem., 1974, 13, 2218. 24 M. A. Hitchman and R. L. Belford, Electron Spin Resonance of Metal Complexes, ed. T. F. Yen (Plenum, New York, 1969), p. 97. (PAPER 5/826) j. A copy of the program, written in ALGOL 3 is available on request from the author.

ISSN:0300-9238    

DOI:10.1039/F29767200054

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Structure of Contact Ion Pairs in the Ground and First Excited States Aromatic Closed-shell Anions containing a Five-membered Ring BY H. W. Vos,* C. MACLEANAND N. H. VELTHORST Chemical Laboratory of the Free University, De Lairessestraat 174, Amsterdam, The Netherlands Received 6th May, 1975 The absorption and fluorescence spectra of the carbanion of indene have been recorded as a function of temperature, solvent and counter ion. In ethereal solvents indenyl-lithium forms contact and solvent-separated ion pairs, depending on the temperature, whereas only contact ion pairs occur when Na+ and K+are the counter ions. The absorption spectra exhibit a blue shift on forma- tion of contact ion pairs, but in the fluorescence spectra a red shift is observed. The results for the indenyl anion are compared with those for the carbanion of fluorene and the anion of carbazole.The differences in cation-anion attraction and fluorescence shift are discussed with the aid of energy level diagrams, and it is concluded that these differences are ultimately determined by the r-electron charge distribution of the anion in the ground and first excited states. Theoretically calculated charge densities conflrm this conclusion ; an exception is the carbazolyl anion, where the nitrogen lone pair electrons play an important role, leading to the formation of a o-complex. The structures of the contact ion pairs of the two carbanions indenyl and Auorenyl in the ground and first excited states are inferred from the calculations. An apparent discrepancy with proton n.m.r.results is resolved by a theoretical examination of the effects of the cationic field on the proton chemicalshifts. It is well known that ion pairs of aromatic anions and alkali metal cations can exist in two forms: contact ion pairs (Ar-, M+) and solvent-separated ion pairs (Ar-11 M+); the latter is spectroscopically indistinguishable from the free solvated ions, which are present in strongly polar solvents. From many investigations 2-9 it has become clear that the change from solvent- separated ion pairs to contact ion pairs invariably leads to a displacement of the absorption spectrum to higher energies. This phenomenon (which we shall call the " ion-pair shift ") has been explained by the redistribution of the negative charge of the anion on excitation; it leads to a relative destabilization of the first excited (S,) state with respect to the ground (So) ~tate.~-~ Until now ion-pair structures in the S, state, which can be studied by emission spectroscopy, have hardly been investigated.Both high energy 2* * and low energy lo ion-pair shifts have been observed in the fluorescence spectra of aromatic ion pairs. In this paper we present a detailed investigation of the fluorescence spectrum of the indenyl anion (InH-), which has been reported to be temperature de~endent.~ It will be shown that InH- can form, depending on solvent and temperature, contact and solvent-separated ion pairs in the S, state, and that there is an ion-pair shift to low energy. These results, together with those for some other carbanions and nitrogen-containing anions reported previously,' are analysed with the aid of energy level diagrams, which in turn will be correlated with MO charge density calculations. The discrepancy with a reported structure of the ion pair indenyl-lithium is discussed and resolved.63 CONTACT ION PAIRS EXPERIMENTAL The absorption and fluorescence spectra of InH- have been recorded for solutions in 2-methyltetrahydrofuran(MTHF) from +20 to -180°C and in 1,Zdimethoxyethane (DME) from +20 to -80°C, with the cations Li+, Na+ and K+. The preparation of the carbanions l2and other experimental details have been described el~ewhere.~~ The fluorescence measurements have been performed with an improved apparatus, which makes possible the control of the cell temperature from room temperature down to liquid nitrogen tempera- ture.RESULTS AND DISCUSSION SPECTRA OF THE INDENYL ANION The results of the present investigation are summarized in table 1. Some of the absorption and fluorescence spectra of InH- are shown in fig. 1 and 2. 0/103cm-I FIG.1.-The absorption spectra of indenyl-lithium in MTHF at three different temperatures. TABLETH THE ABSORPTION AND FLUORESCENCE MAXIMA OF INDENYL ANION @/lo3cm-') absorption fluorescence solvent cation +20°C -80°C -16OOC +20°C -8OOC -16OOC MTHF Li+ 29.4 28.8 26.7a 19.7 20.0 21.8f Na+ 28.1 27.9 28.2 20.4 20.7 20.9 K+ 27.6 27.6 27.6 20.4 20.6 20.7 DME Li+ 29.1 26.5 -19.8 21.7 -Na+ 28.1 26.5 -20.4 20.8g -K' 27.8 27.4' -20.4 20.4 -4shoulders at 24.4, 25.5 and 27.2 x lo3cm-' ; shoulders at 26.0, 27.5 and 29.0x lo3cm-' ; c shoulders at 25.4, 26.9 and 28.3 x lo3cm-' ;dshoulders (see a) visible, but badly resolved ;e from ref.(8) ;fshoulder at 23.2 x lo3cm-' ; 20.5 x lo3cm-' at -6OOC. H. W. VOS, C. MACLEAN AND N. H. VELTHORST ABSORPTION SPECTRA The absorption spectra of the samples with Li+ in MTHF and DME and with Na+ in DME show a conversion from a contact to a solvent-separated ion pair as the temperature is lowered. Those with Na+ in MTHF and with M+in both solvents display only contact ion pairs. The "conversion temperature " (i.e. the temperature at which 50 % of both types of ion pair are present) is approximately -120°C for Lif in MTHF, -30°C for Li+ in DME and -50°Cfor Na+ in DME.The spectra of the various solvent-separated ion pairs are almost identical. The absorption spectrum of the free solvated anion in hexamethylphosphoric triamide (HMPT) has a maximum at 26.7 x lo3crn-l.* The absorption maxima of the contact ion pairs shift to higher energy as the cation becomes smaller ;the magnitude of the ion pair shift is slightly dependent on solvent and temperature. The results are in agreement with those found for other carbanions:6-9 the tendency to form solvent-separated ion pairs increases when a more polar solvent is used (DME is more polar than MTHF), the temperature is lowered or the cation is smaller. I .f? 24 22 20 18 16 403cm-I FIG.2.-The fluorescence spectra of indenyl-lithium in MTHF at three different temperatures.The intensities of the spectra at 3-20 and -100°C are multiplied by factors 16 and 8, respectively. FLUORESCENCE SPECTRA Inspection of table 1 shows that the shifts observed in the absorption spectra have counterparts in analogous, but opposite shifts in the fluorescence spectra. The " conversion temperatures " are approximately -110°C with Li+ in MTHF and -50°C with Li+ in DME; these temperatures are comparable with those found in the absorption spectra. The ion pair InH-,Na+ in DME seems to behave differently: a conversion to InH-11 Na+ is reflected in the absorption but not in the fluorescence spectrum. However, in view of the fact that the fluorescence maximum of this ion pair undergoes a shift of 300 cm-l in going from -60" to -8o"C, we suppose that the conversion 11-3 CONTACT ION PAIRS has just begun at -8O"C, the lowest temperature attainable in this solvent.The " conversion temperature " will be about -9O"C, which is 40°C lower than deter- mined from the absorption spectra. Below -160°C the fluorescence maxima of all three ion pairs in MTHF show a further shift to high energy (400-800cm-l at -18OOC), which is accompanied by a considerable change in band structure. We ascribe this effect to the increased viscosity of MTHF at these low temperatures, which may cause incomplete stabiliza- tion of the Franck-Condon excited state ;the result is a blue shift of the fluorescence spectrum.In our previous work 'we could not distinguish between this viscosity shift and the fluorescence ion-pair shift, hence the latter was not recognized. COMPARISON WITH THE SPECTRA OF RELATED SYSTEMS From the results in the foregoing section we may conclude that the tendency to form contact ion pairs is roughly equal for the So and S, states of the indenyl anion. This does not agree with the results for some benzo-condensed indenyl systems :the fluorenyl anion (FlH-) and the 4,5-methylenephenanthrenylanion (MH-) have much greater tendencies to form solvent-separated ion pairs in their excited states than in their ground states.'' lo Moreover, the absolute value of the ion-pair shift for FlH- is considerably smaller in the fluorescence than in the absorption spectrum,' O whereas there is only a small difference for InH-.Nevertheless, in both carbanions a fluores-cence ion-pair shift to low energy has been observed. The results for InH- do not agree either with those for the nitrogen-containing analogues of FlH- and MH-, namely the carbazolyl anion (Cb-) and 4,5-iminophenan- threnyl anion (Im-).' These systems do not show ion-pair conversion when the temp- erature is varied over the usual range. In ethereal solvents contact ion pairs are formed exclusively, which means that the cation-anion attraction is even greater than in the InH-contact ion pairs. The most remarkable difference is encountered in the fluorescence ion-pair shift of Cb- and Im-, which is in the same direction as the absorption ion-pair shift : to high energy. The absorption and fluorescence ion-pair shifts of Cb-and Im- are approximately equal in magnitude.ENERGY LEVEL DIAGRAMS To account for the different results obtained for the fluorescence ion-pair shifts of a number of structurally related carbanions and nitrogen-containing anions, we give a detailed discussion of the various factors which may influence the position of the fluorescence bands of contact ion pairs. We focus our attention on three typical systems, namely InH-, FlH- and Cb-. The spectral positions of the bands of these anions are indicated schematically in fig. 3. In this figure the maxima in different solvents have been averaged for each type of ion pair. THE ABSORPTION ION-PAIR SHIFT In spite of the differences in fluorescence shift and cation-anion attraction, all ion pairs investigated thus far resemble each other in one important respect: the ion-pair shift is always to high energy.Apparently, the formation of a contact ion pair invariably results in a greater stabilization of the ground state with respect to the Franck-Condon excited state which the system is in immediately after excitation [see fig. 4(a)]. The relative destabilization of the latter state has been explained by several authors 3-6 (although somewhat differently formulated) in the following way. In the ground state the cation is located near that part of the anion which has the H. W. VOS, C. MACLEAN AND N. H. VELTHORST highest charge density.Upon excitation the charge distribution changes, while the cation retains its position. Consequently, the electrostatic cation-anion interaction diminishes. THE FLUORESCENCE ION-PAIR SHIFT For our discussion of the various possibilities for the fluorescence ion-pair shift, we start from the incomplete energy level diagram of fig. 4(a). In this diagram arbitrary values have been given to the absorption energies of both ion pairs and the s K+ ~4+Li+ abs. InH' fi. I I I Li+ N~+.K+ s abs. S \ K+k* Li' \I / FlH- f I. Ill /I\\Li+ I& K+ S s~ K+ Li+. _. Cb' abs. m . 20 .2:. 2.4 1 *6j 1. 20 . M. I1 I fl. II 1S K' Li* wavenumber/1O3cm-I FIG.3.-The absorption and fluorescence spectral positions of InH-, FlH-and Cb-.The positions of the spectral maxima (InH-and Cb-) or the 0-0transitions (FIH-) are indicated for the solvent- separated ion pairs (denoted by s)and the contact ion pairs (denoted by the cation involved). energy --so S C S C (Model S C ( intermediate) FIG.4.-Energy level diagrams which describe the absorption and fluorescence ion-pair shifts. S and C denote solvent-separated and contact ion pairs, respectively. fluorescence energy of the solvent-separated ion pair. For convenience we have made the (not strictly necessary) supposition that the Franck-Condon (FC) de-stabilization energies of the ground and excited state of a certain system are equal. CONTACT ION FAIRS Obviously the fluorescence energy of the contact ion pair may have different values.We can distinguish two extreme cases, which we shall refer to as models 1 and 2. In model 1 [fig. 4(b)] we assume that the FC destabilization energies of the contact ion pair are equal to those of the solvent-separated ion pair. This implies that the S1 lowest vibronic state is less stabilized by the cation than the So lowest vibronic state : the equilibrium stabilization energy of the ion pair is larger in the ground state than in the excited state. On the other hand, in model 2 [fig. 4(c)] we suppose that the equilibrium (lowest vibronic state) stabilization energies of the So and S, states are equal, so that the FC energies in the contact ion pair should be larger than they are in the solvent-separated ion pair.These two models represent extreme cases, an ion pair will generally conform to an intermediate model [fig. 4(d)], in which both the equilibrium stabilization energies and the Franck-Condon energies are different for the two types of ion pairs (see table 2). TABLE2.-sUMMARY OF THE RELATIONS BETWEEN THE GROUND AND FIRST EXCITED ST.4TE PROPERTIES AND THE FLUORESCENCE ION-PAIR SHIFT model 1 model 2 general extent of charge localization in So and S1depth of potential minimum equilibrium stabilization energy 1 different equal different in So and S11position of potential minimum area of charge localization Franck-Condon energy equal different different fluorescence ion-pair shift blue red either direction, may be small example Cb- InM- FlH- The fluorescence ion-pair shift can easily be derived from fig.4(b)-(4 : in model 1 it is equal in magnitude to and in the same direction as the absorption ion-pair shift (blue shift). In model 2 the fluorescence ion-pair shift is also equal in magnitude but has the opposite direction (red shift). In the intermediate case the fluorescence ion-pair shift may be small and have either direction. Comparing these results with the data of fig. 3 it is clear that Cb- agrees well with model 1, InH- largely satisfies model 2 whereas FlH- appears to be an example of the intermediate case. FACTORS DETERMINING THE ENERGY TERMS To answer the question why the spectra of different aromatic anions are described by different energy level diagrams one should know which factors influence the equilibrium stabilization and FC energy terms.The equilibrium (lowest vibronic state) stabilization energy of the contact ion pair is largely determined by the electrostatic attraction between the cation and the anion. The cation will be located at a position of minimum potential energy in the electrostatic field of the anion. The position of the potential minimum will be near that part of the anion at which the major part of the negative charge is localized. The equilibrium stabilization energy is equal to the depth of the potential minimum, which will depend on the extent of charge localization in its neighbourhood. It is therefore required for model 1 that the charge of the anion is more localized in the 13.W. VOS, C. MACLEAN AND N. H. VELTHORST So state than in the S1state, whereas for model 2 the charges must be equally localized (see table 2). The FC destabilization energy is related to the difference in geometry between the lowest vibronic So and S, states. The FC energy arising from the change in anion geometry will not increase appreciably on the formation of contact ion pairs. The difference in the FC energies of the two ion pairs in model 2 should therefore arise from an important change in the equilibrium position of the cation with respect to the anion on excitation. This will be caused by a displacement of the position of the potential minimum, which implies a displacement of the area of charge localization.Such a displacement will be absent in model 1, since the FC energies of both ion pairs are equal in this model (see table 2). So far no attention has been paid to the polarizing influence of the cationic field on the charge densities of the anion. The cation will induce such a change of the charge distribution that the electrostatic stabilization of the ion pair increases. As this effect arises in all possible cation-anion conformations, the transition energies will hardly be affected. We will show in a following section, however, that this effect may account for the anomalous chemical shift differences found in the proton n.m.r. spectrum of indenyl-lithium.ll CALCULATIONS METHOD In the foregoing section it has become clear that the energy terms which SUM up to the ion-pair shift are ultimately determined by the charge distributions in the ground and first excited states.To explain the direction of the fluorescence ion-pair shift it is therefore necessary to calculate these charge distributions theoretically. We have used the semiempirical x-electron Variable Electronegativity SCF method,14 which is an extension of the familiar Pariser-Parr-Pople method and gives a more realistic description of the charge distribution.15 We have applied the following modification in the VESCF-method: the one-centre repulsion integrals have been taken proportional to, instead of quadratically dependent on the Slater effective charge. The atomic valence state parameters have been taken from Hinze and Jaff6.17 The two-centre repulsion integrals have been calculated according to Mataga and Nishimoto.* The two-centre core integrals have been given the following values : pCc = -2.26 eV,19 PCN= -2.40 eV. The latter parameter is chosen somewhat arbitrarily, but its value is close to those used in the literature.20 Since the actual geometries of the anions are unknown we have assumed planar structures consisting of regular polygons with sides of 1.39 A. Our program includes an option to incorporate the influence of the electrostatic field of the cation into the calculation. This has been done in a way analogous to the method of McClelland,*l by adding a term to the diagonal elements Hppof the core matrix. These elements represent the energy of an electron located on an atom p in the field of the positive core : Ip denotes the valence state ionization potential of atom p.The second term on the right-hand side gives the energy of an electron on atomp in the field of all other core centres q(#p) ;Zq is the positive core charge of q and ypq is the repulsion integral between electrons on atoms p and q. Analogously, the third term represents the energy of an electron on p in the field of the cation rn. The metal-carbon repulsion integrals yplrl are calculated by the multipole expansion method. 22 We shall refer CONTACT ION PAIRS to the method with inclusion of the cationic field as the "polarizable anion " (PA) approximation ; without this inclusion we have the "rigid anion " (RA) approxi-mation.The cation-anion attraction energy in the ground state Eigis calculated according to Ei, = En-E," +E,,,,. Enis the total n-electron energy of the contact ion pair, E,"is the corresponding energy of the undisturbed anion and E,,,, is the repulsion between the cation and the anion core; all core-centres are assumed to be point charges. The attraction energy in the excited state E$, is derived from ET, = Eip+AE-BEo AE and AEo are the excitation energies of the contact ion pair and the free anion, respectively. The excitation energies and the excited state charge distributions are obtained using the virtual orbital approximation. It has been shown by 'Li n.m.r. chemical shift measurernent~,~~ that in contact ion pairs with carbanions like InH- and FlH-, the Li+ cation is located above the ring system.We have therefore performed the calculation, both with the PA and RA approximations, for a number of cation positions at a constant distance of 3.0 A from the anion (3.0 A is probably a good estimate for the Na+-carbanion distance 6). Equipotential lines have been obtained by interpolation of fitting polynomials. COMPARISON OF THE THEORETICAL AND EXPERIMENTAL RESULTS The theoretical n-electron charge densities in the So and S, states of the free anions InH-, FIH- and Cb- are shown schematically in fig. 5. The equipotential lines of InH- and F1H- in both states, obtained with the PA approximation, are shown in fig. 6 and 7.The calculated transition energies of the free ions InH- and FlH- and of their contact ion pairs with the cation located in the So and S, state potential minima, are presented in table 3. TABLE3.-THE CALCULATED TRANSITION ENERGIES OF InH-AND FlH-AE"/lOJcm-1 AE(s0)a/103 cm-1 AE(S1)b/103 cm-1 InH- 25.7 27.8 23.6 FlH- 23.2 25.4 22.3 aThe transition energy when the cation is located in the ground state potential minimum and b in the excited state. The negative charge of the indenyl anion is largely localized on the five-membered ring in the ground state, whereas most charge has been displaced to the six-membered ring in the excited state (fig.5, top). This is reflected by a corresponding displacement of the potential minimum on excitation (fig.6); we may conclude that the cation will be located above the five-membered ring in the So state and above the six- membered ring in the SI state. The potential minima in both states are approxi- mately equally deep (-4.90 eV in the So state and -4.92 eV in the S1 state). In view of these results InH- may be expected to satisfy model 2, which is in good agreement with the experimental results. The calculated (absolute) values for the absorption and fluorescence ion-pair shifts of InH- are both 2.1 x lo3cm-l (see table 3) ;they are somewhat larger than the experi- mental shifts of the sodium ion pairs, which amount to 1.5 and 1.4 x lo3 cm-l, respectively (see fig. 3). H. W. VOS, C. MACLEAN AND N. €1. VELTHORST In fluorenyl anion (fig.5, middle) the negative charge is spread over three rings instead of two, it is therefore not surprising that the tendency to form contact ion pairs is smaller than in InH-. From fig. 6 and 7 it appears that the cation-anion attraction energy in the ground state is indeed smaller for FIH-(-4.68 eV) than for InH-. In the ground state the charge is slightly localized in the central ring, and consequently the potential minimum is located there (fig. 7) ; we may expect that the cation is positioned above this ring.24 In the excited state the negative charge is InH-s1 -.-SO 1.04 1.17 FLH' 110 I .07 1.17 1.2I i 0.86 104 so ! 116 Cb- 1.096 1 0 7 I/ 1.05- 1.00 107 ! 117 1.06 I. 10 134l 101 FIG.5.-~-Electron charge densities of InH-, FlH-and Cb- in the ground (So) and first excited (St) states.FIG.6.--Equipotential lines for a cation in a plane at a distance of 3A above the ring system of InH-: bottom, ground state ;top, excited state. Potential energies are indicated in eV. predominantly localized in the two side-rings. In the PA approximation there are now two potential minima (fig. 7), which are less deep (-4.56 eV) than the one in the So state. It is likely that the cation will jump back and forth between the two CONTACT ION PAIRS side-ring minima. The changes in ion-pair structure and cation-anion attraction on excitation agree with the experimental result that FlH- satisfies the intermediate model. In the RA approximation the minimum still remains above the central ring, although it has become very broad.The relative importance of the FC energy term in FlH-, which follows from the observed red shift, is not well explained by the RA result. FIG.7.-Equipotential lines for a cation in a plane at a distance of 3 above the ring system of FIH-: left-hand side, ground state ;right-hand side, excited state. Potential energies are indicated in eV. The calculated values for the absorption and fluorescence ion-pair shifts of FlH- are 2.2 and 0.9 x lo3crn-l, respectively (see table 3). The magnitudes of the corres- ponding experimental shifts of the sodium ion pairs are 1.4 and 0.3 x lo3cm-l, in the same order (see fig. 3). It appears that the theoretical results qualitatively account for the differences between the absorption and the fluorescence ion-pair shifts of FlH-.However, the calculated values are consistently too large, just as for InH-. Plodinec and Hogen-Esch lo have explained the red fluorescence ion-pair shift of FlH- in terms of an energy level diagram which is essentially identical with that of model 2. These authors, however, did not note that the red shift in the fluorescence spectra, expressed in energy units, is much smaller than the blue shift in the absorption spectra, whereas model 2 requires that these shifts are (approximately) equal. Also this model does not account for the greater tendency to form solvent-separated ion pairs in the excited state than in the ground state. The pure model 2 is therefore inadequate to describe the energy levels of F1H-.The charge distributions in the carbazolyl anion (fig. 5, bottom) resemble those of FlH- to a great extent, with the difference that more charge is located on the more electronegative nitrogen atom and less on the neighbouring carbon atoms. The result is that the equipotential lines of Cb- are very similar to those of F1H-. It is obvious that the large experimental difference in cation-anion attraction between these two anions cannot be explained by considering the n-electron charge distribution only. The 7Li n.m.r. chemical shifts of Cb-, Li+ indicate that the cation is located near the nitrogen atom, probably in the plane of the anion.25 The strong association H. W. VOS, C.MACIrEAN AND N. Ii. VELTHORST between Cb- and alkali metal cations may therefore arise from an interaction between the cations and the nitrogen lone pair electrons, leading to the formation of a 0-complex rather than a n-complex: We are now able to explain the experimental result that Cb- satisfies model 1 with the assumption that the cation is co-ordinated to the nitrogen atom lone pair both in the Soand S1states, so that there is no displacement of the cation on excita-tion. The smaller cation-anion attraction in the S, state is caused by the greater delocalization of the negative charge. COMPARISON WITH PROTON N.M.R. RESULTS The structures of some contact ion pairs of InH-and FIH-have been inferred from proton chemical shift data by Van der Kooij et aZ.ll*26 They concluded from the chemical shift differences between InH-, Li+ and InH- [I Lif in THF (measured at +40 and -37"C, respectively) that the Li+ cation is located near the six-membered ring of InH-in the contact ion pair, as the differences are larger for the protons 4-7 attached to this ring (see fig.8). On formation of a contact ion pair with the cation positioned above the n-system, the electron density in each C-H bond will be dis- placed somewhat towards the C-atom, resulting in a smaller shielding of the proton ; the proton resonance will consequently be shifted downfield. This shift will be larger the closer is the cation to a particular C-H bond. We shall call this effect the "direct action " of the cation on the proton chemical shift.4 expartrnantal:O.29 0 22 017 0.12 calculatod 10.66 039 0.31 022 diruct action : h-indiract action : --rosult ( f told -) FIG.&-Top, experimental and calculated differences in proton n.ni.r. shifts (in p.p.m.) between contact and solvent-separated ion pairs ;bottom, schematic indication of the effects on the proton chemical shifts arising from a cation located above the five-membered ring. The above conchsion disagrees with the present result that in the ground state the cation is located over the five-membered ring. This discrepancy may be resolved by taking into account another effect, which we shall call the '' indirect action " of the cation on the chemical shift. It is well known that the n-electron charge densities strongly influence the proton chemical shifts of an aromatic system. A cation above the five-membered ring will influence the n-electron charge distribution of the car- banion: a part of the negative charge will be displaced from the six- to the five- membered ring.By a mechanism similar to that described above the decrease of the negative charge in the six-membered ring will cause a further downfield shift of the protons 4-7, whereas the increase of the negative charge in the five-membered ring 74 CONTACT LON PAIRS results in an upfield shift of the protons 1-3. The effects of the direct and the indirect action of the cation are shown schematically by arrows in fig. 8 ;it appears that the sum of these contributions, dependent on their magnitudes, may result in a greater downfield shift for the protons 4-7 than for the protons 1-3.The effect of the cation on the charge distribution of the carbanion has been calculated by the semiempirical MO method described. The cation was assumed to be located in the potential minimum of fig. 6, 3.0A above the ring system. The influence of the charge distribution in the whole ion pair on the proton chemical shift has been obtained using a formula given by M~sher.~' The results are also indicated in fig. 8. It appears that the calculated shifts are of the right order, which justifies our supposition that the inclusion of the indirect action of the cation resolves the apparent discrepancy between the optical and the n.m.r.spectroscopic results. However, the theoretical shifts are about twice the experimental ones. A theoretical overestimate of ion-pairing effects is often encountered in calculations of this type 26p 28 (see also the foregoing section). It may arise from partial solvation of the cation in the contact ion pair, which screens the cation field to some extent. To account for this effect Takeshita and Hirota 28 have added a screening factor E to the de- nominators of the cation-anion electrostatic interaction terms. Following their suggestion, we have obtained a quantitatively more satisfactory agreement with E = 2. Another parameter which greatly affects the results of the calculations is the position of the cation, which is not known precisely. In view of the purely empirical nature of E, the uncertainty of the cation position and the approximate nature of the theoretical method we feel that attempts to obtain quantitative agreement of theory and experiment have little significance.Similar reasoning as has been given for indenyl-lithium resolves an analogous discrepancy for fluorenyl ion pairs. In this carbanion formation of a contact ion pair results in chemical shift displacements which are approximately equal for all 29* 30 This need not be explained by jumping of the cation between the side rings 26 when the indirect action discussed above is included. CONCLUSIONS It has been shown that the various experimental results for the fluorescence ion- pair shift in the systems InH-, FlH- and Cb- may be interpreted by a careful analysis of the different energy contributions to this shift.There are two important factors which determine the absolute value and the direction of fluorescence ion-pair shift : first, the equilibrium stabilization energies in the So and S1states, which depend on the extents of charge localization of the negative charge in both states, and may be enhanced by cation-lone pair interactions ; secondly, the Franck-Condon de-stabilization energies, which are determined by the difference in the areas of charge localization in the So and S1states. The positions and depths of the potential energy minima for the cation, which correspond with the extents and areas of charge localization, respectively, may be predicted theoretically by n-electron MO calculations for carbanions.More satisfactory results are obtained when the polarization of the anion by the cation is included. In nitrogen-containing anions like Cb- the specific interaction of the cation with the nitrogen lone pair electrons is important. Ion-pairing effects on proton chemical shifts may be described qualitatively with the results of similar calculations, provided the polarization of the n-system by the cation is taken into account. H. W. VOS, C. MACLEAN AXD N. H. VELTHORST The present investigations have been carried out under the auspices of the Nether- lands Foundation for Chemical Research (S.O.N.) and with financial aid from the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).The authors thank Mr. A. Th. van Kessel for assistance in performing the experiments. for a review see J. Smid in Ions and Ion Pairs in Organic Reactions, ed. M. Szwarc (Wiley- Interscience, New York, 1972). vol. 1, p. 85. Th. Forster and H. Renner, 2.Elektrochem., 1957, 61, 340. H. V. Carter, B. J. McClelland and E. Warhurst, Trans. Faraday SOC.,1960, 56,455. K. H. J. Buschow, J. Dieleman and G. J. Hoytink, J. Chem. Phys., 1965,42, 1993. N. H. Velthorst and G. J. Hoytink, J. Amer. Chem. Sac., 1965, 87, 4529. T. E. Hogen-Esch and J. Smid, J. Amer. Chem. Sac., 1966, 88, 307. D. Casson and B. J. Tabner, J. Cltem. Sac. B, 1969, 572. * H. W. Vos, H. H. Blom, N. H. Velthorst and C. MacLean, J.C.S. Perkin 11,1972, 635.H. W. Vos, G. G. A. Rietveld, N. H. Velthorst and C. MacLean, to be published. lo J. Plodinec and T. E. Hogen-Esch, J. Amer. Chem. SOC.,1974, 96,5262. J. van der Kooij, N. H. Velthorst and C. MacLean, Chem. Phys. Letters, 1972, 12, 596. l2 H. W. Vos, Y. W. Bakker, N. H. Velthorst and C. MacLean, Org. Magnetic Resonance, 1974, 6, 574. ''J. J. Dekkers, G. Ph. Hoornweg, C. MacLean and N. H. Velthorst, Chem. Phys. Letters, 1973, 19, 517. l4 R. D. Brown and M. L. Heffernan, Trans. Faraday SOC., 1958,54, 757. G. Karlsson and 0. M&tensson, Theor. Chim. Acta, 1969, 13, 195. l6 L. Paoloni, Nuovo Cimento, 1956, 4,410. ''J. Him and H. H. Jaffk, J. Amer. Chem. Soc., 1962, 84, 340. '* N. Mataga and K. Nishimoto, 2.phys. Chem. (Frankfurt), 1967, 13,40. l9 G. Hafelinger, A. Streitwieserjr. and J. S. Wright, Ber. Bunsenges.phys. Chem., 1969, 73, 456. 2o see, for instance, H. Baba and I. Yamazaki, J. Mol. Spectr., 1972, 44, 118. B. J. McClelland, Trans. Faraday Sac., 1961, 57, 1458. 22 R. G. Parr, J. Chem. Phys., 1960, 33,1184. 23 R. H. Cox, H. W. Terryjr. and L. W. Harrison, J. Amer. Chem. SOC.,1971,93,3287. 24 see also R. Zerger, W. Rhine and G. D. Stucky, J. Amer. Chem. Sac., 1974,96, 5441. 25 R. H. Cox, Canad. J. Chem., 1971, 49, 1377. 26 J. van der Kooij, Thesis (Free University, Amsterdam, 1971). 27 J. I. Musher, J. Chem. Phys., 1962, 37, 34. '* T. Takeshita and N. Hirota, J. Amer. Chem. SOC.,1971,93, 6421. 29 R. H. Cox, J. Phys. Chem., 1969, 73,2649. 30 J. B. Grutzner, J. M. Lawlor and L. M. Jackman, J. Amer. Chem. SOC.,1972,94,2306. (PAPER 5/857)

ISSN:0300-9238    

DOI:10.1039/F29767200063

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Three-dimensional Lattice Model for the Water/Ice System BY G. M. BELL* Mathematics Department, Chelsea College (University of London), London SW3 AND D. W. SALT Mathematics Department, Portsmouth Polytechnic, Hampshire Terrace, Portsmouth Received 12th May, 1975 The sites of a body-centred cubic lattice are either vacant or occupied by molecules with four hydrogen-bonded “ arms ”, two positive and two negative, directed tetrahedrally. Appropriately oriented nearest-neighbour molecules can form bonds by linking bonding “arms ” of opposite polarity. There are also interaction energies for unbonded nearest neighbours and second-nearest neighbours. As well as disordered states, two long range ordered structures are possible, one an open bonded network on a diamond sub-lattice, like ice I(c), and the other a close-packed structure with intertwined bonded networks, like ice VII.A free energy, depending on three order variables, is derived using a zeroth-order statistical approximation modified to allow for asymmetrical bonding. For suitable energy parameter values, there exist four phases, vapour, liquid, open ice and close- packed ice. There is a critical point, an open ice/liquid/vapour triple point and a close-packed ice/open ice/vapour triple point at a lower temperature. Between the triple points increase of pres-sure causes the open ice to transform into a liquid of higher density than the ice while below the lower critical point it causes it to transform to close-packed ice. 1. INTRODUCTION The structure of both fluid water and ice at low pressures depends on the water molecule’s capacity for forming tetrahedrally directed hydrogen-bonds [for more recent discussions see ref.(2) and (3)]. The resulting open four-coordinated network becomes less stable than closer-packed forms as the pressure increases. The present paper follows a number of studies of lattice fluids which show water-like properties resulting from the competition between regions of open structure and regions of high density and energy. These have been ~ne-dimensional,~* two-dimensional 6*’* and three-dimensional,8* the three-dimensional work being most relevant here. Several more recent lattice fluid models of water have, like that of been based on the body-centred cubic (b.c.c.) lattice.Lavis l3 has introduced long-range order into the two-dimensional model of Bell and Lavk7 The aim of the present paper is to introduce long-range order into a three-dimensional model similar to Bell’s and hence consider ice phases as well as fluid phases. The molecule is regarded as having four “ bonding arms ”, two of positive and two of negative polarity, and a bond is formed between nearest neighbours when a positive arm links with a negative arm. The lattice is of the b.c.c. type on which two forms of ice structure are possible. These are ice I(c), an open structure with half the b.c.c. lattice sites vacant, and ice VII, a close-packed structure consisting of two intertwining ice I(c) networks. The use of the b.c.c.lattice thus implies the approximations of replacing hexagonal ice I(h) with cubic ice I(c), which is close to it in local structure and free energyY3 and of replacing all the various close-packed 76 G. M. BELL AND D. W. SALT forms of ice by ice VII. A limitation is thus, in effect, imposed on the configuration space over which statistical averages are taken. However, the results of both the previous work on the fluid phases of water and that presented here indicate that enough configuration space is retained to represent some of the most important characteristic phenomena displayed by water. Since the possibility of long range ordering makes the analysis much more com- plicated than for the fluid phases alone,* the statistical treatment is simplified by using the zeroth-order (molecular field) approximation instead of the first-order approxima- tion.The zeroth-order method is adjusted to allow for the formation of bonds between arms of opposite polarity only and it is found that the entropy of a completely bonded network is then equal to that given by Pauling’s approximation. The zeroth- order approximation smoothes out the anomalous features of the liquid phase but does enable us to consider long-range ordered (ice) phases. Another modification here of the fluid phase theory concerns the “ close-packing ” energy parameter which is necessary to ensure that open ice is more stable at low pressures than the intertwined form. Bell imposed this close-packing “ penalty ” by assigning a posi-tive energy to the smallest fully occupied triads of sites on the b.c.c.lattice. Here better results were obtained by assigning a positive (i.e., repulsive) energy to second- neighbour pairs on the b.c.c. lattice. There may be some connection between this and the change of statistical approximation. After minimising the free energy with respect to order parameters we plot the equilibrium free energy against number density for various temperatures. For suit- able sets of parameter values we then derive phase diagrams involving the four phases, vapour, liquid, open ice and close-packed ice. The well-known experimental property of the low pressure form of ice melting into a denser liquid is reproduced theoretically. 2. THE MODEL The b.c.c.lattice can be divided into four facecentred cubic (f.c.c.) sub-lattices labelled 1, l’, 2 and 2’ respectively (see fig. 1). A pair of diamond sub-lattices can be constructed by grouping 1with 1’ and 2 with 2’ or alternatively by grouping 1 with 2’ and 2 with 1’. When preferential distribution on diamond sub-lattices is considered we shall choose, for definiteness, the first of these groupings. The tetrahedrally directed bonding arms of a molecule on any site are regarded as pointing towards four of the eight nearest neighbour sites on the b.c.c. lattice. Since two of its arms are positive and two negative a molecule on, say, a site of sub-lattice 1 has twelve distinct orientations in six of which its arms point towards 1’-sites (as shown in fig.1) and in the other six to 2’4tes. We denote the fractions of occupied sites in the four f.c.c. sub-lattices by pl, PI., p2 and p2. respectively and the overall number density p is then given by P = $(Pl+Pl,+P,+P,f). We denote the proportion of 1-sites occupied by molecules oriented towards 1’-sites by p11. and the proportion occupied by molecules oriented towards 2’-sites by p12.. Similarly defining p21p, p22J, plt1,plg2,p2t1and p2f2we have P1 = Pll’+P12‘1 Pz = P21’+P22’, P1* = P19+P1‘2’ P2‘ = P2’1+P2‘2. The basic parameters will now be introduced, starting with the volume uo per b.c.c. lattice site. This is regarded as a constant determined by the distance of closest approach of two molecular centres, which in our model is the distance between LATTICE MODEL FOR THE WATER/ICE SYSTEM nearest-neighbour lattice sites.An unbonded nearest-neighbour pair of molecules has energy -Eand a bonded pair -(E+w) (E > 0,w > 0), w being the bonding energy. Experimentally, forms of ice, like ice VII, with intertwined bonded networks only occur at high pressures and hence must involve high energies. In our model this requirement is met by assigning a positive energy u2 to each second-neighbour pair of molecules on the b.c.c. lattice. We now consider possible configurations on the lattice at zero absolute temperature (T= 0). .* FIG.1.-Sites on the body-centred cubic lattice :a molecule is placed on site 1 with bonding arms directed towards 1’-sites. OPEN ICE In perfect ice I(c) all sites of one diamond sub-lattice (taken as 11’) are occupied by a completely bonded network while all sites of the remaining diamond sub-lattice (22’) are empty.Hence, p1 = p11. = p1. = p1.1 = 1, p2 = p2’ = 0. (2.3) Eqn (2.3) implies that all molecules on I-sites are oriented towards 1’-sites and vice-versa. However, for complete bonding it is necessary, in addition, that each positive arm be directed towards a negative arm. The latter condition still allows a large number of orientational states, giving rise to the well-known zero-point en- tropy.14*l5 The value of the latter given by the approximate statistical theory of the present paper is discussed in section 3. There are no occupied second-neighbour pairs of sites (e.g., 12 or 1’2‘in fig.l), each molecule is bonded to four others and the volume per molecule is 2v0. Hence the configurational enthalpy of an assembly of A4 mole-cules is given by H,= ~M(~v~-E-w) (2.4) The perfect crystal to which eqn (2.4) applies exists only at T= 0 and in section 4 we introduce ordering parameters which enable us to consider ices (long-range ordered states) at T> 0. CLOSE-PACKED ICE In perfect ice VII there are two diamond sub-lattices occupied by completely bonded networks. Hence, p1 = PI11 = p1t = p101 = 1, p2 = p22. = p2. = p2.2 = 1. (2.5) The zero-point entropy per molecule is the same as in perfect open ice. As in the latter, each molecule is bonded to four others but it now has four unbonded G.M. BELL AND D. W. SALT 79 nearest-neighbour and six second-nearest neighbour molecules on the other diamond sub-lattice (e.g., in fig. 1, the molecule on site 1 is bonded to nearest neighbours on 1’-sites, has unbonded nearest neighbours on 2’-sites and second nearest neighbours on sites such as 2). Since the volume per molecule is now u0, the enthalpy of a close- packed assembly of M molecules is given by H, = M(Pv~-2~ -4~+3U2). (2.6) Like eqn (2.4), eqn (2.6) applies only at T = 0. SEPARATION PRESSURE Since enthalpy and Gibbs free energy are equal at T = 0, open ice is more stable than the close-packed form if the expression (2.4) is less than the expression (2.6). Now, they are equal when p = po,po being given by pouo = 3U,-2&. (2.7) Hence, if 3u2 > 28 there is a range of pressures 0 < p <po where open ice is more stable than close-packed ice at T = 0.The close-packed form is more stable at T = 0 for p > po. In our model it would also be possible, with an extra energy parameter, to consider ice VIII. The latter has a structure like VII but with the zero-point entropy eliminated by long-range ordering of the proton positions.2* However, we do not introduce this type of ordering, both for simplicity and because it is not likely to be important in the region of the triple points involving ice I. In the present model there is a possible ordered interstitial state, with, say, diamond sub-lattice 11’ occupied by a bonded network and f.c.c. sub-lattices 2 and 2’ respectively empty and occupied by unbonded molecules.It can be shown that there is a pressure range at T = 0 in which this form is stable if --E > w. This is undesirable since all forms of ice known experimentally are fully bonded, apart from imperfections. In fact, we take E =-0 in all calculations so that an ordered phase of this type is unlikely to occur. 3. FREE ENERGY Since a molecule on, say, f.c.c. sub-lattice 1 has twelve distinct orientational states, six with its bonding arms directed towards 1’-sites and six with them towards 2’-sites, the zeroth-order entropy of distribution on the lattice is -+p12.In P12’-+(l-pl) In (l-p,)+ 6 N being the total number of sites on the b.c.c. lattice. It is now convenient to introduce the term “ oriented pair ” which denotes a pair of nearest-neighbour mole- cules each with a bonding arm directed towards the other.Now AS is not the entire entropy since there is also a contribution due to the fact that the arms which an “oriented pair ” direct at each other are more likely than not to be of opposite polarity. Each 1-site, say, has four nearest-neighbour 1‘-sites and in the zeroth order approximation there are aN(4plplf)= Nplplf nearest-neighbour 1-1’ pairs of mole- LATTICE MODEL FOR THE WATER/ICE SYSTEM cules, of which Npl1~p,~,are oriented pairs. Denoting the total number of nearest- neighbour pairs by NAg and the total number of oriented pairs by No,, Nk2 = N(P1P 1 + P 1P2 + P2P1 + P2 P2J (3.2)and Nor = N(p11'PI'1 + ~12'~2'1+ + ~22'~2'2)* (3.3)~21'~1'2 Since each site has six second-neighbour sites, the number NA: of second-neighbour pairs of molecules is NIn",!= W(PlP2 + PlJP24. (3.4) The configurational energy contains terms --N~~E+N~~U~,due to the first and second neighbour energies, as well as a bonding energy contribution. To evaluate the latter, note that each oriented pair has 36 orientational states in half of which the bond- ing arms have opposite polarity.Since there is a bond energy -w we assign a free energy 18+ 18 exp(w/kT) 1+ exp(w/kT)-+(T) = -Win 36 = -kT In 2 (3 5) to each oriented pair. It is not difficult to show that, when kT/w < 1, $(T) = w -kT In 2 + 0{kTexp(-w/kT)) (3.6) while when kT/w % 1, 4(T)= +w+O(w2/kT).(3.7) We may now write, for the configurational Helmholtz free energy F,, Fc = -N(I)&+p)Umm mm z-Nor4(T)-TAS (3.8) where the terms on the right-hand side are given by equations (3.1-5). It is useful to define a free energyf, per site by fc(p, T)= FCm (3.9) Then the pressure p and the configurational part p, of the chemical potential are given by luc = (?fm)T, PUO = PPc-fc. (3.10) In the fluid state there is no long-range order and hence p11. = p12' = p21. = p22. = p1.1 = p1.2 = p2'1 = p292 = +p. (3.11) Hence, substituting in (3.1-4)and (3.8), fc(P, T) = -{d)(T)+4~-3~~)p~+kT Y2p In -+(l-p) ln(1-p)} (3.12){and then, from (3.10) pvo = -(~(T)+4~-33~~)p~-kTIn (3.13)(1-p). The critical-point conditions are apiap = 0, a2PpP = 0, (3.14) which, from (3.13) are satisfied by the critical values pc,pc and T,, given by pc = +, pcc0 = kT,(ln 2 -+), kT, = +(4(T,)+ 4~-3u2).(3.15) The first two relations of eqn (3.15) are the same as in the standard zeroth-order lattice fluid model but the last relation, which is an implicit equation for T,, is char- acteristic for the present model. From (3.7) the entropy contribution from +(T) G. M. BELL AND D. W. SALT disappears when kT/w $ 1 and hence, for high T, the configurational entropy S, is given by S, = AS = kN p In -+(l-p) In (1-p) .{pz I This is correct since it corresponds to a random distribution of orientations and justifies the factor 2 in the last expression for 4(T)of eqn (3.5). In the ice structure at very low temperatures it can be seen from eqn (3.6) that there is an entropy contribution of -k In 2 from &T) for each of the 3M oriented pairs.Hence, from eqn (2.3) and (3.1) the configurational entropy is S, = AS-2Mk In 2 = Mk(1n 6-2 In 2) = Mk In (3/2). This is just the Pauling approximation l4 which is, in fact, a reinarkably good estimate. 4. ORDER VARIABLES AND EQUILIBRIUM RELATIONS Choosing 11’ and 22’ as diamond sub-lattices of the b.c.c. we define order para- meters a, m, and m2 by the following relations, which satisfy eqn (2.2) : P1 = P(’+O) = P1’7 p2 = P(l-= p2.7 (4.1) pllt = +p(l+o)(l+m,) = plP17 plZf = +p(l+a)(l-nzl) = plI2, (4.3) p221= &1(1--0)(l+n2,) = p2p2, p2,’ = 3p(1-a)(l-nz2) = p2r1. (4.3) Both the open and close-packed forms of long-range order can be treated with the aid of these parameters.Non-zero values of a imply preferential concentration of molecules on one sub-lattice and positive values of m, and m2 that oriented pairs are more likely when both molecules are on the same sub-lattice. Substituting into eqn (3.1-3.4), (3.8) and (3.9), the free energy per site becomes fc = -p2[4& -h2(I -a2)+$$(T)(4+ (in, +rn2)2+20(m1-nz2)(nz1+inl +2) + ~~(m;-2mlm2 +4m1 +4m2)>]++in; P3kT[2p In -+p((l +a) In (1 +a)+(1-a) In (1-0))+ l2 (1 -p-pa) In (I -p-po)+(l -p+pa) In (1 -p+po)+ +p(l +a){(l +ml)In (1 +m,)+(l -ml)In (1 -inl)> + +p(l -){(I +m,) In (1 +m2)+(l-mJ In (1 -m,)> . (4.4)1 The free energy has a stationary value with respect to the order variables if the equilibrium relations are satisfied. After some manipulation these relations can be expressed in the form 6u2p0 = 0, (4.6) kT Ifin, -2 In 1-m, --p{!??1(1 +a)+m,(l -o)+2o)$(T) = 0, (4.7) -1n--kT l+m, p(1111(1 +0)+1722( 1-0) -2O)#( T) = 0.2 3 -11112 a2fIaa2 ayC/aaam, a2y~aQam, A = a2f,/aaam, a2f/am; a2f/dmidm2 > 0. (4.14) a2f,/aadm2 a2f,lam,am, a2ffclam; A = AF = (kT-2p4(T)){ -$u,kTp+-(W2 -P~~~(T)}.(4.15)4(1 -P) The boundary of stability between states F and C is given by equating the first factor in AF to zero, which yields eqn (4.12). The boundary of stability between states F and 0 is given by equating the second factor in AF to zero. At high values of p the first factor reaches zero before the second as the temperature is decreased, but for lower values of p the situation is reversed.It should be understood that even though, at a given point (p,T)a certain type of solution gives the lowest free energy it does not follow that the corresponding state will exist there since it may well be possible to further reduce the free energy by phase separation. 5. RESULTS AND DISCUSSION To perform numerical work values of the energy parameter, ratios E/W and u2/w must first be selected. The ranges of suitable parameter values are fully discussed by G. M. BELL AND D. W. SALT Salt? The main criteria are that the separation pressurep,, given by eqn (2.7),shall be positive and that the liquid/vapour critical temperature T,, given by eqn (3.15), shall be greater than the highest temperature at which equilibrium relation solutions of type 0 can exist.0:I 0,2 0:3 0.4 0:5 0,6 0,7 0,.8 0.9 1.00-I I I I I 13 I I I !z-0.1 -P FIG.2.-Curves of reduced Helmholtz free energy per site,f,/w, against number density p for u2/w = 1.0, E/W = 1.45. pm denotes the boundary between close-packed ice and the liquid phase. The broken lines are double tangents. (a)kT/w = 1.1, (b) kT/w = 1.15, (c) kT/w = 11275,-(d)kT/w = 1.45. To investigate the system at a given value of T,it is first necessary to find the ranges of p in which solutions of types F, C and 0 respectively give the lowest free energy LATTICE MODEL FOR THE WATERIICE SYSTEM per site fc. The next step is to plot this value of fc against p.Finally the range or ranges, if any, of p in which phase separation occurs are found by drawing double tangents to the (&I) curves. The full phase diagram results from repeating this procedure at different temperatures. Fig. 2 shows the type of (f,,p) curve occurring at various temperatures if a suitable choice of parameters is made. In fig. 2(a) the left-hand trough of the (f,,p) curve occurs in a low density range where only a disordered state exists and thus corresponds to the vapour phase. The middle trough occurs in a range round p = 4where solu- tions of type 0 give the lowest free energy and thus corresponds to open ice. Hence the left-hand double tangent represents a vapour/ice I phase equilibrium. On the right the free energyf, in the range (pm,l) is for a solution of type C, but for p < pm it is for a disordered state F.Hence the right-hand double tangent represents an 1.8-(0) . --.-.' * '-VAPO U R 1.4-i 3. /// ---___5 1.2-ii li 1.0-// CLOSE -PACKEDOPEN ICE ICE 0.8' I I l > I 5, 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.9 0.9 1.3 '-I1.0I I \ OPEN ICEL11 ,I0.80 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.e 0.9 1.0 e FIG.3.-Density against temperature phase diagram for (a)uz/w= 1.0, E/W = 1.45 ; (b) u2/w= 0.633, E/W = 0.9. The horizontal lines connect coexistent phases. open ice-close packed ice (ice VII) equilibrium. Fig. 2(b) corresponds to a higher temperature and the important change from 2(a)is that the right-hand double tangent now touches the high-density trough where p < pm.Hence there is equilibrium between open ice and a denser liquid phase. The ice I/ice VII/liquid triple point has thus occurred in the temperature range between fig. 2(a)and 2(b). Fig. 2(c) is drawn for a higher temperature at which the two double tangents coincide ;this is the ice I/ liquid/vapour triple point. In a temperature range above the triple point solutions of type 0 still give lower values offc than F or C in a range around p = 3. However fig. 2(d)shows that the corresponding trough in the (f,,p) curve lies above the double- tangent drawn between the liquid and vapour troughs. Hence phase-separation into liquid and vapour gives a lower free energy than formation of ice I.Fig. 3(a) is a density against temperature phase diagram for the parameter values used in the free energy curves of fig. 2 while fig. 4(a) is the corresponding pressure against temperature phase diagram. Fig. 3(b)and 4(b)are respectively density against G. M. BELL AND D. W. SALT 85 temperature and pressure against temperature phase diagrams for a different set of parameter values. The phase diagrams show a liquid/vapour critical point, a liquid/ vapour/open ice triple point and a liquid/open ice/close-packed ice triple point. Between the two triple points there is, at each temperature, an open ice phase in equilibrium with a denser liquid phase. The model thus reproduces the phenomenon of “ice floating on water ”. At temperatures lower than the lower triple-point tem- perature, increasing the pressure sends the open ice phase over into a close-packed ice phase.It is of interest to note that, in contrast to the “ second-neighbour exclu- sion ” quadratic lattice melting model discussed by Runnels l7 the existence of a zero-point entropy does not prevent liquid/solid transitions. The difference is pre-sumably that in the present model the structure of the solid “ices” is as much determined by the attractive directioiial bonding forces as by the “ hard-core ” repulsion. I 0.4 0.4 II CLOSE -’I 0.35 0.35 PACKED’ CLOSE, PACKED 0.3 ICE 0.3 I il 0.25 0.25 LIQUID 3 I -1 0 f$ 0.2 @. 2 ”./c.-j/ UR ”OPEN, VAPOUR 0.15 0.15 ICE/ i/ 0. OPEN-0./ /I I / 0.05 /” ,3.05 / I , .1---0.8 1.0 1.2 1.4 1.6 1.8 0.9 1.0 1.1 1.2 1.3 1.4 (a> kTlw (b) FIG.4.-Pressure against temperature phase diagram for (a) u2/w = 1.0, EIW= 1.45; (b) UZ/W = 0.633, E/W = 0.9. As well as the satisfactory features just mentioned there are some unsatisfactory features in the results. The transition between close-packed ice and liquid is second- order, which results from these two states not giving separate ‘‘troughs ” on the (f,,p) curve as seen in fig. 2 above. Second-order transitions are quite common in lattice melting models when the ‘‘hard core ” of the molecule occupies only a small number of sites,l7-I9 the small number being unity in the present case. Although the open icelliquid coexistence pressure decreases with temperature near the lower triple point the curve turns upwards near the higher triple point. Where this coexistence LATTICE MODEL FOR THE WATER~ICE SYSTEM curve has a positive slope the dense liquid must have a lower molar entropy than the open ice.The positive slope is very small in the case shown in fig. 4(a), but we have not been able to find parameter values where it completely disappears. It is hard to tell whether this results from deficiencies in the lattice model or the zeroth-order statistical approximation and we are currently attempting to apply a first-order approximation to the problem. However, in spite of these difficulties this simple model agrees well enough with experiment to confirm that the water phase diagram depends on the possibility, due to hydrogen bonding, of an open four-coordinated structure occurring.The non-bonding energy parameters E and 2i2 which have been used rep- resent, in a schematic way, a wide range of real interactions due to electrostatic, induction and dispersion forces as well as steric effects for unbonded molecules.2 The fact that the ranges of these parameters must be carefully chosen to give reasonable results indicates that many characteristic phenomena in water depend on a rather delicate balance between different types of interaction energy as well as on the essential structural factors. D. W. S. thanks Chelsea College for a research studentship. J. D. Bernal and R. H. Fowler, J. Chem. Phys., 1933, 1, 515.D. Eisenberg and W. Kauzmann, The Structures and Properties of Water (Clarendon, Oxford. 1969). N. H. Fletcher, The Chemical Physics of Ice (Cambridge University Press, London, 1970). G. M, Bell, J. Math. Phys., 1969, 10, 1753. G. M. Bell and D. W. Salt, MoZ. Phys., 1973, 26, 387. G. M. Bell and D. A. Lavis, J. Phys. A, 1970,3,427. G. M. Bell and D. A. Lavis, J. Phys. A, 1970,3,568. G. M. Bell, J. Phys. C, 1972, 5, 389. G. M. Bell and H. Sallouta, MoZ. Phys., 1975, 29, 1621. lo 0. Weres and S. A. Rice, J. Amer. Chem. SOC., 1972, 94, 8983. l1 D. E. O’Reilly, Phys. Rev. A, 1973, 7,1659. l2 P. D, Fleming and J. N. Gibbs, J. Stat. Phys., 1974, 10, 157, 351. l3 D. A. Lavis, J. Phys. C, 1973, 6, 1530. l4 L. Pauling, J. Amer. Chem. SOC.,1935, 57, 2680. l5 E. H. Lieb and F. Y.Wu, in Phase Transitions and Critical Phenomena, ed. C.Domb and M. S. Green (Academic Press, London, 1972), vol. 1. I6 D. W. Salt, Ph.D. Thesis (London, 1974). l7 L. K. Runnels, in Phase Transitions and Critical Phenomena, ed. C. Domb and M. S. Green (Academic Press, London, 1972), vol. 2. D. M. Burley, in Phase Transitions and Critical Phenomena, ed. C. Domb and M. S. Green (Academic Press, London, 1972), vol. 2. l9 R. D. Kaye and D. M. Burley, J. Phys. A, 1974, 7,843. (PAPER 5/889)

ISSN:0300-9238    

DOI:10.1039/F29767200076

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Quantitative Chemically Induced Nuclear Polarization (CIDNP) Study of the Kinetics of the Photolysis of Pivalophenone in Various Solutions BY P. G. FRITHAND K. A. MCLAUCHLAN* Physical Chemistry Laboratory, Oxford University, South Parks Rd., Oxford OX1 342 Received 19th May, 1975 A simple mechanistic study of the photolysis of pivalophenone in a series of solvents showed that under suitable conditions a unique reaction mechanism obtains. In all the solvents investigated benzoyl and t-butyl radicals are produced which undergo two competitive cage reactions, recombina- tion and disproportionation. In chloroform at 310K 15 % of the radicals recombine and 85 % disproportionate ; in the fluorocarbon PP9 with added tetrachloromethane the figures at 310 K are 17 % and 83 % respectively and the activation energies of the two processes differ by 16.322.0 kJ mol-'.This observation is used to rationalise unusual behaviour in the electron polarization phenomena of the radicals. The overall pseudo first-order rate constant for the scavenging of t-butyl radicals in neat chloro- form was measured directly by flash-photolysis e.s.r. techniques as 5.44f 0.13 x lo3s-' ; CIDNP exposed that two scavenging routes occur: hydrogen abstraction with a rate constant kk = 2.54 -+0.07 x lo2dm3 mol-' s-l and chlorine abstraction with a rate constant kE, = 1.842 0.07 x lo2dm3 mol-' s-l. In PP9 the second order rate constant for reaction of t-butyl radicals with tetrachloro- methane is 4.9 x lo4 dm3 mol-i s-l, with an activation energy of 13.844.2kJ mol-'.These kinetic results are compared with literature values and shown to be consistent with previous observa- tions of the t-butyl radical. The spin-lattice relaxation time of the protons in this radical is 2.22 0.7 x s in chloroform solution and its diffusional correlation time is 41t_:1 x lo-'' s ; this is consistent with theoretical predictions of the time required to produce polarization in a solution having the viscosity of chloro-form. The whole provides a stringent test of kinetic CIDNP theory as applied to radical reactions in solution. In a recent paper we demonstrated that a kinetic analysis of the time-dependence of CIDNP signals treated on the diffusion model produced sensible results ; it is the object of this communication to investigate this model in more depth and to expose the detailed kinetic information which can be obtained from CIDNP studies of care- fully-chosen systems under controlled conditions.As before, a quantitative analysis is possible only after the mechanism is established and this results from an initial qualitative study. The molecule investigated was pivalophenone in which reaction proceeds via an excited triplet state produced on light absorption. Although photoreduction occurs in hydrogen-donating solvent^,^ Norrish Type 1scission of the carbonyl-t-butyl bond occurs with high quantum efficiency in other solvents. A time-resolved e.s.r. study identified the benzoyl and t-butyl radicals which result and confirmed the triplet as their source by using naphthalene as a quencher.The electron polarization behaviour was complex but the CIDNP observations can be interpreted in terms of S-To mixing alone (at high magnetic field values). Pivalophenone constitutes a suitable molecule for detailed study since t-butyl derivatives give simple n.m.r. spectra and reaction can be restricted to one well-defined mechanism. In chloroform solution the rate of dis-87 CIDNP STUDY appearance of t-butyl radicals was determined by direct observation from flash- photolysis e.s.r. studies and with the CIDNP results this led to a value of the proton spin-lattice relaxation time in the free radical. This provided an internal time- standard and allowed a detailed investigation of the rates of competitive atom abstrac- tion reactions in this solvent.The activation energies for these processes were obtained from a study of the temperature-dependence of the time-evolution of nuclear polariza- tions observed in fluorocarbon solvents with added scavengers. Two competitive cage reactions of the radicals were also studied. The experiments have been performed at low concentrations of reactant and radical to ininimise radical re-encounter reactions and reactions of photolysis products. EXPERIMENTAL Experiinents were performed using the n.m.r.5 and e.s.r. spectrometers described previously. Pivalophenone was prepared by a standard method and vacuum-distilled, and its purity (>98 %) was confirmed by g.1.c. and n.m.r.observations (refractive index, ,u = 1.510 8) ; a small quantity further purified by preparative g.1.c. gave identical results to the bulk. The e.s.r. observations showed the presence of peroxyl radicals in oxygenated solu- tions and so all samples were purged of oxygen, manipulated under nitrogen and irradiated (with unfiltered high pressure mercury light) as described before. The quantitative studies were performed in chloroform at pivalophenone concentrations of 0.01 mol dm-3 at 310 K, in the fluorocarbon solvent PP9 at 0.004-0.007 mol dm-3 (in the presence of 0.005-0.120 rnol dm-3 tetrachloromethane) at temperatures 243-343 K, and in silicone oils (y = 1-1000cP) at 0.003mol dm-3 in the presence of 0.02-0.07 mol dm-3 tetrachloromethane.The probe temperature was obtained by chemical shift measurements before and after each experiment by substituting a tube containing ethylene glycol, the two samples each being allowed 10min to attain thermal equilibrium before each measurement. The rate of disappearance of the t-butyl radical was measured by flash-photolysis e.s.r. using a 0.01 mol dm-3 solution of pivalophenone in chloroform; a pseudo first-order rate constant k, = 5.44k0.13x lo3s-l was obtained. Low solubility of reactant and scavenger precluded direct observation of this decay in the fluorocarbon solvent. RESULTS QUALITATIVE OBSERVATIONS The polarized spectrum observed on irradiation of pivalophenone in chloroform is shown in fig. l(a). The lines were assigned on the basis of their chemical shifts and by the sign of polarization predicted from S-To mixing in a benzoyl (g = 2.000 8, A,(meta) = 0.15 mT)-t-butyl (g = 2.002 7, AH = 2.3 mT) radical pair obtained from a triplet precursor.The methyl groups of pivalophenone formed by cage recombina- tion are in enhanced absorption as are the aldehyde groups of the cage-disproportiona- tion product benzaldehyde and the methyl and methylene protons of the other disproportionation product, isobutene. The methyl groups of the two scavenged products isobutane and t-butyl chloride formed by abstraction of hydrogen and chlorine atoms respectively from the solvent are in emission. The reactions are suminarized in fig. 2. Experiments in tetrachloromethane [fig. 1(b)] and PP9 with added tetrachloro- methane confirm this overall mechanism but yield additional information.Thus the absence of the isobutane methyl doublet confirms its production by hydrogen- abstraction from the solvent and a decrease in the absorption at 6 7.2 due to produc- tion of negatively-polarized aromatic protons from pivalophenone and benzaldehyde in the cage is observed. Apart from this region, whicli is obscured, the polarizations observed in the irradiation of solutions in benzene, paraffins and silicone oils are also P. G. FRITH AND K. A. MCLAUCHLAN CHC l3 FIG.1.-The polarized n.m.r. spectrum of photolysed pivalophenone : (a)in CHC13, (6) in CCI, and (c) in C2HC13. CIDNP STUDY consistent with the mechanism given in fig.2. An important absence from the spec- trum obtained from the reaction in tetrachloromethane is that of chloroform which might be formed from disproportionation reactions between, for example, t-butyl and trichloromethyl radicals. Similarly a pair of t-butyl radicals might produce multiplet polarization in the disproportionation product isobutane and isobutene ;the lack of such behaviour confirms that in our experiments, in contrast to those reported with the use of higher irradiation intensities,* the polarizations observed derive solely from interactions of the primary radicals. This is an important simplifying feature in the extraction of kinetic information. Thus polarized benzaldehyde originates in a spin-conserved hydrogen transfer within the cage from polarized t-butyl radicals to benzoyl, and this also produces polarized isobutene.hv, I.s. c. C H COC(CH3)3--b5 ** C6H5COC (CH3) * C6H5COCl recombination diffusion disproportionation CHCl (CH;) 3CH (E) CHCl * 3-(CH3I3CC1 FIG.2.-The reactions of pivalophenone in chloroform. Protons which are polarized are marked with an asterisk. In PP9, increasing the concentration of tetrachloromethane increases the intensity of the t-butyl chloride line as the scavenging rate increases and competes more effec- tively with spin relaxation. Use of the alternative scavenger bromotrichloromethane produces t-butyl bromide in emission but the line coincides with an absorption from the isobutene methyl protons and makes the system unsuitable for kinetic study.With no scavenger added, no polarized lines are observed from scavenged products, which confirms the inertness of the fluorocarbon solvent. However cage polarizations are still observed and this indicates that a competitive reaction pathway still exists; this may be reaction with a photoproduct. A complete absence of multiplet polariza- tions implies that formation of secondary radical pairs is insignificant even in the absence of scavenger. Final confirmation of the reaction scheme was provided in deuterochloroform solution [fig. l(c)]. On irradiation the intensity of the aromatic region was reduced from its equilibrium value but it was restored on cessation of irradiation :molecules with negatively-polarized aromatic protons were produced in the reaction.The iso- butene doublet of the spectrum in chloroform appears in deuterochloroform as a triplet due to coupling to the deuteron, thus confirming that isobutane is produced by hydrogen abstraction from the solvent. P. G. FRITH AND K. A. MCLAUCHLAN QUANTITATIVE STUDIES IN CHLOROFORM On photolysis, pivalophenone yields two radicals which undergo two cage reactions (recombination and disproportionation) and two scavenging reactions, in each of which each product derives from a single reaction pathway. The simultaneous obser- vation of the time-dependence of the methyl polarizations from isobutene, t-butyl chloride, pivalophenone and isobutene provides sufficient information to test the most general form of our kinetic model and, in conjunction with the flash-photolysis e.s.r. rate constant for the disappearance of t-butyl radicals, to evaluate all the kinetic parameters.As in our previous study of benzaldehyde,' the triplet molecule is formed with unit quantum yield at the rate of absorption of photons. In the low concentration ~~limit (ECZ -g 1)chosen in our experiments (E nm~= 2.2 m2 mol-', c < 0.01 mol dm-3) the rate of formation of triplet is once again first order in pivalophenone. Radicals generated in pairs from the triplet (3) diffuse apart and re-encounter with a nuclear-spin (j)dependent probability 3w, of being in a singlet state; we suppose a fraction of fR undergoes re-combination whilst the fraction fT( = 1-fR) experiences the dis- proportionation (hydrogen transfer) reaction.We assume that the overall probability of reaction of radicals which re-encounter in a singlet state is unity. Radicals escape from the cage with an overall probability (1 -3wj) and abstract hydrogen or chlorine from the solvent with pseudo first-order rate constants kHand kcl respectively. These are related to the rate constant for the overall disappearance of t-butyl radicals ks by the equation ks == kH+kcl. (1) Since ks was determined from e.s.r. measurements, it follows that a measurement of the ratio kH/kC1 suffices to determine them absolutely. From the reaction mechanism given in fig. 2, the rates of change of the nuclear spin state populations of the methyl groups in ground-state pivalophenone (A), its triplet (B), t-butyl radicals (C), isobutene (D), isobutane (E) and t-butyl chloride (F) are given by Eqn (2)-(7) contain relaxation terms which compete with reaction in removing polarization, with spin-lattice relaxation times TIA,etc.; the concentration terms with superscript O denote the thermal equilibrium populations of each substance in nuclear spin state j. The factor 2/3 occurs in eqn (5) to allow for the fact that only six of the nine protons in the t-butyl radical become methyl protons in isobutene. With a steady state concentration of triplet, [Aj] = kl[Bj]/kZ and This equation concerns the time-evolution of the concentration of a specific nuclear state of (A); the change of its overall concentration is given by 92 CIDNP STUDY where the relaxation term disappears in the summation because it causes no change in the total number of molecules.The reaction probabilities 3wjcan be calculated approximately from the known e.s.r. parameters of the radical pair and an estimated correlation time (z, "N s) and are found to be very small :fR3wj+ 1 and so d[A]/dt = -kl[A]. (10) The apparent difference between this equation and eqn (8) arises because although the selective population of nuclear states in the reaction may create a population difference large compared with that at thermal equilibrium, the total number of radi-cals which recombine is so small (<1 %) that no significant deviation of the decay of (A) from first-order kinetics is obtained.The rate of depletion of pivalophenone was measured in a series of experiments in which separate aliquot portions of solution were irradiated for different lengths of time and the pivalophenone concentration was measured directly from the n.m.r. spectrum at equilibrium with the light off. Four determinations consisting of fifteen observations at one minute intervals gave kl= 2 1 x s-l. (This justifies our previous calculation of the corresponding rate constant for benzaldehyde : under the conditions of our experiments and at equal concentrations the ratio of the first-order decay constants in the two solutions should be equal to the ratio of the extinction coefficients of benzaldehyde and pivalophenone ; this yields for benzaldehyde kl z s-'.) The half-life of pivalophenone molecules in solution (-350 s) is much greater than the nuclear relaxation time of the methyl protons (4.7 s) and the polarization decays completely before further light quanta are absorbed.Consequently, CAjl = gj[Al/G (11) where gj is the degeneracy of the level and there are G levels. Hence from integration of eqn (9) PI = [A10 exP(-k;o (12) where [A], is the initial concentration of yivaloplienone and The intensity of a transition between states i andj is proportional to their popula- t icn difference, which can be obtained from the appropriate differential equations (2)-(7). The total intensity (I)obtained from a t-butyl group in a product is calculated by summation of these population differences over all the states i and j connected by a spin transition.The spin-dependent part is contained in a quantity D, which is common to all the polarized intensities. With nine equivalent protons there are 29 nuclear hyperfine states which can be divided into ten groups of overall magnetic quantum number m,= 9/2, 7/2, . . . -7/2, -9/2, which we label 1 to 10. Taking into account the selection rule Am, = &I and the degeneracies of the ten groups of Zeeman states we obtain 9 where Ci-* is the coefficient of the term in x9-"in the binomial expansion of (I +x)~, and we have neglected the small coupling to the meta protons in the benzoyl radical. Using our previous methods and eqn (2)-(7) the polarized intensities of the four products (A), (D), (E,) and (F) are given as a function of time by FRITK AND K.A. MCLAUCHLAN where Q is an instrumental constant. Similarly for the aldehyde protons (G) in benzaldehyde, and for the olefinic ones (D') of isobutene, These equations contain statistical factors of 1/9 and 2/9 for similar reasons to that in eqn (5). Evaluation of the parameter D, requires the intensity of the pivalophenone methyl line at equilibrium; it is given by I,(A)" = 36Qno[Alo exp(-kit), (21) where 36n, represents the total thermal equilibrium polarization of three methyl groups each with four Zeeman states and twelve degenerate transitions. These equations contain six quantities of interest,fR,fT, kH,kcl,TICand D,,which can be obtained by taking appropriate ratios of the foregoing equations.Thus from eqn (15) and (16), which yieldsfJf, if TIDand T,, are known. Similarly the ratio of eqn (17) and (18) gives a relationship between the two scavenging rate constants : TIC, the nuclear spin-lattice relaxation time in the t-butyl free radical, appears in the equations together with k, in the term (Tc2+k,) which can be obtained from any of the ratios It(F)/It(D), It(F)/lt(A), It(E)/It(D) and 1,(E)/It(A). Typically, In general an independent measurement of either k, or TI, is required to evaluate the other ;in this case k, is known from the e.s.r. determination. D, is proportional to 72 (see below) and should allow its determination. It can be obtained from ratios of polarised intensities recorded during reaction and the intensity of the same transi- tion at equilibrium. In practice only the intensity of the pivalophenone methyl signal CIDNP STUDY could be measured accurately throughout the experiment and D, was evaluated from intensity ratios of the polarized lines of species (A), (D), (E) and (F) with respect to it.Again a typical equation is The form of eqn (22)-(25) shows clearly the role of spin-lattice relaxation in these kinetic experiments : the relaxation times provide internal standards of time with respect to which the reaction rate is measured. In general the observed intensity of a polarized line should be corrected for a contribution due to equilibrium absorption ;in our experiments the cage products benzaldehyde and isobutene were not observed at equilibrium and no correction was necessary, but the converse was true for the scavenged products t-butyl chloride and isobut ane.KINETIC PARAMETERS The rate constant k; can be evaluated from many of the expressions listed above by fitting experimental line intensities to an exponential time dependence; this is particularly straightforward for the cage products. It also follows from eqn (15) and (21) that the observed intensity of the pivalophenone methyl signal, made up of both equilibrium and polarized contributions, varies with the same rate constant. The time-dependences of the methyl polarizations in isobutene, t-butyl chloride, pivalo- phenone and isobutane are shown in fig. 3 and values of k; derived from various sources are listed in table 1, together with statistical factors of curve-fitting.0 30 60 120 I80 240 300 360 420 480 tls FIG.3.-The time-dependence of resonances observed in the methyl region of the spectrum. Since depletion of pivalophenone is rate-determining, scavenged products accumu- late with the same rate constant k; and the equilibrium absorption intensities I,(P)" of product P increase with time according to the expression I,(P)" = I,(P)(l-exp[-k;t]) (26) where I,@) is the intensity observed after total photoconversion of the reactant. In principle I,(P)" could be measured, as for pivalophenone, but at the low conversions P. G. FRITH AND K. A. MCLAUCHLAN of our experiments the measurement is inaccurate ;rather we have made one measure- ment of I,(P) at the end of a period of irradiation and used an iterative fitting procedure to determine k; and Iw(P).A trial value of k; was inserted in eqn (26) to give a value of IJP) and then the correction for equilibrium absorption was calculated as a function of time to obtain the intensity due to polarization, which decayed with a rate constant k; if this was correctly chosen. Iteration was continued until the decay constant agreed to within 1 % of the trial value. Table 1 contains values obtained in this way from the time behaviour of the t-butyl chloride and isobutane resonances and they are in good agreement with the others. These two estimates do not, how- ever, constitute a test of the kinetic model since it is assumed implicitly in calculating PI,although of course the other estimates do. An independent check is obtained from the ratio Iw(E)/Zm(F)which should equal kH/kC;values of loovaried substantially (25 %) from sample to sample but the ratio remained at 1.3k0.1 whilst kH/kC1is reported below as 1.38 +0.08. TABLEVALUES OF THE FIRST-ORDER KINETIC PARAMETER k', IN CHC13 SOLUTIONS average resonance observed k;/10-3 s-1 correlation coefficient averagestudent's t number of determinations CsHsCHO 3.4+ 0.5 0.996 47 6, (cH3)2 C=CH2 (cB3) 2=CH2 C6H5CO(CH3)3 (CH3)3CH (CH313CCl 3.6k0.3 3.4k0.2 3.1& 0.3 3.1 k0.7 2.7+ 0.6 0.989 0.999 0.997 0.999 0.999 35 375 177 133 217 4 5 5 5 5 The relaxation times of diamagnetic products, needed to evaluate other kinetic parameters, can be determined in principle from observations of polarized intensities at times shorter than TI.If this is itself short few measurements can be obtained and the accuracy of the determination is impaired. Consequently values were also obtained directly in CDC13 solution using a 180-90" pulse sequence on a Bruker HX-90 spectrometer and these are contrasted with the CIDNP values in table 2. Overall the relative magnitudes obtained by the two methods are in reasonable agreement but, even in chloroform, the pulse values were taken as likely to be the more accurate. TABLE2.-sPIN-LATTICE RELAXATION TIMES IN DIAMAGNETIC MOLECULES protons C~HSCZO (CH3)2C=CH_r (C_H3)2C=CH2 (cz3)3ccf (Cg3)3CH C~HSCOC(C~~) CIDNP, t/s 2454 8+2 8+2 9+ 1 7f 3 S+l pulse, t/s 26.1k0.1 4.1k0.1 5.1k0.1 5.0k0.1 -4.7k0.1 The ratio of the fractional cage reaction probabilities in chloroform at 310 K can now be obtained from eqn (22)as 5.9k0.2, givingfT = 0.85 andf, = 0.15.It is of interest that this ratio is sensitive to the ratio of relaxation times but not their absolute magnitudes. In more complex kinetic cases, such as that observed in PP9 with tetra- chloromethane as an added scavenger, such a ratio is implicit in an integration term which must be evaluated numerically (see below). Eqn (1 5), (16), (19) and (20) were derived on the assumption that nuclear spin, and hence polarization, is conserved in hydrogen transfer. This implies that It(G):It(D'):It(D):I,(A)= fTTiG:2fTTiD*:6fTTiD:9fRTIA.Experimentally the ratios observed were 3.7 :1.2 :4.4 :1 whilst those calculated from the determined values offT& and the spin-lattice relaxation times were 3.6 :1.1 :4.2 :1. Agreement is excellent. CIDNP STUDY From eqn (23) and the known values of k; and relaxation times we obtain kH/kCI= 1.38k0.08. Since ks = 5.44+0.13 x lo3 s-l and the concentration of neat chloro- form is 12.4mol dm-3 at 310 K, the second-order rate constant k: = 2.54k0.07x lo2 dm3 mol-1 s-l and kF1 = 1.84+0.07 x lo2dm3 mol-l s-l. We defer discussion of these values until later. Four independent determinations of the proton spin-lattice relaxation time in the radical, Tlc,were made from each set of data using the four equations similar to eqn (24) and the rate constants ks, kHand kcl.Within each set, data values were consistent within 2 % but greater variation was observed between sets : an average of 24 values gave Tlc= 2.2k0.7 x lop3 s. This differs substantially from the value obtained by Fischer et aL9 (2.4 x s) but is that expected for an isolated radical of this size if its relaxation is dominated by dipolar coupling. Fischer’s experiments were performed at higher light intensities, in more concentrated solutions, and pro- duced much higher radical concentrations, as evidenced by observation of polarization from secondary radical pairs. At high radical concentrations, the relaxation rate is roughly proportional to the concentration of paramagnetic species and if this is 40 times that in our system, which seems reasonable from the calculation based upon Fischer’s experimental arrangement, we would expect his value of Tl,to be 40 times smaller than ours, as observed.The translational correlation time zd can be obtained by combining eqn (14) with the expression for the probability 3wjgiven as eqn (26) in our previous paper : where 3w,is a triplet-singlet mixing coefficient which can be calculated from the known e.s.r. parameters of the radical pair; for t-butyl the denominator has the value 1.91 x lo6rad s-l. Four values of D, were obtained from each of six sets of experi- mental data: within each set agreement was within 12 % but the overall variation was 20 %. The mean value of the 24 observations was D, = 2.4k0.4 x giving rd = 4& 1 x lo1 s.This compares well with the correlation time calculated as neces- sary for development of polarization in liquids of the viscosity of chloroform lo (0.5 cP), 2x 10-l’ s. Also, sincef, = 0.15 and xgj3wj/G= 0.091 7, from eqn (13)i we find that k; = 0.99 kl, which justifies the assumption of near-equality. We have seen that in chloroform the product of ksTlc can be obtained from the CIDNP investigation. In inert solvents, the scavenging rate can be expressed as the product of a true second-order rate constant kg and the scavenger concentration [S] and CIDNP then yields the quantity k$[S]Tlc.Its evaluation over a range of scavenger concentration gives an accurate value of the product ktTlc and is a severe test of the kinetic model.Experiments were performed to this end in PP9 solutions with added tetrachloromethane (S = CCl,). QUANTITATIVE STUDIES IN INERT SOLVENTS IN THE PRESENCE OF SCAVENGERS Photolysis in these solutions yielded a single scavenged product t-butyl chloride and the temperature was varied over a wide range to obtain the activation energies (E,) and pre-exponential factors (A) of the cage and scavenging reactions. However the low solubility of reactants prevented e.s.r. observation of radical decay and k: for the t-butyl radical was obtained from the value kgTlc with the value of Tlc calculated from the chloroform results (see below) ;as explained above we believe our value to be that expected for an isolated radical and its value should not change further with P.G. FRITH AND K. A. MCLAUCHLAN falling radical concentration. A further complication is that the scavenger, at con- centrations in the range 0.005-0.10 mol dm-3, was no longer in excess but was con- sumed in reactions with the t-butyl (C) and benzoyl (K) radicals with second-order rate constants kg and kg respectively. The rate constant kg we measure is kc. This yields three additional rate equations for the methyl protons of t-butyl radicals and t-butyl chloride, and the aromatic protons of the benzoyl radical. d[C.jI /dt = kz(1-wj)PjI -k$sI[CjI -([CjI-[CjI")/Tic (28) d[KjI/dt = kz(1- 3~j)[BjI -G[SI[KjI -([KjI -[KjI"] /TlK (29) d[FjI/dt = k2.[SIICjI-([FjI -[FjI")/TlF* (30) In the steady state, and remembering that relaxation does not change the concen- tration of molecules, [C] = ki[AlO exP(-k;t)lk:CSI (31) and [K] = kl,[Al0 exp(-k',t)/kg[S] .(32) The absence of polarization from secondary radical pairs implies that each radical consumes one molecule of scavenger and we eliminate expressions in K by a simple manipulation : d[S]/dt = -kz[S][C]-ki[S][K], = -2k; [AI0 exp(-k;t). (33) Integration over time gives [S] = [Sl0 +2[Alo(exp( -kit)-1). (34) Assuming steady state conditions, ignoring thermal equilibrium population differences, combining eqn (28), (30) and (34) and integrating gives IdF) = -Qk~[Al~Dwyr~xP(-~/TIG) (35) KINETIC PARAMETERS IN INERT SOLVENTS Solutions containing different concentrations of pivalophenone (0.004 4-0.006 6 mol dm-3) and tetrachloromethane (0.004 7-0.12 mol dm-3) corresponding to eighteen different values of [S]/[A] were investigated. It is shown below that the kinetic model was only applicable to a certain range of this ratio.No correction of observed inten- sities was necessary for the cage product isobutene but for pivalophenone the equili- brium intensity at any time It(A)"was calculated from the known initial concentration and the first-order rate constant PI. Values of this parameter obtained from the time evolution of the signals of the isobutene (D) and pivalophenone (A) methyl protons in a series of solutions of different concentration and at different temperatures are given in table 3 ;within a single sample values obtained varied by <7 % but a total variation of 14 % occurred between samples.Good agreement with the data in table 1 is evident. As in chloroform, the observed intensity of the line due to t-butyl chloride was corrected for an equilibrium signal. However, the complexity of the equations describing its time-dependence precluded the use of the iterative method alone. Instead, since the values obtained directly and indirectly in chloroform agree within I14 CIDNP STUDY experimental error, the corrections to I,(F) in PP9 were obtained by using the mean value of ki from above and the equilibrium absorption at one given time to evaluate Im(F),and then calculating I,(F) at any other time from this. The relaxation times of the diamagnetic products in the new solvent and at various temperatures were not determined, for as described above, our equations are sensitive only to the ratios of relaxation times ; since 1/T, cc q/T these ratios are independent of viscosity and temperature.TABLE3.-vALUES OF k', IN PPg/Cc14 SOLUTIONS OF VARIOUS CONCENTRATIONS AND TEMPERATURES average average number [A1I [Sl/ k;/10-3 S-1 correlation student's of deter-mol dm-3 mol dm-3 T/K from Zt(F) from Zt(A) coefficient t minations 0.004 4 0.010-0.043 310 3.5k0.4 3.050.4 0.997 121 12 0.005 2 0.015-0.120 310 3.7k0.4 3.4k0.5 0.996 110 15 0.006 6 0.004 7-0.013 310 3.1k0.3 3.0k0.4 0.995 102 10 0.005 4 0.053 296-3.8k0.5 3.6k0.5 0.997 112 14344 0.006 5 0.061 296-3.450.4 3.550.4 0.998 162 15334 The fractional cage-reaction probabilities fR and fT were obtained from the iso- butene and pivalophenone polarizations, as before, and are listed as a function of temperature in table 4.Their ratio was independent of scavenger concentration over TABLE4.-THE TEMPERATURE DEPENDENCE OF f~/f~ number of TIK fT/fR fT( %) fa( %) determinations 344 2.85 74.0 26.0 6 334 2.97 74.8 25.2 6 325 3.26 76.5 23.5 6 314 3.78 79.1 20.9 6 310 4.75 82.6 17.4 37 307 4.11 80.4 19.6 6 296.5 5.06 83.5 16.5 6 288 5.54 84.7 15.3 6 the range of concentrations investigated and at 310 K the mean values obtained, fT = 0.826 andf, = 0.174 differed little from their values in chloroform. Expressing the rate constants in Arrhenius form, A semi-log plot against inverse temperature is shown in fig.4. It yields the para- meters (ER-ET)= 16.3k2.0kJ mol-' and AT/& = 5.4k2.4x lop3. Thus the re- combination reaction has a higher activation energy than the disproportionation one and requires a smaller entropy decrease in forming the transition state. Two independent determinations of the parameter D, were made in each experi- ment from the ratios of the intensities of the polarized isobutene and pivalophenone methyl lines to the equilibrium absorption of pivalophenone. From these values the diffusional correlation times were calculated using eqn (27) and values are listed in table 5 for solutions of various concentration and temperature. This table also con- tains values calculated as being necessary to observe polarization from the theory of Evans, Fleming and Lawler ;lo agreement is remarkably good. This theory predicts that at high viscosities and low temperatures D, becomes a power series in T$ and P.G. FRITH AND K. A. MCLAUCHLAN attempts were made to observe deviations from a simple square-root dependence in PP9 over a range of temperatures and in silicone oils (y = 1-1OOOcP). Unfortu-nately experimental errors were too large to allow significant conclusions in the former, whilst in the latter, overlap from resonances originating in the methyl groups of the silicones also made accurate measurement impossible. With a knowledge of D, (taken as the average of the two values from the same experiment), intensity ratios can be written to involve the intensity of the t-butyl chloride line in which the only unknown is the product kcTlc: It(D)/It(F)= -2fRTlD/3(1-k;TID)Ytexp(k; -T;$)t (38) 6 4 I 3.0 3.2 3.4 103 KIT FIG.4.-An Arrhenius plot of the cage fractionation ratio.TABLE5.-vALUES OF DwAND rd IN PPg/Cc14 SOLUTIONS [A1Imol dm-3 mol dm-3[Sll TK DUlX 102 2d/10-10 S (obs.) 2d/10110 S (calc.) 0.004 4 0.005 0-0.043 310 5.7k0.5 2.3k0.5 2.4 0.006 6 0.004 7-0.030 310 4.550.5 1.5f0.3 2.4 0.005 2 0.004-0.120 310 5.4f0.5 2.1f0.4 2.4 0.006 5 0.061 296-344 4.8f0.8 1.7f0.5 4.1-1.3 0.005 4 0.053 296-344 5.4k0.5 2.1k0.4 4.1-1.3 kETl,, which is included in Yt,was obtained by minimizing the sum of the squares of the differences between experimental ratios and those calculated at the correspond- ing times by inserting trial values in 9t,which was evaluated numerically and used in eqn (38)-(40) with the values off,, D, and k; taken from the same experiment.Typically 30-40 trial values were used to produce root mean square differences in the range 2-8 %. The values of kzTlc obtained as a function of the ratio of tetrachloro-methane to pivalophenone concentration are shown in fig. 6 which combines the results from three series of experiment ([A] = 4.9 x low3,5.2 x and 6.6 x mol dm-3, [S]/[A] = 0.7-23) with a total of twelve determinations at each concentration ratio. It is apparent that it is constant only over a certain range of concentration ratios, and this defines the solutions to which our kinetic model is applicable. At [S]/[A] > 10, the solubility limit of tetrachloromethane in PP9 was exceeded (sensitivity did not allow work at lower pivalophenone concentrations) and the overestimation of [S] 100 CIDNP STUDY forced kETlc to lower :dues.As the scavenger concentration is decreased on the other hand, radical recombinations become significant but our model required scaveng- ing to consume all the radicals ;in our experiments this is noticeable where [S]/[A] < 2, although it is the absolute concentration of scavenger that is important. This explana- tion is reasonable numerically; in both chloroform and PP9 the systems were both in the fast-motion limit for nuclear relaxation, in which TI cc i?/T:in PP9 at 310 K we calculate T,, as 2.27 x s and hence kz = 4.9 x lo4 dm3 11101-I s-l.If the 'ciom --I ~7-,-7m-,T--*--.---0~r-0 I r1 2 4 fl 8 1'3 13, 14 !e 18 29 22 21 [SI/"AI FIG. 5.-The variation of the parameter kETlc with [S]/[A]. Our kinetic model applies to the concentration range 2-10. 103 KIT FIG.6.-A plot of log (kETlc/T)against 1/T. description above is correct and the abstraction reactions are faster than recombina- tion, k~[Bu*][CCl,] > kz[But][R], where k%is a recombination rate constant w109 dm3 mo1-l s-, and [R] is the total concentration of radicals (N mol dm-3) ; since [CCl,] = 2[A] M 0.012 mol dm-3, k$ > 8 x lo3dm3 mol-l s-l, which is con- sistent with the value deduced. kZTlc was studied as a function of temperature in two separate series of experi-ments ([A] = 5.4 x 6.1 x lov3mol dm-3, [S]/[A] = 9.4, 9.8).Assuming an Arrhenius form of k& with activation energy E,, and fast-motion limit relaxation, with a viscosity activation energy E,, k$Tlc/T = const. x exp -(Ec+ EV)/RT. (41) P. G. PRITH AND K. A. MCLAUCHLAN A plot of log(kETl,/T) against 1/T(fig. 7) gives (E,+E,) = 27.224.2 kJ inol-'. Ev = 13.4 kJ niol-' and so Ec = 13.8k4.2 kJ mol-I ; this implies that the pre-exponential factor Ac = 1.04x lo7 dm3 mol-1 s-l. DISCUSSION The study of the photolysis of pivalophenone under specific experimental condi- tions has provided a detailed test of the diffusion kinetic theory of CIDNP. Frequently a parameter has been obtained from a series of quite separate experimental observa- tions and the consistency of values obtained has provided strong support of the model.Furthermore other checks for consistency have been performed and found good. The care with which experimental conditions must be chosen has been emphasised and it has been shown, not surprisingly, that the detailed kinetics which we have analysed are applicable only over a certain concentration-ratio range in inert solvents with added scavenger. The kinetics have been exposed in a detailed manner and direct measurements made of the fractional probabilities of competing cage processes, which have shown a disproportionation reaction of two radicals to be more favourable than recombina- tion. A variable temperature study has allowed measurement of the difference in activation energies of the two processes and of the ratio of their pre-exponential factors.Two competitive scavenging reactions, namely hydrogen- and chlorine-abstraction, have also been studied in detail. It remains to be discussed whether the values we have obtained are reasonable and what is their significance. The values OffT/Jk, AT/ARand (ER-ET) obtained for the reaction of benzoyl and t-butyl radicals arc consistent with the general trends observed in the data for similar reactions.12 Thus in all cases except those which involve t-butyl,fT/fR < 1, but for t-butyl the ratio is in the range 4.5-27. Although the information is not extensive, fi/fRvaries little with solvent and this implies that the recombination and dispropsr- tionation reactions involving the same species have similar transition states.In all casesfT/fR decreases with increasing temperature, implying ER > ET;for simple alkyl radicals (ER-ET)is small but it increases with substitution and steric hindrance at the radical centre, although the effects of solvation are significant. AT/AR tends to decrease with increasing substitution but it is difficult to assess accurately ;assuming free rotation and that hydrogen-transfer requires a specific relative orientation of the radicals, the smaller Avalues for disproportionation imply this to be a more stringent steric requirement in the transition state than is avoidance of crowding in recom- bination.As the temperature is lowered, the rates of both cage reactions decrease but dis-proportionation becomes relatively more favoured. At some point the more activated process may become so slow that not all the singlet radical pairs produced in the initial S-To mixing react but may emerge from the cage to yield oppositely-polarized transi- tions than those from their triplet counterparts. This implies that in a time-resolved e.s.r. study the polarization might change sign whilst approaching equilibrium when §-To mixing in the singlet radical pairs which do not recombine succeeds in time that in the initial triplet pairs. Electron polarization of this type has been observed below 273 K in viscous solvents but at higher temperatures and lower viscosities only normal polarizations are observed in both e.s.r.and n.m.r. :our previous assumption of unit probability of reaction of radical pairs in singlet configurations appears valid in our experiments. Literature data for the rates of atom abstraction reactions are sparse and those for abstraction of hydrogen and chlorine atoms are summarized in table 6. Most of this 102 CIDNP STUDY information was obtained in the gas phase and it is not possible to estimate the effect of a change of phase with any reliability. However, the values we report for kHand kcl seem reasonable. In the gas phase reaction of methyl radicals and chloroform kH/kC1= 15 and for such a reactive radical this is probably about the same in solution. For plienyl radicals reacting with chloroform in solution the ratio is four.The lower selectivity of t-butyl is surprising at first sight but a study of the relative rates of hydro-gen abstraction from a series of toluenes as a function of the substituents’ Hammett- cr parameter showed that for t-butyl radicals the rate increased with the ability of the substituent to stabilize negative charge; this is opposite to the behaviour of other radicals. The t-butyl radical is unique in possessing a significant contribution from structures such as the following to its transition state :But+ He CCl; ti), But+ H-CCl; (ii), But+ Cl- CHCl; (iii), But+ C1- CHCl., (iv). Structure (ii) is of high energy and its contribution is small. Since CCl; is better able to stabilize negative charge than CHCl;, the contribution of (i) is likely to be greater than that of (iii) but (iv) probably makes a similar one.This may cause a marked increase in the rate of abstraction of chlorine relative to that of hydrogen owing to the juxtaposition of charges in the transition state which is not available to hydrogen transfer. TABLE6.-RATE CONSTANTS FOR ATOM ABSTRACTION REACTIONS atom reactants abstracted k/dm3 molt1 s-1 ref. (CH3),Sn*+ CtjHiiC1 c1 5x lo2 13 n-Bu3Sn-+CH3(CH2)&H2C1 c1 9x lo2 13 CH3*+ CC14 c1 2~ 103 14 CH3*+ CHC13 H 5~ 103 15 But*+CH, H 1 16 But-+cyclohexadiene H 5x lo3 16 But*+ ButCHO H 4~ 103 16 But*+CHC13 I3 1o2 16,17 In the inert solvent, the value of As is comparable with those observed in the gas phase but the activation energy is less and the rate of chlorine abstraction from tetra- chloroniethane is higher than expected by comparison with the chloroform results.Here contributions from the structures But+ C1- CCl; and But+ C1- CCh, are more probable 2o than (iii) and (iv) in chloroform since the stability order is CCl; > CHCl, and CCb, > CHCl-, :the transition state is more ionic and lowers the activation energy for the abstraction of chlorine. This type of explanation has been invoked previously in an analogous case.21 It should not be overlooked that the detailed information reported in this study is the result of several particular features of the experiment. First, all the products of reaction emanate from single reaction mechanisms and give some n.m.r.lines which are singlets. Secondly, this description is true only under certain experimental condi- tions of concentration range and radical concentration. As in our previous work, a major simplification has resulted from the use of low light intensities in illuminations. We thank the Salters’ Company for the support of P. G. F. and Dr. A. J. Dobbs for assistance with the e.s.r. experiments. P. G. Frith and K. A. McLauchlan, J.C.S. Faraday 11, 1975, 71, 1984. P. G. Frith and K. A. McLauchlan, Nuclear Magnetic Resonance (Chem. SOC.Spec. Periodical Rep., London, 1974), vol. 3, p. 378. H. G. Heine, Annulen, 1970, 732, 165. B. G. FIRTH AND K. A. MCLAUCHLAN * P. W. Atkins, A. J. Dobbs and K. A. McLauchlan, J.C.S.Faraday II, 1975, 71, 1269. P. W. Atkins, J. M. Frimston, P. G. Frith, R. C. Gurd and K. A. McLauchlan, J.C.S.Faraday zr, 1973, 69, 1542. P. W. Atkins, K. A. McLauchlan and A. F. Simpson, J.Phys. E, 1970, 3, 547. J. H. Ford, C. D. Thompson and C. S. Marvel, J. Amer. Chern. Soc., 1935, 57,2619. 13. Fischer, Ind Chim. Belge, 1971, 36, 1084. B. Blank, P. G. Merritt and H. Fischer, Report of the 3rd IUPAC Congress (Butterworth, London, 1971), vol. 4, p. 1. lo G.T. Evans, P. D. Fleming and R. G. Lawler, J. Chem. Phys., 1973, 58, 2071. l1 Technical Infornzation " Fluorocarbon Liquids " (Imperial Smelting Company, 1968). l2 M. J. Gibion and R. C. Corley, Chem. Rev., 1973, 73, 441. l3 D. J. Carlsson and K. U. Ingold, J. Amer. Chem. Soc., 1968, 90, 7047. l4 F. R. Mayo, J. Amer. Chem. Soc., 1967, 89, 2654. l5 F. A. Raal and E. W. R. Steacie, J. Chem. Phys., 1952, 20, 578. A. F. Trotman-Dickinson and E. Ratajczak, Supplementary Tables of Bitttulecular Gas Reac- tions (UMIST, 1971). l7 M. G. Evans and M. Polanyi, Trans. Faraday SOC.,1938, 34, 11. l8 W. A. Pryor, Free Radicals (McGraw-Hill, New York, 1966). l9 R. F. Bridger and G. A. Russell, J. Amer. Chem. Suc., 1963, 85, 3754. 2o W. A. Pryor, W. H. Davis and J. P. Stanley, J. Amer. Chem. Sac., 1973, 95, 4754. 21 R. L. Huang, T. W. Lee and S. H. Org, J. Clzem. SOC.C,1969,40. (PAPER 5/947)

ISSN:0300-9238    

DOI:10.1039/F29767200087

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Theory of Non-linear Dielectric Effects in Liquids BYJERZYMALECKI Institute of Molecular Physics, Polish Academy of Sciences, 60-1 79 Poznan’, Smoluchowskiego 17/19, Poland Received 11th June, 1975 A general theory of non-linear dielectric phenomena in liquids is proposed with the fundamental assumption, supported by a large body of experimental data, that such phenomena are specifically sensitive to molecular association processes due to electrostatic and chemical interactions. The reIationships derived provide a good interpretation of the effects observed in dipolar liquids with small dipole moments as well as in strongly polar systems. The theory also holds for solutions presenting tautomeric and arbitrary chemical equilibria involving molecular associates and complexes.Most studies of dielectric polarization in liquids are restricted to linear effects. When performed at low electric field strengths, these studies allow the experimental determination of electric permittivity E, molar polarizability and dipole moment. Numerous papers also deal with dielectric polarization and absorption 14-16 in strong fields. Sufficiently sensitive devices record variations in electric permittivity As caused in a liquid by a strong externally applied electric field. The effect, tradi- tionally referred to as dielectric saturation with regard to its formal description in terms of the Langevin function, will be referred to here in general as the non-linear dielectric effect (NDE). A measure of the NDE is provided by the quantity A&/E2,where A&= cE-& is the variation in electric permittivity of the liquid in an intense external electric field E.Obviously the domain of NDE is an extension of that of dielectric polarization to high electric fields. For theoretical considerations, it is convenient to introduce, in place of the experimentally determined quantity A&/E2,the molar NDE constant S, defined as follows : SE-aPp a(F2)’ where Pdipis the molar dipole polarization, and F the local field component in the direction of E. Often, S is expressed in the form of a dimensionless number R,,the NDE correlation factor :2 47rNptR, = S/S1, S, = -~ 45k3T3’ where N is Avogadro’s number, k Boltzmann’s constant, T absolute temperature and pl the dipole moment of the free molecule, or monomer.Hence S1is the value of Sat infinite dilution, and can be computed from eqn (2) once p1 has been determined from polarization studies of dilute solutions. Applying the definitions (1) and (2), R,can be expressed in terms of experimental quantities. For example, from the generally applied local field model of Onsager,17 one obtains the expression : 104 J. MALECKI 105 where c and n denote, respectively, the electric permittivity and refractive index of the liquid for infinite wavelength, and Vits molar volume. A similar expression holds for so1utions.l Further discussion here does not concern macromolecular solutions or liquid crystals. For molecular liquids only, the experimental results given in the work of Piekara et al.,2-9the early work of Kautsch,l and the papers of Thiebaut and Rivail,l0-l2 can be represented in the form of four characteristic curves of R,as a function off, the concentration of the substance under investigation in an inactive solvent.These functions, for the entire mole fraction interval 0 <f< 1 are represented by the full curves of fig. 1. The straight line (dashed) is the hypothetical result for the case of completely free molecules of the liquid, i.e., those occurring only as monomers throughout the concentration range. The experi- mental results closest to this ideal case are those of curve 1 obtained for simple liquids with a small dipole moment.'. This type of R,(f) dependence is accounted for by Debye's theory, which assumes isotropic interaction between the dipolar mole- cule and its surroundings.Ethyl ether is an example. In strongly dipolar liquids such as solutions of nitrobenzene in benzene, A&/E2is positive and R,accordingly negative at high concentrations and for the pure liquid (fig. 1, curve 2). An attempt 2 t F 0.20 Rs -I -2 -3 f FIG.1.-Experimental NDE results for solutions of molecular liquids : 1, weakly dipolar liquids ; 2, strongly dipolar liquids ; 3, liquids with tautomeric equilibria ; 4, strongly associated liquids. Arrows show the direction in which the molecular interaction energy increases. to explain the inversion in sign of R, was made by Piekara,2 assuming non-rigid antiparallel pairs of dipolar nitrobenzene molecules.Negative values of R,over the entire concentration range (curve 3) have been found for substances exhibiting internal rotation, such as 1,2-dihalogeno derivatives of ethane.4 The effect is well accounted for by the theory of Piekara, Kielich and Chelkowski assuming an influence of the electric field on the trans-gauche equilibrium. Finally, strongly associated liquids, typically represented by solutions of the lower alcohols in hydrocarbons, exhibit very 106 THEORY OF NON-LINEAR DIELECTRIC EFFECTS IN LIQUIDS large negative R,values of about -50 in dilute solutions and positive ones of the order of +50 in concentrated solutions and pure alcohols 6-9 (fig. 1, curve 4). An explanation of these large R,values has also been proposed by Piekara on the assump- tion of influence of external electric field on an equilibrium between two types of the hydrogen bond, a normal bond 0-H .. .0 and an ion pair 0-.. . H-Of. Recent-ly, Rivail and Thiebaut l2 published a NDE theory for systems which involve chemi- cal equilibrium, applying it to chloroform +pyridine mixtures in cyclohexane. The experimental results in this case are well described by curve 1 of fig. 1. As seen from the preceding brief review at least five molecular models have been proposed for interpreting the experimental data. They have played an important role in the development of non-linear studies by stimulating a series of original experi- mental papers. However, it would appear that the difficulties in interpretating the results and the lack of a unique theory are major reasons for the slow progress in the use of this method for studying molecular interaction. The aim here is to derive general relations expressing the experimental quantity R, in terms of molecular structural parameters of the liquid.The fundamental hypothesis of this paper, an extension of that of Rivail and Thiebaut,12 is that NDE are specifically sensitive to molecular association, inherent in the liquid phase. THEORY As an argument in favour of this hypothesis we note (without considering the fore- going suppositions) that, with increasing intermolecular interaction energy, R, decreases (fig. 1). This rule is observed both for increasing concentrations of the dipolar component and for transition from weakly interacting to strongly interacting systems.Generally, intermolecular interactions lead to more or less complex associations, characterized by labile equilibria which are sensitive to variations in the external parameters, and thus to changes in electric field strength E. It is considered here that we are in general dealing with an associated liquid, i.e., strong association, or complexation, e.g. by interaction of the charge-transfer type or hydrogen bond type as in water and alcohols, or much weaker dipole association and states resulting by unspecific electrostatic interactions. Let xidenote the concentration of associates or complexes defined as :I8 iN,xi= -, cxi=l (4)n2 L where Ni is the number of complexes or associates in the solution, each consisting of i molecules of the substance under investigation, and n2 = CiiNithe number of mole- cules of the latter.From definition (1) and without specifying a local field model we can write :11* l6 where summation extends over all associates and complexes present in the solution. (pi)E is the mean value of the projection of the dipole moment of an associate on E. Assume now, as Rivail and Thiebaut l2 did for a similar problem, that the associates undergo no deformation in the electric field, i.e., that we are dealing with rigid associates. This assumption appears to be substantiated by the low interaction energy of dipole moments with strong electric fields amounting to about kT compared to the energy of intermolecular interactions, which is usually 102-104times larger (pi)E can now be expressed simply, by a series expansion of the Langevin J.MALECKI 107 function, and describes usual Debye orientation of a moment in the field F: On similarly introducing to R,a correlation factor of molar polarizability R, = Pdip/ Pyip,we have immediately, with the same assumptions,i8 Assume, albeit as a working hypothesis, that the concentrations xi of the associates are functions of the electric field strength F. In other words, we assume that the field causes a slight perturbation of the association equilibrium. A similar assumption restricted, however, to one case of tautomeric equilibrium, was made by Piekara,3* and was applied to chemical equilibrium by Rivail and Thiebaut.12 With the absence of a privileged direction in the liquid, the function has to be symmetric with respect to the field direction F.Thus, we can express our working hypothesis in the form of the expansion : On rejecting in the expansions (6) and (8) all terms higher than those quadratic in F, and using eqn (5), we obtain directly : From (2), we have : -15k2T2A).ax.R, = P14f: xi a(F2) The problem now reduces to that of obtaining expressions allowing one to deter- mine the derivatives dxijd(F2)numerically. To do this, assume quite generally that the associates (complexes) are formed as the result of n equilibria of the following type : K, m,A,+nz,A,+ --* .+m,A,+n~,+,A,+,+ -* (11) As usual, assume that the right-hand coefficients m2in (11) are positive and the left hand ones negative.Denoting by cz the respective molar concentrations of the substrates and products of the reaction described by eqn (1 1) : = NI/NV (12) we express simply the logarithm of the equilibrium constant as : In K, = 1 rnl In c,, (13) where the summation extends over all complexes of the given equilibrium (1 I). Obviously, the system of equilibria (1 1) must be a complete system, i.e., the n constants KE,a = 1, . . . n, uniquely determine the concentrations of all the associates and complexes. Note that, on omitting the small correction for electrostriction, we have : 108 THEORY OF NON-LINEAR DIELECTRIC EFFECTS IN LIQUIDS From the Boltzmann distribution, we have : Nl = exp (-UJRT), where Uzis the energy of the Zth state, and can be expressed as the sum of the energy V,"at F = 0 and the energy of interaction of the dipole p1and field F: Uz = Ur+~~FcosQl, Ql being the angle between the direction of pz and field F.We find the population of the Zth state by summation of those of all possible orientations Q1: N~ = A exp( -U;/~T)1:exp( -$cos Q,) sin Q, dQl. The constant A is determined from the normalization condition. On integration and using the expansion of the exponential function bearing in mind that plF/kT < 1, we obtain from eqn (12), (13) and (16) : In Ka(F2)= In Ka(0)+xini~ P12F2 6k2T2' from which, with (14), we obtain : Hence, taking into account the dipole-field interaction energy in the total energy of the complex, our working hypothesis is justified.We now introduce, for brevity, the notation : y, = 6k2T2 8x1 XI a(F2)' Hence, the complete set of equations of linear and non-linear polarizability for the case of an arbitrary number n of simultaneous equilibria (1 1) is, finally, of the form : Furthermore, from (10) and (18) : R, = 4 Tp;(pf -2.5~~) f whence we have to find n+ 1 quantities yi by solving a set of equations, in which we have n relations of the type : c m,y1 = c md 1 1 resulting from (17) and (18), and one equation : xxiyi = 0 i resulting directly from definition (4) with the notation (18). Summation over i in eqn (19), (20) and (22) includes all associates and complexes in the solution ; in the equations of the form (21), summation over Z includes all complexes forming in an equilibrium of the type (12).J. MALECKl 109 DISCUSSION The relations (19)-(22) do not contain explicit expressions describing the local field, nor did we specify a local field model when deriving them. They are of a general nature, allowing their application to the description of non-linear effects within a wide class of liquid solutions. In particular, eqn (19)-(22), with the assumption of only one tautomeric equili- brium of the type A’ 21 A” yield for R, and R,relationships identical to the equations of Piekara, Kielich and Chelkowski for trans-gauche tautomerism as well as Piekara’s equations for the process of proton shift in the hydrogen bond bridge.In the more general case of a chemical equilibrium (provided the process is fully described by a single equilibrium constant) the set of equations (19)-(22) gives a relation very similar to the Rivail- Thiebaut equation for isotropically polarizable molecules. As an example, we shall apply eqn (19)-(22) to the case of self-associating liquids, such as alcohols. Hence, we particularize the reactions (11) by assuming that suc- cessive open multimers (Aio)and cyclic multimers (Ai,) arise in the reactions : Kio Kic iA, +Aio and iA, +Aic Assuming that pic = 0, eqn (21) becomes : 2 Yio-iY1 = Pi2 -iP1 and yic-iyl = -ip:. Denoting the respective multimer concentrations by xi, and xicwe have from (22) : i= 1 and can determine all the other yio,yicfrom eqn (23) and (24). Too avoid introducing too many parameters Kio,Kicwe put the free energy of a bond in the linear multimers as AFo and that of a bond in cyclic dimers as AFZc= 0.5AF0,in trimers as AFSc = 0.67AF0,and in a higher multimer as AF, = 0.9AFo.Assuming moreover that the dipole moments of multimers pi have the values resulting from the geometry of the latter, the only parameter in our calculations if AFo, which allows determination of all the constants KiO, Kic and hence xio,xic and yio, yic. In turn, eqn (19) and (20) enable us to calculate the quantities R, and R,. Fig. 2 shows the results, calculated for three values of the parameter AFo.For comparison, in fig. 3 are given the experimental curves for R, and R,for solutions of three alcohols in hexane.6p ’9 From fig. 2 and 3 it can be seen that eqn (19)-(22) provide an accurate description of the experimental facts as observed in associating liquids (the present paper is not aimed at giving a detailed picture of the association processes occurring in these alcohols). Eqn (19)-(22) provide an interpretation both of the very large positive and negative values of R,without the necessity of invoking the improbable process of proton tran~fer.~ We shall deal with the application of non-linear effects to the study of alcohol association, as well as to the question of proton transfer in hydrogen bonds, in separate papers. Along similar lines, with appropriately smaller interaction energies AF for dipolar 110 THEORY OF NON-LINEAR DIELECTRIC EFFECTS IN LIQUIDS liquids one obtains from eqn (19)-(22), curves of R,(f) similar to curves 1 and 2 of fig.1. As an example, we give theoretical curves in fig. 4. This problem, too, will be the subject of further papers. -40 -20 0.2 0.: 0.6 0.3 f f FIG.2.-Theoretical Rp(f)and Rdf) curves for strongly associating liquids for three values of the free energy of the intermolecular bond in open multimers. AF, = -0.5 kcal mol-' (l), -1.8 kcal mol-1 (2), -4.0kcal mol-' (3) (1 cal = 4.184 J). F 1 4 3 RP 2 L Qz 0.1, 0.6 Q8 f f FIG.3.-Experimental Rp(f)and Rdf curves for solutions of n-butanol ( 0),6 t-pentyl alcohol ( x ) ' in n-hexane.t-butyl alcohol (0) J. MALECKJ 111 Curve 3 of fig. 1, which was obtained for trans-gauche tautomerism, is adequately explained by the theory of Piekara, Kielich and Chelk~wski.~ For this case (as already noted) the equations (1 9)-(22) give the relations derived by these authors. The theory proposed provides a simple explanation of all non-linear dielectric effects hitherto observed in molecular liquids. It is interesting that this can be achieved with only the simplest assumption of a Debye effect of reorientation of rigid complexes proportional to pf, whereas the expressions in the form -2.5 p:j)i [eqn (20)], signifi-cantly affecting the value of R,,result from the dependence of the energy of the complex on the electric field strength favouring complexes with a larger dipole moment.The 1 0 Rs -1 -2 f FIG.4.-Theoretical &(f) curves for weakly associating liquids. AF,, = 1.70 kcal mol-' (l), 0.85 kcal mol-' (2), 0.6 kcal mol-' (3). present approach has moreover the important advantage of not requiring the intro-duction of a specific molecular model for each particular case. The foregoing dis- cussion leads to the conclusion that nonlinear dielectric effects are specifically sensitive to very broadly defined molecular association and conformational equilibria in liquids and hence these effects should find application as an accurate method of investigating such processes. Available experimental data indicate that molecular association occurs to some degree in all liquid dipolar systems.F. Kautzsch, Phys. Z., 1928,29, 105. A. Piekara, Acta Phys. Polon., 1950, 10, 37, 107. A. Piekara, S. Kielich and A. Chelkowski, Arch. Sci. (fasc. spec.), 1959, 12, 59. A. Chelkowski, Acta Phys. Polon., 1963, 24, 165. A. Piekara, J. Chem. Phys., 1962, 36, 2145. J. Malecki, Acta Phys. Polon., 1962, 21, 13 ; 1963, 24, 107 ; J. Chem. Phys., 1962, 36, 2144 ; 1965, 43, 1351. I. Danielewicz-Ferchmin, Bull. Acad. Polon. Sci., str. sci. math. astr. phys., 1966, 14, 51. T. Krupkowski, Acta Phys. Polon., in press. J. Nowak, Acta Phys. Polon., 1972, A41, 617. lo J. M. Thiebaut, Ph.D. Thesis (University of Nancy, 1968). l1 J. M. Thiebaut, J. L. Rivail and J. Barrio], J.C.S.Faraday ZZ, 1972, 68, 1253. l2 J. L. Rivail and J. M. Thiebaut, J.C.S. Faraday ZI, 1974, 70, 430. l3 F. Assenegg, Acta Phys. Austr., 1967, 26,43. l4 K. Bergmann, M. Eigen and L. De Maeyer, Ber. Bunsengesphys.. Chem., 1963, 67, 819. l5 K. Bergmann, Ber. Bunsenges. phys. Chem., 1963, 67, 826. 112 THEORY OF NON-LINEAR DIELECTRIC EFFECTS IN LIQUIDS l6 R.F. W. Hopmann, Ber. Bunsenges. phys. Chem., 1973, 77, 52; J. Phys. Chem., 1974, 78, 2341. l7 L. Onsager, J. Amer. Chem. Soc., 1936, 58, 1486. J. Malecki, ActaPhys. Polon., 1965, 28, 891. l9 P. Debye, Phys. Z., 1935, 36, 100, 193. (PAPER 5/1141)

ISSN:0300-9238    

DOI:10.1039/F29767200104

出版商:RSC    

年代:1976

数据来源: RSC