摘要:

Electron Spin Resonance Studies of Cation Radicals of Some Alkylphosphines BY MASAMOTO KISHIAND TAROISOBEIWAIZUhfI,* TAKASHI The Chemical Research Institute of Non-Aqueous Solutions, Tohoku University, Sendai, Japan AND FUMIOWATARI Department of Applied Science, Faculty of Engineering, Tohoku University, Sendai, Japan Received 7th January, 1975 The effect of alkyl substitution on the geometry of the phosphine cation radical has been investi- gated by e.s.r. spectroscopy. It is shown by the use of Huheey’s electronegativity parameters that the geometry of the alkylphosphine cation radicals is related to the electronegativity of the alkyl groups as in other trigonal radicals. The effects of methyl substitution on the geometry of AIH,‘ and SiH3 are also discussed in terms of Huheey’s electronegativity parameters.It is shown, further, that there is a correlation between the 31Phyperfine coupling constants of the alkylphosphine cation radicals and 31Pn.m.r. chemical shifts for the respective neutral molecules. There have been many investigations which concern the structure of AX3 typeradical^.^'^^ It has been shown 1-6 that the molecular geometry of these trigonal radicals is related to the electronegativity difference between A and X, i.e., the radicals adopt a more pyramidal structure as more electronegative substituents are introduced. Meanwhile, interesting features of the effects of methyl substitution on the geometry of PH;, SiH, and AlH; radicals have been observed. Begum et aL5 have shown that PMel flattens more than PH;, while AlMe; bends further than AIH;, and SiMe, takes nearly the same geometry as SiH3.In their study, the importance of the hyper- conjugation effect of the methyl group has been pointed out. However, despite many investigations of trigonal radicals, the effects of alkyl substitution on the mole- cular geometry of the radicals are not really clear. We have examined the cation radicals of some alkyl-phosphines by e.s.r. In this paper, we present the effect of alkyl substitution on the structure of the phosphine cation radicals, and also show there is a correlation between the 31Phyperfine coupling constants of these phosphine cation radicals and 31Pn.m.r. chemical shifts of the parent neutral molecules.EXPERIMENTAL Methylphosphine and dimethylphosphine were prepared by the deprotonation of phos-phine by KOH-dimethylsulphoxide suspension, followed by reaction with methyl iodide, and purification by fractional condensation.15 Trimethylphosphine, triethylphosphine and tri-isopropylphosphine were prepared by the Grignard method from phosphorus trichloride and the appropriate alkyl halide. These were purified by precipitating the phosphines as the silver iodide adducts and subsequently thermally decomposing the adducts in a vacuum line. Tri-n-butylphosphine was of the highest commercial grade and was used as supplied. The cation radicals of these phosphines were produced by 6oCoy-ray irradiation of de-113 E.S.R. OF PHOSPHINE RADICALS gassed sulphuric acid or dichloromethane solutions of the phosphines.E.s.r. spectra were measured with an Hitachi 771 e.s.r. spectrometer with 100kHz field modulation. Both y-irradiation and e.s.r. measurements were carried out at 77 K. RESULTS y-irradiation of sulphuric acid solutions of phosphines produces predominantly monomer cations, but with dichloromethane solutions it sometimes produces dimer cations or other radical species as well as the monomer cations. The identification of the cation radicals was made by reference to assignments by Symons et al. for the spectra of PMe; and PEtJ.’ Representative spectra observed are illustrated in fig. 1. Although the e.s.r. of cation radicals was not always observed for both the solvent systems, as far as was observed, no appreciable differences were found 1.* VALUES, ‘P HYPERFINE COUPLING CONSTANTS AND p/SRATIO FOR ALKYLPHOSPHINE TABLE CATION RADICALS radical medium Q II 91 AII/G Ai/G Aiso/G plsratio PH; a PH2Me+ H2S04 H2S04 1.993 1.987 2.014 2.012 706 634 423 348 517 443 6.43 7.59 PHMeT H2S04 1.989 2.015 624 316 419 8.67 PMe: H2S04 1.984 2.012 616 294 401 9.47 PEt: CHzClz 1.989 2.012 570 273 372 9.42 P(n-Bu)P(i-Pr): CH2C12 CH2C12 1.990 1.986 2.012 2.010 556 534 257 209 357 317 9.88 12.10 aref.(5) 105G j3300G lb FIG.1.-First derivative e.s.r. spectra for (a) PH2CHi in sulphuric acid and (b) P(i-Pr): in dichloro- methane at 77 K. x : signals from the dimer cation radical.between the hyperfine coupling constants in sulphuric acid and in dichlorornethane solutions. The e.s.r. parameters obtained are shown in table 1. Breit-Rabi correc- tions have been made for the determination of g and hyperfine coupling constants. The p and s spin densities were calculated from the isotropic and anisotropic parts of the 31P hyperfine tensors using the atomic parameters given in the literature.16 Table 1 also contains the p/sratio obtained for the unpaired electron orbital. M. IWAIZUMI, T. KISHI, T. ISOBE AND F. WATARI DISCUSSION It is seen in table 1 that there is an appreciable effect of alkyl substitution on the 31P hyperfine interaction. The result shows a marked contrast to SiH3 where no appreciable effects of methyl substitution were observed.If the hyperconjugation effect were dominant as was pointed out by Begum et uZ.,~ the pis ratios of PH;, P(i-Pr); and PMe; would increase in this order. However, the ratios increase in the order of PHT, PMe: and P(i-Pr);, suggesting that hyperconjugation is not necessarily a dominant effect for the determination of molecular geometry. The sequence of the pjs ratio shown in the table rather suggests that the electronegativity of the substituents may be more important. We have examined the relation between the electronegativities of the substit uents and the molecular geometry by using Huheey’s electronegativity parameters. l7 Huheey showed the method of calculating the electronegativity of groups based on the Hinze and Jaffk concepts for the orbital and bond electronegativities.In the calculation, he assumes that the electronegativity of an atom or group is equalized by displacement of charge upon covalent bond formation ; his electronegativity is expressed by the two terms, a+66, where a corresponds approximately to the fixed electronegativity by Mulliken’s definition, b is a constant which may be termed the charge coefficient, and 6 is the formal charge on the atom or group resulting from I I I I I I I .-~ -0.1 0.0 0.1 FIG.‘.-Plot of the pis ratio for the unpaired electron orbital against the formal charge on phos-phorus, Sp, of some alkylphosphine cation radicals. replacement of the charge by bond formation. By the use of his electronegativity parameters, and solving the following simultaneous equations, the formal charge on phosphorus in the radicals of the type of PR’R”R”’+ can be calculated.ap bp8p = a,. + bRfdRt = aR*p+ bRntdR** = + bR~~~dR~t~ 6p + &*+6R)f + 8R”’ = + 1. Fig. 2 shows a plot of pls ratio of the unpaired electron orbital against dp obtained in this way.lg The figure shows well the correlation between the p/s ratio and aP, and indicates that as the density of CT bonding electrons decreases on the phosphorus, the phosphorus 3p orbitals are more favoured for CJ bonding. This result is consistent with Pauling’s explanation of the structure of trigonal radicals and apparently 116 E.S.R. OF PHOSPHINE RADICALS indicates that the electronegativity of the alkyl groups makes a dominant contribution to the determination of the molecular geometry of the alkylphosphine cation radicals as in the cases of other trigonal radicals.It can be shown that the effects of methyl substitution on the structures of AIH;, SiH, and PEP: radicals, which are mentioned above, can also be well explained in terms of Huheey’s electronegativity parameters. Table 2 shows the formal charges on the central atoms for these radicals calculated by a similar method to that used above.2o It is seen that for AlHF the central atom loses electrons by methyl sub- stitution but in PH; it gains more electrons and it neither gains nor loses in SiH,, showing good correspondence to the observed changes in the p/sratios of the radicals by methyl substitution.It is apparent, therefore, that the structures of these radicals are also explained by the electronegativity difference argument. TABLE2.-p/S RATIO, FORMAL CHARGES ON CENTRAL ATOMS, &, AND CHANGES IN 6, BY METHYL SUBSTITUTION AIH; AlMe; SiH3 SiMe3 PH; PMeS pls ratio 3.37a 2.3ga 5.2a 5.7 a 6.43 9.47 &Vf -0.25 +0.14 -0.03 -0.02 +0.15 -0.06 SM(Me)-BM(H) -k0.39 +0.01 -0.2 1 a from ref. (5). On the other hand, it was found that there is a linear relation between the p/s ratios of alkylphosphine cation radicals and 31Pn.m.r. chemical shifts for the corre- sponding parent neutral molecules (fig. 3). It is known that the 31Pn.m.r. chemical 0 50 100 150 200 (ppm) 31Pchemical shift FIG.3.-Plot of the p/sratio for the unpaired electron orbital of some alkylphosphine cation radicais against the 31Pn.m.r.chemical shifts @.p.m. from H,P04) for the parent neutral molecules. shifts for trivalent phosphorus compounds are determined mainly by the bond angles at the phosphorus and the ionic character in the phosphorus-ligand bonds.21 To see the effect of ionic character in the phosphorus-ligand bonds, calculation of charge distribution was attempted for PH3 and PMe, by the use of Huheey’s electronega- tivity parameters, and by assuming 95 and 86 % p characters, respectively, for their c bonds based on the observed bond angles.22 The result showed that the effect of M. IWAIZUMI, T. KISHI, T. ISOBE AND F. WATARI methyl substitution on the charge distribution is very small in such neutral molecules ; the change in the formal charge on phosphorus by methyl substitution was only -0.06, suggesting that the changes in the n.m.r.chemical shifts may not be due to the changes in the ionic character for the present case. The correlation shown in fig. 3 may imply rather that the geometry of the parent neutral molecules varies in similar manner to the cation radicals, and the resulting changes in the hybridization of the phosphorus make a contribution to the observed changes in the 31Pn.m.r. chemical shifts. Quantitative explanation, however, is difficult at present. L. Pauling, J. Chem. Phys., 1969, 51, 2767. D. L. Beveridge, P. A. Dobosh and J. A. Pople, J. Chem. Phys., 1968, 48,4802. L. J. Aarons, I.H. Hillier and M. K. Guest, J.C.S. Faraduy 11, 1974, 70,167. I. Biddles and A. Hudson, Mol. Phys., 1973, 25, 707. A. Begum, A. R. Lyons and M. C. R. Symons, J. Chem. SUC. A, 1971, 2290. A. Begum, J. H. Sharp and M. C. R. Symons, J. Chem. Phys., 1970,53,3756.'J. H. Sharp and M. C. R. Symons, J. Chem. SOC.A, 1970,3084. A. R. Lyons and M. C. R. Symons, J. Amer. Chem. Suc., 1973,95,3483. T. Cole, H. 0.Pritchard, N. R. Davidson and H. M. McConnell, Mol. Phys., 1958, 1, 406. lo K. Morokuma, L. Pedersen and M. Karplus, J. Chem. Phys., 1968, 48, 4801. R. W. Fessenden, J. Phys. Chem., 1967, 71, 74. l2 C.Hesse, N. Leray and J. Roncin, J. Chenz. Phys., 1972, 57, 749. l3 R. L. Morehouse, J. J. Christiansen and W. Gordy, J. Chem. Phys., 1966,45,1751.l4 T. A. Claxton, M. J. Godfrey and N. A. Smith, J.C.S. Faraday 11, 1972, 68, 181. l5 Inorganic Synthesis XI, ed. W. L. Jolly (McGraw-Hill, New York, 1968), p. 124. l6 P. W. Atkins and M. C. R. Symons, The Structure of Inorganic Radicals (Elsevier, Amsterdam, 1967)."J. E. Huheey, J. Phys. Chem., 1965, 69,3284. (a)J. Hinze and H. H.Jaffe, J. Amer. Chem. SOC.,1962,84,540 ; (b) J. Hinze, M. A.Whitehead and H. H. Jaffk, J. Amer. Chern. Soc., 1963,85, 148; (c) J. Hinze and H. H. Jaffk, J. Phys. Chem., 1963, 67, 1501. l9 Inthe calculation, orbital hybridization the same as that estimated experimentally was used for phosphorus. 2o For each central atom, the same hybridization as that estimated experimentally was used. 21 N. M. Crutchfield, C. H. Dungen, J. H. Letcher, V. Mark and J. R. Van Wazer, Topics in Phosphorus Chemistry, vol. 5, 31PNucIeccr Magnetic Resonance (Interscience, New York, 1967), and references therein. 22 G. M.Kosolapoff and L. Maier, Organic Phosphorus Compounds (Wiley-Interscience, New York, 1972), vol. 1. (PAPER 5/025)

ISSN:0300-9238    

DOI:10.1039/F29767200113

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Valence Orbital Photoelectron Spectroscopic Studies of Free Molecules with Zirconium Mc Soft X-Ray Excitation BY DAVIDA. ALLISON G. CAVELL*AND RONALD Department of Chemistry, The University of Alberta, Edmonton, Alberta, Canada T6G 2G2 Received 20th February, 1975 Zirconium Mc X-rays (151.4 eV) have been used to produce valence shell photoelectron spectra of the noble gases Ne and Ar and the molecules CH4, N2 and HzO. This radiation, which appears to be free of nearby satellites, gives good count rates and clean spectra. Since the orbitals with a large proportion of p character have a higher cross-section for photoemission with Zr My than with Mg K,, the spectra show prominent peaks for molecular orbitals with dominantp components whereas Mg K, spectra emphasize those orbitals which have dominant s character.The difficulties of interpreting the valence region photoelectron spectra of free molecules obtained with Mg K, or A1 K, soft X-ray excitation, even when helium(1) spectra are available, have long been apparent because of the great difference in photo- ionization cross-sections of these disparate radiations. Some of the difficulties can be alleviated by the use of intermediate energy radiation sources to provide different cross-sectional photoionization efficiencies. Following the work of Krause who surveyed the available M5 X-ray emission characteristics of the second row transitional metals, we have developed an inter- mediate energy radiation source, Zr MC (151.4 eV), for routine use and we are presently utilizing this radiation to study valence orbital photoelectron spectra of simple molecules.We illustrate our results herein with spectra of CH4, N2and H20, as well as the noble gases Ne and Ar. EXPERIMENTAL Ail work was done with a McPherson ESCA-36 photoelectron spectrometer. The X-ray power supply of that instrument was modified to operate with the anode of the X-ray tube at a positive potential (rather than at ground potential as in the original design) and the cathode near ground as described by Krause.' The anode assembly was insulated from the body of the instrument and cooled with ordinary tap water which was satisfactory for the low (5 to 8 kV) potentials employed. Two anode systems have been designed. In one, the anode material was fabricated in the form of a separate screw cap which can easily be interchanged to permit the exploration of a variety of metals as alternate X-ray sources.This design had limited heat transfer capabilities but could satisfactorily be operated up to 90 W (15 mA, 6 kV). An alternative anode design in which Zr metal was brazed to a copper base provided superior heat transfer capability and permits operation at more than 30 mA of beam current. All of the spectra shown herein have, however, been obtained with the screw-cap anode at lower power. The entrance slit assembly of the spectrometer was modified to provide a better seal between the standard McPherson cubical gas cell (equipped with a thin (-20 pg cmq2) poly- styrene window), the slit and its housing in order to reduce contamination of the open X-ray tube which can occur in the McPherson design.Gaseous samples were introduced alone or premixed with a reference gas through Granville-Phillips leak valves and sample pressures 118 D. A. ALLISON AND R. G. CAVELL were measured with an MKS-Baratron capacitance manometer. The X-ray chamber was separately pumped with a cold (-12K) surface provided by a CTI Model 20 “Cryodyne” refrig- erator. Under these conditions, the X-ray chamber pressures were typically less than Torr with sample cell pressures of the order of 0.2 Torr. The analyser chamber pressure was approximately Torr. RESULTS AND DISCUSSION Zirconium M, X-rays are free of significant interference from satellite contribu- tions, as shown by the extended Ne 2p spectrum illustrated in fig.1, an important I I I I 1 12000 -4 I.9E u 6000302 h Y.CI 8{ 4000 .C( 25 0 50 40 30 20 !3 binding energy /eV (uncorr.) FIG.1.-Neon 2s and 2p orbital spectra excited with Zr Mc radiation at a power of 90 W. The sample pressure was about 0.2 Torr. F.w.h.m. of the 2p peak is 1.3 eV. feature for valence orbital studies. The initial linewidth available in the system is about 1.0 eV (f.w.h.m.) without any preconditioning or special treatment of the anode surface and this linewidth usually broadens to about 1.3 eV after 30-60 min of TABLE1 .-RELATIVE PHOTOIONIZATION INTENSITIES components experimental theoretical c experimental d molecule of ratio (zrMy) (Zr Mg) (Mg K=) Ne PIS 5.97 4.68 4.02 2.13 9 1.28 20.3 0.33 0.28 0.74 0.14 0.53 1.29 0.79 0.10 0.81 0.24 0.52 0.10 a Observed at an angle of 90” with respect to the incident radiation. b Intensities are given as the proportionate areas (obtained by non-linear least-squares fit of the experimental data) of the Zr Mc photoemission peaks. No adjustment for the degeneracy of the orbital has been introduced.C Inter-polated from the tables given in ref. (4). d Spectra taken from ref. (3) and areas evaluated graphically to obtain relative intensities (without adjustment for orbital degeneracy) unless otherwise noted. e Ratio computed from data given in ref. (3), p. 31. f Inverse of the ratio given in ref.(2). Ratio computed from data given in ref. (3), p. 40. h The Mg K, spectrum is given in ref. (8). VALENCE ORBITAL P.E. SPECTRA operation whereupon it remains constant for many hours. No attempt was made to reproduce the optimum linewidth of 0.85 eV reported by Krause since we were interested in the long term prospects of this source. Our spectrum of Ne is similar to that obtained by Wuilleumier and Krause with Zr Mc radiation and the 2p/2s intensity ratios of both studies (table 1) are in good agreement. Note that the Ne 2p linewidth shown in the earlier study is comparable to that obtained by us and is also significantly greater than 0.85 eV. The pressure dependence of the photoelectron emission intensity is similar to that obtained with Mg K, excitation in our system (fig.2) indicating that the lower kinetic 800 700 -!A 600 Y 5 .8 500 h +a .d8 400 U 3._ % 300 ua z" 200 100 0 0 0.2 0.4 0.6 0.8 1.0 1.2 pressure/Torr FIG.2.--Intensity (as count rate) plotted against gas pressure for neon with Mg Kz(Is,300 W) and Zr Mc (2p,80 W) at a 90" observation angle in the McPherson ESCA-36 instrument. The count rate for Zr radiation has been multiplied by a factor of two. energy of the electrons does not lead to a significantly greater absorption in the gas cell at useful pressures. The maximum photoelectron yield occurs at nearly the same pressure for both radiations but, significantly, and as expected, the intensity decreases more rapidly with increased pressure in the case of electrons excited by Zr Mr radia-tion relative to those excited by Mg K, because of the lower kinetic energies of the former.The spectra of Ar, CH4, N, and H20 are shown in fig. 3 to 6 respectively. Com-paring these spectra to those obtained with Mg K, excitation clearly shows the dramatic changes in relative intensities which arise from a change in excitation energy. The most striking change is seen in the spectrum of Ne (fig. 1) where photoelectron emission from the 2p orbital is much more intense than that from the 2s, (by a ratio of 5.63 : l), reversing the intensity distribution obtained for Mg K, e~citation.~ Similar enhancement of 3p intensity relative to 3s is observed for Ar, however, the change is not as pronounced as in the case of Ne.This relative intensity distribution prevails in the molecular orbital spectra of CH4, N2 and H20 where the bands from the deeper molecular orbitals appear with greatly reduced intensity relative to the outermost molecular orbital bands in contrast to the Mg K, spectra where the deeper molecular orbital bands are generally very strong even though these deep J3. A. ALLISON AND R. G. CAVELL molecular orbitals do not have the completely localized 2s character that prevails in the noble gases. Since we cannot at present measure absolute photoemission intensities, direct comparisons of spectra excited by different radiations is not possible. We can, however, make limited and useful comparisons of the relative intensity distribution of valence shell spectra excited with different radiations.Accordingly we have com- 5000 7 ! -1 , ~@L------l-~--~L-_l . -1--_1 I 35 25 15 5 binding energyleV (uncorr.) FIG.3.-Argon 3s and 3p orbital spectra excited with Zr My radiation at a power of 90 W. The sample pressure was about 0.2Torr. F.w.h.m. of the 3p peak is about 1.6 eV. 7---I rl I 18001-1 h “2 1 Y 12004 s x Y .3 600 u8 c .d binding energy/eV (uncorr.) FIG.4.-Valence band spectra of gaseous CH4 excited with Zr Mr; radiation at a power of 90 W. The sample pressure was approximately 0.30 Torr. puted orbital band intensities as ratio values relative to the intensity of the innermost valence shell orbital band to evaluate the effect of employing different radiations for the excitation of molecular orbital photoemission spectra.We have confined our attention to those valence shell bands in the present series of molecules which are accessible with Zr Mr excitation. Relative peak intensities obtained with Zr Mc X-rays and comparison values derived from the Mg K, X-ray spectra of others are presented in table 1. The Zr Mr;ratios were obtained by curve fitting our raw VALENCE ORBITAL P.E. SPECTRA experimental data whereas the Mg K, ratios were obtained by a graphical integration procedure using the published figures as the source and are therefore less reliable. The latter ratios, in spite of their deficiencies, illustrate the differences in cross-section observed with the different radiations.The quoted ratios represent the ratio of areas of the particular photoelectron peaks and do not contain any weighting for the degener- acy of the orbital which is assigned to the band. 3000 3 In 08-2000 U0 8---. x*.+$ I000 U.* C binding energy/eV (uncorr.) FIG.5.-Valence band spectra of gaseous N2excited with Zr Mc radiation at a power of 90 W. The sample pressure was approximately 0.20 Torr. 1050 4 In 3 700 42 c 8 x U.+ Y$ 350 .* 0 I I I I I 4s 35 25 15 5 binding energy/eV (uncorr.) FIG.6.-Valence band spectra ofgaseous H20excited with ZrMc radiation at a power of 90 W. The sample pressure was about 0.20 Torr.The change in relative photoemission band intensities with the energy of the incident radiation appears to be well understood for atomic species such as the noble gases, as shown by the excellent agreement of experimental results for Ne and Ar 9with theoretical calculation.2 We have estimated approximate theoretical p/sintensity ratios for Ne and Ar by interpolation, at the Zr Msenergy, of the computed cross-sections given by Kennedy and Maiis~n.~ The estimated ratios are in very good agreement with experiment in spite of errors inherent in an interpolation pro- D. A. ALLISON AND R. G. CAVELL cedure. It is worth noting that theoretical photoemission curves for the valence shell components of Ne over the accessible energy range (0 to 1500 eV) are in good agreement with present and previous experimental results and clearly show that the s orbital exhibits a larger photoemission cross-section than the p orbital at higher excitation energies whereas the converse is true at lower energies.The resultant predicted cross-over in relative s and p photoemission intensity occurs between Zr Ms and Mg K, energies and the effect is clearly seen in the experimental behaviour of Ne. For molecules, quantitative understanding of the phenomenon has not yet been achieved although the method described by Ellison would appear to have some promise and our studies are presently directed toward the evaluation of this approach. Qualitatively, however, it is clear that the change in relative intensities from Mg K, to Zr Me radiation will likely depend on the atomic character of the molecular orbitals in the valence shell in a similar fashion to the changes observed in Ne and Ar spectra.In the spectra of N2,for example, the In, and 30, orbital bands are weak and the 20, is strong relative to 20, when excited by Mg K, radiation. With Zr MCexcitation, however, the In, orbital band shows a substantial increase in relative intensity (compared with 20,) and the 171, band becomes a prominent feature in the outermost set of orbital bands. The 20, orbital band suffers a substantial decrease in relative intensity (compared to 20,) when Zr Meexcitation is employed in contrast to the high relative intensity of this band in the Mg K, spectrum. The relative intensities of the 30, and 2a, bands appear to be similar for Zr and Mg excitation as illustrated by the small change in the relative intensity ratio with these two sources.To relate these observations to the molecular orbital character we note that the reference orbital, 20,, possesses a large fraction of N 2s character whereas the orbital exhibiting the largest change in relative intensity, the ln,, possesses exclusively N 2p character by symmetry and, in keeping with the behaviour exhibited by neon, this band should show relatively enhanced intensity with Zr Mt; radiation because of its exclusive p molecular orbital character. Similarly excitation by Zr Ms rather than Mg K, causes a large decrease in the relative photoemission intensity of the 20, orbital, which is dominated by a large N 2s character,6 whereas the 30, orbital, which is composed of approximately equal parts of N 2s and N 2p components, is relatively unaffected by this change in excitation energy. Relative intensity variations of the valence orbital bands of H20 are similar. Here all orbitals show substantially increased relative intensity (compared to 2a1) upon excitation with the Zr Me source with the greatest changes in relative intensity exhibited by the lbl and lb, orbitals which are, by symmetry, exclusively oxygen p in character.The 3al orbital which is a mixture of oxygen s and p, but dominated by the latter,' also shows a significant increase in relative intensity under Zr Mt; excitation, in keeping with expected enhancement of the p type molecular orbitals.The spectrum of CH4 with Zr Mr excitation shows a substantially enhanced It, orbital band relative to the 2al band as expected because of the dominant C(2p) atomic orbital character of the former. In contrast the Mg K, spectrum * of CH4 is dominated by the large C(2s) character ofthe 2a, band; the 2a, band is strong and the It, band is almost undetectable because, in addition to its low intensity, the Mg Ka3,4satellite from the 2al band occurs very close in energy to the It, band. With Zr Meradiation, however, the 1 t, band is clearly observed with good relative intensity. It is also noteworthy that the It, band is relatively broad and possesses a double maximum similar to that observed in the helium(1) spectrum of CH4.I0 A previous empirical relative intensity model developed for Mg K, spectra was based on the premise that H(1s) orbital character in molecular hydrides did not VALENCE ORBITAL P.E.SPECTRA contribute to observed photoemission intensity because no Mg Kaspectrum can be observed for H2. Preliminary work indicates that a spectrum can be obtained from H2 using Zr Ms radiation although we have not yet obtained reliable intensity data. If hydrogen orbital contributions are substantial, then the observed relative intensity variations cannot be wholly ascribed to the differing proportions of central atom s and p orbital character. Proper understanding of molecular orbital intensities of hydrides such as H20and CH4 will require inclusion of H(1s) orbital contributions, especially in the ultra soft X-ray and U.V.energy regions. It is unfortunate that the relatively large linewidth obtained with Zr Mr does not permit clear resolution of some of the valence shell component peaks, however, the significant changes in photoionization cross-sections are sufficient to provide useful qualitative assistance to aid the interpretation of Mg Kaspectra which can be obtained with greater resolution at the present time. We are attempting to improve the line- width of the Zr Mr source and investigating alternative sources of similar energy which may have narrower linewidths. The broad line character of the excitation source does not unduly hamper measurements of relative photoemission intensity at 151.4 eV and studies are presently in progress to investigate larger molecules and to apply and test the theoretical procedure described by Ellison. CONCLUSION The use of Zr Mr X-rays as a practical radiation source may aid the interpretation of valence shell photoelectron spectra because of the substantially different cross- sections of this source relative to Mg K4.The 2p character contributes the major portion of the peak intensity in spectra of compounds containing second period ele- ments whereas the 2s contribution dominates in such spectra excited by Mg Ka radiation. We thank the National Research Council of Canada and the University of Alberta for financial support.We are also indebted to Mr. Ron Cox and his colleagues in the Chemistry Department Machine Shop and Mr. Rudy Kenwell for extensive assistance with the electrical modifications. M. 0. Krause, Chem. Phys. Letters, 1971, 10, 65. F. Wuilleumier and M. 0. Krause, Electron Spectroscopy, Proc. Asilomar Conference, ed. D. A. Shirley (North Holland, Amsterdam, 1972), p. 259. K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P. F. Hedin, K. Hamrin, U. Gelius, T. Bergmark, L. 0. Werme, R. Manne and Y. Baer, ESCA Applied to Free Molecules (North Holland, Amsterdam, 1969).D. J. Kennedy and S. T. Manson, Plzys. Rev. A, 1972, 5,227. F. 0. Ellison, J. Chem. Phys., 1974, 61, 507. B. J. Ransil, Rev. Mod. Phys., 1960, 32, 245.’S. Aung, R. M. Pitzer and S. I. Chan, J. Chem. Plzys., 1968, 49,2071. * U. Gelius, Electron Spectroscopy, Proc. Asilomar Conference, ed. D. A. Shirley(North Holland, Amsterdam, 1972), p. 311. W. E. Palke and W. N. Lipscomb, J. Amer. Chem. Soc., 1966, 88, 2384. D. W. Turner, C. Baker, A. D. Baker, and C. R. Brundle, Molecular Photoelectron Spectro- scopy (Wiley, New York, 1970). (PAPER 5/364)

ISSN:0300-9238    

DOI:10.1039/F29767200118

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Antiferrornagnetism in Transiti on Metal Complexes Part 7.'-Critique of the Heisenberg Model and a Re-examination of the System of Three Copper(r1) ions in a Linear Array BY RICHARD F. A. KETTLE"W. JOTHAM,-/-SIDNEY AND JOHNA. MARKS$ School of Chemical Sciences, The University of East Anglia, Norwich NK4 7TJ Methods which have been used to modify the basic Heisenberg-Dirac-Van Vleck treatment of magnetically concentrated systems are discussed. The electronic states arising from the interactions between three collinear copper@) ions are investigated by a variety of methods using orthogonalised atomic or molecular orbital basis sets to examine the explicit form of the exchange integrals involved in the treatment of such systems. The complementary roles of chemical bonding and electron- repulsion effects are illustrated and it is concluded that, in this system, a low-lying doublet state, unrecognised in the spin-exchange treatment, may be of magnetic importance.One aspect of the increasing availability of variable-temperature magnetic suscep- tibility balances in recent years has been the identification of many compounds of the first transition series with sub-normal magnetic moments, which are generally recog- nised to consist of infinite or discrete antiferromagnetically coupled polymetallic units, and a handful of similar systems with an indisputable ferromagnetic coupling.2 This new information has led some workers either to seek more sophisticated models to describe their systems or, alternatively, to question the extant models on essentially empirical grounds.The models available detail spin hamiltonia and assume an effective coupling of the dipolar type as given in eqn (1) and (2). From these, pro- perties of interest, such as the temperature dependence of the magnetic susceptibility, may be calculated. In eqn (1) 2 = %o+A?e+Za (1) X0is the term representing the uncoupled system, Zeis the dipolar coupling term and #'a represents additional coupling terms which may be introduced in an extended model. The dipolar term may be expanded as a summation over metal pairs where Jij is the effective exchange integral and Siz,Six and Si,are the z, x and y components of the spin on the ith metallic centre. k is an essentially arbitrary constant which some workers assign as -2 to emphasise that there is, at least, a formal parallel to true exchange interactions (but which we shall take as -1 precisely to avoid this analogy).The value of y is unity in the isotropic Heisenberg model, the one most frequently considered. This model is a simple and quite good approximation to most real systems and therefore provides a convenient reference against which comparisons withother models are normally made. So, in the example of particular interest in this paper, the case t Department of Science, Kesteven College of Education, Stoke Rochford, Lincolnshire NG33 5EJ. Department of Chemistry, University Pilau Penang, Penang, Malaysia. 125 126 ANTIFERROMAGNETISM IN TRANSITION METAL COMPLEXES of three Cu" ions in a linear array (with negligible coupling between the terniinal atoms), the model predicts a spin quartet of energy Eo-+J and doublet states of energy ,Yo and Eo+ J.From these energies the equation for the molar susceptibility is found to be that given by (3), Ng2P2 1.5+1.5 exp (J/kT)+15exp (1.5JIkT)+ Na& = -3kT 2+2 exp (J/kT)+4exp (1.5JlkT) (3) in which all symbols have their usual meaning. Five main methods have been used to modify the basic Heisenberg model in the interpretation of the bulk susceptibility of some magnetically concentrated compounds. In the first of these, workers have allowed quantities other than the exchange integrals to be treated as variables to improve the agreement between observed and calculated susceptibilities, a procedure difficult to justify for those quantities which can be determined or estimated from independent experimental evidence.However, in infinite systems, the introduction of a temperature-correcting factor (T+0) is in the spirit of the Weiss molecular field approximation. 9 A second, closely related, method involves the introduction of temperature- dependent variables. For example, Lines has shown that when large unquenched orbital angular momentum is present, as in some Co2+clusters, the g factor may be replaced by a temperature and exchange-energy dependent factor. The Ising model, in which y in eqn (2) is given a zero value, has been used widely in the interpretation of infinite coupled systems such as linear metal chains, because it can be used to generate analytical equations for the properties of interest, whereas the Heisenberg model can only be used analytically with additional assumptions.6 However, for some cases it appears theoretically preferable to use an empirical equa- tion obtained from the Heisenberg m0de1.~ The fourth approach involves the introduction of other terms into the spin- coupling hamiltonian (1).For example, Sage has given the results of a high tempera- ture expansion of the magnetic susceptibility which includes second and third order terms in the dipolar coupling.8 Lines, Ginsberg and Martin have investigated the role of antisymmetric bilinear exchange terms of the type (4), Xa= C k'[Si.K, . Sj] (4)i,j where K, is an antisymmetric exchange tensor, in addition to the dipolar coupling terms in the Hamilt~nian.~ Evidence for this coupling, which is only significant when the local symmetry of the metal ion is low and the orbital angular momentum is not fully quenched, has been intensively sought.lo It now appears that this antisymmetric coupling can account for the very strange magnetic susceptibility data of the Cu40X6Y4 tetramers such as Cu,0Cl,(OPPh3)4.9~ l1 The final modification abandons the explicit use of eqn (1) and attempts to assess those electronic states which may be thermally populated and their spin characteris- tics.This approach admits the possibility that states arising from a variety of single- metal-ion configurations may be considered, although the extra states which are relevant at temperatures below 500 K are few in number.Using this approach we have shown that there is aprima facie case for the inclusion of a low-lying singlet state arising from partially filled d,2 orbitals for Cu" dimers, such as copper(I1) acetate, in addition to the familiar singlet-triplet pattern given by the dipolar coupling rneth0d.l Because of the relative crudity of the theoretical model, evidence for the existence of such a low-lying singlet state must ultimately be obtained from experimental data. We have therefore reinterpreted all available susceptibility data on dimeric copper@) R. W. JOTHAXI, S. F. A. KETTLE AND J. A. MARKS species using our model. In these investigations g and Nu were taken from experi- mental data so that the same number of adjustable parameters were used as in the conventional singlet-triplet model.In general this reinvestigation supported the postulate that a low-lying singlet state may exist in dimeric copper(1r) species.1* l3 No model which has been reported to date makes more than parametric allowance for the existence of the ligands which, almost invariably, bridge the interacting metal ions. This is an unsatisfactory situation in view of the commonly held view that it is the " through ligand " interactions which play a dominant role in the coupling between magnetic centres. Unfortunately, it is only marginally within the capability of current computers to include the ligands explicitly in an ab initio calculation and, even then, the accuracy obtainable falls far short of that needed to comment in any detail on the magnetic problem.Accurate calculations involving the metal ions only are much more feasible but the numerical accuracy with which the necessary integrals can be evaluated is quite illusory because of the neglect of ligands. In this situation we have preferred to tackle the problem algebraically and to use order-of-magnitude values for integrals. This approach permits an answer to questions of feasibility, of whether it is conceivable that a certain situation can occur. Whether or not it does has, of course, to be determined experimentally. We have been motivated in this approach by the occasional complete failure of the Heisenberg-Dirac-Van Vleck model when applied to some weakly-coupled transition metal ion systems.This may be indicative of a more general weakness in the theory than is immediately apparent, and which could to some extent be concealed by its adjustable parameters, and suggests that a general search for additional low-lying levels may be rewarding. In the present paper we report the results of such a theoretical investigation on a system of three copper(I1) ions in a linear array on a fourfold axis; the molecular symmetry considered is D4,,.This system, whilst having the advantage of being theoretically tractable by our methods, has the disadvantage that relatively few species are known to which the resultant theory may be applied. The level of sophistication is similar to that used by us in our work on dimeric copper(I1) systems.This means that we attempt to identify the major contribution to the state energies, arguing that when these are identical for two states there is a prima facie case that the two states may be of comparable energy. The only states of direct relevance to magnetic measurements are those within a few hundred wavenumbers of the ground state and so our study is aimed at detecting states which have the same major energy determining terms as the ground state. However, an accuracy of a few hundred cm-' is well beyond the limits of the present calculation so that they may be regarded only as indicative. The geometry which we consider permits two arrangements of ligands, is shown in fig.1. The energy level pattern given by the Heisenberg-Dirac-Van Vleck model for the case of three copper(I1) ions in a linear array is given in fig. 2. The corresponding equation for the temperature variation of the magnetic susceptibility is given by (3). In the limit J = 0, xh reduces to the sum of the susceptibilities of three independent copper(I1) ions, but when it is large xh has the form appropriate to a single copper ion. This latter case corresponds, physically, to the spin on the central atom being aligned anti-parallel to the other two. Full details of the present calculations are given in an Appendix and as a deposited document. As for the case of binuclear copper(I1) species, it is clear from our analysis that the ground and any low-lying states are derived from hole configurations involving the dz2 and d,zLY2 orbitals on the three copper atoms and that all major energy terms important for our purpose arise from within these configurations. In both the eclipsed and the staggered configurations (fig.l), the set of three ]z') orbitals transforms as 2Alg+A2uin D4h 128 ANTIFERROMAGNETISM IN TRANSITION METAL COMPLEXES symmetry. In the eclipsed codguration the set of three Ix2-y2) orbitals transforms as 2B1,+B2,,,but in the staggered configuration this set transforms as B1,+B2,+BZu, so that the two cases must be considered separately. The energy level diagrams which we obtained for the case of (a)staggered and (b)eclipsed terminal and central d,~-~2 orbitals are given in fig.3 and 4 respectively and are to be compared with fig. 2. The problem of the involvement of 42 orbitals cannot be unambiguously resolved by the present calculation. In particular, it depends on the relative magnitudes of core and repulsion integrals involving dz2 orbitals as compared with the corresponding integrals involving dX2-,,2 orbitals. However, a similar situation held in the case of binuclear copper(I1) species and, in view of our conclusions for that case which point towards the involvement of such levels, we believe that there is a case for admitting the possibility of a low-lying state in linear trinuclear copper(I1) systems. In the D4,,geometry which we have considered it is clear that any such low-lying state will be 2A2u(for both staggered and eclipsed geometries, as shown in fig.3 and 4). De-noting the energy of this doublet as relative to the ground state, the expression for the temperature variation of the susceptibility which results is Ng2/I2 1.5 + 1.5 exp(J/kT) + 15 exp(l.5J/kT) + 1.5 exp( -A/kT)+Na. (5)xfrl == -2 +2 exp(J/kT) +4 exp(l.SJ/kT) +2 exp( -A/kT)3kT 2 Z 2 t + (a)eclipsed (b) staggered @ Cu(I1) ion ligand site FIG.1.-The idealised geometry of three linear, six-coordinate, copper(@ ions together with the atomic and molecular axes used in this work. -0 (a)trinuclear Cu(I1) complex (6)trinuclear Cu(W complex (equilateral triangle) (linear) FIG.2.-The energy level patterns for three antiferromagnetically coupled copper(I1) ions predicted by the spin-exchange hamiltonian.R. W. JOTHAM, S. F. A. KETTLE AND J. A. MARKS The system of three co-linear copper ions has the advantage that it is one of the very few amenable to our appr~ach.~ It suffers from the disadvantage that there are, presently, few systems which may provide a test of the relative applicability of the susceptibility eqn (5) and (3). The series of compounds which most nearly correspond to the model discussed in this paper is that reported by Gruber et al. and reinvestigated by Figgis and Martin.14* l5 These workers prepared a series of five trinuclear copper complexes in which a central copper ion was further complexed by two copper-Schiffs base groups. In these complexes the central copper atom is linked to each of the terminal copper atoms by a bridging ligand, and very little coupling between the terminal copper atoms would be expected.The bulk suscepti- bility data for these compounds are described rather poorly by eqn (3) and the best- fit values reported for gyromagnetic ratio vary widely in the range 2.10-2.20. (In I I I I I I I I \ \ \ \ \ \ \ \ f t==A2, a I I 31 x2- y2> staggered eclipsed FIG.3.-The low-lying energy levels predicted by this work for three co-linear copper(@ ions. The degeneracy of the lower set of levels in the staggered configuration is to be contrasted with the non- degeneracy of the eclipsed configuration and the predictions of the spin-exchange hamiltonian (fig.2). this context the widely scattered g-values calculated from e.s.r. data seem improb- able.") However, no general improvement results from the application of eqn (5) (see table 1). Either more accurate experimental data, with careful avoidance of paramagnetic or other impurities, are needed for these compounds, or, alternatively, we conclude that they indicate that additional low-lying excited states are not impor- tant in the description of these compounds. A compound of a similar type is Cu3C1,(2-picoline N-oxide),*2H20, which has an almost constant magnetic moment of 1.03 B.M. per copper atom over the range 77-273 K,16*l7 and is presumed to have a linear structure similar to that of Cu3CI,-(CH3Cv2. Conversely, the room-temperature moment of the linear trimer Cu3Cl,(adenine H),4H20 is as high as 1.86 B.M.per copper atom.19 The series of anhydrous N-2-(2-hydroxyethylthio)henylarenesulphonamidatocopper(11) chelates 11-5 130 ANT IFERR0MAG N E TI S M IN TR A N SIT I0 N MET A L C 0MP L E X E S TABLE1.-MAGNETIC PARAMETERS FOR SYSTEMSOF THREE COPPER(II) IONS published data this work no. of 109~~l10901 1094 refer-%IT -J/ m3 m3 -J/ A! m3 compounda ence model * data 9 cm-1 mol-1 mol-1 modeld cm-1 cm-1 mol-1 14[CU(ES)I~CU(C~O~)~.~H~~ L 10 2.10 80 2.27 1.50 L 89.2 -1.98. LE 86.5 279.9 1.oo T 68.1 -1.42 TE 67.1 362.6 1.03 15 L 22 2.09 86 2.27 2.00° Lf 77.9 -13.4 LEf 57.1 11.3 8.92 Tf 60.8 -11.5 TEf 48.4 9.6 7.90 Cu(HPNS)I3 20 T 14 2.10 344 2.27 4.22 L 316.7 -1.23 LE 289.9 197.1 1.09 T 257.8 -1.13 TE 254.1 442.8 1.12 CU(HPTS)I~ 20 T 13 2.15 344 2.27 3.54O L 323.4 -1.890 LE 321.6 709.2 1.89.T 265.9 -2.028 TE 265.9 Q) 2.020 ~,~-PS)J~CU(C~O~)~*~H~O[CU( 14 L 10 2.15 360 2.21 0.47 L 351.7 -0.50 LE 336.6 356.9 0.45 T 286.4 -0.40 TE 286.4 00 0.40 [Cu(EHA)] ~CU(CIO 4) 2'2H20 14 L 11 2.13 400 2.27 0.42 L 413.1 -0.64 LE 413.1 co 0.64 T 341.3 -0.78 TE 341.3 00 0.78 15 L 18 2.04 380 2.21 5.2Se Lf 748.4 -22.08 LEf 748.4 to 22.0 T f 748.4 -22.0 TE f 748.4 co 22.0e [Cu( 1,3 -PH A)] ~CU(CIO 4) 2.3 H20 14 L 9 2.20 480 2.27 0.99 L 432.1 -2.53' LE 360.6 71.2 2.47 T 352.3 -2.48 TE 309.0 61.0 2.45 15 L 10 2.10 472 2.27 2.30e Lf 840.4 -3.25 LEf 840.4 165.9 3.25 Tf 840.4 -3.25 TEf 840.4 165.9 3.25" 14 L 14 2.12 460 2.27 0.39 L 507.2 -0.89 LE 507.2 to 0.89 T 426.2 -0.97 TE 426.2 00 0.97 15 L 12 2.17 474 2.27 2.08e Lf 685.6 -1.10 LEf 685.6 co 1.10 Tf 607.9 -1.14 TE f 607.9 Q) 1.14 16,17 L 5 2.05 00 2.27 3.62= L 00 -7.46 0 LE 00 03 7.46 T 00 -7.46 TE 03 00 7.46 I21 T 16 2.00 00 2.09 0.70 L 00 4.52 LE Q) to 4.52 T 00 -4.52 8 TE 00 to 4.52 e a ES = NN'-ethylenebis(salicylaldimine), EHA = NN'-ethylenebis(0-hydroxyacetophenimine), HITS = N-[2-(2-hydro-xyethylthio)phenyl]-4-toluenesulphonamidato-, HPNS = [2-(2-hydroxyethylthio)phenyl]-2-naphthalenesulphonamidat0-.2-PNO = 2-picoline N-oxide, 1,2-PS = NN'-1,2-propylenebis(salicylaldimine), 1,3-PHA = NiV'-1,3-propylenebis(o, hydroxyacetophenimine), PA0 = deprotonated pyridine-2-carbaldehyde oxime.0 = ( 2. {x(calc.)-$expt.)}2/n . i=l i >' g = 2.130, Noc = 2.83 x 10-9 m3 mol-1 = 225X 10-6 c.g.s. L = linear, LE = linear plus an additional excited state, T = equilatoral triangle, TE = equilatoral triangle and additional excited state. o 3 1.89 X 10-9 m3 mol-1 = 150X 10-6 c.g.s. : this error is such that it seems probable that either the data is unreliable or the model considered is inappropriate. f g = 2.160 used. R. W. JOTHAM, S. F. A. KETTLE AND J. A. MARKS are thought to be triangular systems,20 but we have investigated them utilising both linear and triangular models (in each case with and without the inclusion of additional thermally populated excited states). The trimeric complex Cu,L,(OH) S04*2H20 (where L = deprotonated pyridine-2-carbaldehyde oxime) is known to involve an equilateral triangle of copper atoms from X-ray data on the crystals.21 The magnetic moment is recorded as 1.00 B.M.(mol Cu)-' over a wide temperature range. It is interesting to note that, in the absence of structural data, it appears to be very difficult to distinguish magnetically even between the linear and triangular models for any of these compounds. CONCLUSION Although it is improbable that in the near future it will be possible to carry out detailed calculations on systems such as those considered here, but with explicit inclusion of the ligands, such that the results may be used to discuss magnetic data, it is, nonetheless, of interest to consider the relationship between such an approach and that presented here.Two situations may arise. First, the pattern of low-lying energy levels may be isomorphous with those considered here. This will be the situation if, as generally assumed, the low-lying levels are dominantly transition-metal in parentage. Each integral in our approach would then become a sum of integrals, the additional integrals being ligand or ligand-metal in origin, modulated by products of mixing coefficients. The most that could be anticipated is that the sum of these integrals would be of the same order of magnitudes as those evaluated on our metal-orbital-only basis (hence our philosophy of regarding order-of-magnitudes as of more importance than exact numerical values).Quite new energy terms would arise from extended configuration interaction but we would hope to have included the major contributions on our approach. Secondly, quite new low-lying levels might appear, largely ligand in parentage. If this were the case it is possible that none of the extant models would be applicable. For instance, it might well be essential that spin-orbit coupling be included explicitly. The indication is that future theoretical advances in this field may well depend on the development of simple methods by which more explicit recognition can be given to the presence of the ligands in polynuclear complexes. APPENDIX Although the case of three co-linear copper ions is one of the few which can be treated algebraically, it only becomes so by transformations which bring all large terms to diagonalpositions. The residual off-diagonal terms are then incorporated by perturbation theory.The basis sets which were chosen for each of the cases discussed are given in the appro- priate place in the text. The transformation properties of symmetry adapted combinations, &-A3 are 21 A, 1-3 1Z2> (a13 (QlJ (a2u) Ixz-y2) eclipsed (b1J (big)' (b2J (Al) Ix2-y2) staggered @2g) @Ig) (bJ. The transformation properties of the ten product functions, with S, > 0 for each basis are given in table 1. In each case, application of the projection operator technique gave rise to one quartet and eight doublet states as given in table 2. * The tables referred to in this appendix have been deposited with the N.L.C.132 ANTIFERROMAGNETISM IN TRANSITION METAL COMPLEXES The principal component terms of all of the energy matrices used in this paper are defined in table 3, in which lower case letters indicate Ix2-yz). The rounded and angular brackets have their normal meaning,2 and results obtained by applying the Mulliken approximation 23 for the smaller, but not negligible, terms are also given in table 3. Complete energy matrices can now be written out in block diagonal form, but it is con- venient to carry out a partial diagonalisation in some of the submatrices so that the larger terms are already diagonalised. We give the necessary eigenvectors and the partially diagonalised matrices for the cases of staggered and eclipsed Ix2-y2) orbital bases (QMO) in tables 4 and 5 respectively. The case of 1z2) orbitals is isomorphous to the latter case, and the appropriate matrix may be obtained directly from table 5 by introducing upper case subscripts and the appropriate symmetry labels.Final energy levels were determined by the normal perturbation technique. Terms in higher powers of S than 2 are omitted from the tables. For comparison with the results obtained by the use of the OAO basis it is con- venient to retain the core terms in G,, in table 5. Although this term may be regarded as zero in the OM0 basis, the corresponding term derived from the OAObasis has a significant value. The OAO basis used as an alternative method for the discussion of the eclipsed Ix2-y2) orbitals was constructed in the following way.24 The method consists of orthogonalizing the atomic orbital basis 4 through the matrix transformation where is the new orthogonal basis, and 9is the overlap matrix constructed with the overlap integrals Sii= 1, Sij=S,,,i=jf1,Sij=0,i=j+n,n>2.We have assumed that negligible overlap occurs between non-adjacent atomsz5 The S matrix has the form 9is non-singular and therefore the transformation 9-4 may be easily constructed by first diagonalizing 9to give 9'.If P is an orthogonal transform such that 9'= P9P-1 (A4) then 9-s = p-1p-+p. (A5) Alternatively, one may use a general relation developed for S matrices of dimension N with the form 26 Sii= 1, Si,ifl= S,Si,ift= 0,r > 1, for i = 1,2... N sin(iqn/N + 1)sin(jqn/N + 1)2c (A611J(Csp+)..= N+l -4=1 1+2S~os(qn/N+l)~ ' ldq<N. The resulting orthogonalized basis has the form R. W. JOTHAM, S. F. A. KETTLE AND J. A. MARKS (1+JZS)'] b + [(1+JZS)' +-(1-JZS)' +2(1-2S2)']c) where we omit the suffix from Subas it is the only non-zero overlap integral involved. (XI, XZ, x3) transform isomorphously as (a,b, c) in D4,,symmetry, so that we may replace (6) by A; = &(l+ $x2 -k x3)-A; = $(xi-J2~2+~3) (A8) 1A; = -Jz(x1-x3). The evaluation of the energy matrix for the eclipsed Ix2-y2) configuration is identical to the previous approach as the two bases are both orthogonal. However, a great simplifica- tion occurs when the core and repulsion integrals are evaluated in terms of their components in the QAO basis because of the orthogonality condition (xilxi) = aij.Only eight core and repulsion integrals arise with the OAO basis : J2= (x1x1 IIx1x1) = (x3x3 Ilx3x3) J$ = (XlXl lIx2;c2) J; = (x2x211x2x2) J: = (xi% k33/3) G: = (Xlllxl) = (x3lix3) G$ = (x2 11x2) GZ = (x1Ilxd GZ = (x1llx3). To obtain the Blg and B2umatrices for the QAO basis from those given in table 5, wz merely omit all terms in Sand S2 and replace each other term by its starred counterpart and, following the same diagonalization procedure, the eigenvalues are obtained by substituting the explicit values of the Lowdin integrals which are given in (A10)Jz= Jaa(l+3s')$-*S2Jab JZ = $S2(Jou Jbb) fJab( 1 YS')f $S2Jac JG = Jbb(1 f 5s2)+3S2J0h JZ = Jot( 1 +%S2) $s2Jab GZ = Ga+$S2(3G,+ Gb)+ SG,, (A10) GZ = Gb+iS2(Ga+ 3Gb)-I-2SGab GZ = Gub(l 2S2)+ @(G,+ Gb) G: = $S2(3G, + Gb) sGub-Part VI, J.P. Fishwick, R. W. Jotham, S. F. A. Kettle and J. A. Marks, J.C.S. Dalton, 1974, 125. (a)M. Kato, M. G. Jonassen and J. C. Fanning, Chem. Rev., 1964, 64, 99; (b)G. F. Mokoszka and G. Gordon, Transition Metal Chem., 1969, 5, 181; (c! E. Sinn, Co-ord. Chem. Rev., 1970, 5, 313 ; (d)P. W. Ball, Co-ord. Chem. Rev., 1969, 4, 361; (e)R. L. Martin in New Pathways in Inorganic Chemistry, ed. E. A. V. Ebsworth, A. G. Mad-dock and A. G. Sharpe (Cambridge, 1968). R. W. Jotham and S. F. A. Kettle, J. Chem. SOC.A, 1969, 2821. S. Koide and T.Oguchi, Ado. Chem. Phys., 1963, 5, 189. 134 ANTIFERROMAGNETISM IN TRANSITION METAL COMPLEXES M. E. Lines, J. Chem. Phys., 1971,55,2977. ti M. E. Fisher, J. Math. Phys., 1963, 4, 124. R. W. Jotham, Chem. Comm., 1973, 178. M. L. Sage, Inorg. Chem., 1971, 10,44. M. E. Lines, A. P. Ginsberg and R. L. Martin, Phys. Rev. Letters, 1972, 28, 684. lo P. Erdos, J. Phys. Chem. Solids, 1966, 27, 1705. J. A. Barnes, G. W. Inman, Jr. and W. E. Hatfield, Inorg. Chem., 1971, 10, 1725. R. W. Jotham and S. F. A. Kettle, Inorg. Chem., 1970, 9, 1390. l3 R. W. Jotham, S. F. A. Kettle and J. A. Marks, J.C.S. Dalton, 1972, 428, 1133. S. J. Gruber, C. M. Harris and E. Sinn, J. Chem. Phys., 1968, 49, 2183. l5 B. N. Figgis and D. J. Martin, J.C.S. Dalton, 1972, 2174. I6 E. Sinn, Inorg. Nuclear Chem. Letters, 1969, 5, 193.' H. Miyoshi, H. Ohya-Nishiguchi and Y.Deguchi, Bull. Chem. SOC.Japati, 1972, 682. l8 R. D. Willett and R. E. Rundle, J. Chem. Phys., 1964, 40, 838. P. de Meester and A. C. Skapski, J.C.S. Dalton, 1972,2400. 2o S. Emori, M. Inoue, M. Kishita, M. Kubo, S. Mizukami and M. Kono, horg~Chem., 1968, 7, 2419. 21 R. Beckett, R. Cotton, B. F. Hoskins, R. L. Martin and D. G. Vince, Austral. J. Chem., 1969, 22, 2527.'* P. W. Anderson, Phys. Rev., 1959, 115, 2. 23 J. B. Goodenough, Magnetismand the Chemical Bond (Interscience, New York, 1963). 24 P. 0.Lowdin, J. Chem. Phys., 1950, 18, 365. 25 R. S. Mulliken, J. Chim. phys., 1949, 46,497, 675. 26 G. W. Wheland, J. Amer. Chem. SOC.,1941, 63, 2035.

ISSN:0300-9238    

DOI:10.1039/F29767200125

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Kinetic Effects in the Electron Spin Resonance Spectra of some Semiquinones BY WILLIAM T. DIXON*AND DAVIDMURPHY Bedford College, Regent's Park, London NWl 4NS Received 14th March, 1975 Protonation of semiquinones gives rise to kinetic effects in their e.s.r. spectra; these have been analysed by making use of changes in relative line heights with acid concentration. Supported bydata which indicate that radical ions cannot be present in significant concentrations over the relevant range of acidities, it is shown that in each case studied, only the neutral species need be considered in the interpretation of the kinetic effects. Rate constants for the proton exchanges are calculated and coupling constants for the fully protonated forms (positive ions) are given for the semiquinones from hydroquinone, catechol, resorcinol and from phloroglucinol.In our studies of substituted phenoxyl radicals we have found that it is often difficult to obtain good e.s.r. spectra of ortho-, meta-or even para-semiquinones in acid solution. This problem was also encountered by Stone and Waters 2-4 who reported that they could not obtain analysable spectra from the oxidation of phloro-glucinol by CeIV in acid solution. These difficulties arose from protonation of the semiquinones, a subject which was studied in detail by Carrington and Smith 59 who considered the radicals from catechol, hydroquinone and froin pyrogallol in particular. We have now studied the radicals from phloroglucinol and resorcinol, and have also re-examined the 0-and p-semiquinones, since interpretation of the earlier results was complicated by the lack of clear supporting data.EXPERIMENTAL The radicals were generated in a flow system by the oxidation of the phenolic compounds by cerium(1v). Using cerium(1v) ammonium nitrate as oxidant the acidity of the solutions could be varied from pH N 2 down to 9 mol dme3 sulphuric acid. The latter concentration was limiting in the flow system due to high viscosity. To obtain spectra from solutions which were nearer pH 7 the Ti'", EDTA/H202 system was employed as oxidant. DERIVATION OF FORMULAE At some stage, in each case studied, lines were broadened due to exchange of protons which was not rapid enough to give sharp, averaged spectra. Due to difficulties inherent in our experimental set-up, it was not practicable for us to measure line widths per se so we used relative line heights in our analysis of the kinetic effects.Using the standard approach for a two-site problem, starting from the Bloch we arrive at the following general formula for the out-of-phase component of the magnetization, v : E.S.R. SPECTRA OF SEMIQUINONES where the symbols have their usual connotations '-lo and where : Pab = probability of change from state A to state B; Pba = probability of change from state B to state A; Pa = probability of a transition between state " A " electron spin levels due to thermal motion (1/Tza); pb = probability of a transition between state '' B " electron spin levels due to thermal motion (l/Tz,); P = pbapa+ PabPb + papb-For our purposes, states A and B can be thought of as being defined by their common "vnuclear spin configurations. " is proportional to the absorption and under appropriate conditions eqn (1) simplifies to the following : (i) Near cu = G = (PbaCL)A+PbaCDB)/(Pab+Pba) when rate of exchange is fast enough for a line to be seen Pa,Pb4Pab, Pba >, IcuA-cuBI At V= a2 + (0-w)2 (14 where Po = (average of Paand P,,) = + pbpab Pab + pba (ii) Near o = W, when a line can still be clearly seen, i.e., " slow " rate of exchange, Pa,Pb < ImA-aBI 2 Pab,Pba v= -A, = Po; to= m'M.A;(w-w)' In all three cases the line shape is Lorentzian with a half-width of " A, ", and the height of the derivative curve is given by : RESULTS RADICALS FROM PHLOROGLUCINOL Same of the spectra obtained from the oxidation of phloroglucinol are shown in fig.1. A neutral species in which the protons were relatively fixed was obtained at pH -2, the coupling constants being similar to those for 3,5-dirnethoxyphenoxyl. W. T. DIXON AND D. MURPHY 0.625 0.625 OH Me0 1.075 OH0.6 The exchange in this case is between three possibilities. However, because the three forms, A, B and C are chemically equivalent, the system can be reduced mathematically to an oscillation between two alternatives. If the probability of a change is P we can write A B+C P Now we consider the nuclear spin states, the individual proton coupling constants being a, or ap.I FIG. 1-Some e.s.r. spectra of the radical from phloroglucinol. *Lines due to some other species. (a)300 dm3 mo1-1 H2S04, slow exchange, (b)50 dm3mol-1 H2S04,intermediate case, (c) 2.5 dm3 rnol-l H2S04,relatively fast exchange. First the states ctaa and /3Pp are unaffected by interconversion of A, B or C. Anystate with a protons on positions 2-and 6-and a p proton on position 4-, (apa) will give rise to lines at +(a2 -a,+a,) ; i.e., for A: +(2a,-a,) } drCi) = la,-a,l B: +(a,) = Aw = la,-a,lIC: +(a,) During the exchange, A, B and C interconvert with equal probability. We can there- fore regard B and C as being effectively the same state “B ” and take Pba = P, Pa,, = 2P. The population of this composite state is twice that of A.Analogous E.S.R. SPECTRA OF SEMIQUINONES arguments apply to the other nuclear spin states. One immediate prediction from formula l(b)is that the lines, near the centre of the slow interconversion spectrum, which have relative intensities of unity, will broaden more quickly than the lines of intensity 2, as the rate of interconversion increases. The interconversion seems to be acid rather than base-catalysed, when the simplest assumption is of initial addition of a proton followed by, or synchronous with, proton loss : A-+H+ + [A.H]+ +B. +H+. The partial rate of loss of A is given by two equations : --=dCA1 K[H'][A] and --dCA1--P,,[A]dt dt i.e., Pa,, = K[H+] using formulae (lb, c) and (2) with data from the spectrum of the radical in 300 dm3 mol-l H2S04a value K -200 Po = 1.1 x lo9 dm3mol-1 s-l, is obtained in agreement with results calculated from the "averaged " lines using formula (la).-4 I 30E rn z/(hOlh) FIG2.-Graph showing variation of line height with acid concentration;A from phloroglucinol,B from catechol, C from resorcinol, D from hydroquinone. From eqn (la) we obtain for this system-;1 2Am2 +l 260.1~- 27KP,[H+] +'* W. T. DIXON AND I). MURPHY Where “ lz ” is the height of the line which has a relative intensity 3 in the limit of fast exchange, ho = height of the sharp lines, relative intensity one, and Po is the natural line width, assumed to be the same for all of the lines. From fig. 2 we see that a plot of J(ho/h)against l/[H+] gives a straight line as required by the formula.To calculate the rate constant for the protonation reaction we take the point where the averaged line has reached half maximum relative intensity. At this point the formula reduces to : The measured line width of the sharp lines in the flow system is about 0.03 mT and [H+] is found to be 0.3mol dm-3. In this particular case we know Acofrom the spec- trum at pH 2, ie., Am = 0.575 mT. Converting to radians/s the true units for Aw, we obtain: K = 1.1 x lo9 din3 mol-I s-l. From fig. 2 it would appear that the results can be explained iii terms of three species A, B and C. The obvious dependence of the exchange on acid strength precludes the intermediacy of anions which would not be expected to appear in the acidic condi- tions. However, the radical cation of phlorogluciiiol is expected to be the intermediate between the three “ isomers ”.The overall width of the spectrum changes in solu- tions stronger than 2 mol dm-3 H2S04,reaching a limiting width of 2.175 mT in 8 mol dm-3 sulphuric acid. We attribute this change to increasing concentration of the intermediate cation and are, therefore, justified in neglecting its effects over the range of kinetic interest (pH 0-2). RADICALS FROM CATECHOL As with their treatment of p-semiquinone, Carrington and Smith treated the o-semiquinone problem in terms of four species : @OH OHk5Jo-0‘ They estimated “ Au” for the different broadened lines (a) by comparison with corresponding o-methoxyphenoxyl radicals and (b)by assigning the sharp-line spec- trum, observed in their more acidic solutions, to the fully-protonated species.In the event, both sets of estimated coupling constants were erroneous. The most important error was that the quartet splitting from the methoxy protons in (a) was mistaken for a doublet splitting, due to the relatively poor quality of the spectrum obtained. The effect of this was to make it difficult to interpret the series of spectra in a simple manner, because the relative broadening of the various lines depends on E.S.R. SPECTRA OF SEMIQUINONES the square of the ratio of (a,-a,) to (a,-a5). This ratio is almost a constant for all o-substituted phenoxyl radicals,l i.e.for o-H, (ratio)2 x (12/8.4)2 x2;for o-OMe, (ratio)2 = (8.5/6.2)2x 2. With the erroneous assignments this ratio was (10.6/ 6.2)2 = 3. Our results indicate that over the region of line-broadening, the charged species do not affect the spectra at all, except possibly indirectly in that they can be short- lived intermediates. The key points are as follows : (A) At pH 4.5, a spectrum of a mixture of the anion and singly-protonated species was obtained, the former consisting of sharp lines showing that it was playing no part in the broadening process (see fig. 3). +. TABLERESULTS FOR RATE OF REACTION Ar+H+ + (ArH) AND COUPLING CONSTANTS OF RADICALS starting material K/109 dm3 mol-1 s-1 ~rqi0-4Tproton splitting for (~r~f.yi0-4Tproton splittings for phloroglucinol resorcinol catechol hydroquinone 1.1 1.o 1.1 4.4 11.75 (1) 10.0 (2)* 6.0 (2) 3.9 (l), 2.3 (1) 0.95 (2)* 4.1 (2)* 2.4 (4)* 7.25 (3) 10.75 (2) 2.0 (l),0.4 (1) 4.75 (2) 0.0 (2) 2.25 (4) * averaged values.(B) The positions of the sharp lines in the spectrum attributed to the singly pro- tonated species is constant over the whole range until at a hydrogen ion concentration of about 2 mol dm-3, the small triplet splitting starts to decrease with increasing acid concentration to a limiting value of zero, and the larger triplet splitting increases at the same time to 0.475 mT. It seems probable that over this region we are observing the change (with rapid exchange so that the lines stay sharp) from the mono- to the di- protonated species. The implication of this is that, in solutions of lower acidity than that of 2 mol dm-3 H2S04,the positive radical ion can be present only to a very slight extent, and that it need not be considered explicitly in the discussion line broadening.The order in which the various broadened lines sharpen as the proton exchange rate increases confirms the relative signs of the various coupling constants.In fig. 4 we show how the lines have to change assuming that one of the coupling constants (a,)is of opposite sign to the others. The formula corresponding to eqn (2) in this case (Pa,,= Pba= P)is : Am2 Am2ho-= -+I = h 8PPo 8KPo[H+] (3) and when the broadened lines are at half their maximum relative heights 8(J2 -1) KPO = Aco'//IH+].From this, the degree of broadening at a particular acidity depends on Am2, and the acid concentrations at which different lines have corresponding heights is given by : Aof At$ L-H+ll -CH" The observed concentrations of acid, at which appropriate parts of signals were at half maximum heights, were : 2 mol dm-3 ; 1.1 mol dm-3 ; 0.5 mol dm-3 ; 0.03 rnol drn-,, W. T. DIXON AND D. MURPHY 141 from which we deduce IAOiI < IAW3I < IAu4I < IAWzI i.e., la,-a,l -la6+a,l < la,-a31 < la4-a,l < la4-a51 4-b6-ad. The ratio (a, -a5/a6-a# comes to 2.1, which is exactly that expected, within experi- mental error. The value of the rate constant K, given in table 1, was calculated using an estimated value of 0.5 mT for (a6--a3).;&;+sharp lines in spectrum of neutral species FIG3.-Mixed spectra of protonated and unprotonated forms of o-semiquinone. (a) pH -5.0, (6) pH N 4.5, (c) pH -2.0. RADICALS FROM RESORCINOL The e.s.r. spectrum of rn-semiquinone is strongly dependent on hydrogen ion con-centration and the relative intensities of the centre lines only reach 2 at [Hf] -3 mol E.S.R. SPECTRA OF SEMIQUINONES dm-3. The positions of the sharp (outside) lines in acid solution were constant until the acid concentration reached about 4mol dm-3, when the total width started to de- crease. Again we attribute this to the formation of a positive ion, for the splitting in 8 mol dm-3 H2S04 were reminiscent of those in the corresponding negative ion, (see table 1).The value of K was calculated from the estimated coupling constants a, =1.1, (26 =0.9 mT, so that Am =0.2 mT. RADICALS FROM QUINOL In this case it appears that there is no decisive way of telling whether or not the radical ions play significant parts in the line broadening process since the positions of the lines would not be much affected and the total width of the spectra hardly changes. However, the centre line is of relative intensity 5 when [H+I2=0.3 mol dm-3, whereas the lines ultimately of relative intensity 4, have half this value at [H+Il =0.07 m~ldm-~. This agrees exactly with the expected ratio :Aw1/Aw2 =2/1, for these lines, in the most simple picture of the exchange. gg:g+&fggg$ApgT%YYVU c1. c c c1.-2 B&B P-slow exchange 1I !lllIlllllll ALQ fast exchange FIG4.-Schematic reconstruction of spectra of o-benzosemiquinone with nuclear spin configurations assuming la41>lasl>la3J>la~I and that a3 is of opposite sign to the others.DISCUSSION It seems that the method of analysing the spectra in terms of relative line heights, has some advantages, especially when the spectra have to be obtained relatively rapidly (as in flow experiments) or when access to a computer is limited. Investigation of the relative broadening/shortening of lines in the same spectrum provides further confirmation about the rate processes involved and interpretation by this method is straightforward and can be easily and rapidly checked. Information about relative signs of coupling constants can also be deduced from the relative broadening of different lines.In the particular cases we have investigated here, the situation is rather more simple than has previously been s~ggested.~ W. T. Dixon, M. Moghimi and D. Murphy, J.C.S. Faraday 11, 1974, 70, 1713. T. J. Stone and W. A. Waters, J. Chem. Soc., 1964, 213. T. J. Stone and W. A. Waters, J. Chem. Soc., 1964, 4302. T. J. Stone and W. A. Waters, J. Chem. SOC.,1965, 1488. I. C. P. Smith and A. Carrington, Mol. Phys., 1967, 12,439. I. C. P. Smith and A. Carrington, MoZ. Phys., 1964, 8, 101.'W. T. Dixon, Theory andhterpretation of Magnetic Resonance Spectra (Plenum, New York, 1972), chap. 7, p. 135. A. Carrington and A. D. McLachlan, Introduction to Magnetic Resonance (Harper and Row, New York, 1967), chap. 12, p. 206. J. S. Leigh, J. Magnetic Resonance, 1971, 4, 308. lo J. Granot and D. Fiat, J. Magnetic Resonance, 1974, 15, 540. (PAPER 5/504)

ISSN:0300-9238    

DOI:10.1039/F29767200135

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Continuous Charge Distribution Models of Ions in Polar Media Part 1 BY PARBURY P. SCHMIDT"? Department of Chemistry, The University, Southampton SO9 5NH AND J. M. MCKINLEY Department of Physics, Oakland University, Rochester, Michigan 48063, U.S.A. Received 20th March, 1975 In this paper we discuss a number of charge distribution models of ions in polar solution. In particular, we examine models based on Born, Slater and gaussian charge distributions as well as some combinations of these charge distributions. Self and interaction energy quantities are derived for the various ionic models. In addition, self energies are calculated for some typical examples. The self and interaction energies are of particular importance in connexion with the electron transfer process.One object of this work is the search for relatively simple and accurate charge distribution representations which take account of the major physical and chemical characteristics of ionic systems. Our investigations indicate that the gaussian distribution gives the best overall representa- tion. This is a great advantage, as the gaussian representation of an ionic charge system is the sim- plest mathematically. Specifically, we find that for cations a double charge distribution works well. The ionic radius marks the region of maximum positive charge density (including the uncovered positive core charge due to subvalence electronic delocalization). A second quantity, an effective Bohr radius which corresponds to the Glueckauf radius, marks the region of maximum delocalized subvalence electronic charge.For anions we find that the atomic instead of the ionic radii give the best results with the use only of a single soft charge representation. Ionic self and interaction energy quantities appear in a number of transport coefficients in connexion with electron transfer transitions. Thus, the activation energy for polaron mobility and the rate constant for an oxidation-reduction reaction depend on these quantities. It is important, therefore, to be able to esti- mate these energies accurately on the basis of the use of simple ion models. Previous-ly, the Born model of an ion has been used almost exclusively. However, for ionic crystallographic radii the values of the energy quantities are excessive.In this paper we introduce new models which provide the opportunity of overcoming many of the difficulties of the Born model, yet do not involve the extensive computation necessary in discrete model^.^ The Born model is the common model of an ion embedded in a polar medium. The ion is represented as a hard charged shell. Landau used such a representation in connexion with the calculation of the self energy of trapped electrons in ionic lat- tices. The concept also finds some application in the description of an excess electron in a polar liquid.6 Quantum mechanically, however, the hard sphere model of an ion is not accurate. The electronic charge distributions in atoms and ions are continuous, but not uniform. The fact that a spherically symmetric quanta1 charge distribution has a significant charge density for radial distances in excess of the Bohr radius has pronounced effects j. On leave of absence from Oakland University, Fulbright Senior Research Scholar. 143 IONS IN POLAR MEDIA which are manifested in molecule formation and in a number of interactions alien to classical physical descriptions. In this paper we examine a number of models for ionic charge distributions. Because of the quanta1 nature of the electronic component of any atom or ion, it is reasonable in the first instance to approximate the ion as a soft charge density similar to that expected on the basis of detailed quantum calculations. Thus, hydrogenic charge distributions seem natural.For simplicity, however, we choose Slater charge distribution^.^ Further, in view of the extensive use of gaussian basis sets in quantum chemical calculations, it is also reasonable to construct charge distribution representa- tions of gaussian form. The general model of the solvated ion which we introduce consists of a double charge distribution, one for the positive core charge and one for the delocalized electrons. If, for example, one considers a simple alkali cation, the net charge on the ion is 1 + e.s.u. However, the likelihood is that there can be delocalization of the outermost subvalence electronic charge(s). The effect of this delocalization is to expose additional core charge. Thus, if only one electron is considered to be effec- tively delocalized, the ion is a composite of 2fe.s.u.due to the core and 1-e.s.u. due to the electron. The net charge remains 1+ e.s.u. However, the charge den- sities interact individually with the polar medium as well as with each other. For an anion, on the other hand, the effect of a single electronic delocalization is to tend to return the system to the atomic, neutral state. Thus, in this case a single mono- negative soft charge distribution can be used. In this paper we consider the following general representations of the ion : (1) a Born-Slater representation (here the core charges are represented as Born spheres surrounded by a soft electronic charge); (2) double Slater distributions; and (3) double gaussian distributions.The interaction of all these charge distribution models with the medium is accounted for with the use of Dogonadze's phenomenological theory of the polar medium.8'1 The Dogonadze theory accounts for the effects which spatial dispersion of the medium has on the energy quantities. Because the Dogonadze treatment is basically an elec- trostatic theory, there are certain restrictions on the nature of the soft charge distribu- tions. The soft charge distributions are allowed to penetrate into the dielectric. However, it is important to note that no account is taken of quantum mechanical exchange interactions. Thus, the nature of the mixing is purely classical. Electron exchange between the ion and the solvent molecules in the first solvation shell is probably significant in many cases.However, at this point classical electrostatic theory of the ionic self and interaction energies cannot account for this. I. MATHEMATICAL PRELIMINARIES The general formula for the self energy of an ion dissolved in a polar solvent, taking into account spatial dispersion in the solvent medium, has been derived by Dogonadze and his associates.8* The formula is w,= --' jd'kC,(k)E*(k). E(k) (1.1)q244 where CN(k)is related to the Fourier transform of the permittivity of the medium (cf. eqn (1.9)). E(k) is the transform of the electrostatic field associated with the ionic charge distribution. The function CN(k)is given by N-1 P. P. SCHMIDT AND J. M. MCKINLEY and the E, are static dielectric constants associated with particular elementary excita- tions in the polar medium ;A, is a correlation length associated with a particular class of elementary excitations.The dielectric constants used in this work are associated with the vacuum and three types of excitation in the liquid :optical, infrared, and Debye librational modes. The index n labels the transparency band.9 They are identified as ~c0 = cst, E, = E ~e2 ,= E,, = n2, c3 = 1. (1-4) Thus, the index Nis 3in eqn (1.2). In eqn (1.4) n is the refractive index of the medium. The functionf,(k, A,,) is the Fourier transform of the spatial correlation function for the medium. For strongly non-magnetic systems, which we assume here, three forms of the correlation function have been used.An exponential approximation to the correlation function yields 8, ' ' fFP(k, A,,) = (1 +/~~;1:)-~. A gaussian approximation yields fz(k, A,) = exp(-k2A:) (1.6) and finally, a Heaviside step function approximation yields where jl(x) is the spherical Bessel function of the first kind of order one, cf. eqn (1.15). Dogonadze and Kornyshev have shown lo that the interaction energy for two ionic charge distributions dissolved in an isotroptic, homogeneous dielectric with spatial dispersion taken into account is 1 r wi = -d3kc,'(k)E"(1,2(k)4 J k) .E(2, k) exp(ik .R). (1.8) Here the transform of the permittivity is lo E,;'(k) = 1-C,(k). (1-9) The field transform quantities E(1, k) and E(2, k) are associated with charges 1 and 2.In the expressions for the self and interaction energies the transform of the electro- static field is used. In this paper we consider only the longitudinal components of the field. Dogonadze and Kornyshev have considered transverse contributions. * The modelling of an ionic system is manifested in the charge distribution chosen to represent it. It is, therefore, convenient to obtain the field directly from the func- tional form of the charge distribution. This can be done with the use of the Gauss law. In fact, the Fourier transform (in momentum space) of the Gauss law directly relates the transform of the field (which is needed in eqn (1.1) and (1.8)) to the trans- form of the charge distribution function. The transform of the Gauss law is ik E(k) = 4np(k) which can be shown to be equivalent to kE(k) = -4ni -p(k) (1.11)k (The proof follows trivially on taking the scalar product with k.) It is seen from the IONS IN POLAR MEDIA form of the transform (1.11) that the factor p(k), the transform of the charge density function, acts as a structure factor.The charge distribution functions used in this work are simply spherically sym- metric. They are normalized to the net charge contained within the individual distribution jd3rp(r) = 2. (1.12) The Fourier transform in momentum space is defined by the integral p(k) = 1d3r exp(ik r)p(r) (1.13) where exp(ikmr) is given by the Rayleigh expansion l2 exp(ik r) = 4n i'j,(kr)Y&(P)Y,,(k). (1.14) l,m In eqn (1.14)jl(x)is the spherical Bessel function of the first kind 2-14 (1.15) and Ylm(g)is the spherical harmonic function.The above formulae form the basis of the analyses of the following sections. In the succeeding sections self and interaction energy expressions are provided for the model charge distributions mentioned in the introduction. In the body of the paper we present only the expressions for the charge density functions and the final results for the self and interaction energy quantities. The mathematical details of the derivations are presented in a series of appendices at the end of the paper. 11. BORN-SLATER IONIC SELF ENERGIES The Born-Slater charge distribution representation of an ionic system is an amalgamation of two individual representations. We assume that the ion consists of a positive core of radius a,.The charge density is represented as a delta function, i.e., the Born m0de1.~ This positive core is surrounded, and to an extent penetrated, by a soft electronic charge distribution. The electronic charge distribution is repre- sented as an s-type Slater distribution. Hydrogenic representations, which contain nodes in the distribution, can also be used. However, for 2s and higher type charge distributions the derivation of the self energy expressions is greatly complicated. The normalized delta function distribution for the underlying core charge is given by z PBW = 4na,26('-ac) (2.1) and 2 is the net core charge contained in the distribution. A normalized Slater distribution for the electrons is given by where denotes a gamma function and the screening parameter p is related to an effective Bohr radius by the following expression p = 2(n+ l>la,.(2.3) This quantity marks the region, measured radially from the origin, of maximum electronic charge density. N is the total electronic charge contained in the distribu- tion. In this paper we use only a 1s-type distribution for which n = 0. P. P. SCHMIDT AND J. M. MCKINLEY The Fourier transform of the electrostatic field associated with the combined, and interacting, charge distributions is given by Note that in our notation the elemental charge, e, is contained in 2 and N, both of which are positive quantities-i.e., the total number of charges in the distribution.The formula for the self energy is derived with the use of eqn (1.5) ; the choice of correlation function for the medium is discussed in section VII. We find (Appendix A) N2 29 7x,+3-IDn--y,,+7)2--+xn -2% 16 (x,+ 1)214d 4 x,+l where in@)and kn(x)are modified spherical Bessel functions l2 (Appendix M1) and where x, = PI,,, y, = ac/l.n, z = pa,. (2-6) The above expression has been written in a form which depends only on the core radius a,. In fact, this expression can conveniently be written as (2.7) 1-X2, N2 I--.-- z 2x,-1 (2.8){ 16 (.,"+ 1)' The first term is the Born self energy which we have separated from the remainder. It is clear that the effect of the second term is to decrease the absolute value of the simple, dispersion-free Born self energy.This occurs both through the effect of spatial dispersion in the medium and as a result of the soft electronic charge distribu- tion. The mixed term in ZNin both eqn (2.5) and (2.8) is finite as x,-+ 1 in the denomi- nator. The limiting form of the contribution is found easily by examining the limit in the defining integral-cf. Appendix A. We find lim w$$) = ZND, (2.9) xn+ 1 IONS IN POLAR MEDIA 111. DOUBLE SLATER SELF ENERGIES The double Slater charge distribution representation, and the double gaussian to follow, is a model motivated by the fact that even the underlying core charge is not physically well-characterized by a Born charge shell. Clearly, if there is delocaliza- tion of the outermost subvalence electrons on a cation, for example, the exposed core charge is not distributed on an infinitely thin shell underneath.The remaining sub- valence electrons shield the core charge and in effect delocalize it. A general double Slater charge distribution has as its field transform the following expression (the charge distribution itself is given by eqn (2.1)) L b2+k2 (3.1) where q and p respectively are related to the effective Bohr radii for the electrons and core charge. We restrict our attention only to a double Is-type charge distribution for which rn = TI = 0. The following expression for the self energy results 1 W, = --D,[z~F(x,, 1)-~ZNF(X,,zj +N~ZF(X,,Z, I)] (3.2)2% where xn = PA,, = qlP (3.3) and p = 2/a,, q = 2/a,.(3-4) The general function F(x,y)is given by +y(x(y l)+ l) x2(y+1)+y(xy +1)+(x +I) +F(x, y, = 2(y+Q2(X +1)2(xy { (y+l)(x +l)(x-y+ 1) +xy [1+x+2xy+l +2x+xy+1 ]+y(x(;;:)f?)). (3.5)x+l xy+1 The advantage of this formula is its simplicity for computational purposes. Calcula-tions are readily worked out on desk and hand calculators. It can be seen from eqn (3.2) that the combination of the charge numbers 2 and N together with the function F(x,y) act as an effective charge on the ion. Indeed, this self energy can be written as where Ze2ffis Zzff = E,,C D,(Z2F(~,,,1)-2ZNF(xn,Z) +N’zF(x,z, 1)) (3.7)Est-1 and it is clearly a function of the screening parameters p and q and of the dielectric constants and correlation lengths of the medium.An alternate approach, of course, is to write the self energy in terms of an effective dielectric constant. This was Booth’s approach.15 In this case it is possible to make the identification N E,ff = {1-1 D,[F(q2,1)--2--F(x,, 2)+(fv/z)2F(x,z, 1)Z IF1 P. P. SCHMIDT AND J. M. MCKINLEY for a cationic species, as an example. A similar expression for an anionic species can be derived. IV. DOUBLE GAUSSIAN SELF ENERGIES At this point we introduce a double gaussian representation of the ionic system. As indicated in the Introduction, and as is illustrated later in section VII, this represen- tation not only seems to give the best overall results both for anions and cations, but it is easily the most mathematically and computationally tractable.It is possible to derive a completely general expression for the self energy valid for any combination of orders of s-type charge distributions. This we have derived. The general self energy expression is easily programmed for computer calculations. An individual gaussian distribution is given by z2np3 PG(Y) = (2n+1)!!71 +ufnexp(-u2) where u = pr and p = (n+I)+/a. Again, a is an effective Bohr radius. The field transform for a double gaussian charge distribution is where Hn(x)is the Hermite polynomial of order n-see Appendix D. With the definitions of the quantities xn = PAn, z = q/p = ((n+ 1)/(m+ l))+(ac/a,> (4.5) and the use of eqn (1.6) in the derivation, the self energy is 1 w, = --Dl(Z2G(x1,1, m, w)-2ZNG(x1, z, m, n)+N2zG(x,z, 1, n, n)] (4.6)2% The general function G(x,y) is G(x,y, m,n) = (--l)m+n + (24 !!(2n)!!A:(nl+n)+2mfn X where The expression (4.7) for G(x,y) is extremely simple, although for high order gaussian representations (i.e., higher order s-type distributions) the expansion can be a tedious exercise.Specific expressions need not be worked out, as the general function (4.7) can be programmed with relative ease. For routine, low order s-type representations, however, it is fairly costly in computer time to use the general expression. The specific formulae in these simpler representations are easy to obtain. IONS IN POLAR MEDIA Again, as with the case of the double Slater representation, the combination of charge numbers 2 and N with the various G-functions acts as an effective charge function.V. INTERACTION ENERGIES FOR BORN, SLATER AND BORN-SLATER COMBINATIONS We turn now to the matter of presenting the expressions for the energies of inter- action between ions in polar media for the various model representations chosen. In this section we present formulae for the interaction energies which arise from the various parts of the composite models, i.e., the contributions from the individual charge densities. The interaction energy for two Born model ions has been derived previously by Dogoiiadze and Kornyshev." It is listed here again, partly for reference, but more iniportantly it is an essential component in the Born-Slater representation ;it provides the contribution of the interacting cores.The interaction energy expression is rela-tively simple. It is ZZ w~(~) *[1/~~~+$C Dll(R/lb,)S6(2;l/A,*; a; b; R)). (5.1)= R The general form of the function Ss is given in the table in Appendix M2. In this particular case S6 is sufficiently simple to write out explicitly : S6(2 ; 1/An ;a ;b ;R)= ~o(xn~)io(xn~)~~(xn) -xn1 il(xn1) io(xn2)ko(xn)-xn2 io(xn1 ii (xn,)ko (xn> +x,io(x,l)io(xn2)kl(x,) (5.2) with xnl= all,,,axn2= b/lLn,x, = RIA,. (5.3) Dogonadze and Kornyshev lo have argued that for R I,, (for any of the system N elementary excitations labelled n provided A, % a, b) the interaction can be consider- ably larger than that predicted on the basis of the Born (or simple Coulomb) interac- tion in a dispersion-free system alone.The consequences for the magnitudes of medium repolarization energies in the case of the electron transfer reaction are obvious. For the Born-Slater model, there is a contribution which arises from the inter- action of the soft electronic charge distribution on one site with the core of the ion on a second site. Such an interaction energy is of course familiar in molecular quantum mechanics. We refer to this as a mixed interaction. An example is [S,,q(2,2; q, 1/Ai; a;X)+(qAJ3S8,,1,n(2, 2; 4, l/Jii; a; R)J). (5.4) The above expression is valid for a11 values of p and i.,. However, it is of interest to compute the limiting form as pi,, -+ 1. It is The above formula is useful for purposes of computation as computers have no ability to recognize the limit in eqn (5.4).P. P. SCHMIDT AND J. M. MCKINLEY Finally, the interaction energy for two soft charge Slater-type distributions is Again, the two cases First, for identical ions we find PR 1 R 1/cSt-48s,(4; p; R)+% D,(1 x P=4 J Second, when p = q = 1/An for a particular An, we find The above expressions can be collected to yield the interaction energies for the composite models-Born-Slater, etc. Thus, for the Born-Slater model, one writes Wi(B-S) = Wi(B)+Wi(e)+”i(m)(a, q)+wi(rn)(b,P) (5.9) For a double Slater representation one uses combinations of Wi(,) with N replaced by 2 in the appropriate places. VI. GAUSSIAN DISTRIBUTION INTERACTION ENERGIES As with the self energies, here as well for the interaction energies, the gaussian model stands alone in versatility.A single expression can be derived to handle any situation. Consider two sites a and b; let a and b also represent the effective Bohr radii : where IONS IN POLAR MEDIA and D,(x) is the parabolic cylinder f~ncti0n.l~ In eqn (6.3) D-l(x) is related to the error function : l4 D-,(x) = exp(x2/4)(71/V(1-4(4> (6.5) with the error function $(x) given by Px +(x) = L dt e-". 0 For n >, 0 the remaining D,(x) are related to the Hermite polynomials : l4 D,(x) = 2-"12 exp(-x2/4)~,(x/J2) (6.7) and H,,(x) is defined in Appendix D. More complicated double gaussian representatians of ionic systems can be handled with combinations of the general interaction energy, eqn (6.3).VII. DISCUSSION In their work on ionic solvation, Dogonadze and Kornyshev 8* l1 used the Born representation of a charged ion, but immersed the charge distribution in a polar medium and took account of dispersion effects. As already indicated, the Born model consists of a delta function charge distribution located on the surface of a metallic sphere. If one makes use, as did Dogonadze and Kornyshev,ll of ionic crystallographic radii to represent the radii of the Born spheres, then, even with the adjusted radii or the Gourary-Adrian radii (cf.ref. (1 l)), the charge distribution essen- tially is that of the ion and includes no account of the spread of charge over solvent molecules in the nearest neighbour solvent shells. It is, therefore, necessary to account for the polarization effects arising from the optical and infrared modes in the medium.As the correlation lengths for these particular excitations are of mon- atomic or monomolecular dimensions, an accounting is given of the polarization response in the first coordination layer. This is so, even in the dielectric continuum limit. The tacit assumption is that dielectric saturation does not occur in this region. Optical and infrared polarization response is still a manifestation of continuum behaviour. Dogonadze and Kornyshev's calculated results, ionic solvation energies, seem to be in good agreement with the experimentally determined quantities.l' The approximations used by them are consistent with the use of the Born model.On the other hand, the soft charge distribution models of the ions we have used account for the phenomenological distribution of electronic and core charge over a considerable range in the solvated ionic system. Thus, the model of the source charge distribution accounts for at least optical polarization effects in a dielectrically saturated system. The extent of the system is the ion plus its first solvation layer. To account for dispersion in the optical modes in addition to using the soft charge distribution, appears from our calculations to over-estimate the dispersion contribu- tion, and thus the values of the self energies obtained are uniformly too small. In the following calculations we have compared our results with the absolute values of the ionic self energies calculated from ionic solvation energies in the manner of Noyes.l6, l7 Even though the experimental "absolute " self energies may not in fact correspond to the true self energies (indeed, at this point they cannot be mea- sured directly), the trends certainly are illustrated. At this stage, this is more revealing than calculations of the solvation energies of ion pair systems.In the past, of course, calculations based on the Born ion in a dispersion-free system using crystallographic ionic radii have consistently over-estimated the self energies both of the single ioiis and of the pairs. P. P. SCHMIDT AND J. nf. MCKINLEY I.OC -' 75 \-.5.. b.27 > 1 2 4 ka FIG.l.-Plots of the integrands in the self energy integrals for various models.Account of medium dispersion is not considered. (a) Slater distribution. (b) Gaussian distribution. (c) Born distribution. I.0C 0.75 0.5C 0.25 0 ka FIG.2.-A comparison of the self energy integrands for the Slater distribution (a)with medium spatial dispersion taken into account and (b)in the dispersion-free limit. IONS IN POLAR MEDIA XLi+ +A % . --16.4.3. 14.0. 12.01 .3.'? 1 0.8 1.0 1.2 1.4 1.6 C.8 i.3 1.2 1.4 FIG.3.-(A) Double Slater distribution model :monovalent cationic self energies. Curve (a) infra-red and librational (Debye) dispersion contributions taken into account. Curve (6) only the libra- tional contribution to the medium dispersion is considered.(B) Double Slater distribution model : divalent cationic self energies. Curve (a) infrared and librational dispersion contributions taken into account. Curve (b) only the librational contribution to the medium dispersion is considered. 2.n 0.75 I.CQ 1.25 1.50 A FIG.4.-Single Slater distribution with librational contribution from the medium. This distribution is suitable for anions. P. P. SCHMIDT AND J. M. MCKlNLEY In particular, we find that with the use of the single and double Slater charge distributions good results are obtained when we use the dispersion free limit for the optical contribution (essentially a delta function spectral function) and exponential representations for the spectral functions associated with the infrared and librational (Debye) contributions.For single Slater distributions, which represent the anions, we find that consideration of dispersion arising only from the librational modes yields good agreement. However, in connexion with the anions we have found, unusually, that any soft charge representation agreement with experimental self energies can only be obtained with the use of atomic radii (or radii close to the atomic ones), not ionic radii. This we believe to be reasonable in view of the fact that the radial quantities which appear in the self energy expressions act as effective Bohr radii. There is, therefore, no peremptory reason why an anionic radius should correspond to the region of maximum electronic charge density.In fact, the ionic radius in the case of an anion probably is the radius for which the charge density is l/e of its maxi- mum value, i.e., rion = l/p. For chloride with a = 0.98 A we find rion = 2.08 A ; this is reasonably close to the Goldschmidt radius, 1.81 A. -0.6 I0 1.2 I.4 1.6 A FIG. 5.-Monovalent cationic self energies with a gaussian distribution model. Infrared and librational dispersion contributions are considered. In the Dogonadze-Kornyshev treatment, and in ours, the polar dielectric and the induced polarization charge densities are allowed to penetrate the source charge distribution. The induced polarization charge density reduces the absolute magnitude of the charge density at a particular point, i.e., distributes the charge over a region of space, and thereby decreases the local field.The effect is one of screening which is of much the same form as the screening encountered in the Debye-Huckel theory.l8 Only the origin of the screening charge is different. In the complete space of the ion- solvent system, however, the total charge remains the same. In fig. 1 the integrands associated with the self energy integrals are plotted. The area under the Born inte- grand is the greatest; thus, as expected, the Born self energy is the largest. The Slater self energy integrand decreases roughly as k8.It can be seen from fig. 2 that IONS IN POLAR MEDIA if we use the Lorentzian representation of the structure factor for the polar medium, the self energy decreases roughly as k-I2.The extent of the spread of the charge distribution is, therefore, great and the self energy is small, smaller than it should be. Basing our considerations on the estimated magnitudes of the correlation lengths given by Dogonadze and Kornyshev," it seems reasonable to us to expect that the effective Bohr radius associated with a particular soft charge distribution contains an account of the dielectrically saturated response as well as the ionic source charge in the inner region. In fig 3 and 4 we have plotted these self energy quantities. An interesting feature of these calculations is illustrated by the fact that good agreement is obtained when the following relations are used for the connexion between the core and electronic effective Bohr radii for cations :for monovalent species a, = l.la,-0.425 and for divalent species a, = 2.9~~-1.375.Thus, effectively as the size of the core increases, the strength of binding of the sub- valence electron diminishes to an increasing degree. 22.c 2C.' 5 I2.C P 16.c I4.T FIG.6.--Divalent cationic self energies with a gaussian distribution model. Infrared and librational dispersion contributions are considered. The results using the gaussian representation are presented in fig. 5 and 6. They are similar basically to the results obtained with the use of the Slater distribution. One object of this work is to develop representations of ionic charge distributions which can be used to calculate reasonable estimates of self and interaction energy quantities.Since we found that the optical and infrared dispersion contributions adversely affect the self energy, it seems reasonable to test the complete dispersion-free limit. The librational contribution is not great in most cases. We find in the dis- persion-free limit that it is possible to represent a given ion, either anionic or cationic, by a single soft charge distribution. The 1s and 2s Slater and gaussian dispersion- P. P. SCHMIDT AND J. M. MCKINLEY free charge distribution self energies are w,(ls) = -(l-l/Est)-- 5 z2 8 2a (7.3) Z2 ~~(2s)= -(1 -1/~~J(93/1X9-(7.4)2a and for the gaussian L WS(2S) = -(1-1/&,J(3/27C+)-.2a The respective self energy curves are plotted in fig.7. It can be seen that, with the use of the Goldschmidt radii, Li and Na agree with the use of the Is Slater distribution and that K, Rb and Cs agree with the use of a 2s representation. With the use of a gaussian representation, on the other hand, it is necessary to increase the atomic 0.8 1.0 1.3 1.6 A FIG.7.-Dispersion-free single charge distributions. (a) 1s Slater type distribution. (b) 2s Slater type distribution. (c) 1s gaussian type distribution. (d) 2s gaussian type charge distribution. radii by about 30 % for Li and Na, but for K, Rb and Cs the increase is only about 12 % with the use of a 1s gaussian. With the use of a uniform 30 % increase in the Goldschmidt radii, Li and Na agree with a Is gaussian, as noted, and K, Rb and Cs agree with the 2s gaussian, similar to the case with the use of the Slater distribution. On the other hand, for anions we find that reductions in the ionic radii are needed to get agreement.This reduction is in the direction of the atomic radii, and is con-sistent with what we have found earlier. Single charge distributions can be used effectively in the dispersion-free limit in IONS IN POLAR MEDIA connexion with the evaluation of self and interaction energies connected with more complicated systems. This is illustrated in the following paper, Part 2.l' In this paper we have not sought to be exhaustive in the presentation of numerical examples of self energy quantities. Rather, our intention has been to derive these expressions and to examine same trends associated with the models.We believe there is considerable computational utility in our expressions, especially when discrete, discontinuous solvated ion models do not warrant the investment of time and effort in order to obtain indications of trends or dominant behaviour. We express our appreciation to Edwin Power for a number of useful comments. We also thank R. R. Dogonadze and A. A. Kornyshev for many comments, much stimulation, and the encouragement to complete this work. A large part of the work was completed with the help of an S.R.C. grant. One of us (P.P.S.) expresses his appreciation to Martin Fleischmann for his generous assistance. APPENDIX A DERIVATION OF BORN AND SLATER ENERGY QUANTITIES The soft charge distributions used are based on analogy to the atomic electronic distributions.The hydrogenic wavefunction for a single 1s electron is = 2(zp/aB>f exp(-Zpr/aB)yOO(P) (Al) where Zpis the core (protonic) charge and aBis the Bohr radius. Ylm(iC)is the spheri- cal harmonic function. As we use this type of wavefunction to define a soft chagre distribution, clearly aBbecomes an effective Bohr radius ;it is arbitrary and its value is fixed by the type of ion and number of valence electrons in the ion. We write the charge distribution for the delocalized electron(s) as where T(n) is the gamma function 1 r(n) = (yt-l)! The screening parameter p, given by eqn (2.3), is determined from the maximum of 4nr2p(r). The spherical harmonic function is introduced for two reasons : (i) to be consistent with the Ray- leigh expansion, eqn (1.14), and (ii), in general, more complicated charge distributions have spatial orientations which can be expressed (usually) in terms of sums of spherical harmonic functions and radial factors.An example of this is reported in another paper (Part 2) dealing with charged ring systems.lg The Fourier transform of eqn (A2) for a 1s charge distribution is and for a general charge distribution it is At this point with the use of eqn (1.11) we can find the electronic component of the field transform given by eqn (2.4). The Born charge distribution is given by eqn (2.1). The Fourier transform is found to be P&) = -Z(4n)+j&aC) yo&* (A5) P.P. SCHMIDT AND J. M. MCKINLEY The transform of the electrostatic field due to a delta function charge distribution leads to a self energy term of the form W@) = -5z2 J*dk CN(klji(kac). CA6)710 Here, and elsewhere in this section, we use 2 C,(k) = c Dn(1+k2&y. n=O Thus, The integral in (AS) can be written as * j@aC) =+Re O3 j,(ka,)h~')(ka,)s dk (1/A,"+k2)2 Jew dk (ljn,"+ k2)2 where hp)(x)is the spherical Bessel function of the third kind : l2 It can be seen that there are singularities at k = 0, +i/A,. A relatively simple way to separate the principle value from the I' = 0 pole, for these integrals only, is to write CN(W as CN(k)= (1 -1/csJ -CD#(k, A,) (A1 1)where Thus, the integral W 4;Re dk jo(kaC)h~')(kac) = n/2ac -00 gives the dispersion-free contribution. Many similar integrals are standard and are tab~1ated.l~~ The remaining principal value integral is evaluated with the use ofl4 the simple contour discussed in Appendix M1.The result of carrying out the indi- cated integrations is the first contribution to eqn (2.5). Similarly, for the Slater function representation of ions we find Although it is not necessary to use eqn (A1 1) and (A12), they can be used to separate a dispersion-free contribution from the remainder. We have not done this in eqn (2.5). In connexion with the double Slater representations, one encounters integrals of the form Ai4 swdk P4q4 0 (1/Az + k2)2(p2+ k2)2(q2+k2)2 00 1 = (S4J-dk (l/A," +k2)2(p2+k2)2(q2+k2)2' IONS IN POLAR MEDIA It is clear that from the above integral all cases encountered in this paper can be derived.Eqn (A14) is, therefore, the most general integral form for the Slater-type Is charge distribution. The integral can be written as The integral (A15) is easily evaluated by means of the residue theorem : rm 1 I=J Idk (a2+k2)(b2+k2)(C2+k2j n(a+b+c)--2a bc( a+b)( b +c)(a+c) When we carry out the indicated differentiations in eqn (A15) and make use of the definitions we find xn = PAu zxnq = zln,4 = q/p (A17) I= x2z(z+ 1)+ z(xz+ l)+x+ 1 + 2x+xz+l (2+l)(x +l)(xz +l)+z'x[1 +x+2xz+1 +x+l xz+1 Thus, with the definition of the function F(x,z) as z(x(z +1)+1)F(x, Z) = (22 +1)"x +1)2(xz +1)2{* -1 where { .. .1 is all that is contained within the curly brackets (braces) in eqn (A18), we find the self energy expressed by eqn (3.7). Finally, in connexion with the Born-Slater representation we examine the mixed contribution. After carrying out the trivial angular integrations, we find The general result can be obtained with the use of eqn (A6) and (A7) : {S4,p('? 2; P; 1IAn; ac)+(~An)~S,,,/R,(2,2;P,1/An; aJ>. (~21) As indicated in section 11, this result is valid in the limit as pl, 3 1. However, the simplest means of obtaining the limiting expression is to set pl, = 1 in eqn (NO) with the result 1 P. P. SCHMIDT AND J. M. MCKINLEY 161 APPENDIX B EFFECT OF A CUT-OFF ON A SLATER CHARGE DISTRIBUTION SELF ENERGY The charge distributions for the electrons so far considered (except, of course, for the Born distribution) have been continuous from the origin.In most ionic solution theories, for example, the Debye-Huckel theory,l* it is assumed that there is a minimum distance, a cut-off, below which the field vanishes. Physically, in the case of the Debye-Huckel theory, this cut-off is the distance of closest approach to which two ions can come without configurational change in either species. A cut-off limit in any theory seems warranted. It is, therefore, necessary to determine what effect a cut-off will have in the case of the solvation self energy of an ion. As the calculations below indicate, the introduction of a cut-off greatly complicates the derivation of the self energy expression.Therefore, for purposes of comparison we consider the dispersion-free limit for the solvent. For reference we write the Is Slater dispersion-free self energy : z2 \vS = -(1-1/~~~)-(5/3) (B1)2% where a, is the effective Bohr radius. The normalized 1s charge distribution in a cut-off approximation is m3Y +2x +x2) exp[ -p(r -a)]@ -a)p(r) = 4~(2 and a is the cut-off radius. O(x) is the Heaviside step function. The quantity x in (B2) is defined as x =pa. (B3)The Fourier transform of p(r) is XP" p'k) = (2+2~+x2)(p2+k2)2 {(p2(x+1) -k2(1-x))j,(ka)+k(p(2+x)+ak2)j-l(ka)). (B4) As a result, the self energy now is 11;dk (p2 +k2)4 {(p2(1+X) -k2(1-~))~jc(k~)+2(p2(i+X) -k2(1-x)) x (p(2+x-)+ak2)kj,(ka)jj_I(ka)+(p(2+x)+ak2)2k2j2_,(ka)). (B5) This expression is simplified with the use of two relations : h(4.L-=j@)/xand j?,(ka) = l/x-j;(ka).The straightforward, but lengthy, integrations can be carried out. The final expres- sion for the self energy is +2~(2 x)~ -(5(2 +x)~ +X) +x2+16~(1+ +16(1+ x)(2 +x)-x2(1+x)~B~~J x2(x2-4x -6)B@ +x2((x2+6x -1)BitA-x4B(')4,6-2x(x+l)(x +2)B3-2x(x2+x-1)Byl-2x2(1-x)ByJ (B8) 11-6 IONS IN POLAR MEDIA where the quantities Bgk,’,are &A = ko(x)(io(x)-3xil(x)+x2i2(x)-x3i3(x)/3) -2xi1(x) +x2i2(x))+xkl(x)(3io(x) +x2k2(x)(iO(x) (B9) -xil@))+x3io(x)k3(x)/3 Bit1 = ko(x)(io(x)-xil(x) +x2i2(x)+x3i3(x)/3) +xkl(x)(io(x)-2xi1(x)-x2i2(x))+x2k2(x)(iO(~)+xil(x))-x3iO(x)k3(x)/3 Bgb = ko(x)(io(x)-3xi1(x)-3x2i2(x)-x3i3(x)/3) +3xkl(x)(io(x)+2xil(x)+x2i2(x)/3)-3x2k2(x)(i0(x)(B11)+xil(x)/3)+x3io(x)k3(x)/3 B4,6 = k,(x)(8io(x)+ 6xi1(x)+2x2iZ(x) (l) +x3i3(x)/15) -xkl(x)(12io(x)+4xi,(x) +x2i2(x)/5) (BW+x2k2(x)(2i0(x)+xi1(x)/5)-x3i,(x)k3(x)/15 BitA = 5k0(2x)+6xk1(2x)+4X2k2(2X)+8x3k3(2x)/3 (B13) BL?; = ko(2~)(2~+4x2+8x3/3) W4) Bi?: = k0(2X)(9+2X+4X2 -8x3/3).(B 15) A plot of the self energy, eqn (B8) together with (Bl) is given in fig. 8. The self energy can be written as z2 w, = -t1-li&st>F(x)~ tB16) where the function F(x) can be identified from eqn (B8). The range of validity of x is 0 < x < 2. For radii a less than the Bohr radius the self energies are not defined. Fig. 8 illustrates the behaviour of the self energy for the ion cut-off model for two values of the hard sphere radius, 0.98 and 1.33 A,corresponding respectively to the A FIG.8.-Dispersion-free self energies for a charge distribution with a cut-off.(a) Core radius : 0.98 I$. (b)Core radius : 1.3A. (c) Dispersion-free 1s charge distribution, eqn (7.3). Goldschmidt radii for sodium and potassium. At x = 2the hard sphere and effective Bohr radii are equal. Thus, for sodium we find ws = -4.7 eV and for potassium W, = -3.64 eV. From eqn (Bl) we find for sodium -4.45 eV and for potassium -3.34 eV. As the differences are rather small, of the order of 0.2 eV, there seems little reason to consider the introduction of a cut-off. P. P. SCHMIDT AND J. M. MCKINLEY APPENDIX C BORN AND SLATER INTERACTION ENERGIES The Born-Slater model of the ion contains all the various forms of the interaction energy quantities likely to be encountered.Thus, for a general case, we can write The first term leads to eqn (5.1), the second gives eqn (5.4), and the last gives eqn (5.6). The general integral is of the form 1 I = Jfa dk f k')'0 where f(k) depends on quantities of the form (p' +k2)-' as well as on spherical Bessel functions. When f(k) does depend on (p' +Ic')-~, the easiest way to obtain the limiting expressions for pLn = 1 is to examine the integrals of the form In this manner eqn (5.7) and (5.8) were derived. APPENDIX D DERIVATION OF THE GENERAL GAUSSIAN SELF ENERGY FORMULA The normalized gaussian charge distribution is given by eqn (4.1).The Fourier transform p(k)is --(2n+1).I I .71:+(p/k)J dk du u~~~~e-"* sin(ku/p) 0 -+I)! exp(-k2/4p2)H2,,,l(k/2p).-(--'1" ~(2n With this expression the field transform is found to be that given by eqn (4.4). In eqn (Dl) and (4.4) Hn(x)is the Hermite polynomial. The definition is l2 "21 n! Hn(x) = C (-1)' (n-2s) !s !(2X)"+ (W s=o and the above expression is employed in the derivation of w,. As indicated in section VII, for the gaussian system we use eqn (1.6) to represent the Fourier transform of the polar medium correlation function. Thus, wsis IONS IN POLAR MEDIA Define the following quantities We now write exp[ -k2(1+2p2@/2p2] = exp[ -y2(1+2x31 exp[ -k2(1 +2q2n:)/2q2] = exp[ -y2(1+2x7z2)/z2] -($-$ +2:))exp( = exp( -y 2 (4x:+l)z2+l) 2z2 With the use of these definitions it is necessary only to evaluate one integral, eqn (D6).All other integrals can be derived from it, as is shown. Define the integral With the use of eqn (D2), we write (2m+1) (2n+I)! Cm++lIn++] Z2r z2n + 1 2 s=o t=O (-lY+'2"+5!t!(2(m-s) +1)!(2(n-t)+l)!X Cm++l Cn+tl -(2m+1)!(2n +I)!Z~~+~A;(~+~)+~= Jx c c (-1)Sf'xs=o t=O (2(m+n-s-t)+I) !! where A1 = {(4x:fl)li+l2>' ' The substitution of (D7) into (D6) yields +I = (-1)"+"(1/2a)(2(rn+n)/x)' (2m)!!(2n)!!A:'" +n) 2m-t-n X P. P. SCHMIDT AND J. M. MCKINLEY Thus, if we define G(xl, z, m, n) = 2aI the self energy eqn (4.6) follows. APPENDIX E INTERACTION BETWEEN GAUSSIAN CHARGE DISTRIBUTIONS The interaction energy can be expressed as The evaluation of the above interaction energy expression can be carried out with the investigation of one general integral.Thus, we evaluate the following : In view of the definitions of the Hermite polynomials and of the spherical Bessel function,j,(x), there is only one general integral type encountered and it is known l3JS dx x2n+1 exp(-b2x2)sin(ax) With the use of the expression for the Hermite polynomials, eqn (D2), the integral (E2) transforms to 1 (2m+1)!(2n+1)! c c (-l)S+fxI=-----. z2n+ 1 s=o t=O z2t2m+n+1 X2s+fS!t !(2(m-s)+1)!(2(n-t)+1)!Jm 0 dy y2(m+n-S-f)-l exp{ -y2 (4x' 4-'Iz2 4-'}sin(yJ2pR).2z2 On coinparison with eqn (E3), we write 1 (2m+1)!(2n+1)! c c (-l)m+nxI=---J2pR z2n+1 s=o t=O 22t2m+n+2 nl+ If -s-f 2"+'~!t!(2(m-s)+ 1)!(2(n-t)+l)! p2R2z2 }D2(-+n-s-r)-1{2((4x; +1)z2+l)%-pRz >.(E5)exp{-t 2 (4x; +l)z2+1 The use of eqn (E5) in eqn (El) leads, after a certain amount of rearrangement, to eqn (6.3). I66 IONS IN POLAR MEDIA APPENDIX M1 In this Appendix and the next we consider the integrals encountered in the text. Here we present a representative calculation of one of the eight basic integral types encountered : 14. The evaluation of the remaining integrals is roughly similar. Consider the integral of the form 03 j&R)I4 =I , dk (a2+ k2)m(b2+ k2)n* An integral of this type can be evaluated directly with the use of the residue calculus.However, in view of the order of the poles, and the nature of the residue theorem, it is expedient to consider means of formally reducing the order of the poles and then applying the residue theorem. If one replaces a2 by a and b2 by fl, then by means of a formal differentiation of the parameters a and p after the use of the residue theorem the desired expression results. This procedure is legitimate as all the integrals are Riemann-integrable. Thus, we write (-l)m+n-2 am+n-2 hb’)( kR)14 = (rn-l)!(n-l)!aam-’ap-’ * where hbl)(x)is the spherical Bessel function of the third kind : ,ix h&”(x),. = 1x Because the function hbl’(x)has a pole at x = 0, there are in toto five poles to consider : k = 0, +i Ja, *i Jp. It makes sense in the evaluation to separate the principal part and treat it separately.The result for the pole at k = 0, evaluated in the upper half plane (complex), is We now write where 9indicates the principal value of the integral. The function k,(x) is the modi- fied spherical Bessel function of the third kind : k,(x) = e-x/x In obtaining (M5) we made use of the residue theorem by evaluating the residues in the upper half of the complex plane. It remains necessary to evaluate the derivatives of (M5) as indicated in (M2). As there are two contributions from the residue theorem in (M5), there are two major contributions to the total integral (excluding (M4)). We consider only one of these contributions for this representative calculation. The other contribution follows on permutation of the indices m and n and the quantities a and b.Considering the first term in (M5), we find for the derivative with respect to p P. P. SCHMIDT AND J. M. MCKINLEY The remaining derivative with respect to a is of the form where (r) is the binomial coefficient and d"A,(x) = -A(x).dx" Let A(@)= ko(a*R) and set a a -= (R2/2),-. aa z az Thus, with (&Y k-o(z) = (-l)ilZ-nk,(z), (M12) we obtain A,(a) = (-1)"(R/2a)"kn(aR) and we have set a = a2 after completion of the differentiation. A similar result holds when A(@ = k@R). Next, let B(a) = (M 14)Thus, (nt+ n +1) !Bn(a) = (b2-a 2)--m-,, (m-l)! * Finally, let C(a) = a-m'2. The differentiation yields C,(a) = (-l)n2-na-(2n+1)(2n -1) ! ! We now combine these results to write m-1 m-s-1 m-171 --7-c 1 in-s-1 '4 = 2Ra2mb2n 2m a2m-1 (m-1)!(n-1)!(b2-a2)" s'zo( )( t )'s=o + terms with permuted indices and a and b quantities.(M 18) In the above expression we have used eqn (M17) with m = 1. In Appendix M2 we write the integrals in the following form of which I, is given as an example : IONS IN POLAR MEDIA where the function S4,a(m,n ;a, b ;R) is defined by m-1 m-s-1 m-1 m-s-1c c (----)(7)(-l)r(m+t+l)!xS4Jm, n; a, b; R) = s=o t=O -(aR)"ks(aR)[2(m-s-t)-3]!! (b:ya2)I with a similar expression for S4,b(n~,n ;a, b ;R). APPENDIX M2 11;dk (a2+ k2)"(b2+ k2)nI m-171 2"(m -1)!(n-1) !a2m-'( b2-a2)n s= [2(m-s)-3]!! -(b::;2y+ n: 2"(m-l)!(n-l)!b2n-1(a2-b2)m [2(n-s)-3]!! (4.>2b2 a2-71 2"(m -1)!(n-1)!a2m-'( b2-a2)nsz,o(m n; a, b)+ n 2"(m-1)!(n-1)!b2"-'(a2-b2)mS2,b(m,n; a, b) 13(m;a; R); [2(m-s)-31 ! ! (a R)'kS(aR) 71 71 -2Ra2"' 2"(m-l)!a 2m-,S,(m; a;R) 14(m,n;a,b;R): co j&R)14=s dk (a2+ k2)"(b2+ k2)" P.P. SCHMIDT AND J. M. MCKINLEY 71 71 ni-1 m-s-1 -c cx2Ra 2mb 2n -( yn -1)!(n-1)!2"a 2m-'(b -a2)n = = (rn ;'>( -;-l)(-l)yn +t -1)! [2(rn -s -t)-31! ! (b;ya2>'!.-R)skS(a R, -71 n-1 n-s-1 (1n-l)!(n-l)!2"b~"-~(a~-b2)ms=o t=O (rn +t -l)![2(n-s-t)-31! ! (a ;Tb2)-R)"kS(b R, jo(kR) I5 = so dk (a2+k2)'((b2+k2)"(c2+k2)n ;rl 71-2Ra2 1b2mC2n P%: 2'a2'-'( b2-a2)m(c2-a2)"(1-1)!(m-1)!(a-1)! X m-1 m-s-1 m-s-t-1 >(-l)f+u(m+t-l)! x s=o (n+ -I)!p(~-~ -(aR)"k,(aR) -t -Zk) -31! (b'::2) (c2-a2)U (m-sit-')(-1)t'"[2(nz-s-t-u)-3]!!(aR)Sks(aR)(ar)fit(ar) x (as)uiu(m) 71 71 = --2Ra2"' 2"aZm- '(m- l)!S,(m; a; r; s; R).In the above expiession in(x)is the modified spherical Bessel function of the first kind : sinh(x)io(x) = -. X 170 IONS IN POLAR MEDIA 17(m;a;r;R): [2( rn -s-t) -31!!(ar>'i,(ar)(aR)'kk,(aR) n; n =--2Ra2"' 2"'a2"'-'(m -I)!S,(rn; a; r; R) 18(m,n ;a, b ;r ;R): x n 1 --2Ra2mb2n-(m-l)!(n -I)! 2ma2m-1(b2-a2)n m-1 m-s-1 m-s-t-1 (-l)r+"(n+u-l)! x s=o [2(m -s -t -u) -31!!(____ R)"kS(a R)(ar)ti,(ar)+b:ya2y(a term with permuted indices .1 7L 7L Ss,,(m, n; a, b; r; R) S,,,(m, n; a, b; r; R)-2Ra2mb2n-(n~-l)!(n-l)! (b2--2)n + 2nb2n-1 (a2-b2)m H.Frohlich, Ado. Phys., 1954, 3, 325. R. A. Marcus, J. Chem. Phys., 1956,24,966 ;V. G. Levich and R. R. Dogonadze, Coll. Czech. Chem. Comm., 1961, 21, 193. M. Born, 2.Phys., 1920, 1, 45. J. D. Bernal and R. H. Fowler, J. Chem. Phys., 1933, 1, 515. L. D. Landau, Phys. 2.Soviet Union, 1933,3, 664. J. Jortner, Radiation Res. Suppl., 1964, 4, 24. H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry (Wiley, New York, 1944). R. R. Dogonadze, A. M. Kuznetsov and V. G. Levich, Doklady Akad. Nauk. S.S.S.R., 1969, 188, 383. R. R. Dogonadze and A. A. Kornyshev, Phys. stat. solid. (b), 1972, 53,439 ;1973,55, 843. lo R. R. Dogonadze and A. A. Kornyshev, Doklady Akad. Nauk S.S.S.R., 1972,207,896. l1 R. R. Dogonadze and A.A. Kornyshev, J.C.S. Faraday 11, 1974, 70, 1121. l2 G. Arfkin, Mathematical Methods for Physicists (Academic Press, New York, 1970). l3 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970). l4 I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1965). l5 F. Booth, J. Chem. Phys., 1951, 19, 391, 1327, 1615. l6 J. Muirhead-Gould and K. J. Laidler, Trans. Faraday Sue., 1967, 63, 944. l7 R. M. Noyes, J. Amer. Chem. Soc., 1961, 84, 513 ;1963, 86,971. P. Debye and E. Huckel, Phys. Z., 1923, 24, 185. l9 P. P. Schmidt, J.C.S. Faraday II, 1976, 72, 171. (PAPER 5/545)

ISSN:0300-9238    

DOI:10.1039/F29767200143

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Continuous Charge Distribution Models of Ions in Polar Media Part 2.-Self and Interaction Energies for Soft Charged Ring Systems Dissolved in a Polar Medium BY PARsURY P. SCHMIDT? Department of Chemistry, The University, Southampton SO9 5NH Received 20th March, 1975 This paper is concerned with the calculation of self and interaction energies for ring type charge distributions as representations of a class of ions (namely, radical ions of the aromatic ring type). The ionic systems are considered to be dissolved in a suitable polar solvent medium. The energy quantities are useful in the calculation of solvation energies. In particular, the interaction energy enters into the consideration of the repolarization energy for an electron transfer transition between donor and acceptor species.The Born model of a solvated ion has been most often used in attempts to calcu- late ionic self and interaction energies in polar media. Apart from its conceptual disadvantages, i.e., a hard metallic shell enclosing a point charge source, it is not particularly well suited to use for the analysis of more complicated charge distribution geometries. This arises partly from the difficulty in carrying out integrations for non-simple charge distributions, even where spherical symmetry can be applied, and partly from the metallic-sphere character of the model. In contrast, in a previous paper an alternate, but still phenomenological, ionic model was introduced which is based on the use of soft ionic charge distributions which are continuous to the coordinate origin.2 The results found on the examina- tion of simple monatomic ionic systems are encouraging.Fairly good agreement with experimental self energies follows with relatively little computational effort. In this paper the previous treatment of soft charge distribution systems is extended to cover cases of more complicated molecular ionic geometry. In particular, simple charged ring systems are considered. Such systems are frequently encountered in connexion with radical ions.3 The self energy figures in the solvation energy, and both the solvation and interaction energies enter the repolarization energy, a quantity which is an important component of the activation energy for an electron transfer transition between a donor and acceptor.CHARGE DISTRIBUTION The form of the charge distribution for the ring system studied in this work is p(v> = P5 sin2 or2exp(-p2r2)n2 where P = J2Ia t permanent address : Department of Chemistry, Oakland University, Rochester, Michigan 48063, U.S.A. ;Fulbright Senior Research Scholar. 171 172 CHARGED RINGS IN POLAR MEDIA and a is an effective Bohr radius.2 It is not convenient to work with the charge distribution expressed in terms of trigonometric functions. Hence, we convert to a form expressed in terms of spherical harmonic functions, Yl,m(P): A further conversion is useful in considering the interaction energy expressions. Specifically, it is convenient to refer the ring system to a space fixed axis system.In this case one most easily does this with the use of the spherical harmonic addition theorem : Thus, another form of p(r) is where f refers to the space fixed axis system ;it is the unit vector in that system. In order to evaluate the self and interaction energies in a polar medium we need the Fourier transforms of the electrostatic fields. These we get from the transforms of the charge distributions through the use of the transformed Gauss law : 1%E(k) = -p(k). (6)k The transform of p(r) in momentum space is given by p(k) =Jd3rp(r) exp(ik .r) (7)with exp(ik r) =471 i'j,(kr)Yz1,(3)Y,,,(ft)l,m andjl(x) is the spherical Bessel function of the first kind.4 Thus, with eqn (3), we find and with eqn (5) p(k) =4712 Y&(f)Yoo(k) (1--k2 -c,y~m~)y~m(1%)exp(-k2/4p2)* (10){The equation for the field transform using eqn (9) is :;2)+ E(k) = -(4z)*Zi-{ Yoo(k)(1-')+(1/5)'Y20(ft)~} exp(-k2/4p2) (11)ft k 6P2 6P2 and with eqn (10) it is P. P.SCHMIDT SELF ENERGY The self energy of an ion immersed in a polar medium can be derived most simply in the momentum space representation 2* ws= --/d3k C,(k)E*(k) 4(2~)~ .E(k) (13) where 2 n=O CN(k) = D, exp(-k2Li). (14) In eqn (14) the A, are correlation lengths associated with spatially disperse fundamental excitations in the polar s01vent.~ The D,depend on dielectric constants associated with each mode of the solvent : D,= l/En+l+l/&n* (15) The above expression gives the self energy with account taken of medium dispersion effect^.^^ In the dispersion-free limit this reduces to W,= -(1-l/&st)-4 d3k E"(k) .E(k) (16)4(24's and ESt is the static dielectric constant.In order to obtain the self energy we evaluate eqn (13) with the use of eqn (11). We find Y&(L)Yoo(&)(l- k2/3p2 +k4/36p4))+-1 k4 -Y* (L)Y,,(L) +180p4 2o J D, fudk(1-k2/3p2+k4/30p4)exp Dn(4/n)*(A-A2/3+A5/5) where and XI =p&. In the dispersion-free limit A = 1, and z2 W, = --(1-1/~,~)(26/15Jn). (19)2a The above results may be compared to the simple spherical case for which one finds z2 w;Ph = --CDn(4/z)'(A-A3/3+A5/12).2a n CHARGED RINGS IN POLAR MEDIA The ring system self energy is larger than that of the simple spherical system by an amount Aw~= -(Z2/2a)CDn(1/~)'(7/30)A5 n which can be about 20% greater.In the dispersion-free limit the difference is Aw:') = -(Z2/2a)(1-1/~,t)(7/30Jn). Thus, one readily sees that as far as self energy quantities are concerned, more com- plicated ionic geometries only change slightly the result one would obtain using a spherical model representation. This result, however, is strictly valid only for vacuum charge distributions. It is expected to be approximately true in polar systems -see, the discussion section. ELECTROSTATIC INTERACTION BETWEEN THE CHARGED RING AND A SPHERICAL CHARGE DISTRIBUTION The formula for the interaction energy for two charges is '(k)E"(l, k) .E(2, k) exp(ik .R)wi = 2(244 d3k~,t~ where E(1, k) and E(2,k) are the Fourier transforms of the fields arising from the electrostatic charge distributions at sites R, and R, : R = R2-R1.(22) We let E(1, k) be given by eqn (11) and E(2,k) by &E(2, k) = -(4n)%-YOO(k)Zsphexp(-k2/4p2) (23)k which is the transform of the field of a spherical charge distribution of the form p(r) = Zsph(q3/7+) exp(-q2r2) (24) normalized to the net charge in the distribution : Sd3rp(r) = Zsph. (25) In eqn (21) ~,t'(k)is the Fourier transform of the medium permittivity : 2-5 E, '(k) = 1-C,(k). (26) The specific expression for the interaction energy is ~wi = ""Jd3k ~,t'(k)(Y~~(k)Y~~(k)(2- k2/3p2)+2(2n)4 (1/ 15)) Ygo(f)Yz'( &)(k2/p2) k -ex p( -k2(p2+q2))exp(ik .R).4p2q2 (27) In order to evaluate this expression we use the Rayleigh expansion exp(ik .R) = 4n iyl(kR)qtlt(l?)Y&(&)1,m together with the spherical harmonic integral relation /4z d~ky:rn,<k>K,nt,(k>~linl(~ where C(Zl, Z2, Z3 ;m,, m,, m,) is the CIebsch-Gordon c~efficient.~ P.P. SCHMIDT Carrying out the angular integrations in eqn (27),we get the radial integral for wi : k2(P2+q2>(k2/p2)j,( kR)( 1-n Although there are six integrals in (30), only three of these need to be evaluated. If we write I-J dt t2"j(tR/a(x,, z))e-", (s = 0, 1); (Z = 0,2) (31)a(x,, z)2s+i 0 where t = a(x,, z)k, z = p/q, x,,= pi,, and then three of the integrals follow with x, = 0. The last definition, eqn (34),is useful in the final expression of the interaction energy. Define the function F(xn,Z) = +(p2R2A2(xn, 4PRz)>--A3(x,, z) x 3Jn ~XP(-P2R2A2(xn, ~ ) ) -1~ ~ 2 cos e-j1) x ~~2 ~-(3 {$4(p2R2A2(xn, a))-FA(,,,,z)J2 The above function F(x,, z) follows naturally from the integrals in eqn (30).The interaction energy can now be written as ZZ ~i = "-'{F(O, z)-C D,F(x,, z)]. (37)R n In the dispersion-free limit this reduces to CHARGED RINGS IN POLAR MEDIA The first two terms in F(x,,, z) constitute the contribution to the interaction from a Is and 2s type Gaussian radial charge distribution.2 The last term gives the angular contribution due to the distortion of the charge from a sphere into a ring. INTERACTION BETWEEN TWO CHARGED RINGS In this section the interaction between two charged rings separated by a distance R is considered.The form of the interaction energy again is given by eqn (21). However, now the field transforms given by eqn (12) are used for both charge distributions. Consider first the problem of the angular integrations : dokE*(k).E(k) exp(ik .R). 14n The product of the field quantities is E*( 1, k).E(2, k) = (~Tc)~Z,Z~~-~{ Yoo(I)Y$o(2)Y&(ft)Yoo(E)x (1 -k2/6p2)(1-k2/6q2)+ Yoo(I) Ygo(k)iYtI(2)Yzm(k) x -2 With the use of the integral (29) together with eqn (28), one finds the interaction energy reduces to the k-space radial integral (1 -k2/6p2)(1- P. P. SCHMIDT I77 The quantities I,,,(x,,,z) in eqn (41) and (42) are defined by the following (with a(x,, z) defined as before by eqn (33) and (34)) : 7c Il(x,,,z) = dt jo(tR/a) e-" = -4(R/2a) (43)a0 2R -Sm Jndt t2j,(tR/a) e-f2 = -exp(-R2/4a2) (44)4a ne-'2 = T(3-R2/2a2) exp( -R2/4a2) (45)8a 14(xn, 2) = -fodt t2j2(tR/a) e-I2 a3 0 = 4R25{6@(R/2a) -(R/a)(6 +R/a)exp(-R2/4a2)) (46) 15(xn, 2) = -{a dt t"j,(tR/a) e-" = @ exp(- R2/4a2) (47)as 0 16 a7 = ~{105,/?~(R/2a)-(R/a)[IO5+Y(R/a)' + 2~5 i(R/a)6]exp(-R2/4a2) (48) (The arguments of a(x,, z) have been suppressed for convenience.) In the dispersion-free limit the interaction energy can be expressed as 1-[f2(3 COS~8, -i)+(3 COS~e2-I)]I~(o,12p2 2)- CHARGED RINGS IN POLAR MEDIA Z2-4(sin2 sin2 e2cos[2(+, -42)][13(0, Z)++QI~(O, Z)++I,(O, z)] -240p sin 28, sin B2cos(q51-(p2)[13(0, z)-+I5(O, z)-?I,(O, z)] + The multipole moments of the charge distribution are defined by qlm= Jd’y’r’’ Y~(~’)P(Y’) (50) and the components of the quadrupole tensor by Qfj = Jd3x‘(3xtx;-r’26ij)~(~’).(51) Thus, with the use of the charge distribution, eqn (5), it is possible to write the function G(xn,z) as This expression clearly indicates the dependence of the interaction energy on the mono- pole and qzadrupole components of the individual ring charge distributions. More-over, it reveals a marginal dependence on the quadrupole components as compared to the monopole terms. DISCUSSION It is essential to point out that strictly speaking the theory for the solvation and interaction energies derived in this paper is invalid.Nevertheless, it is expected to be reasonably accurate under certain conditions. The lack of validity of the theory is a result of the fact that for a homogeneous, isotropic dielectric the self energy can only be calculated for a uniform charged sphere or spherical charge distribution. In connexion with the interaction energy quantities, all theories based on Maxwell’s static equations and the constitutive equations for isotropic dielectrics are invalid. For large inter-ionic separations, however, one reasonably anticipates that the inter- action energy quantities will differ from whatever the true quantities are by a very small amount. The homogeneous, isotropic dielectric is defined only for systems devoid of external field source charge distributions embedded within the medium.Thus, as soon as a test charge is introduced into the dielectric medium, the condition of isotropy is violated.” This causes no problems for the calculation of the self energy for spherical charge distributions. However, for the ring systems considered here there will be an error. In terms of the k-space integrations used in the derivation of the self energy, * I am indebted to A. A. Kornyshev for a discussion of this problem. P. P. SCHMIDT eqn (13), the expression may be valid in the range 0 < k < kmax is the inverse of the distance, measured from the centre of the charge distribution, less than which the isotropy of the medium ceases to be a reasonable approximation. If we limit the range of integration as indicated, the contribution to the self energy which can be expected to be accurate is where Y Yh4, B3A(2m-1, x,,,a: amax)= (I +2x3"-3 (54) and y(m-3, B:) is the incomplete gamma function ?(in -3,B,f)= (nz-3/2) y (m-3/2, Bi)-Bim-3exp( -Bi) (55)m# 1 and finally, B, is a 1+2x3'B, = amax( (57) The inner region possibly can be treated discretely, that is, in the manner of Bernal and Fowler or Muirhead-Gould and Laidler lo to fill in the remaining contribution.The equipotential energy surfaces of the charge sphere are concentric shells (an infinity of them) around the source charge sphere. For a non-spherical charge distribution the equipotential surfaces are neither concentric around the source nor are they even spherical themselves.Thus, it is conceivable that the value of amax = necessary TABLEVA VALUES OF THE FUNCTION G(0,l) (WHICH IS A FUNCTION OF R)FOR THREE RING-RING CONFIGURATIONS : A : (~,O)(O,O)B : (90,0)(0,0) C : (90,0)(90,0) WHERE THE NOTATION IMPLIES (e,,~l)(e,,~,)FOR RINGS 1 AND 2. G(0,l) x 10-1 RIA A B C 4 8.67 9.37 10.01 5 9.21 9.74 10.34 6 9.46 9.85 10.30 7 9.60 9.89 10.21 8 9.70 9.92 10.16 9 9.76 9.94 10.13 to approximate the distance from the origin at which the equipotential surface is essentially closed may be extremely large. Such an enclosed spherical region easily can contain such a large number of molecular species and ions as to preclude any economic and reasonable computer analyses.We know that the results we have obtained are accurate in vacuum. It is also suspected that the correction for dielectric anisotropy and inhomogeneity may be reasonably small. Therefore, we proceed to make several observations concerning the results as they apply to the electron transfer problem mindful of the restrictions which must be placed on the credibility of the numbers obtained. CHARGED RINGS IN POLAR MEDIA In table 1 is found a representative list of the values of the G-function for three distinct angular configurations calculated in the dispersion-free limit. It is immediately clear that the quadrupolar component contribution to the interaction energy over that of the monopole contribution is marginal.The three configurations for which data are given are illustrated in fig. 1. Rotations of the individual rings about the x-axes by their angles q51 and 42 do not produce any configurations with energies higher or lower than the limiting forms given in the table. This result may seem somewhat surprising. However, a similar situation exists with respect to the interaction between two dipoles. FIG.1.-The three limiting orientations for the ring-ring interaction energy. (a) (0,O)(O, 0); (b)(90,0)(0,0); and (c) (90,0)(90,0). Concerning the electron transfer reaction, the repolarization energy, which enters into the activation energy, can be written as ’ Jd3k Cn(k)E*(l, k). E(1,k)+E*(2, k). E(2, k)- Er = 404 2(2n)4 d3kCn(k)E*(l, k) .E(2, k)exp(ik.R).--s This expression applies to the radical anion-neutral ring system electron transfer. Using the results of the previous sections, this expression can be evaluated. In the dispersion-free limit (which is used here merely for ease in illustration) one finds The example envisaged is benzene and its radical anion. If one assumes that the effec- tive Bohr radius is 2A, and that the separation distance is 7 A, then the repolarization energy in hexamethylphosphoramide with a refractive index of n = 1.55 and a static dielectric constant of 30 would be (0, O)(O, 0) ;E, = 4.86 eV (90,0)(0,0) ;E, = 4.85 eV (90,0)(90,0) ;E, = 4.83 eV for the three distinct configurations.* As the difference between these energies is * These repolarization energies are somewhat high and probably reflect the fact that no account has been taken of ion pair formation between the radical anion and alkali (or other) counter cations in the solution.P. P. SCHMIDT very small, it is reasonable to expect that electrostatic factors alone play only a mar- ginal role in the determination of the inter-ring configuration, or orientation, in the electron transfer transition state. Rather, this work suggests that ring-ring quantum overlap factors will predominate. The electronic symmetry representations of the initial and final states must be the same according to the von Neumann-Wigner rule.’ This will remain true even with the modifications of Herzberg and L~nguet-Higgins.’~ Thus, the conformation of the transition state should be determined by examining both the ring-ring overlap quantities and the electrostatic matrix elements between the initial and final states as functions of the relative ring-ring orientations.This should enable one to decide the nature of the transition state for the electron transfer. .85 I I I I I I 4 5 6 7 8 9 A FIG.2.-The behaviour of the function G(0, 1) for the ring-ring orientations (a), (0, O)(O, 0); (b), (90,0)(0,0) ; and (4, (90,0)(90,0>. There is, however, one instance in which the electrostatic interactions can play a predominant part in the determination of the activation energy and hence the transi- tion state. In the case in which the reacting species must approach sufficiently close together that it is possible to consider only a vacuum interaction, then the inter-ionic interaction becomes iniportant.The repolarization contribution due to the medium degrees of freedom will be accounted for in terms of the charge density of the aggre- gate. This term may not be greatly dependent on the ring quadrupole moments. Obviously, these considerations, including the inter-species attraction or repulsion energy, will not be very important for radical anion-neutral species transfers. Hovv-ever, for disproportionation reactions involving two radical anions, for example, the inter-ionic repulsion will be a major contribution to the activation energy if the species approach sufficiently close together at the instant electron transfer takes place.It is a relatively simple matter to extend these calculations to include naphthalenic molecules. Such a system would be represented as a charged ellipse. Although it would seem that because of the smallness of the quadrupole components of the poten- CHARGED RINGS IN POLAR MEDIA tials and fields, the value of this analysis is limited, this is not the case. The equili- brium charge distribution of an ionic system is important in the determination of the modes of excitation of the ionic inner sphere.14 It is quite likely, therefore, that the nature of the ionic charge distribution, which includes a distribution over inner sphere solvent molecules, will be important with respect to inner sphere reactions. This work was supported in part by a grant from the S.R.C.I wish to thank Prof. M. Fleischmann for his hospitality and support given to me. I have derived considerable stimulation and guidance from conversations and correspondence with Dr. A. A. Kornyshev for which I am thankful. 'M. Born, 2.Phys., 1920, 1,45. P. P. Schmidt and J. M. McKinley, J.C.S. Faraday ZI, 1976, 72, 143. N. Hirota, in Radical Ions (Interscience, New York, 1968). G. Arfkin, Mathematical Methods for Physists (Academic Press, New York, 2nd edn., 1970). R. R. Dogonadze and A. A. Kornyshev, Phys. stat. solid (b), 1972,53,439 ;1973,55, 843. R. R. Dogonadze and A. A. Kornyshev, Doklady Akad. Nauk S.S.S.R., 1972, 207, 896; English translation, Proc. Acad. Sci. U.S.S.R.,207, 983.'M.E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957). J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962). J. D. Bernal and R. H. Fowler, J. Chem. Phys., 1933,1, 515. lo J. S. Muirhead-Could and K. J. Laidler, Trans. Faraday Soc., 1967, 63, 944. A. A. Kornyshev, personal communication. l2 J. von Neumann and E. Wigner, Phys. Z., 1929,30,467. l3 G. Herzberg and H. C. Longuet-Higgins, Disc. Faraday Soc., 1963, 35, 46. l4 P. P. Schmidt, J.C.S. Faraday ZZ,submitted. (PAPER 5/546)

ISSN:0300-9238    

DOI:10.1039/F29767200171

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Phase-shift Studies of Hg(3P,,)Reactions Part 5.-Kinetics of Hg, Excimer Reactions in the Presence of Nitrogen BY ONGKEANGHEE,COLING. FREEMAN,* J. MCEWANMURRAY and LEONF. PHILLIPS Chemistry Dept., University of Canterbury, Christchurch, New Zealand Received 28th April, 1975 A modulated source of 253.7nm mercury resonance radiation has been used to excite Hg, excimer bands in mixtures of mercury vapour (at a fixed pressure of 0.27 Torr) and nitrogen at pres- sures between 1 and 200 Torr. Phase shifts of the emission bands at 335 and 485 nm, relative to 253.7 nm atomic fluorescence, have been measured as functions of nitrogen pressure at 373 K. The results are consistent with a mechanism in which Hg(3P0), produced by spin-orbit relaxation of Hg- ('P1)in collision with nitrogen, reacts in a termolecular reaction with a ground-state (ISo) mercury atom and a nitrogen molecule to form Hg2(31u).The Hg2(31,)then either radiates the 335 nm band or is converted to Hg2(30;) by collision with nitrogen (k7 = 4.8 L 0.5 x 10-15 cm3 molecule-' s-l). The 485 nm emission arises, under the present experimental conditions, exclusively from a collision- induced process involving Hg2(30J and a nitrogen molecule (klo = 5.7+ 0.3 x cm3 molecule-l s-'). The effects of wall processes are of minor importance except at the lowest pressures used. In a previous paper in this series,l a phase shift study was reported of Hg, excimer bands at 335 and 485 nm excited by irradiation with 253.7 nm atomic resonance radiation of a system containing pure mercury vapour at high pressures.There have also been a number of investigations 2-7 in which the same emission bands have been produced from systems containing very low mercury pressures in the presence of nitrogen gas. The added nitrogen has the effect of greatly increasing the rate of production of metastable Hg(3P0) (the precursor to Hg, excimers) by spin-orbit relaxation of Hg(3P1). It is well established that the 335 and 485 nm bands arise from the 31, and 30, states of Hg, respectively, and that the Hg, molecule is formed by the termolecular association of metastable Hg(3Po) and groundstate Hg(lSo) atoms.'. 3* The49 third body in this association is another mercury atom or a nitrogen molecule. The band at 335 nm is the result of spontaneous emission from the 1 state of Hg, while the 30, state is populated from 1 by a collision-induced transition.The band at 485nm has been shown to arise from Hg,(30;) by means of a collision-induced emission process in the pure mercury system 1* although Penzes et aL6 have assumed it to be the result of a spontaneous emission process when nitrogen was present. We report here the results of the application of the phase shift method to measure rates of processes leading to Hg, excimer emissions from a system containing a small but constant low pressure of mercury (0.27 Torr) and nitrogen (pressures between 1 and 200 Torr). Because of the very rapid rate of removal of Hg(3Po) by oxygen," the results are critically dependent on the purity of the nitrogen used.To obtain reproducible results, the oxygen impurity in the nitrogen had to be less than 0.1 p.p.m. Even when the nitrogen was of the highest possible purity with the mer- cury pressure at 1.8 x Torr (saturated vapour pressure at room temperature) the intensity of banded emission was insufficient for accurate phase shift measure- ments to be made, although the bands could readily be observed using d.c. detection 183 PHASE-SHIFT STUDIES OF Hs(~P*)REACTIONS under these conditions. For this reason all phase shift experiments were carried out at 373 K (saturated v.p. of mercury 0.27 Torr ll). This temperature represented the best compromise between the need for increased intensity of emission and the require- ment of avoiding kinetic complications resulting from competing reactions of Hg atoms which become more important at higher temperatures.Our results have veri- fied that the mechanism is analogous to that reported for the pure mercury systern.l It has also been demonstrated that the 485 nm band is due almost exclusively to a collision-induced emission process of N2 with Hg,(30,). EXPERIMENTAL The apparatus and procedures were similar to those described previously.'. lo The quartz reaction vessel included a side-arm containing mercury and was enclosed in an oven at 373 K. The monochromator, photomultiplier and methods of detection of both the a.c. and d.c. signals were the same as reported previously.' The variation of sensitivity with wavelength for the monochromator/photomultipliercombination was recorded at 335 and 485 nm by measuring the response to a tungsten strip filament in a quartz envelope and subsequent reference to the emissivity data of de Vos.12 Matheson pre-purified grade nitrogen (manufacturers specification oxygen impurity <20 p.p.m., typically 8 p.p.m.) was used for the bulk of experiments.Without further 60 45 M z G -", 30 2 I:P 15 0 0 50 100 350 200 nitrogen pressure/Torr FIG.1.-Phase shifts at 358 Hz between 253.7 nm atomic fluorescence and (i) the 335 nm emission band (O), and (ii) the 485 nm emission band (a),as a function of nitrogen pressure (mercury pressure = 0.27 Torr, temperature = 373 K). ONG K. G., c.G. FREEMAN, hi. J. MCEWAN AND L. F. PHILLIPS 185 purification of the nitrogen it proved almost impossible to detect any Hg, excimer emission. For this reason careful attention was given to the problem of removing oxygen impurities. The most successful arrangement consisted of two purification stages. Stage (a) was model- led on that employed by Berberet and Clark ,with the nitrogen being cycled continuously by convection around a closed loop which included Cu/CuO heated within a furnace to 820 K and a P205trap. Samples of -10 dm3atm of nitrogen were left in stage (a)for at least a week before small portions were taken into purification stage (b)immediately prior to use. Stage (b)comprised a toroidal quartz reactor in which nitrogen was saturated with mercury vapour and then irradiated with an unfiltered low-pressure mercury lamp.The same method of purification has been used by Callear l3 and probably involves the conversion of oxygen to solid HgQ, presumably through reaction of O2with metastable Hg(3P2) atoms which are formed during simultaneous irradiation by 253.7 and 404.7nm mercury lines. Irradia-tion in stage (b) was continued for at least 4h before the nitrogen was admitted to the reaction cell. No analyses of the purified nitrogen were performed but a comparison with results reported by Callear 137 l4 indicates that purification stage (a) was effective in reducing the oxygen impurity to less than 1 p.p.m. and that stage (6) lowered this to below 0.1 p.p.m. The criterion for purity in the present work was the ultimate intensity of 335 and 485 nm excimer bands produced from mercury+ nitrogen mixtures in the reaction cell.Nitrogen purified by stage (a)alone gave significantly greater band intensities than cylinder nitrogen, and irradiation in stage (b)gave a further marked increase in band intensities during the first 4 h. Irradiation beyond 4 h gave no further change in band appearance and this was taken to be the ultimate level of nitrogen purity obtainable in the present system. As a check of the above estimates of purity a few measurements were made using Matheson re- search grade nitrogen (oxygen content guaranteed by the manufacturers to be less than 1 p.p.m.). With no further treatment this gas gave results comparable to those reported above after purification stage (a).Subsequent irradiation of the research grade nitrogen in stage (b)gave identical results to those mentioned above after two purification stages. RESULTS AND DISCUSSION Curves showing the variation with nitrogen pressure of the phase shifts between the Hg2 emission bands and the scattered 253.7 nm line are given in fig. 1. Phase shift measurements were made at nitrogen pressures down to about 1 Torr for the 335 nm band but were limited by emission intensity to pressures above 20 Torr for As mentioned previously,l phase shift measurements yield information only about the rate of removal of an emitting species and not about its formation rate. For this reason it is possible that reactions (4) to (12) may possess different products and all rate measurements discussed below represent the total rate for all reaction channels 1S6 PHASE-SHIFT STUDIES OF Hg(3P0)REACTIONS available to a particular set of reactants.In addition to the reactions listed above, processes analogous to (5), (7) and (10) but involving ground-state mercury atoms instead of nitrogen are possible Hd3PO)+ 2Hg('So) 13+ Hg('So) (13)-+ Q2(31.) +Hg(lS0) 3 Hg2(30J+Hg(1S0) (14) Hg2(30,)+Hg(1So) + 3Hg(lSo)+hV485. (1 5) Rate constants for reactions (13) to (15) have been reported for 563 K and the assumption that these values are upper limits for the rates in the present system at 373 K indicates that these processes are all of minor importance only. The proposed mechanism predicts that the phase delay (p degrees) of the 485 nm band relative to the 335 nm band arises from the relatively slow removal of Hg2(30;) by reactions (lo), (ll), (12) and (15).Reaction (12) will be of little importance at pressures of nitrogen above 20 Torr and thus we should have tan P = 2~f/(klO"2l+kll +k15[HgI) (16) wherefis the modulation frequency. Fig. 2 shows a test of eqn (16) for both directly 24 18 3 I -s s 12 m25 N, 6 0 0 50 100 150 nitrogen pressure/Torr FIG.2. Test of eqn (16). 0,points read off the smoothed curves of fig. 1. 0,directly measured phase shift values. /3 is the phase shift between 335 and 485 nm emissions, measured data points and also points read off the smoothed curves of fig.1. The slope of the line in fig. 2 yields a value for klo of (5.7k0.3) x cm3 molecule-l s-l ONG K. G., C. G. FREEMAN, M. J. MCEWAN AND L. F. PHILLIPS 1 S7 and the small intercept indicates an upper limit of 500 s-l for the sum of kll and kls[Hg]. McCoubrey has reported a value of 20 s-l for k,, at 473 K and a value of (1.06k0.2) x cm3 molecule-I s-I has previously been measured for k15at 563 K., The use of this value of k15as an upper limit for the rate at 373 K together with the mercury pressure in the present system gives a maximum value of 745 s-l for k,,[Hg]. It is clear that the intercept of fig. 2 is consistent with the above values of k,, and k,, when allowance is made for the probable temperature dependence of k15. Penzes et aL6have assumed that the emission of the 485 nm band is due entirely to reaction (1 1) and report a value of -3 x cm3 molecule-1 s-l for klo(leading only to non-radiative products) at 297 K.It is demonstrated in fig. 2 that, at pres- sures above 20 Torr of nitrogen, 485 nm emission is exclusively due to reaction (10). This is similar to the mechanism reported previously 1i * for emission of 485 nm radiation from systems containing pure mercury. The reason for the low value of lc,, reported by Penzes et al. is discussed below. The phase shift u of the 335 nm band relative to the 253.7 nm line is comprised of two parts, a. and ul. a, depends on the rate of removal of Hg(3Po) by reactions (4), (5), (6) and (13) and alon the removal of Hg2(31u) by reactions (7), (8), (9) and (14).To evaluate the contribution due to a,, values of a, were calculated from the expression tan a, = 2nf/(k4",1 + k5[HglCN21+ k6[N21-l+ kl3[HgI2). (17) The following rate constants were used in eqn (17) : k4 = 4 x cm3 molecule-1 s-I ; l5 k, = 1.32 x cm6 s-l [this is the mean of the two values of k5, 1.05 x and 1.60~ derived in ref. (5)]; k13 = 3.4~ cm6 s-I ; k6 = 6 x lo1' molecule ~m-~ k6 was initially chosen ar-s-I. bitrarily and adjusted by an iterative procedure until the value of k6 and the value obtained experimentally for k9 (see below) were consistent with each other on the basis of the following assumptions : (i) that both reactions (6) and (9) are diffusion controlled with rates inversely proportional to nitrogen pressure ; (ii) that deactiva- tion of both Hg(3P,) and Hg2(31u) occurs on every wall collision; and (iii) that McCoubrey's values for the relative diffusion coefficients of Hg(3Po) and Hg,(30J are applicable to Hg(3P,) and Hg,(31u) in the present system.Of the various terms contributing to the calculation of a, by far the most important is that due to process (5) which contributes over 95 % of the denominator on the right-hand side of eqn (17) at all nitrogen pressures above 10Torr. At lower pressures the wall reaction (6) becomes increasingly important. Values of CI, calculated by the above procedure account for only a small fraction of the total measured phase delay a. At a nitrogen pressure of 10 Torr u0 is only 16 % of a, at 20 Torr 10% and the contribution steadily decreases as the pressure is further increased. For this reason the ultimate accuracy of a, is not greatly affected by the choice of rate constant for reaction (5) and by the assumption that temperature variation of k5is small between 297 and 373 K.For a, we have tan = 2~f/(k7[N2]+k8 +kg[N~]-'+k14[Hg]). (18) At high pressures the term due to the wall reaction (9) becomes negligible and thus a plot of 2i~fltan a, against nitrogen pressure should be a straight line of slope k, and intercept kg+k14[Hg]. A test of this result on high pressure data for 01, calculated as described above is shown in fig. 3. The line drawn in fig. 3 yields a value for k7 of (4.8 k0.5) x cm3 molecule-I s-' and an intercept of (3.0k0.4) x lo3 s-l.Variation of the value assumed for k5in the evaluation of 01, between the limits 1.05 x and 1.60 x cm6 molecule-2 s-' given in ref. (9,ultimately has a negligible effect on the slope in fig. 3 but leads to a further uncertainty of k0.2 x lo3 s-l in the PHASE-SHIFT STUDIES OF Hg(3Po) REACTIONS 'I 6 ~ ~~ 0 30 60 90 120 nitrogen pressure/Torr FIG.3.-Test of eqn (18) for high pressure values of eel. 3 I I I I 0 3 6 9 12 nitrogen pressure/Torr FIG.4.-Test of eqn (18) for low pressure values ofal. The solid line is the same as that drawn in fig. 3. The broken curve is calculated using the solid line together with kg = 3 x 1019molecule s-l. The points (0)~rn-~ are calculated from experimental values of tc using k6 = 6 x lOI9 molecule ~rn-~s-l.intercept. The final value of the intercept given by the plot in fig. 3 is thus (3.0k0.6) x lo3s-l. If the value of k14 = (6.4k0.4) x cm3 molecule-1 s-l at 563 K is taken as an upper limit for the rate at 373 K then the maximum contribution to the intercept from k14[Hg] is 450+30 s-l. The value of k8 in the present system is therefore estimated as (2.8 k0.8) x lo3s-l. A previous phase shift measurement of k8 of (2.0f0.5) x lo4s-1 was reported for 563 K.l ONG K. G., C. G. FREEMAN, M. J. MCEWAN AND L. F. PHILLIPS 189 At low pressures (less than 10 Torr) of nitrogen the neglect of wall reaction (9) is no longer possible. In fig. 4 the full version of eqn (IS) is tested against experimental phase shift data for the low pressure region.The value of ]isused in calculation of ct0 for the data shown was 6 x lo1' molecule ~m-~ s-I and the value of kg correspond- ing to the broken curve is 3 x lo1' molecule ~m-~ Because of the method used s-l. in the evaluation of Jcg and kg, the uncertainty in both of these rate constants is of the order of +30 %. Previously a value of 1.7 x 1020molecule ~m-~ s-l was estimated for k6 at 563 K in the same reaction cell as used in the present study. The present value for k6 is consistent with this when allowance is made for the temperature varia- tion of the diffusion coefficient. Fig. 2 indicates that there is no evidence for reaction (12) being of importance at pressures above 20 Torr of nitrogen.McCoubrey has reported that the wall deactivation of Hg2(30;) is only 10 % efficient. If it is assumed that both states of Hg, have the same diffusion coefficient, then the rate of reaction (9) will represent an upper limit for the rate of reaction (12). The use of this value justifies the neglect of reaction (12) in eqn (16) above. During the present study measurements were also made of the relative intensities of 335 and 4S5nm bands under d.c. illumination. The results of these measure- ments confirmed that the ratio of band intensities, 1485/133s, was directly proportional to nitrogen pressure over the range 5 to 80 Torr. This is in agreement with the find- ings of Penzes et aZ.6 A steady-state treatment of the proposed mechanism (ignoring wall processes) shows that the slope of a plot of 1485/1335 against nitrogen pressure is equal to k7/ks.The slope found in the present work was (1.16-tO.12) x 10-l8 cm3 molecule-1 at room temperature, which is significantly less than the value of (4.3fr0.9) x 10-l8 cm3 molecule-I reported by Penzes et aZ.,6 but agrees within experimental error with the ratio k7/k8 determined by phase shift measurement. Table 1 summarizes the rate constants determined in the present work. Of the TABLERA RATE CONSTANTS FROM THE PRESENT WORK reaction no. rate constant (6) (6+2)x 1019molecule ~rn-~s-' (7) (4.8+ 0.5)x 10-15 cm3 molecule-l s-l (8) (2.850.8)x lo3s-l (9) (3+ 1)x 1019molecule cmd3 s-I (10) (5.7k0.3)~ cm3 molecule-' s-' (11) <500 s-l values measured here Penzes et aL6 have previously reported k, = (4.3k0.9) x lo-' cm3 molecule-l s-l and klo = 3 x cm3 molecule-' s-l.Both of these values were calculated with the assumption of k, = 1.0 x lo7 s-l and this is clearly the main reason for the difference between their results and those reported here. The method of determination of k7 and klo in this work does not rely on any independently measured value of k8. If our value of k8 = (2.Sk0.8) x lo3 s-l is used together with the relative band intensity results reported by Penzes et al. a value of k7 = (1.2f0.6) x cm3 molecule-l s-l is obtained. The remaining discrepancy in values of k7 may well be due to the effect of trace impurities in the nitrogen used by Penzes et al.Since the preparation of this paper the authors' attention has been drawn to two recent papers by Krause et all6* In the experiments of Krause et al. the method of delayed coincidences was used to study the formation and decay of Hg, molecules in mercury+nitrogen mixtures at 432 K. They report no detectable difference in the decay rates of 335 and 485 nm bands in the nitrogen pressure range 35 to 760 Torr. This is at variance with our observation of a phase delay between the two PHASE-SHIFT STUDIES OF Hg(3Po) REACTIONS bands. Of the rate coefficients reported by Krause et al. with which comparison can be made, k7 is given as 4 x cm3 molecule-' s-' (i.e. around lo3greater than that reported here) and kI4is estimated as 1.8 x cm3 molecule-1 s-l (cf.the value of 6.4 x 10-14 cm3 molecule-l s-l of the previous paper in this series l). The reason for the large differences probably arises from the combined effect of (a) the assump- tion of a value close to lo7 s-l for the radiative removal of Hg2(31u) (k8) in the evaluation of the other rate parameters, and (b)the use of Matheson research grade nitrogen with no attempt at further purification by Krause et al. The present study has demonstrated that this grade of nitrogen is not of sufficient purity for reliable results and it is likely that the residual impurities were responsible both for the appa- rent lower band intensities observed by Krause and the failure to detect a difference in decay rates for the two bands when a relatively high pressure of quencher is present.This work was supported by grants from the New Zealand Universities Research Committee and the United States Air Force Office of Scientific Research. A. G. Ladd, C. G. Freeman, M. J. McEwan, R. F. C. Claridge and L. F. Phillips,J.C.S.Fnraday II, 1973, 69,849. J. A. Berberet and K. C. Clark, Fhys. Rev., 1955, 100, 506. S. Penzes, H. E. Gunning and 0.P. Strausz, J. Chenz. Phys., 1967,47,4869. J. E. McAlduff, D. D. Drysdale and D. J. LeRoy, Canad.J. Chem., 1968,46,199. J. M. Campbell, S. Penzes, H. S. Sandhu and 0.P. Strausz, Int. J. Chem. Kinetics, 1971,3, 175. S. Penzes, H. S. Sandhu and 0.P. Strausz, Int. J. Chem. Kinetics, 1972, 4,449. A. C. Vikis and D. J. LeRoy, Phys. Letters A, 1973, 44, 325. A. 0. McCoubrey, Fhys. Rev., 1954, 93, 1249. L. F. Phillips, Rev. Sci. Instr., 1971, 42, 1078. lo C. G. Freeman, M. J. McEwan, R. F. C:Claridge and L. F. Phillips, Trans. Faraday Soc., 1971, 67, 2004. l1 Landolt-Bornstein (Springer-Verlag, Berlin, 1960), Band 11, Teil 2, Bandteil a, p. 19. l2 J. C. de Vos, Physica, 1954, 20, 715. l3 A. B. Callear and J. H. Connor, J.C.S. Faraday II, 1974,70,1767. l4 A. B. Callear, personal communication. l5 J. E. McAlduff and D. J. LeRoy, Canad.J. Chem., 1965,43,2279. l6 R. A. Phaneuf, J. Skonieczny and L. Krause, Phys. Rev. A, 1973,8,2980. l7 J. Skonieczny and L. Krause, Phys. Rev. A, 1974,9, 1612. (PAPER 5/793)

ISSN:0300-9238    

DOI:10.1039/F29767200183

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Kinetic Spectroscopy in the Far Vacuum Ultraviolet Part 2.-Fluorine Atom Resonance Spectrometry and the Measurement of F 2P Atom Concentrations BY PETER P. BEMANDAND MICHAEL A. A. CLYNE" Department of Chemistry, Queen Mary College, Mile End Road, London El 4NS Received 9th May, 1975 A technique for the first atomic resonance detection of ground state F 2p5 "PJatoms in a flow system is described. Collimated hole structures were used in place of windows to separate the lamp, flow tube and spectrometer. Resonance absorption measurements of the 3s-2p5 multiplet transition of F, between 95 and 98 nm, were used to obtain the first values for the oscillator strengths jjk of the 97.39 nm "Pp'P3 and the 97.77 nm "P+-"P+lines of these multiplets. The values, which have an estimated standard error of & 30 %, are :Ak(97.39) = 4 x ; fik(97.77) = 1 x Fluorine atom resonance fluorescence in the 3s "PJ-2P5 "PJmultiplet, and in one line of the 3s "Pj-2~' "PJ multiplet, was observed for the first time ; it was used to measure concentrations of F 'PJ atoms down to 1 x 10" ~m-~.Measurement of absolute [F'PJ] was by means of the rapid reactions (1) and (2), ki F+Br, -+ BrF+Br (11 F+C12 + ClF+Cl.(2) Attenuation of F atom resonance fluorescence was used to measure the rate constant kl at 300 K. Combining this value with another (mass spectrometric) determination of kl, the mean value kl = (2.2k1.1) x lo-'* cm3 molecule-' s-' from 298 to 300 K was obtained. Although ground state fluorine atoms may be directly detected and determined by e.p.r.spectrometry,' mass spectrometry 2-4 and atomic resonance ~pectrometry,~ few reliable kinetic studies of elementary reactions of F 'PJ atoms have been reported until recently. There are several major problems: (i) a satisfactory source of F 2PJ atoms, (ii) the rapidity of most reactions of F 2P,, even at 298 K, which requires high sensitivityof detection and/or rapid time response of the kinetic technique, (iii) mechan- istic complications due to rapid secondary reactions, particularly in the case of hydrogen-containing reactants, and (iv) the irreproducibility of surface reactions of F atoms. At present, problem (i), the requirement for a satisfactory source of F 'PJ atoms, restricts available kinetic techniques to the discharge-flow method.In this method, the simplest source of F 2PJis a microwave discharge in dilute F2+ He(Ar) mixtures,6 whilst similar discharges in CF4 or SF6have also been The sensitivity of e.p.r. spectrometry at present is probably insufficient for systematic kinetic studies of F 2PJ, which has effectively restricted such studies to mass spectrometry in a discharge-flow system. The use of NO + F chemiluminescence appears promising, although the precursor of that chemiluminescence still remains to be positively identified as FNO. It remains to be seen whether the sensitivity of the NO + F technique is adequate for kinetic measurements of very rapid bimolecular rate constants. We therefore attempted to develop a technique for systematic kinetic measure- ments of F 2P, atom reactions using atomic resonance spectrometry, preferably in fluorescence rather than in absorption, to determine F 'PJ atom concentrations.The 191 VACUUM U.V. KINETIC SPECTROSCOPY experimental prodems are compounded by the necessity to work in the far vacuum ultraviolet near 95 nm, below the shortest wavelength (105 nm) possible with windows. However, our attempts were largely successful, and F 'PJ atoms were detected by atomic resonance fluorescence at concentrations down to 1x loll ~m-~.We were able to establish a value for the rate constant of the F+Br, reaction near 298 K. Reactions of F with C12,43 '* Br2,9 I2 and ICl all proceed at rates close to the hard-sphere bimolecular collision frequency at 298 K.It will be possible to make extensive direct measurements of the rate constants for somewhat slower reactions than these, using a resonance fluorescence technique similar to that described here. EXPERIMENTAL A system for atomic resonance fluorescence studies of atoms generated in a low-pressure discharge flow system has been described previously.l0S l1 The present flow system was similar to the previous ones,1o- l1 but important modifications were necessary to allow the detection of radiation near 95 nm where all solid window materials absorb strongly. The following account emphasizes these modifications. PRODUCTION OF FLUORINE F 2p~ATOMS A 2.45 GHz discharge in F2+He or F,+Ar mixtures in an uncoated silica tube (10 mm i.d.) was used as the source of F 2PJatoms.Fz was obtained as a mixture (4 mol %) with helium and was used direct from the cylinder, except for the addition of further dry He or Ar diluent. The typical degree of dissociation of F2was 0.8, and it was insensitive to inci- dent discharge power.4 The major impurity in the dissociated F2was SiF4,3s which arises mainly from attack on the SiOz discharge tube.4 Oxygen atoms, 0 3PJ,were also detected, using atomic resonance spectrometry at A 130.2nm. Use of a discharge bypass for produc- tion of low F atom concentrations led to typical 0 3PJimpurity atom concentrations around under conditions where [F] z1x 10l2~m-~ [He] z 5x 10l6~m-~.2x lo1' ~rn-~ and These relatively high 0 "PJatom concentrations are thought to arise from partial dissociation of O2liberated in the attack by fluorine or fluorine atoms on the SiOz material of the discharge tube.The reaction, O+Fz 3FO+F, is slow at 298 K; therefore 0 3PJ atoms are not scavenged by undissociated F2,and persist as a major impurity in the F+Fz stream. How-ever atomic hydrogen could not be detected under the same conditions, from which an upper limit [HI < 1 x loll ~m-~is derived. For the determination of F atom concentrations, the F+ Br2 and F+ Clz reactions were used. To avoid the use of the necessarily very low flow rates of pure halogens required to measure [F] as low as 10l2~m-~,larger flows (-10l8molecule s-l) of dilute mixtures of Br2 or C12 (-1 mol %) with argon were used.The mole fraction of the halogen in these mixtures was measured by absorption spectrometry at 414nm (Br2) or 350 nm (C12). DETECTION OF FLUORINE ATOMS BY ATOMIC RESONANCE SPECTROMETRY The source of F atom resonance radiation, 3s-2p5, which was a microwave discharge in F2+He mixtures, has been de~cribed.~ Three collimated hole structures (CHS) were used to separate this lamp from the flow tube and the vacuum spectrometer (fig. 1). The CHS, consisting of a stainless steel grid of many parallel channels, 0.1 mm in diameter and 1 mm long, had a light transmission of 40 %, but showed a low gas conductance. Having an aperture of about f/lO, CHS offer a considerable advantage in light gathering power over the long capillaries conventionally used to separate source, absorber and spectrometer in far vacuum ultraviolet spectroscopy.Fig. 1 shows the four distinct regions of the window- less detection system; the lamp L, the flow tube R, the buffer chamber B and the spectro- meter V, with each pair separated by a CHS, C. To minimize diffusional mixing of the regions, argon (rather than helium) was used where possible, and the total pressure in all four regions was as high as possible. The upper limit was set by the maximum gas load that could be handled by the spectrometer diffusion pump (at spectrometer pressure pv) without stalling, and by the fall in lamp intensity with increase in lamp pressure,-pL. The P. P. BEMAND AND M. A. A. CLYNE 193 pressure in the buffer chamber pB was slightly greater than that in the flow tubep~ to prevent F or F2 entering the spectrometer.Also, p~ was always greater than p~ to prevent F or F2 diffusing from the flow tube into the lamp, which would invalidate the atomic resonance results. The typical pressure parameters thus adopted were :p~ = 400, p~ = 220, p~ = 310 andpv = 0.1 N m-2. Net flow into the flow tube from the lamp and from the buffer cham- ber could not be entirely prevented, and the above typical conditions led to an increase in flow rate of 50 pmol s-' due to the lamp and buffer chamber together. This was equivalent B;Ec+ 's " exhourt mom exhaust-to pump-!o pump zone FIG.1.-Windowless system for resonance spectrometry of F 2P~atoms. B, buffer chamber ; C, collimated hole structures, two of which were mounted on specially-fabricated glass discs, D ; K, inlet to lamp L ; M, MI, differential manometers ;R, section of flow tube ;S, spectrometer slit, sealed via an O-ring seal to the silica apparatus ;V, vacuum spectrometer.to an increase of about 6 % in total mass flow rate in the flow tube. An uncertainty in the atom and reagent concentrations, and in the linear flow velocity, of the order of 6 % was thereby introduced, leading in turn to an uncertainty in derived second order rate constants of about +12 %. Replacement of the CHS with 0.1 mm diameter holes by other CHS with 0.02 or 0.01 mm holes would be advantageous in reducing diffusional mixing still further. (However, this would inevitably lead to some loss in detected photon flux).The flow tube was provided with large pumps, (-17 1 s-l nominal displacement) giving a maximum linear flow velocity of 2000 cm s-l, whilst two small rotary pumps were used for the lamp flow and for the buffer volume flow. The spectrometer was pumped by a baffled 10 cm oil diffusion pump, having an actual speed of 80 1s-l, backed by a rotary pump (2.5 1s-'). The spectrometer, a 1 m normal-incidence mounting (Hilger E760), was provided with a concave aluminized replica grating overcoated with Pt (56 x 96 mm), 600 line mm-l, blazed at 90 nm in the first order (Bausch and Lomb). The detector was an E.M.I. 9789QA photo- multiplier cell whose end (silica) window was coated with sodium salicylate as a phosphor. A photon counting system was used to detect and record individual photoelectrons.The photomultiplier cell showed a counting plateau at around 100OV e.h.t. overall, at which potential its dark count rate was about 20 Hz. However, it was found that with a reduction of e.h.t. to 850 V, the dark count rate of this particular photomultiplier fell to 2.5 Hz, whilst the corresponding photon count rate was still two-thirds of that observed at 1OOOV. It appears that, at a given e.h.t., the mean pulse height due to dark electrons was less than that due to photoelectrons originating at the photocathode. The photomultiplier cell was cooled with circulating water from room temperature, typically 299K, to that of the laboratory water tank (290 K), leading to a further useful reduction in mean dark count rate from 2.5 to 1.5 Hz at 850 V.This method of using the photomultiplier, leading to an unusually low dark count, was essential to the measurement of the very low photon count rates due to F atom fluorescence encountered in our work. The advantage of the low dark count of the II--7 VACUUM U.V. KINETIC SPECTROSCOPY E.M.I. 9789QA photomultiplier cell (1.5 Hz) used in this work should be stressed. Use of our previous "best choice " of cell (E.M.I. 6256SA) typically gave a dark count of 20 Hz, detectable with the 6256SAwith a slightly higher quantum efficiency. The minimum [F 2PJ] cell would have been >3x 10'l ~rn-~,a factor of three higher than that possible in the present work. RESULTS AND DISCUSSION F RESONANCE TRANSITIONS An analysis of the emission between 95 and 98 nm from a microwave-operated lamp using F2+He mixtures has shown that the major features are the nine multiplet lines of the lowest-energy 3s 2*4P-2p52P transition of F.Moore's energy level table for F (I) lSa is very fragmentary, but the more recent data of Lidh lSb and extensions and corrections due to Palenius lScpermit a fairly complete and reliable set of energy levels and transition wavelengths to be drawn up (fig. 2). -1/2 2p 2pY3P)4s32 2D -Y2,3/2 2p4('D)3s' 2P5OJ -FIG.2.-Atomic resonance transitions of F(1). The wavelengths and wavenumbers of the lowest energy resonance transitions of F(1). The data are those of ref. (18a)and (18b),corrected where necessary according to ref.(18~). Fig. 3 shows a spectrum of the main 3s-2p5 lines, obtained with the best available resolution of 0.04nm, together with the remaining (much weaker) F atom lines observed between 58.4 and 95 nm. After the 3s-2p5 multiplet, the next most intense feature was a doublet (A 80.70, 80.96 nm), due to the 3s'(lD) 2D*,s-2p5 2P3-and 3s'('D) 2Dj-2p5 2P+transitions (fig. 2). A very weak, but well-defined multiplet due to the 4s 2PJ-2p52PJtransition of F, between 79.00 and 79.44 nm, was also identified in fig. 3. P. P. BEMAND AND M. A. A. CLYNE The relative intensities of the 3s-2p5, 3s'(lD)-2p5 and 4s-2p5 transitions of F in the resonance lamp parallel those of the analogous 4s-3p5, 4s'(lD)-3p5 and 5s-3p5 transitions of Cl emitted from a similar C1 lamp.In fact, the background spectrum in the chlorine resonance lamp was too intense to permit unequivocal identification of the very weak 5s-3p5 lines of C1 near 109 nm. However, only the first 3s-2p5 multiplet transitions of F were intense enough for resonance fluorescence studies of F 2p5 'PJ atoms, unlike the situation for C1 3p5 2Pr atoms, where the 4s'('D)-3p5, as well as the 4s-3p5 transition, is intense enough for detection of C1 atom fluore~cence.~~ 3s1-q,5 2~ -2~ Fl 1 1 JLk MEASUREMENT OF F 2P~CONCENTRATIONS Methods based on rapid "titration "reactions l2 for the measurement of abso-lute fluorine atom concentrations have been reviewed [e.g., ref. (4) and references therein]. Reaction (2), k2 F+C12 -+ClF+Ci, (2) VACUUM U.V.KINETIC SPECTROSCOPY has been used for this purpose in mass spectrometric kinetic studies of F atom reac- tion~.~* Either the concentration of Clz removed, or the concentration of ClF f~rmed,~in reaction (2), was used to measure [F]. Reaction (2)proceeds at a rate approaching the hard-sphere bimolecular collision frequency at 298 K with kig8 reported to be 0.86 x 10-10,8 (1.1 k0.3) x 10-lo and (1.6k0.5) x lo-'' cm3mole-cule-' s-~.~Since the possible secondary step, C1+F2 3 ClF+F, is believed to be slow [kZg8< 5 x cm3 molecule-l s-I (ref. (13) and this work)], reaction (2) is a nearly ideal titration reaction for F. However, careful consideration of the nearly- balanced thermochemistry of reaction (2) is necessary.In the following discussion, the major reacting species is taken to be F 2P,,since F zP+ accounts for only 7 % of X[F] in a Boltzmann distribution at 298 K. Coxon l4 has reviewed the dissociation energies l6 of ClF, BrF and IF, and he concluded that DZ (ClF) definitely lies between the limits 246.7 and 252.4 kJ mol-I. As he pointed out, recent measurements of the equilibrium constant for reaction (2),' proceeding presumably mainly to C1 'P3,support the higher value of 252.4 kJ mol-1 for D;(ClF). A short extrapolation (from 306 IS)of the results of Nordine gave K2eq--(355k50) for reaction (2) at 298 K. The absolute minimum value for KzeQ, based on the (less likely) value D6 (ClF) = 246.7 kJ mol-l, is K,,, > (5.1 k0.8) at 298 K.Even this low value for Kzeqis sufficient to give %90 % reaction in studies, such as those cited,4* where a considerable excess of either F, or of Clz, was present [Br2] added/1012 cnr3 FIG.4.-F+Br2 titration for the measurement of [Fl. Plot shows the variation of F 2P~-2P~ fluorescence intensity with [Br2] added. Note that for complete reaction [q= [FIo-([Br2] added), where [F], is the abscissa intercept corresponding to the F+Br2 titration endpoint. in the F+Cl, reaction. However, in a conventional titration,12 where approximately equal initial concentrations of reagents are used to give an endpoint, i.e., [go-[Clzl0, the reaction (2)would proceed to only -70 % completion if Kze,is as low as 5.1. Reaction (l), ki F 2P3+Br, + BrF +Br ,P+, (1) is an alternative titration reaction for F, with k:98 reported to be (3.1 kO.9) x cm3 molecule-l s-~.~As in the case of DE (ClF), the upper and lower limits for D;(BrF) are well defined, in this case as 261.4 2 DE (BrF) 3 245.6 kJ rn~l-~.~~ Using D;98(Br,) = 192.9 kJmol-l, the absolute minimum value for Kleu is lo9 at P.P. BEMAND AND M. A. A. CLYNE 197 298 K. Reversibility of reaction (1) is therefore not a problem in F atom titrations with Br,. In this work, both reactions (1) and (2) have been used to measure absolute F atom concentrations. The intensity, IF,of F atom resonance fluorescence (see below) was used to monitor relative [F]. Usually, a simple titration of F 2Pr atoms with Br, was carried out, estimating the endpoint to be critical extinction of the F atom fluorescence. Since it was never practicable (for our kinetic studies) to use initial F concentrations less than 1 x 10l2 c1r3, and since kl is reported to be (3.1 k0.9) x 10-lo cm3 molecule-l s-~,~the typical reaction time of 10 ms would be sufficient to ensure >95 % completion of the F+Br2 reaction. In studies using relatively low [F], not much greater than 1 x 10l2 ~rn-~, it was advantageous to obtain endpoints by extra- polating plots of 1, against (added [Br,]) or (added [Cl,]).In these cases, within the admittedly considerable scatter of the results (fig. 4), no significant difference between endpoints measured by the F+Cl, and F+Br, reactions could be detected. F ATOM RESONANCE ABSORPTION Observation of F atom resonance absorption by both F2p5 “p, ground state atoms, and by J-excited F 2p5 2P&atoms, has been reported briefly by us previ~usly.~ It was necessary to use relatively high F atom concentrations (-1014 ~m-~) in the / I O.61 /I / 0.41 0.3! 0.21 [F 2P+]/10’3atom CM-~ FIG.5.-Variation of resonance absorption intensity with [F‘Jfor several transitions.(a) F zPs absorption at A: A, 95.48 nm and 0,95.19 nm; (6) F ’P+absorption at A 95.85 nm; (c) F 2P+ absorption at A 97.39 nm. Broken line in (b)is a Beer-Lambert law plot for the X 95.85 nm absorp- tion. resonance lamp in order to obtain adequate intensity of the fluorine atom resonance lines, which led to strong reversal of those lines having the highest oscillator strengths fik, i.e., the 3s 2Ps2p5 ,Pr multiplet. As expected, therefore, the sensitivity of reso- nance absorption for detection of F2p5 2Pr atoms was low.All three lines of the 2PJ-2PJmultiplet which were investigated showed resonance absorption [fig. 5(a), (b)]. VACUUM U.V. KINETIC SPECTROSCOPY TABLE1.-WAVELENGTHS AND TRANSITION PROBABILITIES FOR SELECTED (n+ 1)s-np5 TRANSI-TIONS OF F, CI, Br AND 1 F 3s c145 2P3 2pi 4p* 2pB 2P+ 4p4 95.483 95.187 97.390 134.724 133.572 137.953 104 731 105056 102 680 74 221 74 861 72 484 10-lc 2.33~10-Zc 3.1 x 10-3c--4.0 x 10-5b 1.14~ --3.5 x 105 4.19~108 1.74~108 1.1 X 107 rips 2P+ l/nm 95.852 95.555 91.775 136.345 135.166 139.653 (x??k-E$)/cnI-' 104 327 -104 652 102 276 73 340 13 980 71 603 1O-Zc 8.8~ lO-dC$.OX 10-sb 4.2~ lO-Zc 8.8~ffk Akils-1 --4.4~104 7.5 x 107 3.2~108 1.5X 106 Br 5s 165 __~ zpa 2p+ 4p3 2p3 2p+ 4p3 /Ips 2P$ Ilnm 148.861 145.004 154.082 178.276 158.361 161.760 (Ek--Ei)/cm-l 67 177 68 964 64 901 56 093 63 187 61 820 fLIE 6.1 x 10-zd 2.0~ 10-1 3.9 x 10-2 5.3 x 10-2 10-2 6.1 x 10-Zd 1.2~ 108 2.56~ 108"AktIS-' 1.8X 108 1.3 X loge 1.7~ logd 2.07~loge 1.34~ np5 2P+ I/nm 157.500 153.190 163.357 206.229 179.909 184.445 (&-Ei)/cm-' 63 492 65 279 59 745 48 490 55 584 54 217 10-3 1.0~fik 1.5 x 10-Zd 7.1 x 10-2 1.3 x 1O-zd 3.8~ 10-1 7.1 x 10-3 Aki IS-2.0~ 107 2.96~106c 2.11 x lose 6.92~10Gu107 2.0 x 108 1.7~ 0 transition energies from ref.(18b) ; b this work; C ref.(20), which is based on data of ref. (21) and (22) ; d mean vaIues from ref. (21) and (27) [see ref. (27) for discussion of data] ; e ref. (21). TABLE2.-KINETICS OF THE F+Brz REACTION AT 300 K [Fjo/lOl? cm-3 [Br&,j1012 cm-3 k'js-1 = -d In [F]/dr kl/lO-lO cn13 molecule-1 s-1 A B A B 1.2 8.75 420 514 0.51 0.74 1.6 6.1 1 264 434 0.48 0.79 1.o 3.12 181 269 0.66 0.98 1.3 4.58 690 895 1.58 2.05 1.3 4.06 474 693 1.34 1.96 1.4 3.46 366 552 1.26 1.90 1.4 2.53 320 468 1.64 2.40 1.2 2.88 404 474 1.68 1.97 1.2 3.51 359 535 1.18 1.76 1.5 4.39 312 476 0.82 1.25 1.7 4.80 417 615 1.01 1.49 1.4 5.1 1 327 536 0.72 1.18 1.3 3.95 308 486 0.90 1.42 1.2 4.62 375 536 0.91 1.30 1.1 5.01 385 577 0.84 1.26 1.3 5.45 379 515 0.4 7 1.25 1.o 6.07 284 409 0.50 0.72 mean values 0.90 1.44 A ;preliminary measurement of kl based on IF 0~ [F'P'].B ;measurement of klusing correction for seIf-reversal of F atom resonance fluorescence (see text). P. P. BEMAND AND M. A. A. CLYNE Fig. 5 shows that the 2P3-2P+transition (at 95.85 nm) showed a higher sensitivity to F ’P+atoms in resonance absorption than the sensitivity to F ”P3atoms of either the ’P3-”P+ transition (at 95.48 nm) or the ’P+-’P+transition (at 95.19 nm). Because the Boltzmann population of F ’P+atoms at the lamp temperature (N 600 K), [F ’P+]/ ([F ”P+]+[F’P3]),was only 0.18, the strong 95.85 nm ’P3-’P+line was less reversed than the strong 95.48 nm ’P+-’P+ line, hence leading to a relatively high integrated absorption coefficient for the 95.85 nm line.17 Plots of fractional absorption, A, against [F ’P+]are shown in fig.5(a). These plots showed a greater curvature for the 95.48 nm transition than for the 95.19 nm transition, and comparable sensitivities for both lines, again consistent with the expected self-reversal in the lamp. Table 1 shows values for the oscillator strengths for the analogous transitions of C1 and Br, from which ratios of oscillator strengths [f,k(2~~-2~~)]/~~~(’~+-2~~)]equal to 4.9 for Cl 2o and 3.9 for Br 21 are derived. This analogy with C1 and Br,20-23and simple theoretical consideration^,"^ thus suggest that the 2P+-2P+transition of F would have a higher oscillator strength (by a factor of about 4) than the ’P,-‘P, transition of F.This in turn is consistent with greater self-reversal in the lamp of the former transition. Similar conclusions follow from consideration of the relative emission intensities in the lamp (fig. 3) of the pairs of lines originating from the excited 3s ’P3and 3s ”P+ states and terminating on the ground 2p5’P2 and 2p5’P+states, i.e., 1(’P3-”P+)/ I(’P+-’P+) and I(2P+-2P+)/I(2P+-2P+).The only “pS2P+transition investigated, the 4P3-’P+ line at 97.39 nm, showed weak resonance absorption [fig. 5(c)], but the corresponding 4P+-2P+line at 97.77 nm showed no detectable absorption (A < 0.07 with [F ”P+]< 4 x 1013~m-~). OSCILLATOR STRENGTHS frk FOR F ATOM TRANSITIONS The absorption measurements may be used to derive values for the oscillator strengthshk of the 4P+-2P3,+pair of transitions as follows.The ratio of emission intensities from the F atom lamp, 1(4~~-2P+)/~(4P~-2P~)was about 6 (fig. 3) compared with values of 7 and 10 for the ratios of Einstein coefficients Aki for the same transi- tions of C1 and Br.19-23 Consequently, reversal in our lamp of the “P,-’P+, 97.39 nm, line of F is expected to be low ;and reversal of the ‘P%-’P+, 97.77 nm line should be negligible. For the 97.77 nm line, we thus assumed the source of resonance radiation to be a non-reversed Doppler line profile at a temperature of 600 K. (This tempera- ture is typical of values used previously 25 for low-power microwave plasmas of the type used here).A simple model 179 23 to convert integrated absorption, A, into the corresponding optical depth in the absorber, kJ, leads to the result for the 97.77 nm line, koZ < 0.15 for N = [F ’P4] = 4 x 1013~m-~.With I = 3 cm and AvD = 0.29 cm-1 for the 97.77 nm line absorber, we deduced the upper limit Ak< 4 x for the 97.77 nm line of F, using eqn (I): f = +AvD(n/ln 2)3(mc/ne2)(k0/N). (1) A similar procedure with similar assumptions, was used to analyse the absorption data of fig. 5(c) for the 97.39 nm ‘P3-’P+ transition of F. This resulted in the values kol/N= 3.0 x cm3 and J;’k = 3.3 x for this transition. Because of self-reversal in the lamp, this value offik is a lower limit. Arguments presented above show that self-reversal of the 97.39 nm line is low.To estimate the magnitude of this self-reversal, it is assumed that the ratio of Einstein coefficients Aki for 97.39 nm and 97.77 nm is 8 (the mean of the ratios quoted above for C1 and Br). The experimental lamp intensity ratio for these lines is 6, which then leads, via the usual mode1,17 to an optical depth in the source kom 21 0.5. When this value, kom 2: 0.5, is used in the model used to obtain koZ from A, the slightly higher values kol/N = 3.5 x VACUUM U.V. KINETIC SPECTROSCOPY cm3 and fik = 4.0 x lo-’ were obtained for the 97.39 nm ‘P*-’P+ transition of F. The value& = 4.0~lo-’ clearly cannot be regarded as an accurate measurement, but it would be most surprising if its magnitude differed from the true value by more than a factor of 2.To estimatefi7-77 we take, as before, A,9i7*39/Az{-77= 8, i.e.,f27.3g/hz7.77 This leads to the re~ultfi;~.~~ = 4. = 1.0 x considerablylower then the upper limit value of 4 x lo-’ deduced at the beginning of this section. In the determination of oscillator strengths from absorption measurements, it is necessary to consider the hyperfine structure of the resonance lines. In the case where the hyperfine components have a wavenumber spacing greater than, or comparable to, the Doppler width of the line, AvD, neglect of hyperfine splitting leads to a significant under-estimate 0ffik.” Fig. 6 shows the nuclear hyperfine structures of the 97.39 nm 5,~* 5!2 A 97.390 h 97.775 II I I I I I I I 0 0 I 0.2 0.3 0 0.1 0.2 0.3 0.4 %/cm-’ FIG.6.-Nuclear hyperfine structure of the 3s 4P+-2p5 “P+,+transitions of 19F.Wavenumber split- ting data for ground states from ref. (30, 31) and for excited state from ref. (186). Relative intensities within multiplets from ref. (30) on the assumption of a Russell-Saunders coupling scheme for the states involved. The line centres are shown in the figure. and the 97.77 nm lines of 9F, which have been calculated from the results of LidCn,l 8b Harvey 28 and Radf‘ord, Hughes and Beltran-L~pez.~~ Each energy level of 19Fis doubled (F = J++) on account of the nuclear spin (I = *). The 4P3-2P+(97.39 nm) transition is a quartet consisting of two close pairs of lines; the overall width of the quartet, 0.168 cm-l, is less than half the Doppler width of 19Fat 600 K (AvD = 0.412 cm-l).The overall width of the quartet is also less than AvD at 300 K for the absorb- ing F ’P+atoms. In view of the approximate nature of the measurements offik here, no correction tofik (97.39) for hyperfine structure is thought to be necessary. On the other hand, the overall width, 0.375 cm-l, of the triplet formed by the 4P+-2Pp,(97.77 nm) transition, is comparable to AvD even at 600 K. Consideration of the wave- number separations and relative intensities 30 of the components of the triplet (fig. 6) shows that the 97.77 nm transition for the 300 K absorber will appear as a partially resolved doublet with relative intensities of 3 :1. However, sincefi, (97.77) was esti- mated from the relevant emission intensity ratio, rather than from line absorption, no correction to the value A, (97.77) = 1.0 x is required.These values offikfor the ‘P+-’P+,+ transition of F appear to be the first published data. They fit in well with the trend established for the 4P+-2P3,ttransitions by the P. P. BEMAND AND M. A. A. CLYNE 201 series Cl, Br, I (see table 1). Due to increasing breakdown of the spin selection rule as a function of increasing atomic mass,24 the fik values for the spin-disallowed (n+1)s'P3-np5 'P+,+transitions increase markedly (by -lo3 to 10') in going up the series from F to I. The magnitudes of fik for the 3s 'P4p52P+,+transitions of F are similar to those for the spin-forbidden (n+ 1)s 'S-np4 3PJtransitions of sulphur, and greater than those for the same transition of oxygen,20 as would also be expected.hk for the 3s4P$p5 'P+transition is between lo3 and lo4 times less than that expected for the 3s "P3-2p5 2P+transition (i.e., about 0.1-OS), and the radiative life- time for the 3s 'P3 excited state is deduced from our fikdata to be 2.3 (us. The rela- tively high emission intensity from the lamp of the 3s 'PJ-2p5 2PJmultiplet, compared with the 3s 2PJ-2p5 2PJmultiplet (fig. 3), is surprising, in view of the presumed suscep- tibility of the long-lived 3s4PJ states to electronic quenching. In that context, we note that the radiative lifetimes of the 'P+ and 4P3states of the 3s configuration are likely to be considerably longer than the value of 2.3 ps estimated for 'P+.Analogy with the 4s 'PJ states of Cl 20* 22 suggests lifetimes of the order of 10ps for F 3s 'Pt and the order of 100 ps for F 3s 4P3. Nevertheless, emission from both these excited states in the lamp was quite intense (fig. 3). It appears that for F, as for C1 and Br,199 23 formation of the lowest energy 'PJ multiplets of the ns configuration is strongly preferred in microwave excitation with He carrier gas. For C1 and Br, a similar propensity to populate the lowest energy atomic states is found in the collision of He 23S1metastable atoms with C1, and Br2.31 F ATOM RESONANCE FLUORESCENCE A preliminary survey showed weak resonance fluorescence due to both the 2Pp2PJ and the 4Ps2PJ multiplets of F. The intensity of the fluorescence (with [F 2P+]-5 x lOI4 ~m-~)was then maximized by increasing the microwave lamp power to 70 W, and by increasing the flow rate (mol fraction) of F2 in the F2+He lamp.These changes obviously led to greater self-reversal of the F atom lines emitted from the lamp than those shown in fig. 3 and used for the resonance absorption studies. Fig. 7 shows the spectrum of F atom resonance fluorescence, excited in a concen- tration of F 2PJ atoms of 3 x 1014~m-~,with resolution of 0.2 nm (100 pm spectro- meter slits). As expected from the resonance absorption results and the above discussion offik values, the major intensity of resonance fluorescence was emitted in the 2Ps2PJ multiplet, with the 2P+-2P+95.85 nm line as the most intense single feature (N 11 Hz of detected photoelectron flux rate).The relative weakness of the strong 2P+-2P+ 95.48 nm transition in resonance fluorescence (fig. 7) is noted. This is un- doubtedly due to extensive self-reversal of 95.48 nm resonance fluorescence at [F 'PJ]= 3 x lo1' ~m-~, since the ratio of Einstein coefficients for spontaneous emission, Ak,(95.48)/Aki(95.85),is of the order of 5-10 (by analogy with C1).20 The relative intensities of transitions in fluorescence broadly parallel those in absorption, with little light being emitted in the spin-forbidden 'Pr2PJ multiplet. Fig. 8(a) and (b) show the dependences of the weak fluorescence intensities, IF,at 95.85 nm and 97.39 nm upon [F 2PJ],using the F+ C12 reaction to determine [F 2PJ]. The detected photo- electron flux rates were too low, with [F2PJ]< 5 x 1013~m-~,for systematic measure- ments.Because of their weakness, the 4PJ-2PJmultiplet transitions were not investigated further. Attention was directed to maximizing the detected fluorescence in the 2PJ-2PJ transitions, by increasing the spectrometer slit width to 1mm, centred on the 95.85 nm line. This spectrometer band width, about 2 nm, now allowed transmission of fluorescence due to the entire 2PJ-2PJmultiplet. Fig. 8(c) shows the resulting VACUUM U.V. KINETIC SPECTROSCOPY n II I __ I 95 96 97 98 Xfnm FIG. 7.-Spectrum of F atom resonance fluorescence. [F ‘PJ]= 3 x 1014 ; 100 pm slits. Observed photon flux rate of 95.85 nm line = 11 Hz. Note the relatively high intensity of the “PS-~P*97.39 nm Iine, due to very strong reversal (and Ioss of intensity) of the 2P~-2P~lines with [Fl = 3 x 1014CM-~.150 6 2 100 50 [F]/l0l4~m-~ FIG.8.-F atom resonance fluorescence. Variation of IF with [F ‘PJ]for various transitions. (a)A 95.85 nm, 3s 2~;-2p5 *P& ; (6) A 97.39 nm, 3s “P3-2~~ ’PJ muItipIet.”P+; (c) entire 3s ’Pj-2~~Inset shows measurements at low [F ’PJ]. variation of I, with TF 2P,lover a wide concentration range (0.01 to 6.3 x lOI4 ~m-~). again using the F+ C1, titration reaction to measure TF 2P.rl. The dot ofI’ against-Y[62PJ][fig:S(c)] shows an apparently linear range at-the lowest [F ;PJ],and a maxi-.I. I *,.I 1, 1 . dnrrr . h-----a A *allN 4 x IW--cm-. 7mum cietectea nuorescence count rare equal to 133 HZ at Lr &uJl Above this concentration, IF decreased slightly with increasing [F2P,].This beha- viour is analogous to that found previously for oxygen atom resonance fluorescence, P. P. BEMAND AND M. A. A. CLYNE 0 3s 3S1-2p43PJ.17 It is consistent with the onset of self-reversal of fluorescence at relatively low [F 2PJ], followed by radiation trapping and quenching of excited atoms at high [F 2PJ].17 In order to establish the variation of IF with [F 2PJ]at lower concentrations (<1 x lo1 ~rn-~),the background count rate (dark count +scattered light count) was reduced from 4 Hz to 2.5 Hz by cooling the photomultiplier (see Experimental). Also, the F+Br2 reaction was used to measure [F 2PJ], for the reasons mentioned earlier regarding measurements of low F atom concentrations. This calibration of IFagainst [F2PJ] [inset of fig.S(c)] was carried out in the concentration range 1.3 x lo1, < [F 'PJ] < 9.4 x 10l2~m-~.The onset of curvature on the plot [fig. 8(c)] is difficult to locate because of the inevitably large statistical scatter of the low counts measured, but it appears to occur below 2x 10l2CM-~,as would be expected (see below). The lowest concentration of [F 2PJ] that could be measured in this way by reso- nance fluorescence was limited by the magnitude of the corresponding photoelectron count rate. The detection system showed evidence of only infrequent interference spikes. In the absence of such spikes, and using an integrating time of 50 s to count photoelectrons, the minimum measureable count rate (with signal-to-noise of unity) was 0.2 Hz.The corresponding F 2PJatom concentration was -7 x 10" CM-~,so a conservative lower limit of detection would be [F 'PJ]> 1 x 10'' ~m-~.We empha- size that this limit could be severely degraded upwards by appreciable rf interference or by drift in the photomultiplier dark count due to temperature variations in its photocathode, which were minimized in our work (see Experimental). ki RATE CONSTANT kl FOR THE REACTION F+Br, + BrF+F Detection of F 2PJatom by resonance fluorescence was used to measure the rate of their reaction with molecular bromine. Because of the high magnitude of kl and because of the limited sensitivity of Fatom resonance fluorescence, the range of condi- tions for the kinetic study was limited.The range of initial [F 2PJ]was 0.98 to 1.70 x 10l2 ~1n-~, The [Br210/ and initial [Brtlo was between 2.5 and 8.8 x lo1, ~m-~. [F 2P,]0stoichionietry ranged between 1.8 and 7.3 with a median value of 4. Pseudo first-order kinetic analysis was used, eqn (11), with correction for Br2 consumption during the reaction, ~n([Flo/[FI!= M[Br2lmean)t. (11) To obtain values for kl,the assumption was made initially that 1, wz [F 'P,]over the F atom concentration range used ;this assumption is considered below. Fig. 9 shows typical first order decay plots for attenuation of F atom resonance fluorescence in the presence of Br,. The scatter of the results, due to the low count rates measured, was much greater than that observed in similar C1 atom kinetic studies using resonance fluorescence.lg 17 independent runs were made and the results are summarized in table 2.Data reduction was carried out computationally, with a least mean squares fitting program, which allowed for any drift (usually <10 %) in initial [F ,PJ]concentration. The mean value obtained for kl,on the assumption of propor-tional dependence of I, upon [F 2Pr], was (0.9k0.2) x 10-lo cm3 molecule-l s-l at 300 K. Several of the experimental difficulties have already been referred to, includ- ing the presence of 03PJatoms whose concentrations are estimated to be -0.1 [F,PJ]. Removal of these 0 3P, atoms by addition of NO2 as scavenger was found to be not feasible, because addition of NO2 considerably increased reaction of F atoms with the walls.Since 0 3PJ atoms react fairly rapidly with Br,, it is possible that our value of kl,which neglects such reaction, is a slight underestimate. VACUUM U.V. KINETIC SPECTROSCOPY A more serious cause for underestimation of kl is self-reversal of resonance fluorescence. The following considerations show that such self-reversal is not negli- gible, even in the range of N 10l2 ~m-~ used for typical [F 2PJ]0. The dominant transitions contributing to the detected fluorescence are the 2P+-2P+line at 95.48 nm and the 2P3-2P+line at 95.85 nm (cf. fig. 6). Since Aklfor the 95.48 nm line is prob-ably between five and ten times greater than that for the 95.85 nm line (see above), the 95.48 nm line contributes the dominant intensity at sufficiently low F 2PJ con-centrations.As [F 2PJ] is increased, the 2P+-2P+ 95.48 nm line becomes strongly self-reversed while the 2P+-2P+95.85 nm line is little affected, because [F 2P+]/([F 2P+] + [F2P+]) is only 0.07 at 300K. Hence, at high [F 2PJ],eg, about 1014cm-3, the 95.85 nm line is much more intense than the 95.48 nm line in resonance fluorescence. This simple concept was used to set up a model for the variation of IF with [F 2PJ] in the concentration range <2x 10l2 ~m-~, assuming that the only emitter was the F 3s 2P+state referred to above. The further assumption was made that the oscillator strengthshk for the 2P3-zP3,+ lines of IF were the same as those for the similar transi- tion of C1 (see table 1 for values).Clyne and T~wnsend,~~ among others, have described a method for relating integrated line absorption, with an optically-thin Doppler source, to the corresponding value forfik. The same computer programs 23 // I I I I I I time/ms FIG.9.-Kinetics of the F+Br2 reaction using fluorine atom resonance fluorescence. Typical data for F atom decay are shown in the form of a first order plot. [Br& = 4.39x loi2CM-~ ;[ao= 1.51 x 1OI2CM-~. 0 and broken line show results of preliminary analysis, assuming IF cc [F'PJ](see text). 0 and full line show results of final analysis allowing for reversal of fluorescence (see text). were used to form a polynomial function relating IF to [F 2PJ].This polynomial function did not approximate well to a straight line above [F 2PJ]= 3 x 10l1 ~m-~, indicating that correction for self-reversal to IF at higher values of [F2PJ]was indeed required. The polynomial was used in an iterative routine to obtain improved values of [F 2PJ]corresponding to each measured IF. The new values for [F 2PJ]for each kinetic run were then reanalysed computationally, as before, to give first order (-d In [F]/dt) and second order (kl)rate constants. The results are given in table 2. The mean value for k, from the improved data reduction procedure was (1.4k0.3) x cm3 molecule-l s-l. The main error in this corrected value for kl was introduced by the uncertainty in the oscillator strength for the 95.48 nm line, F 2P3-2P3.Therefore, the calculation P. P. BEMAND AND M. A. A. CLYNE 205 of kl allowing for self-reversal of resonance fluorescence, was rerun usingf,,(95.48) = 0.228 andf,,(95.85) = 0.084, i.e., exactly twice those used in the above calculation which gave kl = (1.4f0.3) x 10-lo cm3 molecule-l s-l. The mean value for kl, using f,,(95.48) = 0.228 and f,,(95.85) = 0.084, was determined to be kl = (1.6+ 0.4) x 10-lo cm3 s-l. The value for kl, corrected for self-reversal, there- fore is not sensitively dependent upon the magnitude Offik. The most probable value forf,k(95.48) is considered to be around 0.1 to 0.15, so the best estimate of k, that allows for some additional error due to the uncertainty in f,,(95.48) is kl = (1.4+ 0.5) x 10-lo cm3 molecule-l s-l at 300 K.This value for kl may be compared with that reported by Appelman and Cl~ne,~ using mass spectrometric analysis in a fast flow system to measure the rate of consump-tion of Br, in the presence of excess F; k, = (3.1kO.9) x lo-'' cm3 molecule-l s-l at 298 K. As in the present study, the extreme rapidity of reaction (1) severely restricted the maximum stoichiometry [F]o/[Br2]o that could be used in their study.9 On the other hand, the problem of irreproducible surface reaction of F at the flow tube walls (see below) led to less difficulty in the mass spectrometric work. This is because F atom concentrations, and their decays, were determined directly through the reaction zone using mass spectrometric measurements.The presence of 0 3PJ stream would have led to an overestimate of kl in the mass spectrometric study, by accelerating the decay of [Br,]. However, the same impurity leads to a drop in -d In [F]/dt in the resonance fluorescence study. These considerations suggest that the best estimate of kl at present is obtained by taking a mean of the two determina- tions, giving equal weighting to the mass spectrometric study and to the present study. The result of this procedure is kl = (2.2 1.1) x lo-'' cm3 molecule-' s-l at 298-300 K. The most useful improvement in the resonance fluorescence method for kinetic studies of F 2PJatom reactions would be reduction of wall reaction of F. The sensi- tivity of resonance fluorescence for F is sufficiently high for direct pseudo first-order kinetic studies of F atom reactions whose rate constants are as high as 5 x 10-l1 cm3 molecule-l s-'.Such reactions would include F +H,, F+CH4 and other significant elementary processes concerned with the HF chemical laser. However, rather rapid loss of F 2PJatoms at the silica walls was found in our work. The rate constant for this loss process was found to be markedly affected by addition of moderate concentra- tions (>1013 ~m-~) of many reactants (particularly NOz, O,, 03).Pseudo first-order kinetic analysis [eqn (I)] is only valid when first-order wall loss is unaffected by the molecular reactant, as in the F+ Br, reaction where very low concentrations of Br, were added. Future work in this laboratory will be directed (typically 4 x lo1 ~m-~) towards replacing the silica surface of the flow tube by an inert teflon film,32 in order to reduce (and stabilize) surface loss of F 2P atom^.^ We thank David Gutman for introducing us to collimated hole structures.We thank the S.R.C., the Ministry of Defence and the Central Research Fund Committee of London University for support. A. Carrington, D. H. Levy and T. A. Miller, J. Chem. Phys., 1966, 45,4093. 'K. H. Homann, W. C. Solomon, J. Warnatz, H. G. Wagner and C. Zetzsch, Ber. Bunsenges. phys. Chem., 1970,14, 585. C. E. Kolb and M. Kaufman, J. Phys. Chem., 1972,76,947. M. A. A. Clyne, D. J. McKenney and R. F. Walker, Canad.J. Chem., 1973,51, 3596. P. P. Bemand and M. A. A. Clyne, Chem. Phys.Letters, 1973, 21, 555. D. E. Rosner and H. D. Allendorf, J. Phys. Chem., 1971, 75, 308.'T. L. Pollock and W. E. Jones, Canad.J. Chem., 1973,51,2041. J. Warnatz, H. G. Wagner and C. Zetzsch, Ber. Bunsenges.phys. Chem., 1971, 75, 119. VACUUM U.V. KINETIC SPECTROSCGPY E. H. Appelman and M. A. A. Clyne, J.C.S. Faraday I, 1975, 71,2072. lo P. P. Bemand, M. A. A. Clyne and R. T. Watson, J.C.S. Faraday Z, 1973, 69,1356. l1 P. P. Bemand, M. A. A. Clyne and R. T. Watson, J.C.S. Faraday I, 1974,70,564. l2 See, for example, M. A. A. Clyne in Physical Chemistry of Fast Reactions, ed. B. P. Levitt (Plenum Press, New York, 1973), vol. 1, 245. l3 G. Schatz and M. Kaufman, J. Phys. Chem., 1972, 76, 3586. l4 J. A. Coxon, Chem. Phys. Letters, 1975, 33, 136.l5 P. C. Nordine, J. Chem. Pliys., 1974, 61, 224. l6 JANAF Thermochemical Tables, NSRDS-NBS 37, 2nd edn., 1970. l7 Similar considerations for 2p4 'P.7 oxygen atoms have been discussed by P. P. Bemand and M. A. A. Clyne, J.C.S. Fnraday IZ, 1973, 69, 1643. (a) C. E. Moore, Atomic Energy Levels, vol. 1, NBS Circular 467, 1949 ; (6) K. Lidkn, Arkiv Fysik, 1949, 1, 229 ; (c) H. P. Palenius, Arkit, Fysik, 1969, 39,425. l9 P. P. Bemand and M. A. A. Clyne, J.C.S. Faraday ZI, 1975, 71, 1132. 2o W. L. Wiese, M. W. Smith and B. M. Miles, Atomic Transition Probabilities, vol. 2, NSRDS-NBS 22, 1969. 21 G. M. Lawrence, Astrophys. J., 1967,148,261. 22 W. Hofmann, 2.Naturforsch., 1967, 22a, 2097. 23 M. A. A. Clyne and L. W. Townsend, J.C.S. Faraday ZI, 1974, 70, 1863. 24 H. G. Kuhn, Atomic Spectra (Longman, London, 2nd edn., 1969). 25 See, for example, D. A. Parkes, L. F. Keyser and F. Kaufman, Astrophys. J., 1967,149,217. 26 A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms (Cambridge Univ. Press, 1971). 27 M. A. A. Clyne and J. Tellinghuisen,J.C.S. Faraday ZZ, in press. 28 J. S. M. Harvey, Proc. Roy. SOC. A, 1965, 285, 581. 29 H. E. Radford, V. W. Hughes and V. Beltran-Lopez, Phys. Reu., 1961,123, 153. 30 E. U. Condon and G. H. Shortley, The Tlzeory of Atomic Spectra (Cambridge Univ. Press, 1951). 31 L. A. Gundel, D. King, W. S. Nip, D. W. Setser and M. A. A. Clyne, to be published. 32 H. C. Berg and D. Kleppner, Rev. Sci. Zmtr., 1962, 33, 248. (PAPER 5/882)

ISSN:0300-9238    

DOI:10.1039/F29767200191

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Rotational Analysis of the 2A’-+2A”Emission Band System of HOa at 1.43 /lrn BYPHILIPA. FREEDMANand W. JEREMY JONES* Department of Physical Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EP Received 28th May, 1975 The HOz 2A’ + ZA”emission band at 1.43 pm has been recorded at a resolution of 0.3 cm-I by use of a SISAM interferometer. Only K’< 2 levels are found. A full spectroscopic study has been carried out and rotational constants obtained for the ‘A’ and ’A’’ states, the latter being in agreementwith those recently obtained using the laser paramagnatic resonance technique. The 1.51 pmemission band previously assigned to HOz is found to be due almost entirely to the OH radical. In recent years there has been considerable interest in the study of free radicals using the technique of laser paramagnetic resonance (LMR).This technique is of high sensitivity and can often give precise values for the rotational constants of small molecules. Among the free radicals detected by this method is HOz, for which Radford, Evenson and Howard 1* were able to detect a number of pure rotational transitions and hence a set of ground state rotational constants. This radical was also detected by Becker, Fink, Langen and S~hurath,~ who reported the observation of some infrared emission bands using the reaction of discharged oxygen with various compounds, similar bands having since been observed in the H + O2 ~ystem.~Hun-ziker and Wendt also using the H + O2reaction have succeeded in recording a num- ber of these bands in absorption.Laser paramagnetic resonance studies, although providing information of great accuracy on the ground states of molecules, provide little or no information on excited vibrational and electronic states. Such information is present, however, in transitions between different vibronic levels and where the detailed structure of such bands can be resolved significant insight into the electronic binding can be obtained. The development in this laboratory of a single pass SISAM spectrometer and its coupling to an intrinsic germanium detector has provided us with an extremely sensitive tool for the study of emission bands in the 1.2 to 1.6 pm region and it was decided to try to record some of the HO, emission bands at a much higher resolution than was previously possible.Only the 1.43 pm and 1.51pm bands were intense enough for study at a resolution of -0.3 cm-1 and it was soon realised that fundamental dif- ferences existed between these bands observed in emission and those seen in absorption. EXPERIMENTAL HO, was formed in a discharge flow system by the reaction of ethylene or ammonia with discharged oxygen at a total pressure of 3-4 The reaction took place within a 2 litre gold coated integrating sphere, the emitted radiation being focused onto the entrance aperture of a SISAM interferometer. This instrument was used in two modes : (i) conven- tionally, using a circular entrance aperture of 9 mm diameter to give a resolution of 0.3 cm-l ; (ii) by chopping the input signal and making use of the diffraction gratings necessarily present 207 I.R.EMISSION OF HOz in this form of interferometer. With the path difference between the two arms of the inter- ferometer held constant the instrument behaves as a conventional slit spectronieter and the spectrum is monitored by recording the ax. component of the emergent radiation. The resolution in this mode could be changed by replacing the circular apertures by slits aligned parallel to the grating grooves. The low resolution spectra were obtained by this method. RESULTS SPECTRA RECORDED UNDER LOW RESOLUTION CONDITIONS The low resolution spectra are shown in fig. 1. The first of these traces displays the emission spectrum obtained from the ethylene +discharged oxygen reaction at a resolution of -200 cm-l, to compare with previous studies.3* The same spectrum at a resolution of -30 cm-l is shown in fig. lb.The sharp high frequency edge of the 1.43 pm (-7000 cm-l) band and the structure in the 1.51 pm (6 650 cm-l) band should be noted and compared with the corresponding bands seen in absorption at about the same res~lution.~ Such a comparison shows that the spectra recorded in absorption are very much more extensive than those displayed here, the 1.43 pm band in particular showing no evidence for a sharp cut-off at the high frequency end. Wclm FIG.1.-Near i.r. spectra of the products of the reaction of discharged oxygen with (a) ethylene ; (b) ethylene ; (c) molecular hydrogen ; (d) NH3, in excess ; (e) ND3 ; (f) nothing.Resolution is 30 cm-1 except for (a) where it is 200 cm-l. In fig. lc the spectrum obtained by reacting molecular hydrogen with the products of the oxygen discharge is exhibited. Within this band the sharp features in the 1.4-1.5 pm region are identified as rotational transition of the v' = 2 -+ v" = 0 (conven-iently abbreviated to 2-0 throughout this paper) and 3-1 emissions bands of the OH P. A. FREEDMAN AND W. J. JONES radical. High resolution studies of this band found no trace of H02. This strongly suggests that the 1.51 pm band seen in emission is composed mainly of the 3-1 OH overtone emission (in fact the peak of the low resolution spectrum corresponds to the Q branch of this band) and that the 1.43 pm band is heavily polluted with 2-0 OH emission.This is confirmed in fig. Id in which the spectrum of H02 found by reacting ammonia with discharged oxygen is shown. If the ammonia concentration is in excess of that necessary to produce the maximum signal, the OH emission is quenched, leaving the underlying HOz spectrum. Unfortunately the total intensity of H02emission is markedly less in this case than in the reaction of discharged oxygen with ethylene so that the former source could not be used for the high resolution spectrum. Fortunately the OH emission lines thus present in our spectrum did not affect the analysis significantly. In fig. le is shown the corresponding low resolution spectrum obtained using deuterated ammonia and in fig.If the spectrum of discharged oxygen with no additive to show the O21As-3Xi band. SPECTRA RECORDED UNDER HIGH RESOLUTION CONDITIONS Comparison of the low resolution spectra with those obtained in absorption strongly suggests that many of the levels in the excited state of H02 are efficiently quenched and do not emit. The high resolution spectra, obtained after much experi- mental difficulty because of the low intensity of the emission, bear this observation out. The spectrum is shown in fig. 2 together with the proposed assignment. The symmetry of the nuclear framework of H02 clearly belongs to the C, point group, states being of A' or A" symmetry with respect to the plane of the molecule. The transitions in the near infrared are considered to be 2A'-2A" in nature 39 5* and, as found from the LMR study, one would expect to observe a small spin-splitting with a quadratic K dependence.In carrying out much of the assignment and analysis of the spectrum, the formulae of Polo for a slightly asymmetric top proved invaluable. These equations are fairly long and since they are easily accessible both in Polo's original paper and in Herzberg l3 they will not be repeated here. Since the 1x1 value for both states of HOz is >0.997, these energy equations converge very rapidly and enable the analysis of each K sub-band to be carried out independently of the others to give "effective " rotational constants which may be related to the true molecular parameters via Polo's formulae.As an example of their use consider the K' = 1 + K" = 0 Q branch. This may be fitted to an equation of the form v = VTb-CLeffN(N- 1)-AD,,N2(N+ 1)2, (Y1) where N is the quantum number for the total angular momentum excluding spin. Use of Polo's formulae then gives CL,ff = A(B"-C")+*(a, +bc), and ADeff = QP'-$P"+ADJ, (Y3) where Pis written for (B-C)2/16[A-t(B+C)] and ADJ = 0;-DI;. This separation into sub-bands proved necessary not only in the initial assignments but also in locating weak perturbations (see below). As may be appreciated from fig. 2, the assignment of the spectrum proved to be a non-trivial operation. The most distinctive feature is the extensive Q branch starting at 7049 crn-l, another starting at 7010 cm-1 being less prominent.These two Q branches have been assigned to the 1-0 and 0-1 sub-bands respectively, the former I.R. EMISSION OF HOa being rather anomalous in intensity. The location of the associated P and R structure proved difficult to determine, only a weak R head being assigned to the 1-0 sub-band whilst a fairly complete P/R assignment was possible in the other case. Our original v/cm-' 7075 7049 7028.7 7013 I I I 1 X I I Y t iGz I -O Q W O-IR w 7013 7000 6986.5 6971.4 6953 I I I I I (4 6953 6935 6911.2 68875 6866I 1 I 1 I FIG.2.-High resolution emission spectrum of H02 ('A' --f 2A"): (a) 7075 to 6953 cm-' region ; (b) 6953 cm-' to 6865 cm-l region. The hot band assignment should be regarded as tentative. Lines marked x are due to OH emission.assignment was found to be in error by one unit in N by means of the ground state LMR constants which became available as this work was being completed. Using these data it was then possible to assign part of the 1-2 Q branch, although only P. A. FREEDMAN AND W. J. JONES 21 1 transitions originating from the lower K’ = 1 level appear to be present. This is, of course, the same component involved in the 1-0 Q branch transitions. No spin splitting could be identified in any of the observed lines. Similarly no transitions originating from K’ = 2 could be identified. The data were analysed in three ways. (a) Each band was analysed in isolation of the others to give “ effective ” con-stants, as mentioned earlier.Although this method was not used in obtaining final sets of constants it proved invaluable for the assignment and the following points should be noted. (i) The 1-0 and 1-2 Q branch data were only consistent if the analyses were carried out over the range of N values common to both bands. This, in practice, meant the exclusion of N 3 26 lines from the 1-0 fit. If N < 33 lines were used in the 1-0 Q branch, although the r.m.s. standard deviation of the fit remained low (0.05 cm-l) definite systematic errors were present in the residuals. All the 1-0 Q data (up to N = 42) produced a very poor quality fit, these final points showing considerable perturbation. (ii) The 0-1 Q branch fit strongly suggests a weak perturbation at about N’ = 10.This is presumably not unconnected with the weakness of the R branch head, which also occurs at about this N’ value (see below). (b) The onlyP/R branch lineswhich we were able to identify positively originate from K’ = 0. Since these obviously share a common upper level with the corresponding 0-1 Q branch lines one may obtain ground state combination differences of the form Q(N)-P(N+ 1) = EY(N+ l)-E:(N) and R(N-1)-Q(N) = Ek(N)-Ey(N-l), (Y3) where Ey(N)is written for the energy of the upper component of the K = 1, N = N level of the 2A”electronic states, and so derive ground state constants. In the least squares fit a direct matrix diagonalisation was used as well as the much simpler (and faster) solution of the appropriate Polo formulae.The results obtained, which were identical, are B” = 1.1161+0.0017 cm-l C” = 1.055,$_0.0017cni-l DI;= (1.4k2.0) x cm-l. Since only combination differences within K” = 1 were obtained, the data showed only the smallest dependence on A”which was fixed at the LMR value of 20.358 cm-*.’ These results are seen to agree very well with the LMR values of B” = 1.1179+0.0005 cm-l C” = 1.0567$-0.0005cm-l D)’N - (4.2k0.8) x lop6 CITI-’. (c) Since it had been shown from the separate sub-band analyses that the upper state is perturbed, a global fit of all the fundamental data had to be carried out with care. Only lines with N’ < 25 of the 1-0 Q branch were employed [followingsection (a)], and constraining the ground state constants to the values found in the LMR study we obtained V~~(~A‘-2A”)= 7029.764 k0.03 cm-l ctA(=AN--A’)= 0.006+0.037 cm-l ccB(=B”-B’) = 0.097 3$-0.000 2, cm-‘ a,( =C”-C’)= 0.088 8 0.000 25 cm-l ADN(=Df;-Dh)= (-2.8k0.6)~ cm-’.The standard deviation of the fit for the 98 lines employed was 0.09 cm-l. 212 I.R. EMISSION OF HOz TABLE1.-MEASURED LINEPOSITIONS fundamental 2A'+ 2A"transition 1 + 0 Q branch N" positionlcm-1 obs-calc. a N" positionIcm-1 obs calc.0 5 7045.995 0.054 24 6987.3761 0.050 6 7044.554 -0.117 25 6982.332 -0.01 8 7 7043.169 -0.023 26 6977.112 b 8 7041.455 -0.049 28 6966.166 b 9 7039.572 -0.036 29 6960.5 1 1 b 10 7037.515 0.009 30 6954.600 b 11 703 5.236 0.039 31 6948.526 B 12 7032.724 0.039 32 6942.3 18 b 13 7030.034 0.063 33 693 5.8 17 b 14 7027.064 0.008 34 6929.398 b 15 7023.962 0.020 35 6922.703 b 16 7020.674 0.044 36 691 5.902 b 17 7017.182 0.058 37 6908.771 b 18 7013.436 0.010 38 6901.457 b 19 7009.580 0.044 39 6894.032 b 20 7005.507 0.048 40 6886.365 b 21 7001.168 -0.029 41 6878.513 b 22 6996.7 80 0.029 42 6870.5 3 6 b 23 6992.133 0.006 0 1 Q branch 6 7007.216 -0.009 24 6962.928 0.014 7 7006.169 0.036 25 695 8.768 -0.092 8 7004.944 0.059 26 6954.600 -0.028 9 7003.492 0.013 27 6950.176 -0.041 10 7001.720 -0.144 28 6945.553 -0.073 11 7000.137 -0.053 30 6935.8 17 -0.073 12 6998.199 -0,109 31 6930.763 0.022 13 6996.202 -0.062 32 6925.264 -0.138 14 6994.043 -0.017 33 6919.770 -0.100 15 6991.681 -0.012 34 6914.175 0.032 16 6989.164 O.OO0 35 6908.209 -0.009 17 6986.457 -0.01 3 36 6902.035 -0.056 18 6983.61 1 0.o00 37 6895.63 3 -0.126 19 6980.600 0.014 38 6889.210 -0.011 20 6977.377 -0.015 39 6882.549 0.077 21 6974.069 0.039 40 6875.655 0.147 22 6970.530 0.033 41 6868.717 b 23 6966.829 0.037 1 += 2 Q branch (from lower K'= 1 component) 7 6966.165 0.090 19 6930.763 -0.082 11 6957.770 -0.150 20 6926.365 -0.046 14 6949.587 0.105 22 6916.752 -0.005 15 6946.183 -0.034 23 691 1.724 0.074 16 6942.603 -0.120 24 6906.234 0.001 17 6939.01 8 0.019 25 6900.490 -0.072 18 6935.035 -0.005 P.A. FREEDMAN AND W. J. JONES 0 +-1 R branch 14 7019.364 0.049 20 70 10.7 10 -0.006 15 7018.396 -0.051 21 7008.600 0.117 16 7017.182 -0.172 22 7006.169 0.149 17 701 6.254 0.21 8 23 7003.492 0.169 18 7014.345 -0.146 24 7000.541 0.149 19 7012.650 -0.068 25 6997.387 0.160 0- 1 P branch 7 6990.602 0.091 18 693 7.557 0.000 9 6983.084 0.232 19 693 1.471 0.053 10 6978.7 1 3 0.017 20 6924.8 3 5 -0.221 11 6974.069 -0.253 21 691 8.457 -0.011 12 6969.675 -0.054 22 691 1.724 0.069 14 6959.975 0.087 23 6904.751 0.135 15 6954.600 -0.037 24 6897.264 -0.084 16 6949.28 6 0.121 25 6889.825 -0.027 17 6943.5 52 0.080 26 6882.167 0.041 a from global fit of all fundamental data, b not used in fit, see text.hot band v~(~A')= 1 + v3('A'') = 1transitions 1-0 Q branch? 0-1 Q branch? N" positionlcm-1 N" position/cm-l 4 6928.689 4 6890.890 5 6927.663 5 6890.124 6 6926.367 6 6889.210 7 6924.835 7 6888.120 8 6923.054 8 6886.812 9 6921.058 12 6879.524 10 6919.020 13 6877.271 11 691 6.752 15 6872.235 12 6914.175 16 68 69.482 14 6908.422 1-0 P branch? 15 6905.512 16 6902.035 7 691 0.348 17 6898.448 8 6906.88 5 18 6894.755 9 6903.250 19 6890.890 10 6899.260 20 6886.812 11 6895.107 21 6882.548 12 6890.890 22 6878.139 13 6886.365 23 6873.457 15 6876.635 24 6868.71 7 16 6871.672 1-0 R branch? N " position/cm-1 N" position/cm-l 13,14 6944.699 22 693 7.552 16 6943.943 23 693 5.8 17 17 6943.391 25 693 1.664 18 6942.603 26 6929.397 19 6941.654 27 6926.964 21 6939.026 28 6924.103 214 I.R.EMISSION OF HOz TABLE1.-continued strong, unassign ed lines position kin-1 position/crn-l position/cm-l 6969.81 2 6913.39 6884.691 6963.858 691 2.3 62 6867.032 6955.850 6907.182 6850.555 6952.71 5 6897.870 6844.796 6947.196 6893.086 6832.643 6933.525 6892.300 6826.265 6923.054 6892.071 6865.231 Lines bracketed together form a distinct sequence. Further unassigned sequences may be obtained by combining some of these lines with those shown above as fully assigned and are therefore not indicated.In addition to this ground state analysis, lines of a hot band were also found, although their assignment is on much more uncertain ground. A weak Q branch with origin at 6931 cm-l, was assigned to the 1-0, v3 = 1, 2A’ + v3 = 1, 2A”transi-tion. A few members of the corresponding 0-1 Q branch, with origin at 6892.5 cm-l, have similarly been tentatively identified. Using some imagination, extremely weak P/R structure with the same origin as the 1-0 Q branch could be identified. Since the assignments are relatively uncertain and the constants obtained of low quality (due not only to the poor quality of the data but also to uncertainty in relative N numbering), apart from checking that the assignment was not inconsistent with the results from the fundamental analysis, no further study was carried out.It is, how- ever, instructive to calculate the v; vibrational frequency using this assignment. Using the value of 1095 cm-l for the v3 (0-0 stretching vibration) frequency of the ground state,’ one obtains v~(~A’)= 983& 5 cm-l. This inay be compared with a value of v~(~A’)= 950+ 30 cm-I calculated from the position of the v3 = 1, 2A’ + u3 = 0, 2A”transition recorded in absorpti~n.~ Table 1 records the measured line positions together with suggested assignments. The residuals shown for the fundamental transitions correspond to those obtained in the global fit [method (c) above]. DISCUSSION Only rotational levels with K’ = 0 and K’ = 1 could be found for the lowest vibrational level of the 2A‘ state.This observation is confirmed from the low resolu- tion spectra which show a sharp high frequency edge at N 1.41 ,urn. This may be compared to the absorption spectrum which shows apparent RR-type band heads to 1.38 pm. It is also interesting to note that the 1.51 pm band, assigned to the (200)- (000) ground electronic state overtone ’9 could not be found in this emission study. The low resolution spectra of DO2 should be compared with these HO, results. In the spectrum of DOz the high frequency edge is much less sharp which, considering the fact that A(D0J x +A(HO,), would suggest that levels of reasonably high K value are present in this case. There would appear to be two possible explanations which could be invoked to account for these observations.(I) By analogy with the situation which exists in HNO,1° one could assume that P. A. FREEDMAN AND W. J. JONES the dissociation energy of HO, (2A”)lies at about the same energy as the K’ = 2 state of HO, (2A’). A predissociation mechanism would then account for our observations since the zero point energy of DO2, being less than that of HO,, would allow many more K levels of DO, to lie below this predissociation limit. However, there appears to be fairly good evidence to suggest that the dissociation energy (of 2A”)is much higher than this (assuming that the lower state of 2A’’seen here, as would appear very likely, is the ground vibrational state). Thus Foner and Hudson l1 give a mass spectrometric value of 16 500+ 1000 cm-1 for the H-0 dissociation energy in H02,the 0-0 value being even higher.12 If this process were possible, crossing to the 2A”states via an a-type Coriolis interaction would be the probable mechanism. (2) If a-or b-type Coriolis interaction between the lower vibrational levels of 2A’and high vibrational levels of 2A” is very strong, due to near coincidence of energies, efficient vibrational quenching within the 2A” ground electronic state may remove those levels with appreciable 2A”character. Such quenching would also explain the apparent absence of the 2v1 overtone emission of ,A”.The difference between the HO, and DO, cases would then suggest that this energy coincidence does not occur for DO, indicating that at least one quantum of either v1 (0-H stretching vibration) or v2 (0-0-H bending vibration) is involved, since only these two vibrations show the necessary isotope effect.To explain the weak perturbation in the K’ = 0 level of 2A’, one must obviously invoke a Coriolis type b interaction with a K = 1 level within 2A’’,whilst the different properties of the two components of the K’ = 1 level of ,A’ is probably a function of the relatively large asymmetry splitting of this level, giving a better energy match for the upper component. Until anharmonic constants become available for this molecule, however, it is a fruitless task to try to identify any of these perturbing states, although a proposed high resolution absorption study may shed further light on this problem.Thanks are due to Dr. I. W. M. Smith for the loan of the intrinsic germanium detector used in this work. We are also extremely grateful to Dr. Jon T. Hougeii for forwarding the LMR results before publication and to Dr. J. H. Carpenter for the use of his asymmetric top fitting program. P. A. F. thanks Trinity College, Cambridge, for the award of a Research Fellowship. H. E. Radford, K. M. Evenson and C. J. Howard, J. Chem. Phys., 1974, 60, 3178. J. T. Hougen, H. E. Radford, K. M. Evenson and C. J. Howard, J. MoZ. Spectr., in press. K. H. Becker, E. H. Fink, P. Langen and U. Schurath, J. Chem. Phys., 1974, 6,4623. D. J. Giachardi, G. W. Harris and R. P. Wayne, Chem. Phys. Letters, 1975, 32, 586. H. E. Hunziker and H. R. Wendt, J. Chem. Phys., 1974, 60,4622. S. M. Till, K. C. Shotton and W. J. Jones, Pruc. Roy. Soc., in press.’J. C. Gole and E. F. Hayes, J. Chem. Phys., 1972, 57, 360. S. R. Polo, Canad. J. Phys., 1957, 35, 880. T. T. Paukert and H. S. Johnston, J. Chem. Phys., 1972,56,2824. lo M. J. Y.Clement and D. A. Ramsay, Canad. J. Chem., 1961,39,205. S. W. Foner and R. L. Hudson, J. Chem. Phys., 1955,23, 1364. l2 P. Gray, Trans. Faraday SOC., 1959, 55,408. G. Herzberg, Molecular Spectra and Molecular Sfvucture 111 (Van Nostrand, Princeton, N.J., 19661, p. 106. (PAPER 5/988)

ISSN:0300-9238    

DOI:10.1039/F29767200207

出版商:RSC    

年代:1976

数据来源: RSC

摘要:

Prediction of Ordered and Disordered States in Colloidal Dispersions BY IAN SNOOK* AND WILLIAM VAN MEGEN Department of Applied Physics, Royal Melbourne Institute of Technology, Melbourne, Victoria, Australia Received 28th May, 1975 Excess pressures and radial distribution functions of a monodisperse colloid are calculated from a statistical mechanical model using the Monte Carlo method to evaluate the configurational integrals. Particular regard is paid to the effects of variation of volume fraction and background electrolyte concentration on the structure of the system. Numerous experimental studies have been reported 1-4 on the structure of colloidal dispersions of spherical particles. These studies indicate the existence of regions of order (solid-like regular arrangement of particles) and regions of disorder (liquid-like, showing no long ranged order).In view of this experimental activity we have attempted a description of this behaviour in terms of the forces between the colloidal particles. Recently we have developed a model for colloidal dispersions based on classical statistical mechanics which explains the main features of the compression of, for example, an aqueous suspension of polystyrene spheres. 9 To further demonstrate the versatility and usefulness of the application of models based on classical statistical mechanics, this paper presents the results of calculations on spherical colloidal particles dispersed in aqueous electrolyte. Systematic variation of the background electrolyte concentration and volume fraction, with all other conditions held constant, predicts some interesting structural effects similar to recent experimental observation.An outline is given of the method and interparticle potentials used and the results are presented and discussed. THEORY As outlined in earlier work 59 we calculate the equilibrium properties of an electrostatically stabilized colloidal dispersion using classical statistical mechanics, the Monte Carlo method being used to evaluate numerically the multidimensional integrals in the canonical ensemble. The Monte Carlo method is adequately de- scribed elsewhere 7-9 and the details and problems associated with its application to colloidal dispersions are discussed in ref.(6). In particular we evaluate the internal energy E, the pressure P and the radial distribution function g(rI2), given by ; E 31 -=-NkT 2+NkT-<w PV 1 -= l--(YN}NkT 3NkT 216 I. SNOOK AND W. VAN MEGEN and where p = N/V is the number density and 1exp(-ON/kT)dr, . . . dr,N! d2)(r,,r,) = -(N-2)! eXp(-@N/kT) dr, . . . dv, (4) is the two-particle distribution function. Vis the volume, T the absolute temperature, k Boltzmann's constant, N the number of particles and DN and YNare the total potential energy and the virial term, respectively. Furthermore, the weighted aver- ages in eqn (1) and (2) are given by and where Uijis the pair potential and rijthe interparticle distance. The interparticle potential Uijused consists of the usual lo double layer repulsion URplus the van der Waals attraction UA,which for spherical particles of radius a are given by; uR(rij)= 271caI,biIn(1 +e-'''j) (7) and In the above expressions, z = k-a, uij = rij/a-2, I,bo is the surface potential, E the dielectric constant of the background, A the Hamaker constant, and l/k-the usual characteristic thickness of the double layer, given by where No is Avagadro's number, IZ the electrolyte concentration, e the electronic charge and z the valence of the ions in the background medium (assuming a sym-metrical electrolyte).Since, in this work, we are only attempting to produce the main features of the experimental results, the use of such simple pair potentials is fully justified.Further-more, their use substantially reduces computer tinie requirements. CALCULATIONS AND RESULTS Unfortunately, in the available experimental papers insufficient data is given pertaining to the exact Eature of the dispersions used. Hence, in our calculations we have chosen the following parameters which appear to be representative : a = 5.95 x loA7m, t+bo = 0.06 V, A = 2.5 x J, z = 1, T = 300 K and E = 80 E~ (E~is the permittivity of free space). Equilibrium properties have been calculated for the above system with background electrolyte concentrations of 1, 5 x lo-', lo-', 5 x and mol m-3. 21 8 PREDICTION OF COLLOID STABILITY rlo FIG. 1.-The interparticle potential U/kTas a function of the particle separation in units of (J = 2a for n = mol m-3 (-), n = 5 x mol m-3 (---), n = 10-1 mol m-3 (-.-.-), n = 5 x 10-1 mol m-3 (-..-..-) and y1 = 1 mol m-3 (..--..).5 4 0 A Q I'I-* 4 bQ-2 A 0 -.-IA I ElI -.-I El h -IC 70 60 50 40 30 20 10 0 4 FIG.2.-The logarithm of the reduced pressure, P* = PV/NkT,as a function of the volume fraction mol m-3, (-t-) y1 = 10-1 mol m-3, (m) n = 5 x 10-1n = mol m-3, (A)n = 5 x4, for ; (0) (:#: andm-3mol ) n = 1 mol m-3. I. SNOOK AND W. VAN MEGEN Fig. 1 shows the total potential energy as given by eqn (7) and (8) for the five electrolyte concentrations. Note the increase in the repulsive range of the potential as the electrolyte concentration is reduced. I 1.2 1.4 1.6 rlu Frc.3.-The radial distribution function g(r) for n = 1 molm-3 with ---, 4 = 5 o/o; --I-, ,$ = 15 %; ....., 4= 25y--.--.-, ,$ = 35 -..-..-, 4= 50 %. UY 0, r.1. FIG. 4.-The radial distribution function g(r) for /I = lo-' mol nr3 with ----, ,$ = 57;;---,4= 15 ;<; ....., 4 = 25 %; -.-.-, 4 = 35 x. In most cases the initial configuration was taken as the usual face centred cubic structure 6* at the number density pof the system NN p=-=-v L3' PREDICTION OF COLLOID STABILITY where L is the length of the cell side, this being related to the usual percentage volume fraction 4 by As in our earlier work, the initial few hundred thousand configurations were rejected to allow the system to equilibrate. Then, at least 700 000 further configura- tions were used to obtain the reported averages.rlu FIG.5.-The radial distribution function g(r),as given by eqn (3) and (4) as a function of the particle separation, for the lowest electrolyte concentration of n= mol m-3 with 4 = 5 % (-) and 4 = 15 %(----). 12-8-Nl 4-04 0 10 20 30 40 50 60 70 4 FIG.6.-The number of nearest neighbours N,, as given by eqn (12), plotted against 4with (x) nn= 1 mol m-3, (0)= 5 x lo-’ rnol m--3,(+) n= lo-‘ mol m-3, (A) n= 5 x lo-* rnol m-3 and n= mol m-3. (The dashed lines are to facilitate reading the results and are not meant (0) to indicate a continuous variation of Nl with d). I. SNOOK AND W. VAN MEGEN 221 Fig. 2 shows the resulting reduced pressure P* = PV/NkT as a function of 4 for the five values of the electrolyte concentration.In each case P*increases quite drama- tically with volume fraction up to a 4 of around 70 %. Beyond this point the system is no longer stable as is evidenced by a sharp drop in the calculated pressure. In fact, these latter pressures are negative due to the infinitely deep potential well [see eqn (7) and (S)] at very small surface to surface separations of the particles. Note that in the results for the lowest electrolyte concentrations the rate of pressure increase with 4 is reduced, from moderate to large values of 4. This effect can be rationalized by noting that the pressure is directly related to the pair virial rdU/dr and that for the lower electrolyte concentrations this virial is smaller.Fig. 3 to 5 show some typical radial distribution functions [given by eqn (3) and (4)] for ranges of 4 covering the transition from disorder to order. Note, for example in fig. 3, g(r) indicates an ordered “ solid-like ” structure for 4 = 50 %, whereas at lower volume fractions no such longer ranged structure is evident. For 4 > 50 %, of course, ordered systems were also obtained with the first peak becoming sharper (and higher) with increasing 4. 60-. *. 0. *. 0. 40 0 00 00 0 DO 002olI 0 00 00 ‘Ol 0-t -2 -I 0 log (4 FIG.7.-A summary of which systems were found to be ordered (0)and disordered (0). For all systems (4 = 5,15,25,35,50,60,70 % and n = 1,sx lO-l, 5 x mol m-3) the number of nearest neighbours Nl was determined, as shown in fig.6. This quantity is determined from the usual symmetrized expression l1 N1 = 8nprmaxr”g(r)dr 0 where r,,, is the position of the first peak in g(r). Note the shift to lower 4 in the transition from order to disorder as the electrolyte concentration is reduced. This is readily explained by the fact that a decrease in electrolyte concentration increases the double layer thickness, hence the effective particle diameter and effective volume fraction also increase. For a perfect hexagonal close packed system of spheres one expects N1 to be 12 as we obtained for higher volume fractions. However, for q5 just beyond the order/ PREDICTION OF COLLOID STABILITY disorder transition, calculated values for N, are slightly less indicating the structure obtained is not quite a perfect close packed solid.Furthermore, since the first peak in the radial distribution functions are not perfectly symmetrical about r,,,, the actual values of N, are approximate. Finally in fig. 7 results of ordered and disordered systems are summarized. The features of this graph seem to agree remarkably well with the experimental results of Hachisu et aZ.,4 except, of course, the region of coexistence of the ordered and dis- ordered phases is not obtained in our work. The reasons being that in our computer experiment the system is not subjected to a gravitational field. Furthermore, the small system of 32 particles as we have treated it, although quite adequate to evaluate bulk properties of systems of particles with short ranged interparticle potentials, is l2quite inadequate for a realistic treatment of two phases in eq~ilibrium.~* DISCUSSION The Monte Carlo method as it stands can, in principle, be used to study phase transitions (e.g.disorder to order in a colloidal dispersion) and even two phases in equilibri~im.~ However in practice, a very large number of particles is needed because of the density gradients and the long range correlations existing at the phase transition. Thus phase transitions are studied approximately in several ways ;for example by keeping the system in one phase beyond its region of stability (e.g. the single occupancy model of Ree et a1.)’ or by placing an external potential on the system.12 These calculations still require large numbers of particles and many more hours of computer time than the normal Monte Carlo method.In this study we merely find the region in which a phase transition occurs, the most stable phase being determined by starting the system initially in either phase and finding the phase in which the system finally settles (after around a million con- figurations). This approach is similar to the early studies of systems of particles with hard sphere square-well and Lennard-Jones 12-6 potential^,^^ and like these studies, our calculations yield only the region of the phase transition to within a range of 5 to 10 % in 4. However, we expect this is quite sufficient for an initial study just as it was in the case of molecular systems,13 particularly as there is little precise experi- mental data on the phase transition region anyway.It may be noted that the region of the phase transition occurs at lower and lower values of 4 as the background electrolyte concentration is decreased. This packing fraction is based on the size of the particle, not on the size of the particle with its attached double layer. The phase of the system was determined from the radial distribution function. For instance, from fig. 4 one can readily observe that the dispersion with an electrolyte concentration of 10-1 mol m-3 is disordered when 4 5 25 % and ordered when 4 2 35 %. This can be checked by calculating the number of nearest neighbours (NJ. Approximate calculations show N1 to be close to 12 for ordered systems and much less than this value for disordered systems.In all cases the initial configuration was taken as the face centred cubic configura- tion. However, attempts were also made to obtain the ordered structures of the n = mol m-3 and 4 = 15, 25 % systems from a random (disordered or liquid- like) initial configuration as generated for the corresponding $ but higher (n = 1 mol m-3) electrolyte concentration. Although ordered structures were obtained in these runs, the exact details of those runs starting with the face centred cubic structure could not be reproduced within a realistic number of configurations (about one million). This situation reminds one of the random hard sphere packing experiments of the last decade in which it was found impossible to obtain packing fractions or I. SNOOK AND W.VAN MEGEN volume fractions in excess of about 65 %.14 Thus it appears impossible to generate a perfect close packed solid randomly, or the probability of doing so is negligible. 9* l3In practical terms, one generates a glassy A very eloquent explanation of this effect has been given by Wood who states that at very high densities (and we are here discussing dispersions of high effective density) trajectories in phase space are located within a pocket around the point corresponding to the initial configuration and the probability of a trajectory to another pocket, around a point corresponding to the close packed hexagonal configuration, is extremely small.At all but the lowest electrolyte concentration, the onset of coagulation was indicated at volume fractions beyond 70 %. Again, the exact point of this phase transition is difficult to locate for reasons mentioned above. Furthermore, the large negative pressures obtained for these partially coagulated states are indeed question- able. This is due to the unrealistic form of the inter-particle potential ;approaching minus infinity as the particle surface to surface separation approaches zero. In reality this narrow infinite well is truncated by the Born repulsion of at least a mono-layer of ions absorbed on the surface of the particles. Since in our model we have not accounted for the structure of this inner region of the double layer, it would be unrealistic to accept any actual values generated by the Monte Carlo calculation for the partially coagulated state.For the lowest electrolyte concentration (n = mol m-3) coagulation is evident beyond 4 = 60 %. The long range of the repulsive potential for this case results in systems (at least at moderate to large 4) with a high free energy. This enhances the coagulation of the primary particles. The authors thank the Royal Melbourne Institute of Technology Computer Centre for the use of their facilities, Dr. R. 0. Watts for his valuable assistance with the Monte Carlo program and Dr. Andrew Homola for his fruitful comments. G. W. Brady and C. C. Gravatt, Jr., J. Chem. Phys., 1971, 55, 5095. L. Barclay, A.Harrington and R. H. Ottewill, Kolloid-2.2. Polyrnere, 1972, 250, 655. P. A. Hiltner and I. M. Krieger, J. Chem. Pliys., 1969, 73, 2386. S. Hachisu, Y.Kobayashi and A. Kose, J. Colloid Interface Sci., 1973, 42, 342. I. Snook and W. van Megen, Chem. Phys. Letters, 1975,33, 156. W. van Megen and I. Snook, J. Colloid Interface Sci., in press. 'W. W. Wood, Physics of Simple Liquids, ed. H. N. V. Temperley, J. S. Rowlinson and G. S. Rushbrooke (North-Holland, Amsterdam, 1968). R. 0. Watts, Rev. Pure Appl. Chem., 1971, 21, 167. F. H. Ree, Physical Chemistry, An Adzanced Treatise, ed. D. Henderson (Academic Press, New York, 1971), vol. VIIIA, chap. 3. lo H. R. Kruyt, Colloid Science (Elsevier, Amsterdam, 1952), vol. 1. l1 C. J. Pings, Physics of Simple Liquids, ed. H. N. V. Teniperley, J. S. Rowlinson and G. S. Rushbrooke (North Holland, Amsterdam, 1968). l2 J. K. Lee, J. A. Barker and G. M. Pound, J. Chem. Phys., 1974, 60, 1976. l3 W. B. Streett, H. J. Raveche and R. D. Mountain, J. Chem. Phys., 1974, 61, 1960. l4 J. D. Bernal and J. Mason, Nature, 1960, 118, 910. (PAPER 5/991)

ISSN:0300-9238    

DOI:10.1039/F29767200216

出版商:RSC    

年代:1976

数据来源: RSC