摘要:

Faraday Symp. Chem. SOC.,1984 19 125-135 Ab Initio Calculations on Weakly Bonded Systems TEUS B. VAN DUIJNEVELDT BY JOOP H. VAN LENTHE,* VAN DAMAND FRANS Theoretical Chemistry Group University of Utrecht Padualaan 8 De Uithof Utrecht The Netherlands AND LOESM. J. KROON-BATENBURG Laboratorium voor Kristal- en Structuur-chemie University of Utrecht Padualaan 8 De Uithof Utrecht The Netherlands Received 16th August 1984 Large-basis CI calculations are performed on the van der Waals complexes Ar-HC1 and (H,O),. It is shown that a reasonable estimate of the CI basis-set superposition error is obtained from a ghost calculation involving the orbitals of the monomer and only the virtual orbitals of the ghost. Both basis-set superposition error corrections and size-consistency corrections are of vital importance to obtain a reliable potential-energy surface.For Ar-HC1 the minima of the potential are predicted within 50 phartree of the experimental surface viz. -804 phartree for the Ar-HCl orientation and -565 phartree for the HC1-Ar geometry. The water dimer van der Waals minimum is estimated to be -4.9 kcal mol-l which is less deep then the experimentally derived minimum of -5.4 kcal mol-l but just within the experimental error limit. The use of supermolecule ab initio CI techniques for the calculation of potential- energy surfaces for weakly bonded (van der Waals) systems has been frought with difficulties.lP6 In 1981 Maitland et a/.’ wrote ‘In general except for the case of He, where significant progress is being made towards the calculation of the complete potential function there is no immediate prospect of calculating an accurate potential.’ Accurate CI calculations require large basis sets for a proper representation of both intra- and inter-molecular correlations so huge numbers of configuration state functions (CSF) are involved. Only in recent years with the advent of the new generation of supercomputers (Cray I Cyber 205) and CI have CI calculations involving lo5CSF and extensive basis sets become routinely possible. It is therefore timely to establish reliable procedures to perform CI calculations on moderately sized van der Waals systems of chemical interest. As one prototype system the Ar-HCl system was chosen. This complex has been extensively studied experimentallyl99 l1 and the uncertainty in the empirical potential- energy surface is thought to be 50 phartree.? The complex is weakly bound the well depth being only 800 phartree (0.5 kcal mol-l) and only a minor part of this is reproduced at the SCF leve1,12 so the system is well suited to test CI schemes.No satisfactory CI surface has yet been published however.’ In order to be useful a scheme has to be applicable to a wide range of van der Waals t 1 hartree = 4.359814 x 10-l8 J. 125 CI CALCULATIONS * /’ / / / CIL (a) (b) Fig. 1. (a)Ar-HC1 coordinates. The HCl centre of mass is the origin for the (R 0) coordinates. (b) (H,O) coordinates. The origin is at the acceptor oxygen atom. systems. Therefore our second test system is the H20 dimer where the interaction energy is an order of magnitude larger and the induction energy constitutes a significant part of the interaction potential.The experimentally derived well depth of 5.4 kcal mol-1 has a large uncertainty of 0.7 kcal m01-l.~ A number of potential- energy surfaces have been calculated using CI the most recent one 37 59 being that calculated by Clementi and Habitz.6 An important systematic defect of supermolecule finite-basis calculations is the basis-set superposition error (BSSE).5yl3 Methods to deal with this problem at the CI level include avoiding it by the choice of CSF,2 ignoring it4 and calculating it using the standard Boys-Bernardi scheme and either correcting for it5 or In this paper we will explore an alternative CI scheme which yields reasonable estimates of the BSSE thus producing reliable results for both the van der Waals systems mentioned above.The importance of either using a size-consistent method or applying size-consistency corrections is demonstrated. For a long time SCF calculations augmented by an estimate of the dispersion energy from second-order perturbation theory14 have been an important ab initio source of potential-energy surfaces. Since these calculations are considerably cheaper than CI calculations it is worthwhile to compare the results of the two approaches. COMPUTATIONAL PROCEDURE The interaction energy for a complex with a geometry defined by a distance R and angle 0 (see fig. 1) is given by where E(R,O) and E(R = 100) are the single- and double-excitation CI (SDCI) energies resulting from dimer calculations at the specified geometry and at large distance and EBssEand E, are the basis-set superposition error and the size-consistency correction respectively.The dimer calculation at a large distance is used instead of the separate monomer calculations to avoid the bulk of the size-consistency error which for our Ar-HCl calculation (basis I) is 0.0107 hartree. E, corrects for the change in the size-consistency error with the geometry in the dimer calculation. EBssE is calculated from size-consistent monomer calculations making the Stone and Price correction15(ca. 50 phartree for Ar-HC1) superfluous. For Ar-HCl we calculated E, using a correction formula due to Pople,16 since it proved to reduce the size-consistency error at infinity to only 250,uhartree an order of magnitude better than the performance of the Davidson f0rmu1a.l~ For the H,O dimer the size-consistency effects were included by employing the CEPA( 1) method as implemented in the ATMOL~direct CI program.VAN LENTHE VAN DAM VAN DUIJNEVELDTAND KROON-BATENBURG 127 Table 1. Basis sets used in the Ar-HC1 and (H,O) calculations ~~~ ~ denotation atom(s) +reference composition basis I" Ar Cl HZ9 basis 11" Ar C1 H29 DZP' 0,HZ1 DZPP 0,HZ1 EZPM 030 H21 7 EZPP 030 H21 7 CH 0,H6 a Bases I and I1 correspond to bases I1 and I11 of ref. (1 2). In all CI calculations the inner shells were kept frozen and the corresponding virtual orbitals were deleted.For Ar-HCl this concerns the Is 2s and 2p orbitals of Ar and C1; for H20only the oxygen 1s orbital was frozen. All CI calculations were performed using the ATMOL program system as implemented on the Cray 1 S in Daresbury/London and Rijswijk the Cyber 205 in Amsterdam and the Cyber 175 of the University of Utrecht. A typical calculation on one geometry of Ar-HC1 took ca. 20 min of Cray 1s C.P.U. time. BASIS SETS The GTO basis sets are characterized in table 1. The Ar and C1 sp basis sets are derived from 12s,9p bases of Huzinagalg and Veillard,20 respectively by replacing the four most diffuse s functions by 6 gaussians and the three most diffuse p atomic orbitals by 4 gaussians in order to obtain a better description of the valence regions and the exchange repulsion between the monomers.The extended-double-zeta basis (EZ) for oxygen is obtained similarly from the double-zeta (DZ) basis.21 The Ar-HC1 polarization functions were spherical harmonic gaussians. The high-exponent C1 d and H p functions were chosen to minimize the HCl Hartree-Fock energy while the diffuse Cl d and H p as well as the Ar d and fand C1 fgaussians were chosen to maximize the dispersion energy at 10 bohrt distance and 0". In basis I1 one d gaussian on Ar is optimized for the intra-atomic correlation while the other as well as the d gaussian on hydrogen is meant to maximize the dispersion energy The H20 dimer calculations employed Cartesian gaussian orbitals. The single set of polarization functions used in the DZP basis is optimized to reproduce good Hartree-Fock multipole moments and polarizabilities.21 The EZPM basis is meant to mimic the basis set used by Clementi and Habitz6 and uses their p exponent of 0.4 for hydrogen.GEOMETRIES The geometry parameters of the dimer are shown in fig. 1. The monomer geometries were kept fixed in all calculations. The HCl distance used is 2.426 bohr,22 the water monomer geometry is taken from e~periment.~~ 1 bohr = 5.2917706 (44)x lo-" m. CI CALCULATIONS BASIS-SET SUPERPOSITION ERROR SCHEMES For the estimation of the BSSE in Hartree-Fock calculations the Boys-Bernardi13 counterpoise (ghost-orbital) method is widely used although there is still an argument as to whether the complete orbital space of the ghost molecule should be included or only the virtual orbitals.For moderately sized van der Waals molecules like the ones studied here it may be possible to circumvent this problem by extending the basis so that the BSSE is negligible. For the large basis sets used in accurate CI calculations the SCF BSSE is indeed very small. However even for our largest basis set the CI BSSE for Ar-HCl is of the same order of magnitude as the well depth of the van der Waals minimum so it is vitally important to obtain a proper estimate. DlMER C1 GHOST C1 (1) I I I I 1 I I occupied A I occupied B r l laTypdl] ;uj+; Fig. 2. Schematic representation of the three CI procedures scheme (1) SCF molecular orbitals; scheme (2) combined monomer molecular orbitals ;scheme (3) combined monomer molecular orbitals.We compare three schemes to perform CI calculations and obtain estimates of the BSSE. They are represented pictorially in fig. 2. (1) The standard Boys-Bernardi recipe the dimer CI calculation is performed in the basis of the dimer SCF orbitals. The ghost-orbital CI calculations employing the complete orbital space are preceded by ghost SCF calculations. (2) In order to be able to separate orbital spaces no dimer SCF calculation is carried out. The dimer occupied space consists of the occupied orbitals of the monomers suitably orthonormalized. The dimer virtual space is obtained by orthogonalizing the combined monomer virtual orbitals onto the occupied space. The BSSE CI calculations employ the occupied space of the monomer augmented by the same virtual orbitals as used in the dimer calculation.In this scheme exactly the same intramolecular excitations that are possible for the monomer in the dimer calculation are also supplied in the ghost calculations. This comes closest to the original Boys-Bernardi concept of employing identical function spaces in monomer and dimer calculations. (3) The dimer CI calculation is identical to that of scheme (2). For the ghost calculations the combined but not yet orthogonalized monomer +ghost virtual spaces are added to the monomer occupied orbitals and then suitably orthogonalized. In this scheme which was originally proposed by Daudey VAN LENTHE VAN DAM VAN DUIJNEVELDT AND KROON-BATENBURG 129 Table 2.Ar-HCl SDCI interaction energies at R = 7.53bohr 8 =0” in various schemes (in phartree)a BSSE scheme pncorrected Ar HCl int 0 -1266 0 0 -261 -1527 -1804 -1 -1266 -906 -237 -261 -383 2 -1107 -655 +lo2 -442 -996 -1063 3 -1107 -761 -142 -442 -646 -785 exptllO -824 a Components are given in basis I. et al.,24the complete unperturbed monomer space is employed to which only the virtual orbitals of the ghost are added. By contrast in scheme (2) the ‘occupied’ orbitals of the ghost are projected out of both virtual spaces. Finally we define a scheme (0) which involves a CI calculation as in scheme (1) and no BSSE correction. The reasoning behind the exclusion of the occupied ghost orbitals in schemes (2) and (3) is as follows.The ghost calculations are meant to simulate the extra freedom (correlation or excitation possibilities) that the monomer obtains in the dimer calculation and so to estimate its artificial energy lowering. Since the occupied orbitals of the other monomer are not available for this purpose in the dimer calculation they should also be excluded from the ghost calculation. The situation is quite different for SCF calculations. It has been 26 that the use of monomer orbitals calculated in the full dimer basis set (large-basis monomers) yields a more consistent first-order interaction energy than when using small-basis monomers implying that large-basis orbitals are the most natural start for a dimer calculation. Furthermore it may be argued that in the dimer SCF calculation at least part of the occupied space of the ghost is available to the monomer since the ghost’s Hartree-Fock function being in its variational minimum is easily ‘pushed away’ during the dimer SCF process.Finally it has been that using the full orbital space in the ghost calculation does yield SCF results that are most independent of the basis set. Considering the argument given above there are two possible choices for the occupied orbitals in the CI calculations in schemes (2) and (3) uiz. either small-basis or large-basis SCF monomer molecular orbitals. In the original Daudey scheme only the small-basis case was considered. The procedure for the large-basis case is exactly as in scheme (3) except that the occupied molecular orbitals are taken from monomer SCF calculations in the full dimer basis.Since the SCF BSSE is small in our Ar-HCl calculations (<50 phartree) there is almost no difference between the two approaches and we will use small-basis monomers. For the H,O dimer there is a substantial difference between the two approaches in some basis sets so the problem will be addressed there. Ar-HCl COMPARISON OF BSSE SCHEMES The results of schemes (O)-(3) for a point close to the experimental minimum are given in table 2. It is obvious that if no correction is applied [scheme (O)] no useful 5 FAR CI CALCULATIONS Table 3. Properties of Ar-HCl obtained with method (3) and size-consistency correctionsa property SCF +dispersion basis I basis 11 e~periment~~*~~,~~ RdO") 7.49 7.73 (0.06) 7.69 (0.06) 7.58 Emin(0") -999 -671 (6) -804 (6) -824 Re(180") 6.64 6.93 (0.00) 7.05 (0.00) 7.04 Emin( 180") -1087 -598 (1) -565 (1) -616 (Pl) 0.38 0.62 (0.02) 0.76 (0.01) 0.67 -32) 0.26 0.35 (0.02) 0.46 (0.04) 0.34 M3OO mb -75.3 -20.0 -25.5 -35 a R in bohr E in phartree and B12in cm3 rnol-'.Fit-error estimates in parentheses. The long-range parts (R > 9) of the fitted CI potentials were replaced by the experimental potential. lo result is obtained. As expected scheme (1) overestimates the BSSE and much too small interaction energies are obtained. Only the schemes where the occupied orbitals of the ghost are omitted yield results close to the experimental value. A fundamental problem with scheme (2) can be observed in table 2.The BSSE for the HCl molecule turns out to be positive i.e. the orbital basis in the BSSE calculation is less suited to describing the correlated monomer than the original monomer basis. This is due to the fact that the occupied orbitals of the ghost are projected out of the monomer virtuals. It is easy to see that in an infinite basis (in which the BSSE ought to be zero) scheme (2) always yields a positive BSSE (the BSSE is defined as the difference in energy between the ghost calculation and the isolated monomer). The problem might be avoided by using as a reference instead of the isolated monomer a monomer whose virtual orbitals are maltreated in the same way as in the ghost calculation. This however would make the calculations even more cumbersome.We therefore deem scheme (3) to be the best procedure to perform CI calculations on van der Waals complexes. At first sight a drawback of schemes (2) and (3) seems to be that the SDCI dimer calculation cannot fully reproduce the mutual polarization effects that are present in the omitted SCF dimer calculation. Indeed in schemes (2) and (3) the uncorrected CI interaction energy is 160 phartree smaller than in scheme (1). However the size-consistency correction (Esc) when added to @z!orrected has the remarkable property that the difference between the schemes is reduced to only 22phartree (table 2). Thus the size-consistency correction seems to be able to replace the SCF relaxation step to a large extent. The same effect is observed in (H,O) calculations.The size-consistency correction is an important contribution to the interaction energy in all schemes. In scheme (3) it is a decisive factor in obtaining a good anisotropy of the potential-energy surface.12 RESULTS We calculated the Ar-HC1 interaction energy using BSSE scheme (3) for three distances (6.6 7.1 and 8.0 bohr) and three angles (0 90 and 180") in both basis I and basis 11. In order to obtain an optimal representation of the potential-energy surface the origin of this grid was laid 0.5 bohr from the Cl atom. In basis I we also calculated three points at 39.2 and at 140.8" the zero points of P3(cos6). The points were fitted with a three-term exponential function for the R coordinate and up to P,(cos 6) and VAN LENTHE VAN DAM VAN DUIJNEVELDT AND KROON-BATENBURG 131 I I I 1 I I 0 45 90 135 180 el" Fig.3. Ar-HC1 potential-energy variation with 6 at R = 7.6 bohr (a)CI (basis 11) (b)HHMS. P2(cos8) in the angle for the basis I and I1 results respectively. Using these fits the lowest vibrational-rotational wavefunction was calculated yielding spectroscopic observables. In table 3 we present a comparison between our CI results and the experimental values. In parentheses we give estimates of the errors caused by the limitations of the fit and the small number of points.12 All errors are surprisingly small indicating that even only 9 points of the potential-energy surface (basis 11) yield a reasonable description of the surface around the van der Waals minimum.Also provided in table 3 are results from SCF +dispersion12 calculations using second-order perturba- tion theory and basis I which being an order of magnitude cheaper were calculated for many points. The 0" well depth and equilibrium distance determined in basis 11 agree well with experiment; the well depth is within the experimental error limits. The remaining difference may be due to the fit error and the lack of polarization functions higher thanf. From data on Ne227 we estimate the contribution to the dispersion energy due to these functions to be ca. 60 phartree at Re yielding an estimated well depth that remains within the experimental error bound of 50 phartree. The basis I1 180" well depth shows a larger difference with the HHM5lO surface; the minimum is even shallower than in basis I which results from a larger exchange repulsion due to the better description of intra-atomic correlation of the Ar atom in basis 11.On the other hand the inclusion of g functions may lead to some lowering. The experimentally determined surface is uncertain in the 180" region so our minimum may well be the more accurate one. The main defect of our calculated potential-energy surface is that the angular barrier is probably too high as may be seen from fig. 3. This would account for the high values obtained for (PI) and (P2) as well as for the fact that the mixed second virial coefficient B, is too small in spite of the essential correctness of the minima. We suppose that the inbalance in the angular behaviour is caused by the fact that our diffuse polarization functions were optimized at O" thus biasing the basis set 5-2 CI CALCULATIONS towards a good description of the 0" region.In both basis sets the perpendicular dipole polarizability of HCl is considerably further from the experimental values than the parallel one. From these polarizabilities we estimate that Ckis 7 % too low with respect to Cl. With a dispersion energy of ca. 1500 phartree near the barrier this accounts for a barrier which is 100 phartree too high. Finally we would like to assess the SCF +dispersion calculations. While they correctly predict two minima one at 0" and one at 180° the balance between 0 and 180" is wrong. The perturbation estimate of the dispersion energy is not sufficiently accurate and yields only part of the total correlation energy.For weakly interacting systems like Ar-HC1 the SCF +dispersion approach therefore yields at best qualitative results. H20 DIMER COMPARISON OF BSSE SCHEMES Since for Ar-HCl scheme (3) was shown to produce the most reliable results all calculations on (H20) were performed according to this procedure. The only variation is in the choice of the occupied monomer orbitals. Both large-basis (L) and small-basis (S) results are shown. Only one CEPA dimer calculation (viz.starting from small-basis molecular orbitals) is used in calculating interaction energies. This is justified because although a dimer SDCI using large-basis monomers is 0.3 kcal mol-1 lower than the small-basis variant the difference between the CEPA calculations is only 0.03 kcal mol-1 in the DZP' basis at the geometry studied.Moreover a CEPA calculation employing dimer SCF orbitals yields an energy only 0.14 kcal mol-l lower than the combined large-basis monomer results. Thus in accordance with the Ar-HCl result the (size-consistent) CEPA calculations are remarkably invariant with respect to the choice of occupied orbitals. In table 4 we present results for a geometry close to the experimentally derived van der Waals minimum. While the uncorrected CEPA interaction energies differ significantly from one another the BSSE correction especially the large-basis variant succeeds well in making the interaction energies more nearly basis-set independent.The BSSE is far from negligible ranging from 2.3 kcal mol-1 in the DZP' basis to 0.55 kcal mol-1 in the EZPP basis. RESULTS The main error sources in our EZPP calculation are the lack off and higher polarization functions and possibly the missing dimer SCF step. As mentioned above the latter step gives an additional 0.14 kcal mol-1 in the DZP' basis. For the Ne dimer it has been demonstrated2' that using a basis up to d yields ca. 80% of the total dispersion energy at the van der Waals minimum implying that addingfand higher polarization functions may deepen the calculated well depth by 0.4 kcal mo1-1.2s We may therefore expect the limiting value for the well depth of the van der Waals minimum of the water dimer to be 4.9 kcal mol-I just within the experimental error limits.Clementi and Habitz6 found an interaction energy of -5.5 kcal mol-l only corrected for the SCF BSSE. The second virial coefficient calculated for their surface is a factor of 2 too high indicating too much attraction. When they decreased the correlation contribution to the interaction energy by an arbitrary factor of 2 the virial coefficient was in good agreement with experiment and the calculated van der Waals minimum was -4.7 kcal mol-l a result supported by our above estimate. Indeed a calculation using their basis set and the present procedure (column headed CH in table 4) gives VAN LENTHE VAN DAM VAN DUIJNEVELDT AND KROON-BATENBURG 133 Table 4. (H,O) CEPA interaction energies at R = 2.96 A 8 = 40” (in hartree)a basis DZP’ DZPP EZPM EZPP CH pncorrected -0.011 504 -0.008 579 -0.008 105 -0.007 824 -0.008 600 int EBSSE (S) -0.002 727 -0.001 212 -0.000 829 EBSSE (L) -0.003 748 -0.001 485 -0.000 969 -0.000 874 -0.001 232 int Eorrected (S) -0.008 777 -0.007 367 -0.006 995 Forrected (L) int -0.007 756 -0.007 094 -0.007 136 -0.006 950 -0.007 369 porrected (L) 74-87 -4.45 -4.48 -4.36 -4.62 int /kcal mol-1 a L and S refer to large- and small-basis monomers respectively.Table 5. (H,O) equilibrium energies and geometries from SCF+dispersion calculations28a SCF +dispersion multipole corrected basis R e E R e E DZP‘ 2.87 52 -5.83 2.9 1 56 -4.97 DZPP 2.86 46 -5.71 2.91 52 -4.93 expt15 --2.98 58 -5.4 a R in A 8 in O E in kcal mol-l. a dimerization energy of -4.6 kcal mol-l much less than their -5.5 kcal mol-l result.The other basis sets in table 4 are seen to reproduce the EZPP result rather closely the largest discrepancy arising for the DZP’ basis. This discrepancy was not expected since calculations at the SCF +dispersion level at a variety of (H,O) geometries had shown excellent agreement between the DZP’ and DZPP results.2s A summary of these results is shown in table 5. Because the SCF multipole moments differ considerably from the far more accurate CEPA ones there is a large error in the electrostatic part of the interaction energy. This error has been compensated in the entries under ‘multipole corrected’. The multipole-corrected DZP’ energies and equilibrium geometries for (H,O) agree well with the corresponding DZPP results.However the resulting Re is much shorter than the experimental value (table 5). Also the resulting DZPP interaction energy is significantly larger than in the present CEPA-CI approach (table 4). These differences seem to be due to the neglect ofintramolecular correlation effects in the SCF +dispersion approach. When we varied R in CEPA calculations in the DZP’ basis at 9 = 30° the well depth came out to be 4.75 kcal rnol-l the same value as in the corresponding multipole-corrected SCF +dispersion calculations (4.74 kcal mol-l) but the equilib- rium distances differed considerably (3.00 as against 2.93 A). This shows that the intramolecular correlation is important in obtaining a correct equilibrium distance.The DZP’ basis through its lack of high-exponent polarization functions under- estimates these intramolecular correlation effects thus leading to a larger CEPA interaction energy than in the EZPP basis. 134 CI CALCULATIONS CONCLUSIONS (i) In dimer CI calculations on van der Waals complexes neither the size-consistency corrections nor the BSSE may be neglected. (ii) The best way to estimate the BSSE in CI calculations is to perform the ghost-orbital calculation in the monomer basis augmented by only the virtual space of the ghost as suggested by Daudey et al.24(iii) If the occupied orbitals in the CI BSSE calculation are obtained in an SCF ghost calculation (large-basis monomers) the resulting interaction energies are more basis-set independent than when using the occupied orbitals of the isolated monomers.(iv) The calculated potential-energy surface for Ar-HCl is well within the experimental error limit of 50phartree at the van der Waals minimum but its anisotropy shows an inbalance. (v) The estimated van der Waals well depth for the H20 dimer is 4.9 kcal molt1 which is considerably less attractive than the experimentally derived result of 5.4 kcal mol-l but just within the bounds of experimental error. (vi) For weak van der Waals complexes like Ar-HCl the SCF +dispersion approach only yields approximate results. For stronger interacting systems this approach after correction for the errors in the SCF multipoles can yield acceptable qualitative results. We thank V. R. Saunders for helpful discussions for making available the ATMOL system and for help with the calculations and we thank G.Chalasinski for helpful discussions. The computer time was made available through financial support of the S.E.R.C. the Dutch Supercomputer project Z.W.O. and the University of Utrecht. C. Votava and R. Ahlrichs in Proc. 14th Jerusalem Symp. Quantum Chemistry and Biochemistry ed. B. Pullman (Reidel Dordrecht 1981). G. H. F. Diercksen W. P. Kraemer and B. 0. Roos Theor. Chim. Acta 1975 36 249. 0. Matsuoka E. Clementi and M. Yoshimine J. Chem. Phys. 1976,64 1351. H. Lischka J. Am. Chem. SOC.,1974,% 4761. M. D. Newton and N. R. Kestner Chem. Phys. Lett. 1983 94 198. E. Clementi and P. Habitz J. Phys. Chem. 1983 87 2815. ' G.C. Maitland M. Rigby E. B. Smith and W. Wakeham in Intermolecular Forces Their Origin and Determination (Clarendon Press Oxford 198 1). P. E. M. Siegbahn J. Chem. Phys. 1980,72 1647. P. E. M. Siegbahn Int. J. Quantum Chem. 1980 18 1229. V. R. Saunders and J. H. van Lenthe Mol. Phys. 1983,48 923. lo J. M. Hutson and B. J. Howard Mol. Phys. 1982,45 769. l1 E. W. Boom Ph.D. Thesis (Amsterdam 1981). E. W. Boom and J. van der Elsken J. Chem. Phys. 1980 73,15. l2 T. van Dam Ph.D. Thesis (Utrecht 1984). l3 S. F. Boys and F. Bernardi Mol. Phys. 1970 19 553. l4 P. H. Smit J. L. Derissen and F. B. van Duijneveldt Mol. Phys. 1979,37,501; E. Kochanski J. Am. Chem. SOC.,1978 100 6971. l5 S. L. Price and A. J. Stone Chem. Phys. Lett. 1979 65 127. l6 J. A. Pople R.Seeger and R. Krishnan Int. J. Quantum Chem. 1577 S11 149. l7 S. R. Langhoff and E. R. Davidson Int. J. Quantum Chem. 1974,8 61. C. Zirz and R. Ahlrichs in Electron Correlation Proc. Daresbury Study Weekend (1979) ed. M. F. Guest and S. Wilson (Daresbury publication DL/SCI/R14 1980); W. Meyer J. Chem. Phys. 1972 58 1017; W. Meyer Int. J. Quantum Chem. 1971 5 341. l9 S. Huzinaga Approximate Atomic Functions IZ (Department of Chemistry University of Alberta Canada 1971). 2o A. Veillard Theor. Chim. Acta 1968 12 411. 21 J. G. C. M. van Duijneveldt-van de Rijdt and F. B. van Duijneveldt J. Mol. Struct. 1982,89 185. 22 G. Herzberg Molecular Spectra and Molecular Structure (van Nostrand New York 1957). 23 W. S. Benedict N. Gailar and E. K. Plyler J. Chem.Phys. 1956 24 1139. 24 J. P. Daudey P. Claverie and J. P. Malrieu Int. J. Quantum Chem. 1974 8 1. 25 Th. P. Groen and F. B. van Duijneveldt to be published. VAN LENTHE VAN DAM VAN DUIJNEVELDT AND KROON-BATENBURG 135 46 M. Gutowski G. Chalasinski and J. G. C. M. van Duijneveldt-van de Rijdt Znt. J. Quantum Chem. 1984 to be published. 27 G.Chaiasinski J. H. van Lenthe and Th. P. Groen Chem. Phys. Lett. 1984 110 329. 28 L. M. J. Kroon-Batenburg Ph.D. Thesis (Utrecht 1985); L. M. J. Kroon-Batenburg and F. B. van Duijneveldt J. Mol. Struct. 1984 in press. 29 F. B. van Duijneveldt ZBM Research Report RJ 945 (IBM San Jose California 1971 j. 30 J. H. van Lenthe and F. B. van Duijneveldt J. Chem. Phys. 1984 to be published. J. M. Hutson and B. J. Howard J. Chem. Phys. 1981 74 6520; S. E. Novick K. J. Janda S. L. Holmgren M. Waldman and W. Klemperer J. Chem. Phys. 1976 65 11 14. 32 B. Schramm and U. Leachs Ber. Bunsenges. Phys. Chem. 1979 83 847.

ISSN:0301-5696    

DOI:10.1039/FS9841900125

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1984 19 137-147 MCSCF Gradient Calculation of Transition Structures in Organic Reactions BY FERNANDO AND ANDREA BERNARDI BOTTONI Istituto di Chimica Organica Universita di Bologna Bologna Italy AND JOSEPH AND MICHAEL J. w. MCDOUALL A. ROBB* Department of Chemistry Queen Elizabeth College London AND H. BERNHARD SCHLEGEL Department of Chemistry Wayne State University Detroit Michigan U.S.A. Received 16th July 1984 The applicability of MCSCF gradient methods to the calculation of transition structures and diradicaloid intermediates is discussed. It is shown how the diabatic surface model provides a useful criterion for the choice of the valence space in the MCSCF method and also provides useful qualitative information about the electronic rearrangement associated with various transition states.These ideas are then applied to the synchronous and asynchronous 1,3-dipolar cycloaddition of fulminic acid to acetylene. One of the major development areas in quantum chemistry in recent years has involved the computation and characterization of the intermediates and transition structures for model organic reactions. Often these structures are diradicaloid in nature and are not described even qualitatively at the SCF level. The investigation of this type of species has been facilitated by the development of the MCSCF method [for a comprehensive review see ref. (l)] and gradient-optimization techniques. In recent work we have been involved in the development of MCSCF2 gradient3 programs and in the subsequent analysis of the reaction profile using diabatic In the present communication we shall attempt to present a methodological strategy that can be used to characterize potential surfaces that may contain diradicaloid structures.The application of this strategy will be illustrated with a study of transition structures occurring in the 1,3-dipolar cycloaddition of fulminic acid to acetylene. The major difficulties in the application of MCSCF gradient methods to the computation of transition molecular structures lie (1) in the choice of configuration space of the MCSCF (i.e. the choice of the orbitals to be used in the CI expansion) and (2) in the determination of regions of the molecular potential surface in which to search for critical points.In this work we use a model for the description of saddle points based upon the surface of intersection of two diabatic surfaces (one associated with the reactants and one with products). Clearly the reference CI expansion must be chosen so that each diabatic surface is represented equally well. The diabatic surfaces can be computed approximately using the methods discussed in ref. (4) and (5). The subsequent geometry optimization is then performed using the methods of ref. (3). 137 CALCULATION OF TRANSITION STRUCTURES THEORETICAL METHODS The standard methods for the optimization of molecular geometries (e.g. the Newton-Raphson method with updating of the Hessian as used in the GAUSSIAN 80 series of programs) work quite well for diradicaloid transition structures (saddle points) provided one uses MCSCF gradient3 methods.However one must start the optimization procedure with a molecular structure where the force-constant matrix has one direction of negative curvature. Given the large number of molecular parameters in typical model organic reactions the location of these regions of the molecular potential surface can be quite difficult. Further it is obviously important not to make apriori assumptions about the nature of the reaction coordinate and thus miss possible transition structures. In recent work we have developed an MCSCF procedure4* for the computation of a diabatic surface in which the adiabatic surface of the reaction is obtained from the interaction of two diabatic surfaces.The main feature of this procedure is that each diabatic surface is associated with the specific bonding situation of either the reactants or products and the transition structure lies on the surface of intersection of the two diabatic surfaces. In ref. (5) we have shown that the geometries of the transition structure of the cyanate-isocyanate rearrangement the 1,2 sigmatropic shift in propene the S,2 reaction of H and CH and the addition of singlet methylene to ethylene correlate very accurately with the intersection of appropriate diabatic curves. In the case of the sigmatropic shift we were able to locate the transition structure a priori from preliminary diabatic-surface calculations. The significant feature of this approach is that not only does it furnish us with an excellent starting transition structure but also it provides some insight into the origin of the reaction barrier and conformational preference of possible reaction intermediates.The latter follows from the model used to define the diabatic surfaces themselve~.~?~ We now give some brief discussion of this model. One assumes that for a reaction X+Y the adiabatic surface can be described in terms of two sets of variables R and R,. We also assume that we can model this reaction with two diabatic surfaces E,(R, R,) and Ey(R, Ry).The surface Excorrectly describes the energy minimum of the species x and represents the bonding situation of the species x at all regions of configuration space (including the minimum of the surface E,).The definition of the two diabatic surfaces themselves involves (1) the use of fragment MCSCF orbitals of x and y and (2) a partition of the full valence CI space (CASSCF) into packets. Each packet consists of an isolated fragment configuration (either a Heitler-London or no-bond configuration in the language of valence-bond theory) plus all possible one-electron- transfer configurations. Thus for example in the cycloaddition of two ethylenes one packet (reactants) corresponds to two ethylene molecules in their ground states (a no-bond configuration) and the second (products) corresponds to two triplet excited Fthylenes coupled to an overall singlet (a Heitler-London configuration). The truncation of each packet at one-electron-transfer configurations will clearly begin to break down at small interfragment separations where more than one electron transfer will become very important and other locally excited configurations will begin to make large contributions.However the truncation at one-electron transfer has the feature that the packets are mutually exclusive and hence approximately orthogonal. With this definition at hand one can compute a diabatic surface4y5 by performing an MCSCF calculation on each packet separately. The possible transition structures correspond to minima on the surface of intersection of these diabatic surfaces. It is F. BERNARDI et al. apparent that the surface of intersection of two diabatic surfaces is essentially a conformational hypersurface thus the conformational minima (corresponding to various possible transition structures) allow a semi-classical analysis in terms of electrostatic polarization exchange repulsion and charge-transfer energies.In ref. (4) we have shown how this type of analysis can be performed for MCSCF wavefunctions. APPLICATIONS During the past two decades 1,3-dipolar cycloaddition reactions have become a general method for the synthesis offive-membered rings. However there is considerable controversy concerning the reaction mechanism.** There are two obvious alternatives a synchronous mechanism involving two-bond cycloaddition via an aromatic transition state or an asynchronous mechanism involving the formation of a diradicaloid intermediate. In the latter case one must find two transition states corresponding to the formation of the first and second bonds.While a synchronous cyclic transition structure can be determined by SCF methods,lo?l1 the possibility of an extended diradical intermediate can only be investigated using methods that include at least the two configurations necessary to describe the diradical. Recently,12 the addition of fulminic acid (1,3-dipole) to acetylene (dipolarophile) has been investigated using multireference CI methods. In these calculations a low-energy extended diradical intermediate was found. In the MCSCF calculations to be discussed here the synchronous and asynchronous processes can be treated with equal accuracy at the same level of theory. This problem is of considerable theoretical interest.First there are a very large number of possible 1,3-dipole/dipolarophile combinations with very different (computed) barriers for the synchronous process. l3 Secondly it should be apparent that the diabatic-surface method discussed above is capable of rationalizing the mechanistic controversy. In this formalism the preference synchronous/asynchronous is simply related to the relative positions of the two possible minima of the surface of intersection of the two diabatic surfaces corresponding to reactants and products. These facts in turn can be rationalized using the methods described in ref. (4). We shall begin our discussions with a qualitative discussion of the diabatic surface model as it applies to the 1,3-dipolar cycloaddition of fulminic acid to acetylene.Initially we are concerned with the choice of the valence orbital space (i.e.those active orbitals whose occupancy must be allowed to have values other than 2 or 0 and which will form the basis of the CI expansion used in the MCSCF calculations). This discussion is most conveniently given in terms of the orbitals of the isolated fragments and formulated in a valence-bond type of language. The fulminic acid can be thought of as having two allyl-like systems of three n orbitals referred to as the in-plane n (incipient 0 bonds) and out-of-plane n (incipient delocalized norbitals) sets as the two fragments approach each other. Similarly the acetylene has two sets of ethylene n orbitals which we shall refer to as the in-plane and out-of-plane set.The isolated fragments must have the orbital occupancy shown in fig. l(a) for the in-plane and out-of-plane n systems. For the product isoxazole the in-plane n-orbital occupancy must correspond to the promotion of one electron in each of the fragments from HOMO to LUMO. The unpaired electrons must be spin-coupled to a state of triplet spin within a fragment and these two triplet states subsequently spin-coupled to a singlet in order to describe two new 0 bonds as a valence bond Heitler-London configuration. For the out-of-plane n system of the product isoxazole the fragment n systems retain the configuration shown in fig. l(a) (ie.there is no requirement to uncouple and recouple the spins to describe the isoxazole n system). From the preceding argument it should be obvious that (in terms of the fragment CALCULATION OF TRANSITION STRUCTURES HCNO HCCH HCNO HCCH $3 -Fig.1. (a) No-bond and (b) Heitler-London configurations for the singlet-singlet and triplet-triplet diabatic surface for the addition of fulminic acid to acetylene. orbitals) one requires a valence space consisting of four in-plane n orbitals (the HOMO and LUMO of each fragment) in order to describe the product the transition state and the reactants with equal accuracy. Thus the valence space to be used in the MCSCF calculations should contain these four cr orbitals at least. Furthermore the product isoxazole the intermediate diradical and the transition state for the formation of the second bond should be dominated by the diabatic surface associated with the Heitler-London configuration (for the in-plane z orbitals) shown in fig.1 (b).We shall refer to this surface henceforth as the triplet-triplet surface. On the other hand the reactants (fulminic acid and acetylene) must be dominated by the diabatic surface associated with the no-bond configuration (for the in-plane z system) corresponding to the configuration shown in fig. 1 (a).We shall refer to this surface in subsequent discussions as the singlet-singlet diabatic surface. The transition structures for the synchronous formation of isoxazole and for the formation of the first cr bond in the asynchronous process should be associated with the intersection of these two diabatic surfaces. We can now outline our computational strategy for the characterization of the critical points on the surface for the cycloaddition of fulminic acid to acetylene.The orbitals for the initial MCSCF calculation for each geometry optimization were obtained from MCSCF calculations on the isolated fragments. These orbitals were then orthogonalized as follows (1) the core orbitals (doubly occupied in all reference CI configurations) the valence orbitals and the virtual orbitals were each symmetrically orthogonalized within each set and (2) the valence orbitals were then Schmidt orthogonalized to the core and the virtual orbitals subsequently Schmidt orthogonalized to the core and then to the valence orbitals. This procedure unambiguously defines the valence-orbital set. Note that the SCF orbitals are not suitable as starting orbitals because the virtual orbitals tend to be very diffuse with extended basis sets and the appropriate weakly occupied orbitals do not usually correspond to the lowest-energy occupied orbitals of an SCF calculation.All of our geometry optimizations are then carried out with a full CI in the space of the four in-plane 7z orbitals (corresponding to HOMO and LUMO of each fragment). In order to improve the energetics MCSCF calculations were also carried out (at the geometry just obtained) with a valence space that consisted of the HOMO-LUMO out-of-plane n orbitals as well corresponding to eight valence orbitals and 1764 configurations. This calculation should account for some of the dynamic correlation of the delocalized n system.The calculations were carried out at the STO-3G and 4-31G basis set level. At the STO-3G level each critical point was characterized by computing the hessian (by finite difference) in the subspace consisting of the interfragment geometrical parameters and F. BERNARDI et al. the CNO angle of fulminic acid which was strongly coupled to these variables. This hessian was then updated numerically in the 4-3 1G geometry optimizations. In fig. 2(a)-(f) we illustrate the geometries of the critical points located by the procedure described above. Table 1 contains the geometrical parameters of fulminic acid and acetylene obtained at the same level of theory. The absolute and relative energetics are summarized in table 2. It can be seen that there is a cyclic transition state [fig.2(b)] for the synchronous two-bond addition to give isoxazole [fig. 2(a)] an extended diradicaloid transition state [fig. 2(c)] for the formation of the first bond leading to a diradical intermediate which exists in a trans form [fig. 2(d)]or a cis form [fig. 2(e)] and a second diradicaloid transition state [fig. 201 which connects the cis form of the diradical intermediate [fig. 2(e)] with the product isoxazole. It can be seen that the STO-3G geometries are in quite good agreement with the 4-31G results indicating that the STO-3G basis is capable of giving a good qualitative description of the surface. The geometry of the cyclic synchronous transition state is in good agreement with that obtained in ref. (1 1) at the SCF level in the same basis.This is to be expected since this species is well described at the SCF level. The description of the diradicaloid region of the surface obtained in the present work is slightly different from that obtained in ref. (12) and deserves some comment. The geometry of the diradicaloid transition state [fig. 2(c)] that connects reactants and the diradical intermediate is in qualitative agreement with the structure obtained in ref. (12) at the UHF and 3x3 CI level (in the 4-31G basis). The major difference is that the interfragment C(3)-C( 1) distance is considerably shorter in the present compu- tations. However in the present calculations we find two different structures for the diradical intermediate a trans form [fig. 2(d)] and at slightly lower energy a cis form [fig.2(e)] that differ only in the arrangement of the C-C-H angles in the acetylene fragment.In the present work we have also searched for a cis form of the transition state shown in fig. 2(c). Since the calculation was clearly converging to the trans form it was abandoned Thus it seems likely that the transition state for the formation of the first bond connects the reactants with the cis form [fig. 2(e)] of the diradical intermediate and the trans form [fig. 2 (d)] represents a subsidiary minimum accessible uia inversion at C(2). The cis form of the diradical intermediate [fig. 2(e)] is again in good agreement with the structure found in ref. (12) where an improved estimate (obtained from large-scale multireference CI calculations) for the transition structure for the asynchronous process (formation of the first bond) was obtained by inter- polating geometries along a path connecting the cis form of the intermediate and the transition structure located at the SCF+ 3x3 CI level.An interpolation based on the trans form would have yielded a structure in better agreement with the present fully optimized transition state. Finally a transition structure has been obtained for the formation of the second bond in the asynchronous process [fig. 2(f)]. This lies energetically below the transition structure for the formation of the first bond in the asynchronous process. The total and relative (to reactants) energies are given in table 2 along with the CI results obtained from ref.(1 1) and (12) for comparison. It can be seen that the relative energetics for the synchronous process are in good agreement with those obtained from ref. (1 1) using a CI treatment that included replacements from two a’ and three aff orbitals. The MCSCF reaction barrier (synchronous process) of 26-28 kcal mol-l will be lowered by the inclusion of correlation in more a’ orbitals.11’l2 However it must be stressed that one must be careful in comparing CI calculations for the synchronous and diradicaloid regions of the surface. In the diradicaloid region of the surface SCF orbitals are far from optimum for use in the CI expansion even if a multireference expansion is used. Thus one must be certain that one is describing both regions of CALCULATION OF TRANSITION STRUCTURES Fig.2. Optimized geometries computed at MCSCF/STO-3G (no superscript) and MCSCF/4-3 1G (asterisk) (a) isoxazole (b) synchronous cyclic transition leading from fulminic acid and acetylene to isoxazole (c) asynchronous transition state corresponding to the formation of the first bond (i.e.connecting reactants and the trans diradical intermediate) (d) trans diradical intermediate (e)cis diradical intermediate and (f)asynchronous transition state connecting the cis diradical intermediate to the product isoxazole. 143 F. BERNARDI et al. 144 CALCULATION OF TRANSITION STRUCTURES Table 1. Optimized geometries computed at MCSCF/STO-3G and MCSCF/4-3 1G for fulminic acid (HCNO) and acetylene (HCCH) (atoms labelled as in fig.2) bond length/A molecule bond STO-3G 4-31G HCNO C(3)-H(3) 1.06 1.05 C(3I-W 1) 1.17 1.14 N( 1)-O( 11 1.29 1.26 HCCH C( 1)-H( 1) 1.07 1.05 C( 1)-C(2) 1.18 1.20 Table 2. Total and relative energies for the reaction of fulminic acid with acetylene MCSCFa MCSCFa MCSCF* structure STO-3G 4-31G 4-31G ref. (1 ref. (12)d total energies/hartreee HCCH +HCNO -241.3130 -244.1360 -244.2394 -244.5444 -244.6181 cyclic synchronous -241.3056 -244.0952 -244.1941 -244.5062 -244.5973 TSf [fig. Wl asynchronous first -241.2950 -244.0871 -244.1841 -244.603 1 TSf [fig. 2(c)l trans diradical -241.37 15 -244.1214 -244.2005 intermediate [fig. 2(d)] cis diradical -241.3746 -244.1246 -244.1977 -244.608 1 intermediate [fig. 2 (e)] asynchronous second -241.3636 -244.1 123 --TSf [fig.2cf)l isoxazole [fig. 2 (a)] -241.521 1 -244.2426 -244.3150 relative energies/kcal mol-1 HCCH +HCNO 0.0 0.0 0.0 0.0 0.0 cyclic synchronous 4.6 26.0 28.4 24.0 13.0 TSf [fig. 2(4l asynchronous first 11.3 30.7 34.7 -9.39 TSf [fig. 2(c)l trans diradical -36.6 9.1 24.4 intermediate [fig. 2(d)] cis diradical -38.6 7.2 26.2 -8.20 intermediate [fig. 2 (e)] asynchronous second -31.7 14.9 --TSf [fig. 2cf)l isoxazole [fig. 2 (a)] -130.6 -66.9 -47.4 -69.1 -63.4 a Full CI (CASSCF) in four a’ orbitals and four electrons. Full CI (CASSCF) in four a’ and four a” orbitals and eight electrons. CI using SCF orbitals all single and double excitations from two a’ and three a’’ orbitals. CI (two reference second-order Moller-Plesset) using SCF orbitals.1 hartree = 2625 kJ mol-l. f Transition state. F. BERNARDI et aZ. 145 the surface with equal accuracy. The MCSCF energetics clearly do not include any true dynamic correlation. Thus the exothermicity and the reaction barriers will not be accurately reproduced. Nevertheless the valence space has been chosen in this work so that the region of the synchronous transition structure and the region of the diradicaloid structures are described with equivalent accuracy. Thus one has some confidence in the relative energetics of the high-energy regions of the synchronous as against asynchronous pathways. From the MCSCF results presented in table 2 it can be seen that the transition state for the synchronous process is predicted to be lower than for the first step of the asynchronous process at all levels of computation.Furthermore the second transition state in the asynchronous processes lies at a lower energy than the first and the second barrier is predicted to be quite small. Thus both the synchronous and asynchronous processes could be consistent with retention of stereochemical information (in the case of substituted adducts given the same relative orientation of approach). Finally we turn to an a posteriori rationalization of some of these results in terms of the diabatic-surface model. This is instructive because it is indicative of the success of the model when used to locate transition structures apriori and it also complements the qualitative discussion given previously.In fig. 3-5 we give cross-sections through the diabatic surfaces chosen to pass through the cyclic synchronous transition structure (fig. 3) the first asynchronous structure (fig. 4) and the trans diradical -240.80 -241.00 -24140 . 2.0 2.5 3.0 3.5 4.0 RXdB Fig. 3. Diabatic surface cross-section passing through the synchronous cyclic transition state (-) singlet-singlet (--) triplet-triplet. intermediate minimum (fig. 5). The figure in each case shows the diabatic energies obtained along the coordinate (RXN)corresponding to a rigid dissociation along the line connecting N(l) of the fulminic acid fragment and the centre of the C(l)-C(2) bond. In fig. 3 (cyclic synchronous transition state) one observes that the singlet-singlet diabatic curve is very flat and begins to become repulsive only in the region of the transition state (RxN= 2.74 A) itself.In contrast in fig. 4 (corresponding to the first diradicaloid transition structure) the singlet-singlet diabatic surface quickly becomes repulsive and intersects the attractive triplet-triplet surface in the region of the transition structure (RxN = 3.41 A). The intersection of the singlet-singlet and triplet- triplet surfaces in the synchronous process occurs in a region where the truncation at one-electron transfer is no longer valid. However in fig. 5 (diradical intermediate) CALCULATION OF TRANSITION STRUCTURES -24 0.80 1 t -241.00 irl -241.20 -241.40 t 2.0 2.5 3.0 3.5 4 .O RXdB Fig.4. Diabatic surface cross-section passing through the asynchronous transition state for the formation of the first bond (-) singlet-singlet (--) triplet-triplet. -240.60--240.80 -a 2 -241.00 s -. t 4; 9 I -241.20 : -241.40 Y 2.0 2.5 3.0 3.5 4.0 RXdB Fig. 5. Diabatic surface cross-section passing through the trans diradical intermediate (--) singlet-singlet (--) triplet-triplet. the geometry of the fragments is now such that the triplet-triplet diabatic surface lies below than the singlet-singlet for all values of the interfragment separation. Thus the synchronous transition state and the first asynchronous transition state correspond to regions of intersection of the diabatic surfaces. However the second asynchronous transition state (from the diradical intermediate to isoxazole) lies (like the diradical intermediate itself) on the triplet-triplet diabatic surface.The synchronous two-bond cycloaddition appears to be feasible in this situation because of the fact that the singlet-singlet diabatic surface is not strongly repulsive due to the stabilization of the extended n system. F. BERNARDI et al. CONCLUSIONS It is clear that SCF methods cannot provide an adequate description of both the synchronous and asynchronous pathways for 1,3-dipolar cycloaddition reactions whereas MCSCF methods will provide an accurate description of both processes. For the asynchronous process MCSCF is necessary both to describe the avoided intersection of the singlet-singlet and triplet-triplet surfaces and because the intermediate is diradicaloid.The use of the diabatic surface model provides a useful scheme on which to base one’s selection of the valence space and also provides some a posteriori rationalization of the topology of the surface. Note added in proof We have also located (at the STO-3G level) the transition state connecting the cis [fig. 2(e)]and trans [fig 2(f)]diradical intermediates. The energy of this transition state is 7.5 kcal mol-l above that of the trans intermediate. All computations were run on the CDC 7600 or the CRAY 1s computers at the University of London national computing centre and on the VAX 11/750 at Queen Elizabeth College. The cooperation of both computer centres is gratefully acknowledged.The MCSCF gradient programs have been installed as part of the GAUSSIAN 8014suite of codes. We also acknowledge the financial support of NATO under grant no. RG 096.81. A. B. thanks the Royal Society for the award of a visiting fellowship. J. Olsen D. L. Yeager and P. Jorgensen Adv. Chem. Phys. 1983 55 1. R. H. A. Eade and M. A. Robb Chem. Phys. Lett. 1981,83 362. H. B. Schlegel and M. A. Robb Chem. Phys. Lett. 1982 93,43. F. Bernardi and M. A. Robb Mol. Phys. 1983,48 1345. F. Bernardi and M. A. Robb J. Am. Chem. Soc. 1984 105 54. F. Bernardi M. A. Robb H. B. Schlegel and G. Tonachini J. Am. Chem. SOC.,1984 106 1198. F. Bernardi A. Bottoni and M. A. Robb Theor. Chim. Acta 1984 64 259. * R. Huisgen Angew. Chem. 1963 2 565. R. A.Firestone J. Org. Chem. 1968 33. 2285. lo D. Poppinger Aust. J. Chem. 1976 29 465. l1 A. Komornicki J. D. Goddard and H. F. Schaefer J. Am. Chem. SOC.,1980 102 1763. l2 P. C. Hiberty G. Ohanessian and H. B. Schlegel J. Am. Chem. SOC.,1983 105 719. l3 M. Sana and G. Leroy J. Mol. Struct. 1982 89 147. l3 GAUSSIAN 80 J. S. Binkley R. A. Whiteside R. Krishnan R. Seeger D. J. DeFrees H. B. Schlegel S. Topiol L. R. Kahn and J. A. Pople Quantum Chemistry Program Exchange 1981 13 406.

ISSN:0301-5696    

DOI:10.1039/FS9841900137

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem. SOC.,1984 19 149-163 Studies of Molecular States using Spin-coupled Valence-bond Theory BY DAVIDL. COOPER Department of Theoretical Chemistry University of Oxford South Parks Road Oxford OX1 3TG AND JOSEPH GERRATT* Department of Theoretical Chemistry University of Bristol Bristol BS8 ITS AND MARIO RAIMONDI Istituto di Chimica Fisica University of Milan 19 via Golgi 20133 Milano Italy Received 3rd October 1984 The spin-coupled wavefunction for an N-electron system is described and the technical aspects (formation of density matrices energy minimisation generation of virtual orbitals) briefly outlined. The physical interpretation provided by this model is illustrated by calcula- tions on the potential-energy surface of the C++H system.The spin-coupled theory is quanti- tatively refined at the spin-coupled valence-bond stage. Results for calculations on CH+ LiHe+ and BH; are presented which show that at least for these systems chemical accuracy may be attained with 20&500 non-orthogonal 'spin-coupled structures ' (configuration state functions). Each eigenvector is dominated by one or two such structures thus providing visuality without sacrificing accuracy. 1. INTRODUCTION The basic idea of valence-bond (VB) theory is very simple. The wavefunction for the electrons in a molecule is constructed directly from the wavefunctions of the constituent atoms. This implements in a clear-cut way a large part of the experience of chemistry. VB theory provides a coherent explanation of a whole range of chemical phenomena the valency of atoms in different states the saturation of valency the directional properties of bonds and so on.Concepts such as avoided intersections between potential curves arising from 'zeroth-order ' states (e.g. ionic and covalent) form a basic part of our general mode of description of fundamental chemical processes.t This qualitative VB theory runs into difficulties as in the well known case of 0 and in the description of conjugated systems where simple molecular-orbital (MO) theory provides a more natural picture. Nevertheless it is clear the VB concepts are full of useful chemical and physical insights which exert a profound influence upon our view of molecular states and their evol~tion.~ However the translation of these attractive qualitative considerations into numerical results has always been disappointing.A very large number of terms is needed in the 7 A recent example is that of the Be molecule,' which possess an unexpectedly deep well of 714 cm-'. This can be understood as arising from the interaction between two structures the ground state (Be('S); Be(lS)} which gwes only repulsion and (Be(sp; 3P);Be(sp; 3P)>,which produces bonding.' 149 STUDIES OF MOLECULAR STATES wavefunction including many ionic structures whose presence in pre-eminently covalent situations is to say the least unexpected. The reason for this is clear. On molecule formation the participating atomic states are deformed. This deformation is expressed by means of ionic or charge-transfer structures as was first demonstrated by Coulson and Fi~cher.~ Consequently as long as one uses linear combinations of undeformed atomic states to describe molecular states convergence of the wavefunction will be very slow.These shortcomings of the classical VB theory are overcome by the spin-coupled VB theory at least for small systems. The spin-coupled wavefunction is of VB type but uses optimised orbitals whose forms vary with nuclear geometry. The wavefunction and its physical interpretation are described in section 2 together with an outline of the technical implementation of the theory. We illustrate the concepts which emerge from this model in a recent study of parts of the potential surfaces of the C++H system.The spin-coupled wavefunction is quantitatively refined by the addition of excited spin-coupled structures to form the spin-coupled VB wavefunction. This extension of the theory is described in section 3. Finally in section 4 the spin-coupled VB theory is applied to a study of the lC+ states of CH+ the lZ+states of LiHe+ and the potential surfaces of the B+ +H system. We show that at least for these systems the present modern form of VB theory is capable of providing results of the same accuracy as the highest quality MCSCF-CI wavefunctions while retaining an essential chemical and physical visuality. 2. THE SPIN-COUPLED WAVEFUNCTIONS5* In order to retain the fundamental physical content of the VB approach while allowing for a compact description of the deformation of atomic states we construct the following wavefunction for a system of N electrons ySM = cSkdN! d{41(1)42(2) **.4N(N)@F~M;L) k {#I 42 ...#N)-The orbitals #,(i) = #,(ri) are a set of Ndistinct non-orthogonal spatial orbitals. They are represented as quite general linear combinations of basis functions m 4 = c cppxp p=1 where m is the number of such basis functions much as in MO theory. The coefficients cppare determined simultaneously with the spin-coupling coefficients CSk by a second-order energy-minimisation procedure described below. The @$ k are a set of orthonormal N-electron spin functions which are eigen- functions of S2and S,with eigenvalues Sand M,respectively as shown. The index k denotes the specific mode of coupling the individual electron spins to form the final resultant spin S,and wavefunction (2.1) is a linear combination of all the modes of coupling that are relevant for a particular problem.The spin functions are usually constructed according to the branching diagram as described in ref. (5) and (6). However an alternative basis may be used if it makes physical sense to do so. The operator d is the usual N-electron antisymmetrising operator. Wavefunction (2.1) is extremely flexible. The practical implementation of the model represented by it rests upon a number of technical developments which we briefly D. L. COOPER J. GERRATT AND M. RAIMONDI 151 describe below. Further details may be found in ref. (5) and (6). The total energy corresponding to YsMis given by The operators hand g are the usual 1-and 2-electron operators and A D(pI v) and D(pvI az) are respectively the normalisation integral and 1-and 2-electron density matrices.These are connected by relations of the form N A = c D(Plv)<4pI4v> v=1 N DblP2 I Vl v2) = "3 z =1 mu,P2 P3I Vl v2 v,) (#p3I 4J N DOIl...pN-lIV1...VN-l)= x D(~~.*.~N-l~~~vl"'vN-lVN)(~p~I~V~)' vN=l It is understood that no index on a given side of the vertical bar in D(... 1 ...) is repeated. Eqn (2.4) provide the basis for the simultaneous computation of all the necessary density matrices by an extremely fast recurrence technique. We begin from the N-electron density matrix D(pl...pN 1 v ...vN) which is a list of purely group- theoretical quantities,8 and form directly from it the 4-electron density matrix.From this we form in succession the 3- 2- and 1-electron density matrices and A as specified in eqn (2.4). This procedure is very convenient since all of these quantities are required in the calculation. Implementation of this recurrence method has led to the development of very efficient methods of list processing which are well adapted to the newer generation of machines with large virtual memory. If we write the set of coefficients (cpp,cSk)as a single vector c the total energy E is minimised by the 'stabilised Newton-Raphson' (C+a1)6c= -g. (2.5) Here g is the vector of gradients of E with respect to the parameters cp and CSk Gis the matrix of second derivatives of EAq with respect to the same parameters and 6cis the vector of corrections to c.The orbital part of the gradient is given by N z cv4"-&p4p (P = 1,2 ...,N 1. (2.6) v=1 The operators RpVcontain 1- 2-and 3-electron density matrices and the parameter cp is interpreted as the average energy of orbital 4p in the presence of the other electrons. The second-derivative matrix possesses the following structure The diagonal block G(p,v) represents the second derivative of E with respect to the coefficients cpp,cvqoccurring in the orbitals 4 and d,, and contains I- 2- 3-and STUDIES OF MOLECULAR STATES 4-electron density matrices. Block G(c, c,) represents the second derivatives of E with respect to the spin-coupling coefficients CSk,and the off-diagonal blocks C@,c,) G+@,c,) contain the cross-derivatives of E with respect to orbital and spin-coupling coefficients.The parameter ain eqn (2.5) is given by a= -e0+0.5(gIg) where e is the lowest eigenvalue of G (assumed negative) and (g Ig) is the length of the gradient vector. As soon as e becomes positive a is put to zero and the procedure becomes the pure Newton-Raphson method. We have now accumulated much experience with the stabilised Newton-Raphson procedure. Our usual strategy is to begin a calculation at a point on the surface where all the atoms are well separated. Using an intuitive starting guess for c convergence is attained in 10-15 iterations. All coefficients are refined to at least i.e.the energy is correct to Convergence at neighbouring points is then reached in 5 or 6 iterations.Local minima are found occasionally but in all cases so far the corresponding wavefunction turns out to possess a clear-cut physical interpretation usually that of a nearby excited state. The orbitals 4pare almost always semi-localised the extent of the deviation from pure atomic form varying with nuclear geometry. As the interatomic separations increase the distortions diminish so that at sufficiently long range and provided that the correct spin couplings have been included the spin-coupled function (2.1) becomes identical to an antisymmetrised product of undeformed atomic states. The physical significance of the spin-coupling coefficients CSk is that they allow for a progressive recoupling of electron spins as the interacting atoms approach.At large interatomic separations the CSk coefficients assume values characteristic of the mode of coupling of the spins in each isolated fragment. At shorter ranges the coupling changes over to one which typifies the newly formed molecule. In the case of a chemical reaction the spin coupling changes from one characteristic of reagents to a coupling associated first with an intermediate (if there is such) and then with products. These variations in the spin-coupling coefficients are accompanied by corresponding changes in the orbitals. The chief qualitative finding to emerge from our work so far is that in nearly all cases the spin recoupling and orbital deformations occur quite suddenly over a narrow range of internuclear distances frequently between ca.4.50 and 5.5a0. Beyond the ‘critical radius’ of ca. 5.5a0 the system essentially consists of reagent atoms and molecules. At distances < ca. 4.5a0 both the orbitals and spin couplings are substantially characteristic of the product molecule or of the intermediate. This phenomenon is clearly shown in a recent study of the reactionlo C++H,-,CHl-+CH++H. (2.9) The mechanism and rate of formation of CH; from C+ ions and H molecules in the gas phase at very low temperatures are of great significance in interstellar chemistry.ll The overall reaction however is endoergic. Sections of the potential-energy surfaces have been previously calculated by Liskow et a2.12 and Pearson and RoueF3 using the MO-CI method. Extensive MCSCF-CI calculations have also been carried out more recently by Sakai et aZ.14Spin-coupled and spin-coupled VB studies of this system are in progress and we here consider the perpendicular approach of C+(,P) to H,.The basis set consists of triple-zeta (1s 12s I2p) functions on C+ and double-zeta (1s 12s I2p) STO on hydrogen atoms H and H,. This should be sufficient to account for the polarisation of H by C+. D. L. COOPER J. GERRATT AND M. RAIMONDI tH’ b H* Fig. 1. Coordinate system for C+ +H,. The coordinate system employed is shown in fig. 1. Ignoring the two core electrons on C+ the spin-coupled wavefunctions are of the form WCW) =@lo2 03 04 451 (2.10) Orbitals o and o2are deformed ls(H) functions and o3and o4are in general two hybridised C+ orbitals whose typical form is (2.11) At short interatomic separations orbital o3 consequently possesses a maximum amplitude along the C+-H direction and o4similarly along C+-H,.The two orbitals are interchanged by the operation of reflection in a plane passing through the C axis perpendicular to the molecular plane. Orbitals o1and 0,are similarly reflected into each other by the same operation. As R becomes large orbital 45assumes one of the following forms (2.12) In the present case of N =5 S =$ there are a total of five spin couplings corresponding to the paths on the branching diagram shown in fig. 2. With the orbital symmetries as above the coupling k = 3 is forbidden for all of the lowest energy states of eqn (2.10). In addition the coefficients c; and ci2 corresponding to the spin couplings k = 1 and k =2 must be related by cil =(1/2/2) ci,.The fulfilment of these symmetry constraints serves as a useful check on the operation of the programs. Cuts through the potential-energy surface for various values of R for the lowest ,B state are shown in fig. 3.15 The minimum-energy path runs in a deep entrance valley almost parallel to R through a long-range attractive well with R remaining close to 1.4uo the equilibrium separation between H and H in the isolated H molecule. At R = 1.85aothe path runs into a steep repulsive wall and turns sharply becoming almost parallel to R and climbs over the energy barrier. Thus as R decreases by just 0.05~~ to 1.80a0the value of R for the minimum increases to 3.80a0 The C+ ion suddenly inserts itself into the H1-H2 bond.Once over the barrier the path plunges directly down into the minimum in the plane which occurs at R =0. Associated with this is a complete change in the spin couplings as shown in table 1. Before the repulsive wall at R = 1.85a0 is reached all spin-coupling coefficients are essentially zero except ci5 which corresponds to the reagents C+(,PP)and H,(lZ;). STUDIES OF MOLECULAR STATES 312 1 1 /2 LLl 12345 so 3/21 (C) 112 'Lu!l 012345 w N Fig. 2. Branching diagrams for N =5 S =a:(a) c, (b)c, (c) c3 (d) c and (e) c,. -3832 ,,I,(,II ,,,,I,,1 ,(,,,,,,,,,,,,,,,,,,,11 1 2 3 L 5 1 2 3 L 5 -3832 -mrr-38487 I I I I I I I I I ,I11 I 11 I I II I I I I I I 1 2 3 L 5 1 2 3 4 5 H-H internuclear distance (a.u.) Fig.3. Potential-energy surfaces C++H, ,B for various values of R, (a) 2.50 (b) 2.29 (c)2.20 and (d) 1.80. D. L. COOPER J. GERRATT AND M. RAIMONDI Table 1. CH;(2B1) minimum-energy path in the region of the energy barriera E Cf1 Cf3 1.90 1.60 -38.359 330 0.022 678 0.032 071 0 0.137 821 0.989 678 1.87 1.60 -38.351 617 0.024 042 0.034 001 0 0.147 378 0.988 203 1.85 1.65 -38.346 074 0.027 991 0.039 586 0 0.172 959 0.983 735 1.SO 3.77 -38.447 038 0.157 325 0.222 492 0 0.806 505 0.524 687 1.50 3.75 -38.491 311 0.143 8 0.203 3 0 0.828 3 0.501 9 a All units are atomic. The coordinate system defining R and R is shown in fig. 1. Spin-coupling coefficients are numbered according to branching diagram (fig.2). As R decreases very slightly and R,simultaneously widens to 3.80a0 the spin-coupling coefficients become essentially q4= d3/2 and = 1/2. This now corresponds to the situation where o1and o3are singlet coupled and similarly o2and 04,i.e. to the formation of two C+-H bonds. We thus see in a particularly striking way the rapid change-over from reagents to intermediate as the system surmounts the potential barrier. It is worth emphasising that these large changes in the wavefunctions never give rise to any discontinuities in the total energy. This last remains smooth although it is frequently associated with a maximum as in the CH; case above. Such rapid changes in the spin coupling coefficients as the nuclear geometry varies are usually indicative of a nearby excited state which interacts strongly with the ground state.The presence of such a state is revealed as the second solution of the secular equation at the VB stage (see next section). The range of application of the spin-coupled theory is determined by the number of electrons N in wavefunction (2.1). The ‘non-orthogonality problem’ occurs in the computation of the density-matrix elements the work for which is proportional to N! This is minimised in the recurrence scheme (2.4). If there are N occupied orbitals of o symmetry and N of 71 symmetry the amount of computation is instead proportional to Nu!+N,! where N = No+N,. Core orbitals in a closed-shell con- figuration of the form {& 4 * * * 4%,,> (2.13) where Nc is the number of such orbitals play no role in this respect and any number of them may be included.Present density-matrix programs are capable of treating a problem with Nu = 8 N = 8 and N arbitrary. The actual time taken to evaluate the density matrices is negligible compared with the time required for all other steps in an iteration and the current limitations on No and N arise from the lengths of lists which it is convenient to process. Nevertheless the present limits on N and N make it possible to apply the theory directly to a wide range of molecular systems with a reasonably small number of valence electrons.* * It is worth emphasising the distinction between the number of electrons N which are treated in a non-orthogonal fashion and the total number of orbitals Nt (occupied and virtual) which might be used in a spin-coupled VB calculation.The computational labour is always proportional to N! and not to Nt!. We have for example used 35 non-orthogonal orbitals without any great difficulty in spin-coupled VB calculations on 6-electron systems. STUDIES OF MOLECULAR STATES 3. THE SPIN-COUPLED VB WAVEFUNCTIONS’ When the stabilised Newton-Raphson procedure converges all gradients are zero and eqn (2.6) becomes Its higher-energy solutions are obtained by reformulating the equation as follows where In eqn (3.3) fiP) stands fo the supermatrix of CVoperators with row and column p missing and similarly &(PI represents a diagonal matrix of all the orbital energies except cP.Eqn (3.3) is now an eigenvalue equation which can be solved in the normal way to produce an orthonormal set of orbitals 4;) and associated eigenvalues c!). Each occupied orbital gives rise to an orthonormal set (or ‘stack’) P 21 (4;) I@’)) = 6.. (3.4) the occupied orbital reappearing as one of the solutions @) say. The solutions stemming from different stacks are not orthogonal to one another and we have (4;’ 14;’)) = Api;vj 01 # v). (3* 5) Each operator Qff)is constructed from quantities involving only N-1 electrons Effectively orbital 4Pis missing and the solutions therefore describe the movement of an electron in an average field arising from just the other electrons. Consequently the virtual orbitals 4;) in general have the same semi-localised form as the occupied orbitals.Higher-energy solutions are of course more diffuse. As internuclear distances become large the virtual orbitals assume the form of excited atomic orbitals. Excited spin-coupled structures are formed by replacing 1,2,3 . . . etc. occupied orbitals by virtual orbitals The final total wavefunction is expressed as c. cc;:::I:) Y? = Wl4Z *.. 4N}+ {@$) ... $q$q (3.6) Pl ... PN i, ..,,i,v where the first term is the spin-coupled function and the succeeding terms represent the excited structures. The coefficients Co,C(::-) are determined by constructing the matrix of the Hamiltonian in this basis of spin-coupled structures and diagonalising by means of a normal VB program. From a formal point of view eqn (3.6) may be regarded as an expansion of the exact wavefunction in terms of N orthonormal sets a distinct complete set is used for each electron the particular expansion being in some sense ‘tailored’ for that electron coordinate.If the N sets coalesce into a single expansion set we regain the MO-CI or MCSCF-CI representation. This indicates that expansion (3.6) may be expected to converge reasonably quickly at least for small systems. Indeed as shown below we have found that we can achieve the same accuracy as the highest quality MCSCF-CI wavefunctions using 20-50 times fewer terms. The price to be paid of course is in the construction of the elements of the Hamiltonian matrix between the non-orthogonal D. L. COOPER J. GERRATT AND M. RAIMONDI spin-coupled structures since this requires ca.an order of magnitude longer than for a similar size matrix constructed from configurations of orthogonal orbitals. However the fact that it is now possible to attain this level of accuracy with VB-based wavefunctions has stimulated a great deal of development of our programs. Even at their present stage the times required by them per energy point are strictly comparable with that taken by large MCSCF-CI programs for closely comparable total energies. The most useful feature in our view of the compact expansion of Y given by eqn (3.6) is the physical and chemical visuality provided by it. The spin-coupled structure by itself reproduces with reasonable accuracy all the features of the ground-state potential-energy surface.Thus for example a spin-coupled wavefunction typically yields 85% of the observed binding energy and equilibrium internuclear separations are accurate to 0.01 A. This function therefore dominates expansion (3.6) at all nuclear geometries. The various excited structures provide angular and other types of correlation as an extra quantitative refinement but do not alter the qualitative picture. The same appears to be true of the low-lying excited states. The evidence we have so far shows that such states are well represented by a single excited spin-coupled structure in which one (or at most two) occupied orbitals are replaced by appropriate virtual orbitals q5t),gb$j). In other words the first few roots of lowest energy of the secular equation are dominated by just one or two structures for all configurations of the nuclei and these provide the essential physical interpretation.4. RECENT APPLICATIONSf6 4.1. ‘z+ STATES OF CH’ As indicated in section 2 CH+ is of considerable importance in interstellar chemistry. It has also been well studied in MCSCF-CI methods.l79 l* The spin-coupled wavefunction is of the form (a,a2a3a4a5o6}.Orbitals o,and a2are core orbitals and may be characterised simply as ls(C+) and ls’(C+) respectively. They change very little as R varies. Orbitals a3and o4are lone-pair orbitals. At large internuclear distances they are of the form 2s(C+) and 2s’(C+) but as R decreases they alter considerably. The same is true of the first member of the bonding pair a,. At large values of R this assumes the form of a 2p,(C+) function but undergoes considerable variation as R decreases.Its partner 06 is almost completely a ls(H) orbital at small values of R there is a significant amount of delocalisation onto C+. The wavefunction {a a 0,a4a5G6} thus furnishes a qualitatively correct description of the dissociation of CH+ into C+(ls 1s’2s2s’2p2; ,P)+H(ls). If orbitals a and a2remain coupled to a singlet there are just two spin function which give a net spin S = 0. We write these simply as 0 and 0,. Spin function 0 describes orbitals a3and a4coupled to a triplet a5and a6similarly the two triplets giving a resultant singlet. Function 0 is the perfectly paired spin function with the orbitals o3and o4coupled to a singlet and the same for a and a6.The basis set used consists of 180,20n and 66 Slater orbitals and is the same as that used by Green et except for the omission of 4ffunctions. It includes the basis functions 2p([ = 1.O) and 3p([ = 1 .O) on the H atom which are necessary to account correctly for long-range induction effects. However no diffuse 3s(C) or 3p(C) functions which would be needed to describe any Rydberg character in excited states are included. Note that at large values of R the C+ wavefunction contains a substantial contribution from the coupling ((2s 2s’) 3S;2p2}2P(spin function Q1), as well as from the expected ((242s’) lS; 2p,} 2Pstate the coefficients of the two spin functions being c 0.36 c2 % 0.93. The role of the (2s 2s’) 3S coupling is clearly to afford some STUDIES OF MOLECULAR STATES -3780 1 x 'I+ C'(*P) H -37851 h -3790 N 2 v x - 2 z -3795: -3800 1 ---- ---3805- I 1111IIIII I I I 11 1111 I11 I 1111III Ir7i111111 I111I 11111 1111II I 11 I I I 11 2 4 6 8 10 12 14 internuclear distance (a.u.) Fig.4. Potential-energy curves for CH+ XIC+ 0,spin-coupled VB 500 structures; 0, MCSCF-CI (Green et aE.17). additional radial correlation between the lone-pair electrons. At shorter interatomic distances this type of correlation becomes much less important. Spin-coupled VB calculations were carried out using a total of 26 orbitals 6 occupied 6 0 virtual and 14 n virtual orbitals. The final wavefunctions consist of 500 structures formed from 286 distinct spatial configurations of C+ symmetry.These consist of the spin-coupled referene function and (1 + 2 + 3+ 4)-fold excitations. No excitations from the (a,,a,) core were included. About half of these structures (single +double replacements) contribute to the ground state the remainder improve the description of the excited states. In fig. 4 the potential-energy curve for the XIC+state is compared with that obtained by Green et al. using an MCSCF-CI procedure. This wavefunction includes core excitations and gives the lowest energy of any calculation on CH+ in the literature. In fig. 4 this potential-energy curve has been shifted upwards so as to coincide with the spin-coupled VB result at R = 2.0~1,(the point closest to the equilibrium value of R given by the authors).The two calculations are very similar over the whole range of R.Computed spectroscopic constants for the spin-coupled VB wavefunction are shown in table 2 where they are compared with other calculations and with experiment. The excited states are shown in fig. 5. They are compared with the results of Saxon et aZ.l*by shifting the spin-coupled VB curves so as to coincide with the MCSCF-CI results at R = 15a,. The two sets of curves are remarkably similar for all values of R.The only exception occurs in the 4 lX+ state at R < 3.5~1,.The discrepancy between the two calculations in this region is due to an avoided intersection with the next highest state. This is Rydberg in character and lies 0.7 eV above the 4 lC+ asymptote yielding C(2s22p3s;'P)+ H+ on dissociation.Since no diffuse 3s or 3p basis functions were included this state is absent in our calculations and the divergence with the MCSCF-CI result occurs just where it would intersect the spin-coupled VB state. D. L. COOPER J. GERRATT AND M. RAIMONDI Table 2. Properties of CH+(XIZ+) 3.63 spin-coupled -37.950 56 1.141 276 1 60 4.14 spin-coupled VB -38.024 26 1.135 2845 69 MCSCF-CI” -38.060 64 1.130 2847 63 4.1 1 4.14 MCSCF-CI” -38.022 33 1.128 2860 59 experimentalz9 -1.131 2858 59.3 4.26 -371 2 -374 0 v ;h C’(‘D ) H 2 -376 4 C(’S) HI + -3 3 -377; 3 3 ~ -378 7 I. I I I I I, I I I I I, 2 4 6 8 10 12 14 internuclear distance (a.u.) Fig.5. Potential-energy curves for excited lZ+ states of CH+ 0,spin-coupled VB 500 structures; 0, MCSCF-CI (Saxon et~11.l~). 4.2. lZ+ STATES OF LiHe+l9 The spin-coupled wavefunction for this system can be written as {‘T~ o3a,}. ‘T~ Orbitals crl and o2 stem from the Li+ ion and can be characterised as ls(Li+) and ls’(Li+) respectively while orbitals ‘T and ‘T can be regarded as ls(He) and ls’(He). There are two spin functions corresponding to singlet states these being the same as those used above for CH+(O and OJ. Consequently the ground state of this ion corresponds simply to the interaction of the two closed-shell species Li+(ls 1s’; ‘S) and He( Is 1s’; lS). However as soon as a single electron is excited out of either atom a whole series of closely spaced states arises.These differ widely in character from one another. Their corresponding potential-energy curves show several avoided crossings and the associated non-adiabatic radial couplings give rise to charge exchange in low-energy collisions. The description of all of these states by spin-coupled VB theory furnishes a test of the quality of the virtual orbitals and affords us valuable experience with a small system in the development of procedures for the selection of the necessary structures and orbitals. STUDIES OF MOLECULAR STATES Table 3. LiHe+‘C + states comparison of experimental asymptotes with calculated energies at 30 bohr from a 23-structure spin-coupled VB calculation as described in the text energy/eV state asymptote experimental2* calculated (1) 1C+ (2) 1E+ (3) 1c+ (4) lC+ (5) 1c+ (6) IC+ Li+(lsls’)He(lsls’) Li(lsls’2s) He+(ls) Li+(1 s 1s’) He( 1 s2s) Li( 1s1s’2p) He+( 1s) Li+(1 s 1 s’) He( ls2p) Li(lsls’3s) He+(ls) -19.190 0 1.425 1.848 2.028 3.373 -18.752 0 1.454 1.843 2.049 3.676 The basis set employed is a large ‘universal even-tempered ’ set of Slater functions,20 comprising 420 and 44n orbitals.Using this converged spin-coupled wavefunctions were obtained at 24 internuclear distances between la and 30a,. Initial spin-coupled VB calculations were carried out usingjust 23 structures formed from 16 orbitals and the asymptotic energies are shown in table 3. Note that since a theorem analogous to that of Brillouin2’ applies to the spin-coupled wavefunctions few of these structures interact with one another and each state is described essentially by just one or two structures.As can be seen the calculated asymptotic splittings are in good agreement with experiment. It is worth remarking that by using such a large basis set the virtual orbitals are more flexible and consequently fewer of them in fewer structures seem to be required. The final calculations include 14 virtual orbitals of CT symmetry and 16 of z. All single and double replacements from o3and o4(Is 1s’of He) only were included giving rise to 192 spatial configurations. The orbitals o1and 0,(Is Is’of Li+) were regarded as core and not excited. However several virtual orbitals stemming from the o1and o2stacks were included in the calculations.The lowest seven lC+ states are shown in fig. 6. There are multiple avoided crossings of states which correspond to Li++He( Is nl) and Li( Isls’ nZ)+ He+(1s). The asymptotic splitting between the first two states is now 19.03eV. The lowest state possesses a shallow minimum 843 cm-l deep with equilibrium internuclear distance of 1.930A owing to inductive and van der Waals effects and which supports six bound levels. The first excited state shows a much deeper minimum of 4979 cm-l with Re = 3.587 A and gives rise to ca. 28 bound states. The use of such a large number of non-orthogonal orbitals to expand essentially just two electronic coordinates brings with it the hazard of near linear dependence. In practice this arises from large overlaps (> 0.998) between virtual orbitals at certain internuclear separations.We have found that the most effective remedy is simply to orthogonalise the offending orbitals amongst themselves using the Schmidt procedure.22 If two such orbitals are denoted by u and v then as long as all structures of the type {... uu ...I,{... u2...> {... u2 ...} are included the total wavefunction is invariant to such a transformation. As a result the final wavefunctions and potential-energy curves are free from any unwanted effects arising from incipient linear dependence. However there remain some small variations in the lowest root (of the order of a.u.) in the region of R = 8.5a0.We ascribe this to basis set superposition error which we have not attempted to correct.D. L. COOPER J. GERRATT AND M. RAIMONDI -90j cJ -92j 41 v x 2 -94 E -96 -981 -100 -102 €I 5 10 15 20 25 30 internuclear distance (a.u.) Fig. 6. Potential-energy curves for excited lC + states 192 structures. of LiHe+ spin-coupled VB 4.3. POTENTIAL-ENERGY SURFACES FOR THE B+ + H SYSTEM PRELIMINARY RESULTS The reaction B++H + BH++H has been studied by Friedrich and Herman23 and by Ottinger and Reichm~th.~~ The incident beam consisted of a mixture of B+ ions in the lS and in the metastable 3P states. Emission from product ions was observed and assigned to the transitions BH+(A ,lT+ X2C+)and BH+(B T++ X2C+).The existence of the bound B zC+ state was previously unknown. It was confirmed by the calculations of Klein et uI.,,~ and correlates well with our own findings for the corresponding state of BeH.g Preliminary ab initio calculations on the BH system have been carried out by Hirst26 using the MRD-CI method and extensive DIM calculations of both singlet and triplet surfaces have been reported by Schneider et uI.,~ The aim of the present spin-coupled VB work is to study the potential-energy surfaces corresponding to the processes /BH+(A2n; B2X+)+H B+(1S13P)>B(2P) + H;(,X;) LBH(x~x+)+H+.The calculations were initiated with all three atoms well separated in order to select orbitals and structures which would reproduce reasonably well the observed energy differences between all the different possible asymptotic states. A large Gaussian set is used consisting of B( 1 1,6,2/9,5,2) and H(6,2 1 /4,2 1) on each H atom.This set is taken from ref. (25) and (30) and is well suited for study of both BH+ and BH. It 6 FAR STUDIES OF MOLECULAR STATES Table 4. Asymptotic splittings for the B++H system separation/eV total energy state (a.u.) computed experimental ~ B+(lS)+H(T) +H(?S) B+(3P)+H(,S) +H(2S) B(2s22p;2P)+H++H(,S) B+('P) +H(,S) +H(2S) -25.308 46 -25.135 76 -25.105 90 -25.104 96 0.0 4.71 5.52 9.44 0.0 4.63 5.31 9.09 also includes diffuse functions on the B atoms which are needed to represent the B(2s23s;2S)Rydberg state. A set of 342 structures was chosen consisting of all single and double excitations from the occupied orbitals and also single and double excitations from certain singly excited configurations which are selected to represent the different excited and charge-transfer states.This set of structures yields values for the asymptotic splittings shown in table 4. The experimental separations are thus reasonably well reproduced. Experience by ourselves and othersls shows that a further increase in the number of structures serves only to lower each state by a uniform amount which is almost independent of nuclear geometry i.e.the remaining deficiencies in the selected set of structures are almost completely atomic in origin and play little or no role in the behaviour of the molecular potential-energy surfaces. Further results on the potential-energy surfaces of this system will be presented at the Symposium.B. H. Lmgsfield A. D. McLean M. Yoshimine and B. Liu J. Chem. Phys. 1983,79 189 G. A. Gallup and J. R. Collins to be published. J. Gerratt in Theoretical Chemistry (Specialist Periodical Report The Chemical Society London 1974) vol. 1. C. A. Coulson and I. Fischer Philos. Mag. 1949 40,386. J. Gerratt Adv. Atom. Mol. Phys. 1971 7 141. N. C. Pyper and J. Gerratt Proc. R. Soc. London Ser. A 1977 355 407. ' M. Kotani A. Amemiya E. Ishiguro and T. Kimura Tables of Molecular Integrals (Maruzeu Tokyo 1963). J. C. Manley and J. Gerratt Comput. Phys. Commun. 1984 31 75. J. Gerratt and M. Raimondi Proc. R. Soc. London Ser. A 1980,371 525. lo S. G. Walters Ph.D. Thesis (University of Bristol 1984). l1 A. Dalgarno and J. H. Black Rep. Prog. Phys.1976 39 573; M. Elitzur and W. D. Watson Astrophys. J. 1980 236 172. l2 D. H. Liskow C. F. Bender and H. F. Schaefer J. Chem. Phys. 1974 61 2507. l3 P. K. Pearson and E. Roueff J. Chem. Phys. 1976 64 1240. l4 S. Sakai S. Kato and K. Morokuma J. Chem. Phys. 1981,75 5398. l5 J. Gerratt S. G. Walters and R. Williams to be published. l6 J. Gerratt J. C. Manley and M. Raimondi J. Chem. Phys. in press. l7 S. Green P. S. Bagus B. Liu A. D. McLean and M. Yoshimine Phys. Rev. A 1972 9 1614. R. P. Saxon K. Kirby and B. Liu J. Chem. Phys. 1980,73 1873. l9 D. L. Cooper J. Gerratt and M. Raimondi to be published. 2o D. L. Cooper and S. Wilson J,Chem. Phys. 1983 78 2456. 21 L. Brillouin Act. Sci. Ind. 1934 159. G.A. Gallup personal communication ;R. Courant and D.Hilbert Methods of Mathematical Physics (Interscience New York 1953) vol. 1. 23 B. Friedrich and Z. Herman Chem. Phys. 1982 69,433. Ch. Ottinger and J. Reichmuth J. Chem. Phys. 1981 74 928. 25 R. Klein P. Rosmus and H. J. Werner J. Chem. Phys. 1982,77 3559. 26 D. M. Hirst Chem. Phys. Lett. 1983 95 591. I). L. COOPER J. GERRATT AND M. RAIMONDI 163 27 F. Schneider L. Ziilicke R. Polak and J. Vojtik Chem. Phys. 1984,84,217; Chem. Phys. Lett. 1984 105 608. C. E. Moore Natl Stand. Re$ Data Ser. Natl Bur. Stand. 1971. 29 K. P. Huber and G. Herzberg Constants of Diatomic Molecules (Van Nostrand Reinhold New York 1979). 30 W. Meyer and P. Rosmus J. Chem. Phys. 1975 63 2356. 6-2

ISSN:0301-5696    

DOI:10.1039/FS9841900149

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

Faraday Symp. Chem.SOC.,1984 19 165-173 Numerical Perturbation Calculations for Diatomic Molecules BY EDWARDA. MCCULLOUGH AND KENTW. RICHMAN JR,* JOHN MORRISON? Department of Chemistry and Biochemistry UMC 03 Utah State University Logan Utah 84322 U.S.A. Received 7th August 1984 A numerical method is described for diatomic-molecule perturbation calculations based on a natural orbital expansion of the pair function. Results are presented for several diatomic molecules and are compared with results obtained with finite basis sets. A discussion of basis-set errors and the advantages and disadvantages of the numerical method is given. For atoms it is common to carry out many-body perturbation calculations entirely numerically. 9 For molecules however nearly all perturbation calculations presently use basis-set expansion^.^ The inaccuracies introduced in this way can be difficult to estimate since the convergence of such expansions is often quite erratic.In this article we report on numerical second-order perturbation calculations for several diatomic molecules. Our purposes are twofold. First we wish to test the feasibility of obtaining numerical solutions of the pair equation fpr diatomic molecules. Secondly we would like to examine some of the best published basis set calculations with the idea of determining the levels of basis-set error in such calculations generally. THEORY In the diagrammatic perturbation expansion of the energy of a closed-shell system2 the second-order direct diagram D,,has the value We consistently shall use a b and c for occupied (core) orbitals r s and t for unoccupied (virtual) orbitals and i and j for unspecified orbitals.Here these orbitals are chosen to be eigenfunctions of the closed-shell Fock operator h. Eqn (1) may be evaluated readily using the first-order pair function lyd(ab-+ rs) = C I rs) (rs 1 r;,l I ab)/(&,+ &t -E -cS). (2) rs An inhomogeneous equation for 'yd is obtained by operating on eqn (2) from the left with E 4-&b-h(1)-h(2). The result is t Present address Computer Science Department Box 1679 Station B Vanderbilt University Nashville Tennessee 37235 U.S.A. 165 NUMERICAL PERTURBATION CALCULATIONS FOR DIATOMICS Similarly the pair function for the exchange diagram satisfies the equation [E +E~ -h(1) -h(2)] W,(ab -+ rs) = -Z I rs) (rs 1 r; I ba).(4) rs If we add eqn (3) and (4) we obtain a determinant Itya ybl,on the right. It is more convenient however to use a symmetry-adapted linear combination of determinants since the spin dependence can then be factored out and spin eliminated from the pair equation. We shall use a bracket notation to denote a symmetry adapted spatial function. For example the functions for i # j are where the upper (lower) sign refers to the singlet (triplet) case. With this notation the symmetry-adapted pair function yUb,satisfies the equation where it is understood that all quantities now are functions only of spatial variables. For atoms eqn (6) can be solved numerically with high efficiency and accuracy.* While the same technique could be applied to molecules the appearance of double partial wave sums would make the calculations quite time consuming.For this reason we shall here pursue another approach. We begin by noting that eqn (6) can be obtained by requiring that the functional be stationary with respect to any first-order variation Styab consistent with strong orthogonality to the core orbitals. The stationary value of the functional cab is the symmetry -adap ted pair energy. One can automatically satisfy strong orthogonality by expanding yabin terms of the virtual orbitals. This approach is often used for basis-set calculations where in the course of obtaining the occupied orbitals one obtains as a by-product an approximate set of virtual orbitals.It is much less attractive for numerical calculations because each orbital (virtual as well as occupied) must be found independently. Also if h is the closed-shell Fock operator all of the virtual orbitals usually lie in the continuum. An alternative approach is to use a natural orbital5 expansion of yUbas suggested by Kutzelnigg,g and make cab stationary with respect to both variations in the expansion coefficients and the forms of the pair natural orbitals (PNO). This leads to a set of equations that are formally quite similar to the multiconfiguration Hartree-Fock equations. The natural orbital expansion is of the form k For a singlet pair in which orbitals a and b have the same spatial symmetry it is possible to choose uk and vk to be identical.This leads to certain simplifications which we shall not consider more fully. Henceforth we shall regard uk and uk as distinct. Substituting eqn (8) into eqn (7) gives where the ck are assumed to be real. E. A. McCULLOUGH JR J. MORRISON AND K. W. RICHMAN The stationary condition with respect to ck gives The stationary condition with respect to say uk consistent with orthonormality of the PNO and strong orthogonality to the core leads to the equation where the d are Lagrange multipliers and Picr)( 1) = vi(1) Idz w*(2) ry,lvi(2) for an arbitrary function cr). An analogous equation can be derived for vk. In addition to variations of individual PNO rotations between pairs of PNO must be allowed. For example consider a rotation of two u uk -+ uk+8uland ul-+ ul-8uk.Expanding &,b in powers of 8 to first order we obtain so that the stationary condition with respect to rotations implies that cf Llk = ct Lkl. If this condition is not satisfied automatically the PNO must be rotated until it is. The optimum rotation angle can be obtained by expanding &abto higher order in 8 and setting deab/d8 equal to zero. One must also consider of course rotations between pairs (uk v,) and (vk vl). The PNO equations also can be derived by substituting eqn (8) directly into eqn (6). This is legitimate only if the PNO expansions could be a solution of eqn (6) i.e. only if the expansion is infinite. Taking the scalar product of the result from the left with say vk(2) we obtain ck(Eu+Eb-h)uk-x Cl((vk~hlvl)ulk(vkIhl ul>vl) 1 = c <'k 1 s> vr (rsI rTtI va vbfvb va>.(14) rs The right-hand-side of eqn (14) can be simplified using the completeness of eigen- functions of h,which allows us to express each sum over excited states Cvt(r)v/F(r') as a delta function minus a sum over core states. After some rearrangement eqn (14) becomes +ck(ca+cb)uk+~ (vcvkIr~~l~avb*vbva) vc. (15) c By comparing eqn (15) with eqn (11) we may obtain expressions for the Lagrange multipliers. In particular we see that ck dlk = -Cl (Vk I hI 01) I # k. (16) It is then shown easily that ci Alk = cf dkl. The Lagrange multipliers also may be evaluated by taking the scalar product of eqn (1 1) from the left with each PNO. Equating these expressions with those obtained NUMERICAL PERTURBATION CALCULATIONS FOR DIATOMICS from eqn (1 5) we can derive consistency conditions for an exact PNO solution of the pair equation such as -C1<UkIhlU1) =Ck<UIIhIUk)+(UIUkIY~~IWaWbfWbWa) Ifk.(17) To the extent that the PNO expansion is exact these consistency conditions are satisfied and cabis automatically stationary with respect to rotations. This does not follow for a truncated expansion however where the effect of rotations must be considered and departures from the consistency conditions can be expected. Our first problem of course is how to solve eqn (I 1) for molecules. For diatomic molecules it is possible to obtain numerical solutions using a partial wave expansion of the PNO in prolate spheroidal coordinates.For instance we write up(<,q,4) =2Xklm(<) qm(v,6) (18) where the qmare spherical harmonics and the Xklm are numerical functions. The symmetry-adapted PNO for diatomic molecules can be labelled with m quantum numbers e.g. upup’. The advantages and disadvantages of this approach as well as computational details have been reported previously for the multiconfiguration Hartree-Fock problem. The principal advantage of eqn (18) is that it converges smoothly which allows us to determine PNO with controlled numerical accuracy. The principal disadvantage is that it is only applicable to diatomic molecules. RESULTS The formal similarity between the pair natural orbital and multiconfiguration Hartree-Fock energy functionals enabled us to use a very slightly modified version of our numerical MCSCF code to solve the PNO equations.All computations were carried out in double precision on a VAX 11/780. Our first example is atomic beryllium which we treated as a heteronuclear diatomic molecule with zero charge on one nucleus and the internuclear distance set arbitrarily to 0.5a0.This is an important test case since very accurate pair energies are available for Be. Table 1 shows the results. The pair energies Eab(mm’),obtained from expansions containing N(mm’) PNO configurations are compared with the essentially exact cab(mm’)calculated from the work of Malinowski et al. (MPJ).8 The MPJ results are equivalent to infinite PNO expansions for each (rnrn’). The results in table 1 are indicative of the general level of accuracy that may be expected from application of the PNO approach to molecules.One sees that fairly short expansions can produce Eab(mm’)correct to d 5 x hartree,* and that the initial convergence rate of the PNO expansion is quite high as csS2(a2)demonstrates. The second-order energy E2(6),obtained by summing our degeneracy-weighted Eab(mm’),is -0.0715 hartree. The MPJ values of E2(6)and E2(w)are -0.0743 and -0.0763 hartree respectively The error in our E2(6)due to truncating the PNO expansions is of the same order as that due to neglect of (rnrn’) >2. Table 2 gives our results for H and LiH. Here only the total pair energies &,b for (mm’)d 2 are shown. For the energetically most important pairs we used two sets of PNO expansion lengths to investigate the convergence with respect to N(mm’).Our best E2(6)for LiH is -0,0696 hartree which may be compared directly with * 1 hartree =4.359814 x lo-’* J. E. A. McCULLOUGH JR J. MORRISON AND K. W. RICHMAN Table 1. Numerical PNO and accurate pair energies (hartree) for Be accurate ab 4 mm’ N(mm’) PNO Eab(mm’) Eab(mm’) 9 -0.0206 -0.0209 6 -0.0164 -0.0168 3 -0.0015 -0.0018 4 -0.0101 -0.0108 6 -0.0163 -0.0168 3 -0.00 17 -0.0020 8 -0.0017 -0.0017 6 -0.001 3 -0.0013 3 -0.0001 -0.000 1 1 -0.0002 -0.0002 1 -0.0004 -0.0005 Table 2. Numerical PNO pair energies (hartree) for H (R= 1.4a,) and LiH (R = 3.015a0) ab -+ mm’ N(mm’) &ab N(mm’) ‘ab H2 10; -+ Cr2,712,s2 9 5 3 -0.0325 13,9 5 -0.0331 LiH 102 -+ 02,712,62 9 6 3 -0.0375 13 9 5 -0.0381 202 48,712 d2 9 6 3 -0.0286 13 9 5 -0.0290 l02o(lC) + u2,n2 4 3 -0.0013 1a2a(3q jog‘ 71.n’ 1 1 -0.0004 the value of -0.0654 hartree obtained by Bartlett and Silverg (BS) using a large Slater basis set.Their basis did not contain any functions beyond 6 but they did sum over all virtual orbitals. Thus their calculation is equivalent to an (mm’)d 2 PNO calculation within their finite basis and the difference between their E2(6)and ours is a measure of the basis-set error in their calculation. We now turn to FH. This molecule is considerably more complex than LiH because of the greater number of electrons and the presence of several spatially proximate pairs. For these reasons FH provides a more realistic and more stringent test of high-accuracy methods.Systematic variation of all PNO expansion lengths was impractical in the case of FH owing to the large number of pairs. For the intrashell ozpairs we chose initial expansion lengths based on our experience with smaller systems. The occupied 7t shell was a new feature; however we were aided by the similarity between correlation in the 71 shells of FH and Ne. We computed several 7t2 pair energies for Ne and compared them with accurate values.l0 We then used these comparisons as a guide in choosing initial N(mm’) for FH 7t2 pairs. One advantage of the PNO approach is that one can estimate roughly the maximum energy that could be gained by adding another PNO configuration from the energy contribution of the least important configuration already present.The relatively greater complexity of FH was revealed clearly in the intershell pair calculations. The iterative solution of the PNO equations requires initial guesses for the PNO and these were often problematic for intershell pairs While the FH occupied NUMERICAL PERTURBATION CALCULATIONS FOR DIATOMICS Table 3. Numerical PNO and basis-set pair energies (hartree) for FH (R= 1.7328~~) ab PNO &,b basis set &,b 202 -0.01 12 -0.0100 2034'Z) 203u(~Z) 302 -0.0156 -0.0028 -0.0273 -0.0138 -0.0026 -0.0251 20ln('I-I) -0.0138 -0.0124 20 174311) 3~17c(~II) 301~(~ll) 17cz (average singlet) -0.0027 -0.0136 -0.0088 -0.0199 -0.0027 -0.0130 -0.0088 -0.0189 1q 3 ~ ) -0.0090 -0.0089 orbitals strongly resemble slightly distorted fluorine orbitals attempts to utilize initial guesses based on atomic symmetry were only moderately successful.For intershell pairs involving 20 or 30 especially the PNO frequently displayed a tendency to move away from the fluorine by large distances in either direction. Plots of these PNO exhibited few vestiges of atomic symmetry. More variation of expansion lengths was carried out for these pairs. In addition to problems arising from choices of initial guesses and expansion lengths convergence of the iterative solution procedure was poorer in general for all aa pairs of FH than for smaller systems. Possibly this was due to the more numerous strong orthogonality constraints. Some pair energies for FH are shown in table 3 where they are compared with the large Slater basis results of BS.Pairs involving la are excluded as the BS basis set contained very few high-exponent functions and could not be expected to represent correlation involving lo very well. Our E2(6)is -0.331 hartree. Based on our experience and limited testing we estimate that this is not less than 95% of the exact E2(6).The BS E2(6)is -0.306 hartree. Almost one third of the difference between their result and ours is due to the la2 pair. This probably is not a serious error since few molecular properties should be sensitive to 1 a2correlation. Discussion of the remaining pairs will be deferred until the next section. For FH we also tested the sensitivity of the pair energies to the accuracy of the occupied orbitals.We chose for this test orbitals computed in a moderate sized Slater basis by Nesbet.ll His wavefunction gives an energy of -100.057 hartree compared with the exact Hartree-Fock energy of -100.071 hartree.12 This basis is of slightly better quality than typically might be used for large molecules although much inferior to the BS basis sets. We computed for Nesbet's wavefunction eight of the energetically most important Eab(mm'),excluding any involving 1a.The largest deviation from our results calculated with the accurate numerical wavefunction was 2% and most differences were smaller than 1%. Basis-set error in the occupied orbitals is unlikely to be important in high-quality molecular-perturbation calculations.DISCUSSION Before embarking on a detailed discussion of the numerical PNO method vis-a-vis basis-set methods we must identify the essential differences between the two approaches. E. A. McCULLOUGH JR J. MORRISON AND K. W. RICHMAN Table 4. Numerical PNO optimized Slater orbital (OSO) and optimized gaussian geminal (OGG) second-order energies (hartree) for H and LiH molecule OSO E2(S) PNO E2(S) OGG E2(cn) H2 -0.0331 -0.0331 -0.0342 LiH -0.0678 -0.0696 -0.0722 Errors in the numerical PNO method are of three types :(a)error in individual PNO (b)error introduced by truncating the PNO expansion for each (mm’) and (c) error due to truncating the sums over (mm’).Any expansion of the pair function in a set of orthonormal orbitals can be reduced to PNO form by a suitable unitary transformation.This includes the conventional sum-over-virtual orbitals basis-set method where both the PNO expansion lengths and the maximum (mm’) are determined by the basis set. The only essential difference then between the numerical PNO method and most basis-set methods is in the representation of the PNO and the only practical advantage of solving the PNO equations numerically is that type (a)errors can be made negligible at least for the relatively short expansions we used. Recently two less conventional basis-set methods have been developed. The method of Adamowicz and Bartlett13 also is restricted to diatomic molecules. In their initial application they employed an expansion of each pair function in simple Slater-orbital products; however they were able to use long expansions and perform complete exponent optimizations.Their results are the best obtainable from a given number of Slater orbitals with fixed (nlm)quantum numbers. The method of Monkhorst and utilizes explicity correlated gaussian geminal expansions of the pair functions while avoiding many of the difficulties normally associated with this approach. Long germinal expansions with extensive optimization are a feature of their work. Unlike almost all other methods this method does not involve truncated (mm’) sums. As yet both of these methods have been applied only to small systems. A comparison with our numerical PNO method is given in table 4. The optimized Slater orbital E2(6)for H is in excellent agreement with ours.Recent work indicates that both are within 1% of the exact E2(6).15The situation for LiH is less satisfactory. Although Adamowicz and Bartlett used 26 completely optimized Slater functionsperpair they only reduced the error in the BS E2(6)by ca. 60%. Note that the PNO representation gives the shortest orbital expansion of a two-electron f~nction.~ Therefore one could not expect to improve on our E2(6)for LiH with any basis containing fewer than 27 functions since that is the number of PNO in our longest expansion. The convergence exhibited by the PNO expansions in tables 1 and 2 indicates that much of the difference between our E2(6)and the optimized geminal E2(co)truly is type (c) error. The type (a)-(b) error in the optimized Slater orbital results for LiH is of the same order as the type (c)error.Type (a)-(b) errors are even more serious in the BS calculations. We emphasize here that we have not singled out the BS calculations for special criticism. On the contrary their work is of very high quality. Basis sets for molecular perturbation calculations are rarely larger and frequently much smaller than those used by BS. Thus it is disconcerting to find that the type (a)-(b) error in their E2(6)for LiH is the most important error in their calculation by a factor of 1.6. Even more disconcerting is the behaviour of some of the FH pairs involving the bonding 30 orbital. Actually BS employed several basis sets. In passing from a 22 NUMERICAL PERTURBATION CALCULATIONS FOR DIATOMICS function basis to 32 functions (their largest) the changes in the 2a3a(%) and 3a2 pair energies were -0.0017 and -0.0020 hartree respectively.As table 3 shows the residual errors for these same pairs are at least -0.0018 and -0.0022 hartree so only about half of the error has been eliminated with 10 additional basis functions. In general triplet correlation is more accurately described than singlet correlation in the BS calculation on FH even when the former is large (e.g. for the 3aln and In2pairs). 10% errors in singlet pair energies are common. The total correlation energy of FH has been estimated to be -0.381 hartree.16 The value for Ne is almost exactly the same and in Ne E2(m)amounts to > 99% of the total.l* Our calculations indicate that E2(6)for FH is not less than -0.331 hartree but probably not more than -0.35 hartree based on our 5% estimate for type (b) errors.Type (c) errors thus should be between -0.03 and -0.05 hartree. The type (a)-(b)error in the BS E2((6)is at least -0.025 hartree and may be as much as -0.04 hartree. Many people have noted that an error in the correlation energy is important only if it affects some molecular property of interest. With respect to this point we would only make two observations. First empirical evidence certainly suggests that some molecular properties can be computed reliably with basis sets much smaller than those used by BS.3 Secondly the difference between E2(m) and the total correlation energy for many light molecules probably does not exceed lo% whereas even the BS basis set for FH leads to second-order energy errors of the order of lo% and the error in the first-order wavefunction itself may be much larger.Extensive error cancellation in higher order seems to be the only way these two observations can be reconciled. Basis-set improvements are straightforward in principle. Type (a)-(b) errors call for more or better functions of a given m.Here the numerical PNO method might make a special contribution since it is possible to project the PNO onto a basis” and determine by hindsight an optimum basis for correlation. Experience gained with a few representative examples might lead to a general prescription for choosing such basis sets. Type (c) errors can be reduced by going to higher m,but the asymptotic convergence rate of the (mm’) sums probably is slow.The evidence for this comes from the slow convergence rate of the analogous (If‘) sums in atoms where extrapolation formulae are often employed to get better estimates of the asymptotic limits.1° Similar formulae might be useful in the diatomic-molecule case. We conclude with some general remarks about the PNO approach in general and the numerical PNO method in particular. The rapid initial convergence of the PNO expansions for fixed (mm’) makes this approach quite attractive if moderate accuracy is the goal. It can certainly be implemented with basis-set methods as well as numerically. The iterative convergence problems we encountered might be overcome with guaranteed convergence algorithms of the type developed for MCSCF calculations.The PNO approach can be extended readily to higher orders. The first-order pair function, eqn (6) suffices for the calculation of third-order energies. This is made more difficult by the non-orthogonality of PNO for different pairs; however the number of PNO is relatively small. An equation for the second-order pair function can be derived for which the first-order pair function is used to calculate a modified right-hand side of eqn (6) in a manner analogous to that developed for atoms.18 This leads to an iterative scheme that could be carried to arbitrarily high order. If one examines the derivation of eqn (1 5) one finds that the right-hand side of eqn (6) is responsible for the terms involving the K operators as well as the terms forcing orthogonality to the occupied orbitals.These terms would have to be re-valuated at each order of perturbation theory. E. A. McCULLOUGH JR,J. MORRISON AND K. W. RICHMAN The PNO approach has certain defects as well. The higher PNO oscillate rapidly which may lead to some difficulty with type (a)errors if very long expansions are used. More seriously the number of PNO giving an energy contribution above a fixed threshold increases as the threshold decreases so that rapid initial convergence is offset by slower asymptotic convergence. Overall truncation error smaller than ca. 5% in E2(m)would probably be difficult to achieve in practical calculations and there is no reason to expect smaller errors at higher orders.Thus iterating to very high order as has been done successfully in atomic coupled-cluster calculation^,^^ seems questionable with the PNO approach. These last remarks of course apply equally well to basis-set methods. For the above reasons we do not view a numerical PNO approach as the optimum method for numerical perturbation calculations even on diatomic molecules. The method is probably most useful in moderate accuracy applications at fairly low orders of perturbation theory. Still there are a number of important problems which could be addressed at this level including basis-set incompleteness questions of the type considered in this paper. We thank the US. National Science Foundation and the Swedish Research Council for partial support of this work.H. Kelly Adv. Chem. Phys. 1969 14 129. I. Lindgren and J. Morrison Atomic Many-body Theory (Springer-Verlag Berlin 1982). R. J. Bartlett Annu. Rev. Phys. Chem. 1981 32 359. J. Morrison J. Phys. B 1973 6 2205 and references therein; A. M. Mirtensson J. Phys. B 1979 12 3995. P-0. Lowdin and H. Shull Phys. Rev. 1956 101 1730. W. Kutzelnigg in Methods of Electronic Structure Theory ed. H. F. Schaefer I11 (Plenum Press New York 1977) chap. 5. E. A. McCullough Jr J. Phys. Chem. 1982 86 2178. P. Malinowski M. Polasik and K. Jankowski J. Phys. B 1979 12 2965. R. J. Bartlett and D. M. Silver J. Chem. Phys. 1975 62 3258. lo K. Jankowski and P. Malinowski Phys. Rev. A 1980 21 45. l1 R. K. Nesbet J. Chem. Phys.1962 36 1518 specifically tables XI1 and XIV. l2 P. A. Christiansen and E. A. McCullough Jr J. Chem. Phys. 1977 67 1877. l3 L. Adamowicz and R. J. Bartlett In?. J. Quantum Chem. to be published. l4 K. Szalewicz B. Jeziorski H. J. Monkhorst and J. G. Zabolitzky J. Chem. Phys. 1983,78 1420; 79 5543. l5 L. Adamowicz and R. J. Bartlett personal communication. C. F. Bender and E. R. Davidson Phys. Rev. 1969 183 23. l7 L. Adamowicz and E. A. McCullough Jr Int. J. Quantum Chem. 1983 24 19. S. Garpman I. Lindgren J. Lindgren and J. Morrison 2.Phys. A 1976,276,167; A. M. Mirtensson Ph.D. Thesis (Chalmers University of Technology Gothenburg 1978). I. Lindgren and S. Salomonson Phys. Scr. 1980,21,335; J. Morrison and S. Salomonson Phys. Scr. 1980 21 343.

ISSN:0301-5696    

DOI:10.1039/FS9841900165

出版商:RSC    

年代:1984

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. I. M. Mills (Reading University) said Prof. Davidson has given us a very balanced and fair summary of the present state of play in ab initio calculations which we all appreciate. As an experimental spectroscopist I would like to make one comment on the need for better communication between experimentalists and theoreticians the value of ab initio calculations to experimental workers would be greatly enhanced if theoreticians would give some estimate of the probable uncertainty in the results of their calculations. Although there are obvious difficulties in doing this even a subjective estimate would be valuable. It is for this reason we particularly value workers who apply their techniques to a wide range of systems at least some of which have been chosen as systems for which good experimental results already exist.Dr R. E. Overill (King’s College London) said In his introductory survey Prof. Davidson reminded us that the development of digital electronic computers has been a crucial factor in the practical application of quantum mechanics to molecular systems. Dr Handy underlined this point in his Lennard-Jones lecture when he stressed the importance of today’s researchers having access to the latest computer hardware. In the light of these remarks it may be worth looking ahead to see what implications future developments in computer technology and architecture are likely to have for molecular calculations. Towards the end of the present decade silicon-based technology will start to be superseded by that based on gallium arsenide with the resulting machines (such as the Cray 3) having projected speeds of up to 10 gigaflops.By the turn of the century the first optical computers based on lasers and non-linear optical materials such as indium antimonide should be under construction with potential speeds in the teraflop region. As processor speeds increase however correspondingly larger memories and data-path bandwidths will also be required to exploit their full potential. Multiprocessor architectures with different processes executing in parallel on different processors (MIMD) are beginning to be developed. Early examples include the Denelcor HEP and the Cray X-MP family. In order to achieve the superior performances possible with such architectures radical algorithm redesign and meti- culous synchronization of parallel processes will be required.A cost-effective software engineering strategy for such systems is to base all algorithms round a small kernel of standard procedures which can be individually optimized for each multiprocessor architecture. Dr G. Hunter (York Uniuersity Ontario Canada) said I would like to ask Prof. Davidson if he forsees any utility for density-functional theory in molecular- structure calculations? Also does he envisage being able to construct wavefunctions for large molecules by assembling structural fragments taken from smaller molecules? Prof. E. R. Davidson (Indiana University U.S.A .) replied Density-functional theory as exemplified by Xaor more refined methods is a useful approximation for large molecules and solids.It is not easily refined however if one wants greater accuracy. Also the Kohn-Sham theorem which justifies using a Fock-like equation 175 GENERAL DISCUSSION to give orbitals whose densities sum exactly to the total density does not at all justify use of the virtual orbitals from this operator for excited states. In practice excitation energies calculated from virtual orbitals are not particularly more accurate than those calculated with SCF improved virtual orbitals. I do not envisage any quantitative scheme based on constructing wavefunctions from molecular fragments. Probably non-orthogonal localized molecular orbitals are qualitatively transferable at the SCF level but correlated wavefunctions are almost certainly not transferable.Also I do not see in an ab initio calculation any advantage to transferring orbitals. It would only be used to construct an initial guess to an SCF procedure but would save almost no computation. Empirical schemes like ‘diatomics in molecules’ which seem to transfer fragment energies are very difficult to convert into feasible ab initio methods which meet the criteria of being refinable and giving the right answer. Dr D. L. Cooper (Oxford University) said In his interesting General Introduction Prof. Davidson considered the usefulness of different ab initio methods. As his yardstick he often took the ability to perform accurate calculations for naphthalene. It is important to bear in mind that many users of the different techniques are interested in the results for a limited set of systems.To take an extreme example there may be little relevance to the observation that a successful method for the study of highly stripped diatomic transition-metal hydride ions cannot also treat naphthalene. Prof. A. D. Buckingham (Cambridge University) said It has been fascinating to hear Prof. Davidson review the past and present position of ab initio computations. It would be very interesting to learn his thoughts about the future. Prof. E. R. Davidson (Indiana University U.S.A.)replied The computer hardware and numerical algorithms are now in place to make computational quantum chemistry a routine tool for experimental chemists. Many successful experimental chemists such as Lipscomb Herzberg and Cotton have always used theory to interpret the results of their experiments.In the future chemists of all types will have available to them semiquantitative quantum-theory results at the press of a button. These calculations will not rely on having a quantum chemist involved in routine applications. There will always remain problems for which the present methods are either inaccurate or too expensive. The job of the theoretician will be to extend the range of applicability of theory. This could conceivably mean abandoning everything that we now do and starting with a fresh approach but it is unlikely that any radically different approach will be more economical while remaining infinitely refineable. One problem with the present methods is that to the novice the wavefunctions seem too complex.There is a continuing search as there should be for ‘simpler’ wavefunctions where simpler seems to be defined as ‘closer to the preconceived ideas of Lewis theory’. This search in my opinion is doomed to failure if wavefunctions giving at least 90% of the binding energy excitation energy etc. are considered. One very important fact emerged from the 1960 decade of quantum calculations. This was that the error in a full CI wavefunction in a near-Hartree-Fock atomic valence basis set (i.e.full CI valence bond with undistorted atomic orbitals) is largest where the wavefunction is largest. This conclusion was reached with difficulty because formal perturbation expansions in terms of eigenfunctions of some H, had taught people to think about term-by-term corrections which were large where Yowas small.Thus the usual calculation of C for two hydrogen atoms is explained in terms of expansion in excited states of the hydrogen atom which are all small where Yois large. GENERAL DISCUSSION A variational calculation of C6,on the other hand can recover 90% of C6 with one p orbital provided that p orbital is the same size as the 1s orbital. One can illustrate this same difference in viewpoint and efficiency by comparing the different perturbation theories for the hydrogen atom in an electric field. One can solve exactly for Yl and obtain a closed-form compact simple expression which is large where Yois large or one can expand Y in the excited states of the atom.In the latter case each term is small where Yois large and one cannot easily grasp the correct picture of Y by examining individual terms. The same criticism applies to modern users of MBPT with a discrete basis set. As long as the basis set is of the typical type say 63 1G*,all virtual orbitals are necessarily of the same size as the occupied valence orbitals and the perturbation sums in any order make sense. In the limit of a complete basis set including very diffuse orbitals the low-energy virtual orbitals will all be diffuse and the perturbation sum for any diagram will become ill-defined (i.e. the limit of a sum truncated after a fixed number of virtual orbitals does not exist as the number of diffuse basis functions is increased).Of course the same comment applies to any attempt to use SCF virtual orbitals in a CI calculation. What is required to improve a full valence-bond calculation is two-fold. First is a relaxation of the valence atomic orbitals in the molecular field (i.e. contraction and polarization). Conceptually this is a small effect but it provides a large fraction of the binding energy. Secondly however is the introduction of ‘non-physical ’ orbitals to describe the remaining electron-correlation effect. These orbitals to be efficient must be the same size as the occupied orbitals but with additional modes. They are non-physical in the sense that they do not correspond to true excited states so are outside of the vocabulary and experience of spectroscopy or semiempirical theory.Any quantitative ‘simple’ theory must somehow admit the existence and importance of these orbitals. Dr P. R.Surjan (Chinoin Pharmaceutical Works,Budapest Hungary) said The fact that silacyclobutadiene has a distorted structure merits some discussion. It is important that the energy separation between the distorted and symmetric (square) singlet structures is much less than that for cyclobutadiene (ca. 4 as compared with 12 kcal mol-I). One can also invoke some qualitative arguments suggesting that this energy difference for silacyclobutadiene should not be too high. In fact cyclobutadiene which is square has a degenerate ground state at the simple Hiickel level. Of course electron interaction (correlation) splits this degeneracy but one can speak of a quasidegenerate ground state so a Jahn-Teller distortion takes place.Silacyclo- butadiene however has a non-degenerate ground state even at the Hiickel level owing to the diagonal perturbation presented by the Si atom. The quasi-Jahn-Teller effect if any is less pronounced and the energy difference between the square and distorted structures is expected to be smaller. It would be useful to see the full potential curve of silacyclobutadiene with respect to distortion. This curve must have two identical minima corresponding to the two identical distorted structures while a third stationary point of saddle-point character corresponds to the symmetric transition state. It would also be useful to determine the effective mass and the zero-point level and compare it to the barrier of ca.4 kcal mol-l. Finally although Born-Oppenheimer calculations predict a distorted equilibrium structure the corresponding states are by no means eigenstates of the full (beyond Born-Oppenheimer) Hamiltonian; it is well known1 that only the superposition of the two equilibrium structures can be considered as a stationary solution. It is 178 GENERAL DISCUSSION therefore possible that while we have an unsymmetric equilibrium structure experi- ments would predict a symmetric ground state as an average. This again depends strongly on the height of the barrier which is rather small in this case. This contribution was stimulated by a discussion with I. Mayer. A. Laforgue Znt.J. Quantum Chem. 1981 19 989. Prof. J. S. Wright (Carleton University Ottawa Canada) said As discussed by Colvin and Schaefer the singlet state of silacyclobutadiene contains a significant contribution from the low-lying configuration corresponding to the 3a” +44”excit-ation. This is to be expected by analogy with rectangular cyclobutadiene where the HOMO and LUMO n orbitals lie very close together. Optimization of the molecular geometry of silacyclobutadiene using one SCF configuration is therefore problematic since the nodal structure of the 3a” molecular orbital shows that it emphasizes the short-bond-long-bond rectangular geometry. The second configuration (doubly occupied 4a”)places the node between the short bonds so that inclusion of the second configuration at the geometry optimization state will shift the geometry toward less dramatic bond alternation.The shift may be considerable. Calculation of the CI energy at this new optimum will then further stabilize the singlet relative to the triplet probably by several kilocalories or more. This suggestion could be given a rough test by increasing the short bonds and decreasing the long bonds by the same amount say 0.04 A. Prof. H. F. Schaefer (University of California Berkeley US.A.) (communicated). We have also determined the equilibrium geometry of singlet silacyclobutadiene at the two-configuration (TC) SCF level of theory using the double-zeta basis described in our paper. The structure is displayed below and should be directly compared with H 1.068 H< 1.772 1.341 n nn 988.3” psi 1.963 1.469 H \H the analogous single-configuration SCF structure seen on the left-hand side of fig.I of our paper. In one sense the TCSCF structure does lessen the degree of bond alternation compared with the earlier reported SCF structure; i.e. the difference in Si-C bond distances is decreased from 0.237 A (SCF) to 0.191 A (TCSCF). How- GENERAL DISCUSSION ever an opposite (although smaller) effect is seen with respect to the C-C distances which differ by 0.203 A at the SCF level and 0.211 A at the TCSCF level of theory. Neither of these structural changes is major confirming our assumption that the structure of ground-state silacyclobutadiene is qualitatively correct at the single- configuration Hartree-Fock level of theory.A truly quantitative (bond distances reliable to 0.003 A) structural prediction would require the use of a large basis set in conjunction with TCSCF-CI wavefunctions. Prof. S. D. Peyerimhoff (University of Bonn West Germany) said I turn to Prof. Schaefer. You have optimized the structures at the one-configurational level. On the other hand it has been known since the first CI-type treatment of cyclobutadiene (1968) that at least the singlet state requires a two-configurational description; this has also been confirmed in your calculation on silacyclobutadiene in which you find two configurations with coefficients 0.94 and -0.33. Hence my question how reliable are the optimized singlet SCF structure and the corresponding vibrational frequencies i.e.how close do you think these data are to physical reality? Is the quite asymmetric structure you find perhaps a consequence of symmetry-breaking? If you would perform a good CI calculation for an optimal symmetric structure and compare this result with the equivalent CI result at the geometry given now do you think the asymmetric geometry would still be preferred? As a sideline I notice that there is a tendency in the literature (for example in your paper and that of Morokuma and coworkers also presented at this Symposium) to give results of gradient calculations (bond lengths and bond angles) to high precision (bond lengths with three figures after the decimal point) whereas I am convinced that the actual accuracy of these results (especiallyif SCF gradients are employed) is considerably lower.Perhaps this practice should be reconsidered. Prof. I. M. Mills (Reading University) said The small deviation from C,,symmetry in the structure calculated for silacyclobutadiene suggests that this molecule might show a tunnelling spectrum between two symmetrically equivalent minima of C symmetry which are illustrated in fig. 1 of the paper. To predict where such transitions might be observed in the spectrum it would be particularly valuable if Prof. Schaefer would calculate the energy in the symmetrical C, structure with average bond lengths and angles on the two sides of the molecule so that we can have some prediction of the barrier to tunnelling between the two minima.Prof. P. Siegbahn (Stockholm University Sweden) said We have been interested in d-shell effects on second-row atoms for some time. The molecules we have studied most carefully are CCl and ClF. For these simple molecules large basis sets and a high-level correlation treatment could be used. In agreement with Prof. Schaefer’s findings we have also found large errors from using incomplete d basis sets. The use of a double-zeta (DZ) basis set gives bond distances which are at least 0.2 au too long at the SCF level. Adding a d set drastically improves the agreement with experiment. Unfortunately further extensions of the basis set does not necessarily lead to better results. We have found that d basis functions with large exponents > 2.0 are necessary for reaching the Hartree-Fock limit which is 0.1 au too short for C1F.A small MCSCF treatment allowing proper dissociation gives an over-long bond distance by 0.1 au and not until we have performed a much larger calculation do we again obtain good agreement with experiments. With these large systematic errors of different signs it is fairly easy to obtain a calculation which will often give reasonable geometries owing to cancellation of errors. One such calculation would be an SCF GENERAL DISCUSSION calculation with a DZ +d basis set. Another such calculation at a higher level would be a one-reference-state SDCI calculation with cluster correction. Summarizing a proper description of resulting geometries at the SCF DZ+d level for molecules containing second-row atoms would be that they contain systematic errors of the order of f0.1 au but that one can assume based on empirical evidence that the geometries are actually much better.I would like to hear if Prof. Schaefer shares this viewpoint. Prof. J. Morrison (Uniuersity of Utah U.S.A.)said McCullough Richman and I have just completed a calculation of the correlation energy for FH. This calculation which was done numerically and will be reported presently at this meeting indicates that the singlet states are more sensitive to basis-set errcr than are the triplet states. For this reason I found it interesting that the addition of d functions affected the geometry calculated for the singlet state much more than it effected the geometry for the triplet.Does Prof. Schaefer imagine that the basis set he used for his singlet calculation was very ‘hungry’? Prof. H. F. Schaefer (Uniuersity of California Berkeley U.S.A.)(communicated).As demonstrated in my response to the question from Dr Wright the asymmetric structure for the ground state of silacyclobutadiene is not a consequence of symmetry breaking. For triplet silacyclobutadiene when the deviation from C, symmetry is much smaller it would be very worthwhile to perform an MCSCF structural optimization to investigate this point. At the single-configuration SCF level of theory the C, and C triplet structures differ in energy by < 0.5 kcal mol-l. In my opinion it is important to distinguish between the precision of the prediction made at a certain level of theory and the reliability of that prediction relative to experiment.If one states clearly the basis set and type of wavefunction selected it should be possible precisely to reproduce a given structural prediction in other theoretical laboratories around the world. This element of reproducibility long common to experimental chemical methodologies has only in the past decade become a reality for electronic-structure theory. Moreover relative errors in theoretical predictions are known to be much smaller than absolute errors. A notable recent validation of this fact is given by McKean et a1.l These authors demonstrate that ab initio bond-distance differences as small as 0.0005A can be physically meaningful if a consistent level of theory is adopted.Prof. Mills made a very appropriate comment. We are in the process of locating the C, stationary points for both singlet and triplet silacyclobutadiene. These stationary points are not only pertinent to the possibility of a tunnelling spectrum but also relevant to the question of the aromaticity of silacyclobutadiene. Our experience confirms Prof. Siegbahn’s statement that the DZ+d SCF level of theory generally provides molecular structures in good agreement with available experimental data. His analysis of the reason for this agreement a cancellation of basis-set extension and correlation effects is excellent. In response to Prof. Morrison I would say that the addition of d functions to the basis set should largely satisfy the increased ‘hunger’ of singlet silacyclobutadiene relative to the triplet state.However I do suspect that further extensions of the basis will lower the singlet state by an additional amount (perhaps 2 kcal mol-l) relative to triplet silacyclo bu tadiene. D. C. McKean J. E. Boggs and L. Schaefer J. Mol. Struct. 1984 116 313. GENERAL DISCUSSION Prof. P. Siegbahn (Stockholm University Sweden) said My comment is concerned with geometry optimizations of molecules containing transition metals at the SCF level. I have recently been involved in several optimizations of metal-ligand bond distances in different molecules. For two of these Fe(C,H,) (ferrocene) and Fe(CO), errors of over 0.4 au are found at the SCF level. This occurs even though the ferrocene bond in particular is very strong.A reasonably small MCSCF calculation corrects for most of this error. For other molecules the SCF approximation does farily well. In particular for molecules containing palladium which has a closed-shell ground state the results are quite good. My question is Does Prof. Morokuma have any rule of thumb empirical or otherwise from which one can distinguish the case where SCF does well from the cases where SCF does not do so well? Dr J. Tennyson (S.E.R.C.Daresbury Laboratory) said I turn to Prof. Peyerimhoff s paper. The calculation of rovibrational spectra of small molecules is of great current interest. However if predictions are to be made to aid experimentalists e.g. in characterisation of novel species it is often necessary for transition frequencies to be computed to high accuracy (< 1%).Whilst the ab initio prediction of accurate diatomic rovibrational data may be considered routine the situation for polyatomics is less clear. Within the Born- Oppenheimer approximation the ab initio calculation of spectra generally has three stages (1) solution of the electronic-structure problem at a grid of points (2) fitting/interpolating these points to obtain an (analytic) potential and (3) performing nuclear-motion calculations to obtain the (low-lying) bound states of the molecule. Of these state (1) is computationally the most expensive but errors will of course accumulate in stages of such a calculation. The freedom to choose how one embeds the coordinate frame in a body-fixed system has led to the development of several exact (within the Born-Oppenheimer approximation) hamiltonians and solution strategies which differ principally in the internal coordinates used.The choice of appropriate internal coordinates can be said to reflect the physics of a molecule as represented by the potential. Comparative studies e.g. on the floppy CH; molec~le,~-~ have shown that agreement to within 0.1 cm-l can be obtained for the same potential function using different hamiltonians. Conversely many rovibrational studies have blamed the lack of agreement with experiment on errors in the potential. Even fur the electronically simple H,+ molecule rovibrational calculations predict the vE bending fundamental 1% (20 cm-l) too low4 for an extensive high-accuracy SD-CI su~face.~ Recent work on H$ has also shown vE to vary by more than this when different plausible fitting procedures are used.6 I would like to ask Prof.Peyerimhoff the following questions. (I) To what accuracy can potential-energy calculations be performed for say the vibrational fundamentals of triatomic molecules especially those with large amplitude modes? (2) What hope is there for improving these potentials so that calculations can be performed to better than I % ? (3) What steps are taken to ensure that fitting does not unduly degrade accurate ab initio data? S. Carter and N. C. Handy J. Mol. Spectrosc. 1982 95 9. J. Tennyson and B. T. Sutcliffe J. Mol. Spectrosc. 1983 101 71. S. Carter N. C. Handy and B. T. Sutcliffe Mol.Phys. 1983 49 745. J. Tennyson and R. T. Sutcliffe Mol. Phys. 1984 51 887. ' R. Schinke M. Dupuis and W. A. Lester Jr J. Chem. Phys. 1980 12 3909. P. G. Burton E. von Nagy-Felsobuki G. Doherty and M. Hamilton Mol. Phys. in press. GENERAL DISCUSSION Prof. S. D. Peyerimhoff (Uniuersity of Bonn West Germany) replied In principle potential-energy calculations can be performed to almost any degree of accuracy (at least for small polyatomic systems) within the Born-Oppenheimer approximation if one is willing to invest enough computer time. Every single calculation requires then a large atomic-orbital basis (up to f functions) and large MRD-CI-type expansions; furthermore a large number of grid points is then needed. I think a 1 kcal mol-1 accuracy over the entire region of large-amplitude motion requires more time than one would generally like to spend with a few representative exceptions perhaps.For diatomics solution of the nuclear-motion part of the Schrodinger equation is easy and it is documented in the literature that co values for diatomics based on purely ab initio potentials are calculated within an accuracy of a few wavenumbers. I do not think one can push the accuracy much higher since diagonal Born-Oppenheimer corrections already approach this magnitude. Generally agreement of co for triatomics seems to be worse; however I do not think this is the fault of the potential-surface calculation but rather of the nuclear-motion treatment. Quite often co is only given from the derivative of the curve at equilibrium (and this number is not necessarily the same as that one obtains from an extrapolation of the vibrational levels actually calculated in the same potential) or co is extracted from solving the Schrodinger equation for nuclear motion in an approximate way e.g.without considering coupling of various modes. Although I believe that quantum chemists do a good job in calculating potential-energy surfaces they now have to learn how to solve the Schrodinger equation for nuclear motion with comparable accuracy. Our group has limited experience in this area of dynamics in polyatomics; better information on this particular problem could probably be obtained from Dr Botschwina. We have studied the Renner-Teller effect for AH systems1? such as NH, but in this case terms beyond the Born-Oppenheimer approximation which are normally neglected in potential-energy work also play a role.This aspect is important for excited states since often measured frequencies and those calculated on Born- Oppenheimer surfaces cannot directly be compared on theoretical grounds. Our experience also indicates that an analytical fit of the entire surface is quite difficult but fitting of E by spline polynomials is quite efficient. Unfortunately one must calculate a considerable number of grid points in this case; furthermore these points should be chosen with an eye towards the dynamics which is often not done. Finally our own philosophy is to obtain with as little computational effort as possible a maximum of useful information on the electronic structure of systems in particular data which cannot easily be obtained by measurements.A 1% accuracy in vibrational frequencies for polyatomics in large-ampli tude motion will probably be cheaper to obtain from experiment for some time at least for most molecules. R. J. Buenker M. Perid S. D. Peyerimhoff and R. Marian Mol. Phys. 1981 43 987. * M. PeriC S. D. Peyerimhoff and R. H. Buenker Mol. Phys. 1983,49 379. Prof. P. Siegbahn (Stockholm Uniuersity Sweden) said I would like to make some comments on the calculation on ScH which illustrates some of the points I am trying to make in my paper. The comment is concerned with the fact that in MRD-CI calculations one generally chooses to expand the configurations in an SCF orbital basis obtained from one of the states of interest.It is interesting to compare the qualitative chemical conclusions drawn from this calculation on ScH with the conclusions from a calculation by Bauschlicher and Walch on the same molecule but using MCSCF orbitals optimised for the different states. It should first be said that the resulting energies and potential curves are of high quality in both these papers. As a GENERAL DISCUSSION background Sc binds to hydrogen in the ground ds2atomic state. The d occupation can be db dn or ds resulting in six fairly low-lying states singlet and triplet C I3 and A. In the MRD-CI paper the reason for the lE+ ground state is said to be a mixing between the open-shell db7a state and the 7a2 state which can only occur for the C.state. In the MCSCF paper the open-shell state is not seen at all and the 70 state has a coefficient of 0.93. Instead important excitations are noted from the 702 to ln2and 8a2,which moves the non-bonding electrons away from the bonding region and reduces the repulsion. The most important origin of the lZ+ ground state is however an orbital hybridization between the 4s and 3db orbitals which can clearly only occur for the C state. The orbital hybridization will improve the overlap of the bonding orbital and move away the electrons in the non-bonding orbital. This general chemical effect will in addition to ScH also explain the ground state for a large number of molecules such as NiCO NiN, NiH,O and NiPH,. The question is Does Prof.Peyerimhoff see any trace of the different hybridizations in the two states of ScH in her calculations using SCF orbitals for one of the states? Prof. S. D. Peyerimhoff (University of Bonn West Germany) replied The actual differences between the MCSCF results you mention and the MRD-CI results are very small. We look at the entire region of internuclear separations and note [see our ref. (33)] that there are o2(d)70 configurations which give rise (as you mention) to singlet and triplet A II and E+ states but that only for the lC+ there is an additional low-energy configuration 6a27a2,Both lX+ states show interaction (or mixing) which depends on the internuclear separation. If the SCF molecular orbitals of the 6a2702 solution are employed the contribution of the 6a27a2 configuration at the equilibrium bond separation is c2 = 0.87 very close to that found by Bauschlicher and Walch (c = 0.93 c2 = 0.869) who looked only at Re.The XIE+ curve rises quite sharply and the contribution is reduced to c2 = 0.79 at R = 5.0 a, at which point XIZ+ is still below 3Z+. As we state in ref. (33) (p. 302) ‘the closed shell 70 molecular is predominantly given by the (4s3da) atomic function’ which is the hybridization between 4s and 3da mentioned by Prof. Siegbahn. The open-shell 70 possesses mixed (4s4p)character around equilibrium as pointed out in ref. (33)(p. 300) and thus has a different hybridization. The relative energy separation between the two lC+ states in ScH also has been calculated in a basis of natural orbitals of the X lC+ state.At equilibrium the 70 SCF molecular orbital of the 6a2702configuration is the dominant part of the 70 natural orbital. The 2 lE+ is expanded in the same set of natural orbitals and at Re the main contribution is 7080 7090 70100 with weights of c2 = 0.55 0.17 and 0.13 respectively and shows that the open-shell 0 is different from the closed-shell counterpart. The 2 lC+ in ScH (apparently not treated in the MCSF work) is also important at bond lengths near Re because we believe that the measured absorption in the 17690-18 350 cm-’ region must be assigned to the X lZ+(7a2)-2 lC+ (70d0) transition calculated in our work to be around 2.1 eV. Dr P. R. Surjan (Chinoin Pharmaceutical Works Budapest Hungary) said The reader is much impressed by the accuracy of energetic-type results discussed in this paper.For transition intensities of course direct comparison with experiments is more difficult mainly owing to the uncertainty of experimental data. However the quality of the wavefunction in this respect can be checked by consistency tests such as the validity of the off-diagonal hypervirial theorem and its consequences (equivalence of dipole-length and dipole-velocity forms sum rules etc.) I would like to ask Prof. Peyerimhoff about this problem concerning large-scale CI calculations. GENERAL DISCUSSION Dr D. W. Davies (Uniuersity of Birmingham) said Prof. Peyerimhoff in her interesting survey of the work at Bonn on molecular spectra has discussed the calculation of transition probabilities for electronic-dipole transitions in diatomic molecules.She suggests that the calculations are usually within 30% or at most a factor of two of the experimental results. She refers to the use of the dipole-length matrix element only for this calculation. It has often been pointed out that the dipole velocity1 form of the transition probability should also be calculated in the hope of providing a check on the accuracy of the result. It is also possible to use the dipole acceleration.2 For atomic transition probabilities a critical discussion has recently been given by Crossley3 who refers to ‘rules-of-thumb’ for deciding whether to accept the dipole- length or the dipole-velocity result,2 and who sets a limit of 6 electrons for ‘very accurate ’ calculations of ‘predictive quality’ and 20 electrons for calculations of ‘any real accuracy ’.I would be interested to hear Prof. Peyerimhoffs further comments on the reliability of electronic transition probabilities obtained from the dipole length particularly as several speakers have raised the general question of the limits of accuracy of ab initio molecular calculations. See for example D. W. Davies Trans. Furaday SOC.,1958 54,1429. R. J. S. Crossley Adv. At. Mol. Phys. 1969 5 237. R. J. S. Crossley Phys. Scr. 1984 T8 1 17. Prof. S. D. Peyerimhoff (University of Bonn West Germany) replied In all of our calculations we routinely evaluate the oscillator strengths on the basis of both the length and the velocity form of the transition-moment operator.We have however never employed the accelerator form. In various cases we have published both values fe(r) and fe(V) but generally we give only theflr) or Rere.value to save space and we feel that the length operator is less sensitive to ‘kinks’ in the wavefunction expansion than is the velocity operator. A representative example1 is SiH in which the ratio betweenflr) andflv) is 0.71 0.92 1.25 1.26 1.48 0.88 1.46 1.37 and 0.78 for transitions between X 211 and A 2A B 2Z+,2 2Z+,3 2C+ 4 2C+ 2 211 D 2A 3 211 and E 2Z+states respectively. Other published data are for example for the inner-shell excitation in ethylene,2 i.e.oscillator strengthsflr) of 0.088 (0.087) for 1 s + n*,0.0032 (0.0025) for 1 s + 3s 0.00051 (0.00049) for 1s-+ 3p0,0.0036 (0.0034) for 1s-+ 3pn and 0.0026 (0.0025) for 1s -+3dn transitions [the f(V) result always given in parentheses] or the corresponding data for the n -+n*excitation evaluated in various natural-orbital basis 0.310 (0.267) 0.260 (0.200) 0.293 (0.227) 0.286 (0.128) 0.482 (0.387) and 0.3 15 (0.209) in the same notation.The latter values also give an indication of changes due to various orbital transformations employing the same atomic-orbital basis. The effect which the truncation of the MRD-CI wavefunction has on the calculated electronic-transition moment has also been investigated for C, for e~ample.~ The quantity C I I2 for the Swan bands changes from 5.33 to 4.74 if the CI expansion is increased from 1000 to 6000 while it stays fairly constant upon further increase to 9000 i.e.the range in which the calculations are generally undertaken. It should be stressed however that the molecular-orbital (or natural-orbital) basis employed for the CI expansion should be such that it represents the charge distribution of the state(s) under discussion already in a relatively reasonable manner. In other words a longer CI expansion would be required than we generally have if a H+F-charge distribution (dipole moment) has to be represented by orbitals which are optimized for a state of H-F+ charge distribution. Unfortunately I have not succeeded so far in obtaining ref. (3) mentioned by Dr GENERAL DISCUSSION Davies and so I cannot comment on his statement of the 20-electron limit. In my experience I do not see such a limit although it is clear that systems with many electrons need a large atomic-orbital basis so that all CI calculations become more time-consuming if the same accuracy as in smaller systems is maintained.’ M. Lewerenz P. J. Bruna S. S. Peyerimhoff and R. J. Buenker Mol. Phys. 1983 49 1. A. Barth R. J. Buenker S. D. Peyerimhoffand W. Butscher Chem. Phys. 1980 46,149. R. J. Buenker and S. D. Peyerimhoff Chem. Phys. 1975 9 75. C. F. Chabalowski R. J. Buenker and S. D. Peyerimhoff Chem. Phys. Letf.,1981,83 441. Dr D. L. Cooper (Oxford University)said :The large-scale computations of the Bonn and Wuppertal groups as described by Prof. Peyerimhoff are indeed very impressive. Nonetheless a few words of caution may be in order concerning the theoretical study of those fine-structure effects which arise primarily from spin-orbit coupling.In a first-principles derivation from quantum electrodynamics the next dominant corrections beyond the Breit-Paul Hamiltonian (order &mc2) correspond to the ‘electron anomaly’ modifications to charge current densities. We have investigated the inclusion of the so-called anomalous magnetic moment in effective Hamiltonians for perturbative (Breit-Pauli) and non-perturbative approaches to fine structure;‘ we find that the anomalous magnetic moment (a.m.) terms which are of the order (a/71)a4mc2 are particularly simple to include in the perturbative approach and that they make significant contributions to fine-structure intervals in high-accuracy computations.We were slightly amused to notice that in the computation of the splitting in the ground state of OH off-diagonal effects were found to be ca. 0.086 cm-l. There is no mention of the a.m. contribution. In an MCSCF study of this system3 we found a value approaching 0.4 cm-l near re whose inclusion would improve the agreement with experiment. We are more concerned by the application of the Breit-Pauli approximation to atoms beyond the second row of the periodic table. This higher-order relativistic contribution (order a6mc2)increases along the second row reaching ca. 1.5 cm-l for sulphur and ca. 5.5 cm-l in chlorine. Very extensive Breit-Pauli computations on molecules containing heavier atoms might not be worthwhile. We have performed numerical comparisons4 of different approaches to fine-structure calculations in the boron and fluorine isoelectronic sequences up to nuclear charge Z =26.An accuracy of one or two wavenumbers is precluded by the higher-order relativistic effects for 2 > 10. The Breit-Pauli approach to fine-structure splittings after inclusion of electron anomaly effects is very accurate for low 2. We would caution that ‘spectroscopic accuracy’ of a few wavenumbers is more than difficult for higher 2 because of neglected higher-order relativistic effects. J. Hata D. L. Cooper and 1. P. Grant J. Phys. B 1985 in press. D. L. Cooper J. Hata and I. P. Grant J. Phys. B 1984 17 L499. D. L. Cooper Mol. Phys. 1985 54 439. D. L. Cooper J. Hata and I. P. Grant J. Phys. B 1985 18 1081.Prof. S. D. Peyerimboff (University of Bonn West Germany)replied to Dr Cooper We are fully aware of the fact that we have neglected higher-order terms in our calculations (based on Breit-Pauli approximations) of spin-orbit splittings in mole- cules containing first-and second-row atoms. We also realize that they will become important (i.e. contribute > 1 %) for splittings in heavier at0ms.l The effect of the anomalous magnetic moment (a.m.) terms which you have investigated is indeed easy GENERAL DISCUSSION to include in the calculations; it is however still quite small (ca. 0.25% in atoms B to C1 inclusive and 2 cm-l on an absolute scale for the 3Psplitting in Cl) and we cannot claim to reach an accuracy of this order. You are right that the calculated second-order contribution to the OH splitting would also fall within our error bars -but this was not entirely obvious from the beginning.Furthermore it is not only an additive term to the first-order splitting. In a rotational analysis of the diatomic spectrum the second-order spin-orbit splitting parameter A,exhibits the same J dependence as the spin-splitting parameter which is different from the J dependence of the (first-order) spin-orbit parameter A. Hence even though the second-order spin-orbit splitting is smaller at Rethan the a.m. term (which has predominantly the behaviour of A)it might be useful information for the analysis of experimental spectra. Finally as soon as elements from the third row of the periodic table are involved we do incorporate higher-order effects by modifying the kinetic-energy term and adding the Darwin term to the standard non-relativistic hamiltonian.Examples for the bromine2 and selenium atom3 are in the literature. In Br the 2P ground-state splitting is calculated to be 3420 cm-l employing the non-relativistic hamiltonians while it is 3655 cm-l if the higher order terms in the hamiltonian are approximated whereby the measured splitting is 3685 cm-l. In the selenium atom3 the corresponding 3P2-3P1energy difference is 1684 cm-l (non-relativistic first-order contribution) 1805 cm-l (non- relativistic first-order plus second-order spin-orbit) 1850 cm-l (first-order spin-orbit but employing a modified relativistic hamiltonian) and 1994 cm-l (first-order plus second-order spin-orbit and modified relativistic hamiltonian) compared with a measured splitting of 1989.5 cm-l.F. Mark C. Marian and W. H. E. Schwarz Mol. Phys. 1984,53 535. B. A. Hess P. Chandra and R. J. Buenker Mol. Phys. 1984 52 11 77. T. Matsushita C. Marian R. Klotz S. D. Peyerimhoff and B. Hess Chem. Phys. in press. Prof. S. D. Peyerimhoff (University of Bonn West Germany) said In responding to Dr Werner's remarks about our paper I would remark that fig. 5 shows rotationless transition probabilities for Av = 1 in the X211 ground state of OH obtained by employing dipole-moment functions from various sources. Unfortunately ref. (57) is not appropriate (it contains dipole-moment functions for HF HC1 and HBr) The original versions of the various data are (1) MRD-CI (W.Quade Diplomarbeit Bonn 1982); (2) MCSCF (W. J. Stevens G. Das A. C.Wahl M. Krauss and D. Neumann J. Chem. Phys. 1974 61 3686); (3) MCSF/CI (S-I. Chu M. Yoshimine and B. Liu J. Chem.Phys. 1974,61,5389); (4) CEPA2 [W. Meyer Theor. Chim. Acta (Bed) 1974 35,2771. The experimental dipole-moment curve is deduced by R. E. Murphy J. Chem. Phys. 1971,54,4852 and A. F. Ferguson and D. Parkinson Planet. Space Sci. 1963 11 149 from measurements. They approximate it by M(r) = Mdr)+ P~XPLD(r -re)] whereby MFpis deduced by Ferguson and Parkinson as 5 MFp(r)= Mi(r-re)i i-1 with M = 1 M = 0.036 M = -0.129 and M = -0.543 and M5 = 0.441 (all in units of A-1). According to Murphy et al. P = -0.276 and p = 2.32 A_'. Two more recent dipole-moment functions (H.J. Werner P. Rosmus and E. A. Reinsch J. Chem. Phys. 1983 79 905) employing an MCSCF-SCEP treatment one by S. R. Langhoff (1983 personal communication to E. F. van Dishoek) and our GENERAL DISCUSSION MRD-CI curve are tested in detail by N. Grevess A. J. Sauval and E. F. Dishoek (in press) who find that the three functions calculated by Quade Langhoff et al. and Werner et al. are in very close agreement. Typical values are 12.2 (MCSF-SCEP) and 12.6 (MRD-CI) for Ah and 15.6 (MSCF-SCEP) as compared with 16.6 (MRD-CI) for AX and 13.2 in contrast to 14.7 for A:. Further comparison between the results of various methods can be found in the work by N. Grevesse et al. and by Werner et al. In summary one then finds that the three most recent calculations with different methods agree even more closely than seen in fig.5.This is of course very encouraging and demonstrates the power of theoretical calculations to obtain absolute data for various quantities without too much effort while experimentally only relative intensities are directly known in this case One should however not forget that a direct comparison with experiment requires also averaging over rotational levels. Finally some additions to references (la) and (52)-(56) of my paper for quite reliable calculations of transition probabilities between various excited states could be made; H. J. Werner J. Kalcher and E-A. Reinsch J. Chern. Phys. 1984,81,2420 on N, P. Rosmus and H. J. Werner J. Chem. Phys. 1984 80 5084 on C; and S.R. Langhoff E. F. Van Dischoek R. Wetmore and A. Dalgarno J. Chern. Phys. 1982,77 1379 on OH and probably many more including results of our own group. Prof. J. S. Wright (Carleton University Ottawa Canada) said Contrary to Dr Werner’s suggestion that basis sets containing bond functions ‘can give results anywhere ’ we find very systematic and understandable results in calculations of the dissociation energy as a function of basis-set composition. These results have been reported recently for two representative molecules HCI and N,.l There we showed that as bond functions are added to the basis set the bond energy will continue to increase. When too many such functions have been added the bond energy will always be overestimated.In the normal calculation involving only nuclear-centred functions the bond energy is invariably underestimated. There is therefore a balance point (balanced basis set2’ 3 between nuclear-centred and bond-centred functions which gives an optimum description of both atomic and molecular regions i.e. a nearly correct bond energy results. In the case of HC1 the optimum balance point occurred with double-zeta plus (DZP) polarization nuclear atomic orbitals plus s and p bond functions. For N, it consisted of DZP nuclear atomic orbitals plus two sets of s p and d bond functions. Thus more bond functions are needed to describe the triple bond as expected. These basis sets gave D,= 4.62 eV for HCl (experimental value 4.62 eV) and 9.96 eV for N (experimental value 9.91 eV).These are still small basis sets by most standards and yet the potential curves and derived spectroscopic parameters are remarkably accurate (e. g. within 1% in we).* Basis sets containing bond functions do not violate the variation principle even though the dissociation energy may be overestimated. The absolute energy of any point on the potential-energy curve still lies above the true value and will converge from above to the true value as more functions are added to the basis set bond functions or otherwise. The point is that addition of mid-bond functions leads to a better description in the region of the potential minimum and is of decreasing importance as the bond is stretched (where overlap with the atomic basis sets tends to zero). Since the potential-energy curve is the difference of the atomic and the molecular energy improving the molecular description leads to a deeper minimum.This is exactly what is needed to correct the inherent bias in a nuclear-centred basis set which provides a better atomic than molecular description thus making it extremely difficult to calculate accurate bond energies. GENERAL DISCUSSION J. S. Wright and R. J. Buenker Chem. Phys. Lett. 1984 106 570. J. S. Wright and R. J. Williams J. Chem. Phys. 1983 79 2893. J. S. Wright D. J. Donaldson and R. J. Williams J. Chem. Phys. 1984 81 397. J. S. Wright and R. J. Buenker to be published. Dr G. Hunter (York University Ontario Canada) said Prof. Wright has told us that the use of bond-centred functions in addition to nuclear-centred functions leads to more rapid convergence with respect to the size of the basis set and that it can even produce dissociation/ionization energies larger than the experimental values.Dr Werner commented that the use of bond-centred functions can produce ‘results anywhere’. These observations suggest that the bond-centred functions do not satisfy the appropriate boundary conditions thus leading to a violation of the variation principle. Dr B. T. Pickup (University of Shefield) said A spin-adapted operator basis was first introduced for the non-singlet electron pr0pagator.l The ‘up’ and ‘down’ operators we defined had the desired property of factoring the propagator equations into blocks describing pure spin-ionized and attached final states.We found however that the resulting matrix elements were complex to evaluate [like eqn (29)-(37) in the paper by Yeager et al.] and had an undesirable M-dependence. We sought a redefinition of the operators which emphasised the spin covariance (coupling) properties. The ‘new ’ spin-shift formali~m~-~ gave general spin-shift operators ensuring spin adaptation for any kind of transition (ionization excitation etc.) and component-free (M-independent) expressions. There is a Wick theorem embracing the spatial unitary group structure of the many-fermion problem and a diagrammatic development which gives a generalization of the standard many-body theory. We have more recently6 defined spin-shift propagators which reduce to the Zubarev definition in the singlet reference state limit.The paper by Yeager et al. describes spin-adapted (T) operators for the non-singlet polarization propagator in the spirit of our old spin-shift forma1ism.l I have no quarrel with the results in columns 3-6 of table 1 for the singlet reference state. These reveal the power and promise of multiconfiguration propagator methods pioneered by the authors. The triplet-state calculations however use the T operators. It is not clear that these operators actually factor the propagator equations into spin blocks To understand this let us consider the action of operators TT(dS 0) [6S = 1,0 in eqn (18)-(20)] which act upon any spin (S M) ket to give a state Tt(6S 0) I SM) with spin (S+dS M> (or possibly a null vector).The Tt operator however does not produce a spin state when it acts upon a state of any other spin. Tt is coupled to a ket. The bra vector (SM I Tt(dS 0) is not spin-adapted either. The adjoint operator (SM I T(dS 0) is (obviously) spin adapted and is bra coupled. In the present work the states (OSM I Tt (dS 0) GENERAL DISCUSSION are actually null (because of orbital-index considerations). This ensures no coupling of different spin blocks occurs in A-matrix elements. In B-matrix blocks (OSM I fTt(dS' 0) [Tt(dS 0) HI) I OSM) = (OSM I HTt(dS 0) Tt(dS' 0) I OSM) is non-zero because the operator Tt(dS 0) acts upon a state of spin (S+ds' M) for which it is unmatched. There are ways round this but the old spin-shift notation is no match for the required subtlety.One actually requires the superior transformation properties of the new spin-shifts and the M-independent propagators of ref. (6). All the complexities of eqn (29)-(37) are then swept aside. The correct operators cannot give rise to any linear dependences. I suspect that the reason why columns 6 and 8 of table 1 are not in better agreement is entirely due to the faulty formalism. I warmly recommend that the authors repeat their calculations with the new definitions. B. T. Pickup and A. Mukhopadhyay Chem. Phys. Lett. 1981,79 109. A. Mukhopadhyay and B. T. Pickup Chem. Phys. Lett. 1982,93,414. B. T. Pickup and A. Mukhopadhyay Int. J. Quantum Chem. 1984 26 101. A. Mukhopadhyay and B. T. Pickup Int. J. Quantum Chem. 1984,26 125.B. T. Pickup and A. Mukhopadhyay in Molecular properties Proceedings of the CCPI Study Weekend Cambridge 25-27 March 1983 ed. R. D. Amos and M. F. Guest (SERC Daresbury Laboratory 1984) p. 146. B. T. Pickup and A. Mukhopadhyay int. J. Quantum Chem. Symp. 1984 18 309. Prof. D. L. Yeager (Texas A & A4 University U.S.A.) and Dr P. J$rgensen and Dr J. Olsen (Aarhus Uniuersity Denmark) said We would like to thank Dr Pickup for pointing out a problem with the generalized multiconfigurational time-dependent Hartree-Fock formulation. He is certainly correct that the B matrix is not quite correctly blocked in spin. However for the Be-atom results that are reported the effect is not significant. The results in column 6 are entirely correct since the initial state is singlet and so no T or T+ operators are used.Our calculations within the same spin manifold (3S-+ 3S) in columns 7 and 8 were reported without including the T or T+operators since initial calculations had indicated that the effect of T+ and T operators was extremely small. For the 3S+ lSresults T and Tf operators are obviously necessary. We have repeated our calculations with the B matrix equal to zero. (Hence there will be no symmetry problem.) The results change only in the second decimal place for the first five entries in column 8. The largest change is from 10.23 to 10.26 eV for the transition (2~3~)~s -,(2s7s)lS.Our numbers are significantly improved over the good results in table 1 by using a set of five Gaussian d functions instead of three.l Our triplet to singlet energies also suffer from the fact that the initial MCSCF state is optimized using Q+(00),Q(O,O) R+(1,O) and R(I ,0) operators and consequently is not the most consistent state to use for linear response to states of different spin symmetry.(Hopefully the MCSCF state is a very good approximation.) We would also like to point out that our T+ operators were derived independently but are the same as those in eqn (28) of the beautiful and very elegant paper of Pickup and Mukhopadhyay.2 We have not as yet seen their paper where they introduce new spin-shift operators but are eagerly awaiting its publication in order to see if these new ope,rators will resolve the B-matrix problems. Finally matrix-element formulae with the old spin-shift operators are not extremely difficult to evaluate.For ionization potentials and electron affinities or excitation energies the formulae can be derived by a good graduate student in a day or two. If they are written in terms of reduced matrix elements and vector coupling coefficients they do not even appear to be very complex. ' D. L. Yeager J. Olsen and P. Jerrgensen In[. J. Quanfum Chem. Symp. 18 to be published. €3. T. Pickup and A. Mukhopadhyay lnt. J. Quantum Chem. 1984 26 101. GENERAL DISCUSSION Prof. J. S. Wright (Carleton University Ottawa Canada) said The question of when an MCSCF should be used was addressed by Prof. Siegbahn in his paper on the MC-CI method. He states that the first area is the (trivial) case where the single-configuration SCF method gives bad results e.g.by predicting a poor energy or an incorrect ground state. However the power of a single SCF multireference CI to regain the correct energetics is quite remarkable. For example using the MRD-CI method in a study of N dissociation,lY the SCF configuration appropriate to the equilibrium geometry can be maintained throughout the potential curve provided that enough reference configurations are supplied in highly stretched geometries (14 are required at large R). This is true in spite of the fact that the SCF configuration which dissociates to N3+and N3- is as much as 1.0 hartree above a more appropriate open-shell SCF at large R. Also calculations of molecular spectra e.g. as in the discussion by Prof.Peyerimhoff at this meeting routinely make use of an SCF configuration corresponding to some intermediate excited state from which both ground and other excited states may be generated with transition energies which are usually accurate to 0.2 eV. We therefore wonder whether many of the problems encountered in both Prof. Siegbahn’s calculations and in the literature are due not to the lack of an MCSCF starting point but rather to the fact that most CI calculations have been of the single reference configuration variety. W. Butscher S. K. Shih R. J. Buenker and S. D. Peyerimhoff Chem. Phys. Lett. 1977 52,457. J. S. Wright and R. J. Buenker unpublished results. Dr J. Gerratt (University of Bristol) said Prof. Siegbahn has spoken of the errors introduced into molecular calculations which are essentially atomic in origin.This is closely connected with the problem of obtaining correct values of asymptotic energy splittings. In many cases unless the asymptotic energy spacings are reasonably accurate we cannot be sure that the surfaces elsewhere do not lack an essential feature such as a barrier or a small well. This problem was considered some time ago by Moffitt,l essentially within the valence-bond framework. Moffitt introduced corrections to the elements of the hamiltonian matrix of the form H +H’ = H++[S(Ei-EO)+(Ei-E,)S]. (1) In this equation Ei is a diagonal matrix consisting of the computed asymptotic energies as given by H as R + rm and E is a diagonal matrix of experimental energies. Consequently at large values of R the eigenvalues of the secular equation become just the experimental energies Ei.Hurley2v modified certain aspects of the method and then used it with very small valence-bond wavefunctions to predict binding energies in two crucial cases those of N,* and C0.5 The spectroscopic data at that time allowed a choice of two or three different values for D,. Hurley’s results showed that the experimental values had to be taken as 9.756 eV for N and 11.24eV for CO. The theoretical calculations with his ‘intra-atomic correlation correction’ were accurate to 0.5 eV a feat which has scarcely been bettered to this day. However in spite of its attractive features AIM has proved to be unworkable for larger calculations mainly because in the conventional GENERAL DISCUSSION valence-bond approach atomic corrections are required for every state included in the expansion of the wavefunction.Nevertheless there now seems to be a case for re-examining this idea in the context of more accurate wavefunctions such as the spin-coupled valence-bond and the MCSCF approaches. In the CASSCF procedure it would be necessary to transform the hamiltonian matrix (or certain blocks of it) to a representation in which coupled atomic states appear explicitly. In any case this occurs numerically at large internuclear distances when solving the secular equation. Hurley gives one example (N,) where a small MO-CI wavefunction is transformed to a valence-bond representation and it seems that there are now sufficient theoretical techniques for carrying out this transformation quite generally.sv A version of the Moffitt-Hurley correction is being implemented for our spin-coupled valence-bond procedure and the results of using it will be reported shortly.W. Moffitt Proc. R. SOC. London Ser. A 1951 210 245. A. C. Hurley Proc. Phys. Soc. London Sect. A 1956 69 49. G. G. Balint-Kurti and M. Karplus in Orbital Theories of Molecules and Solids (Clarendon Press Oxford 1973). A. C. Hurley Proc. Phys. Soc. London Sect. A 1956 69 767. A. C. Hurley Rev. Mod. Phys. 1960 32,400. P. E. S. Wormer and Ad van der Avoird J. Chem. Phys. 1972 57 2498; Int. J. Quantum. Chem. 1974 8 715. ' D. L. Cooper and J. Gerratt to be published. Prof. J. S. Wright and Mr R. J. Williams (Carleton University Ottawa Canada) said In their paper on ionization of the hydrides Pope et al.found that the ionization potentials of the neutral molecules were consistently low whereas the ionization potentials of the diatomic cations were usually too high. They suggested that the latter result was due to the experimental ionization potential originating from vibrationally excited states of the cation leading to a low experimental result. In order to see whether these trends could be due entirely to theoretical shortcomings it is necessary to have more information about the calculation. With three surfaces involved (neutral molecule monocation and dication) there are two major sources of error (1) the error in the dissociation limits which changes the position of the asymptotes relative to their correct values and (2) the error in the dissociation energy of the neutral molecule and cation which shifts the position of the minimum relative to each asymptote.Using nuclear-centred basis sets and CI treatments comparable to the present one it is well known that the dissociation energy of the neutral molecule will be only 90-95% of the correct value i.e. consistently underestimated.l Less is known of potential curves for cations and their basis-set dependences although Meyer and Rosmus did large-basis-set CEPA-CI calculations of both neutral diatomic hydrides2 and mono cation^.^ However their method of calculation of the cation dissociation energy involved use of experimental data and is therefore indirect.We have also calculated potential curves for the neutral diatomic hydrides4 and partially completed work on the mono cation^.^ Consider as an example the series of ionizations OH(,II) -+OH+(3C-) -+OH2+ (unbound). The neutral OH dissociates into O(3P)+ H(2S) OH+ dissociates into O(3P)+ H+ and OH2+ dissociates into O+(,D)+H+. For the first ionization OH + OH+ the asymptotic error will be small since any reasonable basis set will accurately recover the hydrogen-atom energy (typically 13.55 eV as compared with an experimental value of 13.60 eV). Our published CI calculations show D = 4.58 eV for OH (experimental value 4.62 eV) GENERAL DISCUSSION whereas using a similar basis set with ion-optimized bond functions gave D,(OH+) = 5.73 eV (experimental value 5.28) i.e.the dissociation energy of the monocation is overestimated whereas the dissociation energy of the neutral molecule is underestimated (slightly). The repulsive dication curve is unbound so that we may expect the dication calculation to be less sensitive to basis-set deficiencies. Ignoring any asymptotic problems with the dication calculation a schematic view of these results is shown below with the ‘correct’ p.e. curves as the solid lines. As is clear from this diagram Ei(1) will be underestimated and Ei(2) will be overestimated quite independent of any experimental complications. O’(* D)+ H+ 0fP) + ti+ Similar results were obtained for FH which also ionizes to the neutral atom F(2P) and H+ on first ionization. The results are more complicated for NH and CH where the ground-state monocations dissociate (adiabatically) to N+ and C+ so that the asymptotic errors can be significant.It would be interesting to know whether Pope et al. have performed this type of analysis of the apparent discrepancies in their computed ionization energies. J. S. Wright and R. J. Buenker Chem. Phys. Lett. 1984 106 570. W. Meyer and P. Rosmus J. Chem. Phys. 1975 63 2356. P. Rosmus and W. Meyer J. Chem. Phys. 1977,66 13. J. W. Wright and R. J. Williams J. Chem. Phys. 1983 79 2893. R. J. Williams and J. S. Wright unpublished results. Prof. I. H. Hillier Dr M. F. Guest and Dr S. A. Pope (University of Manchester) said :Prof. Wright and Mr Williams suggest that the origin of our discrepancy between the calculated and experimental adiabatic ionization potentials of the cation may arise from theoretical shortcomings.They present calculations on OH and OH+ (but not OH2+)using basis sets involving bond-centre functions. We first note that bond-centre GENERAL DISCUSSION 193 functions may overestimate interactions in the bonding region compared with the dissociation limit and that the molecule OH may not be the best system to examine in view of the larger experimental uncertainty in the ionization potential arising from the repulsive OH2+ curve. Furthermore their arguments ignore the problems associated with calculating the O+(2D)-O(3p)energy separation. Thus their schematic overestimation of Ei(2)relies on the assumption that there is no error in the theoretical ionization potential of O(3p).This is clearly not the case and the schematic AE should be reduced accordingly.Also in their figure they associate Ei(2) with OH2+at the dissociation limit. This is open to question particularly in other cases when the dication is bound. The argument of Wright and Williams would suggest that the discrepancy between theory and experiment will be reduced as the quality of the calculation improves. The extensive work of Siegbahn [ref. (71) of our paper] and Pople [ref. (72)] shows this is not the case. In view of these considerations we believe that particularly when the dication is bound there exists a residual discrepancy between theory and experiment which is not computational in origin.Dr P. J. Bruna and Prof. S. D. Peyerimhoff (University of Bonn West Germany) (communicated).It is of interest to find a simple explanation of the facts that both dications NH2+and OH2+ coming from the first row have repulsive ground states while by contrast the isoelectronic second-row species PH2+ and SH2+ show electronic ground states with local minima. Although the low-lying states of AH2+correlating with dissociation products of the type (A++ H+)are always repulsive for larger internuclear separations it is assumed that the local stabilization of such low-lying electronic states in a given AR is due to a larger contribution (or interaction) of the attractive channel (A2++H) in the corresponding wavefunctions. One might thereafter expect that the stability of AH2+ hydrides depends more or less on the relative energy separation between the attractive (A2++ H) and the repulsive (A++ H+)channels respectively.In order to illustrate this point the results obtained by Pope et al. for those AH2+species mentioned above (as well as the relative energetics of the separated atomic states of interest) are collected in fig. 1. We consider first the four-valence-electron cations NH2+and PH2+.The two low- lying dissociation limits involving the repulsive symmetrical charged products (A++H+) as well as the first attractive (A2+++) asymptote correlate with the following molecular states of AH2+ 3Pg(N+, P+)+ 'S,(H+) + '((n C-);EI (1) 'Dg(N+,P') + 'S,(H+) + '(H A C+); EII (11) 2P,(N2+,P2+)+ 2S,(H) + '(H €+); EIII.(111) '7 This correlation scheme indicates that the 3*l(H C+) states from the dissociation limit TIT can interact with 311(I) as well as with '(I-I C+) from channel 11 respectively. The most unfavourable situation for interaction occurs of course at infinite separations owing to the repulsive character of channels I and I1 and the attractive behaviour of channel TIT. We now define AEI = (E,, -E,) and AE, = (ITIII-E,,) i.e. the relative energies between channel I11 and the two lowest dissociation limits related with (A++H+) respectively. These parameters can be regarded as a measure of the allowance for interaction from the energetical point of view. The experimental values of AE and 7 FAR GENERAL DISCUSSION 16.O 16.0 12.0 12.0 8.0 8.0 4.0 LO > 0.0 0.0 a N*+H’ P*+H* 4 Q 22.0 -‘-,02*+H -16.0 OH ** SH2* / ~ 20.0 -2 n,Lp’ -12.o -12.0 1-LlT.2r\, 52’+H -> I Zn,L1-” -8.0 -Fig.1. Comparison of the relative energy separation (in eV) between the ground state of AH2+ and some dissociation limits. AEII,also for the isoelectronic atomic species from the second and third row are contained in table 1 of this comment. For the nitrogen atom one notes that the values of AE and AEIIlie in an energy range between 16.0-14.0 eV respectively while by contrast the second-row phosphorous atom is characterized by much smaller energy differences namely of the order of 6.0-5.0 eV. These findings suggest that even though the dissociation channel (N2++H) could stabilize high-lying 39 l(ll,Z+) electronic states of NH2+,this stabili- zation process is not strong enough to cause an interaction with similar underlying repulsive (A++++) potential curves.The work of Pope et al. finds a dissociative 311(NH2+)ground state directly connected with the lowest [3Pg(N+)+lSg(H+)] products. Because the relative energetics of channel 111 are more favourable in the case of PH2+,a strong interaction can occur at shorter bond lengths and hence this compound possesses a local-minimum lZ+(a2a2) ground state. Furthermore it is also possible that the lowest 311 (I) state of PH2+also acquires some degree of stabilization at smaller distances owing to interaction with the 311state (111) but this state probably changes GENERAL DISCUSSION its character (and stabilization) more abruptly than the low-lying lX+ state because it correlates with the lowest atomic products.Owing to the fact that the singlet state correlates with channel 11 it must be crossed at larger distances not only by this lowest 31-1 state but also be another strongly dissociative 3C-electronic state (see fig. 1). We now undertake a similar analysis for the 5-valence-electron radicals OH2+and SH2+.The dissociation channels of interest are the following 4s,(o+ S+)+IS,(H+) -+ 4c -;E (1) 2D,(O+,S+)+IS,(H+) -+“ll,A C-);E, (11) 3pg(o2+,s~+)+~s,(H)-+ 4q-1 z-); E,,,. (111) Table 1. Relative energies a (in eV) between some dissociation channels of AH2+ and AH:+ first row second row third row N 0 P S As Se AEI (AH2+) AEII (AH2+) AE (AH:+) 16.00 14.10 15.40 21.54 18.22 11.40 6.04 4.94 5.10 9.79 7.95 6.30 6.59 5.26 - 7.89 6.70 - a Atomic excitation energies taken from ref.(3). One finds again that channel I11 can interact with the other two dissociation limits and therefore the relative energies A& and AEIIplay a key role in rationalizing the differences between OH2+ and SH2+.The larger values of AE for the oxygen atom (i.e.,21.5-18.0 eV Table 1) suggest that there is little chance of the attractive (02+ +H) levels stabilizing the underlying dissociative states. As reported by Pope et al. the OH2+radical has a repulsive 4Z-state directly correlated with the first dissociation asymptote. The data for the sulphur atom contained in table 1 reveal that the AE values are ca.11.8-10.3 eV smaller in magnitude when compared with the corresponding results for the oxygen atom. One again finds a situation in which the attractive (S2++H) structure can interact with low-lying unstable states in a similar way to that discussed before for the PH2+cation. The theoretical study of Pope et al. indicates an X211(0202n) ground state and this means that the second channel becomes more stable than the lowest in accordance with the ordering of states predicted for the isovalent SiH radical.’? It is also possible that the lowest 4X-state acquires a partial stabilization but it crosses the low-lying 211state along the dissociation path (see fig. 1). Another point which reinforces this energetical argument is the following :because both stable PH2+ and SH2+ dications correlate with the corresponding second atomic limit it is of interest to compare their AE, values and the respective activation barrier for the dissociation process (AH2+-+A++H+).The results of table 1 assign to the sulphur atom a AE, = 7.95 eV while in the case of the phosphorus atom the corresponding A& is equal to 4.94 eV i.e. ca. 3.0 eV smaller and therefore implying a larger stabilization interaction for PH2+when compared with the SH2+species. This behaviour is clearly reflected in an activation barrier of 2.80 eV (table 11 of Pope et al.) for the process PH2++P++H+ while the similar dissociation of SH2+ is characterized by a potential depth of only 1.60eV. 7-2 GENERAL DISCUSSION Along the same line of reasoning one expects that the third-row (and higher) dications AsHz+ and SeH2+ also have local minima in their ground-state curves because the AEparameters given in table 1 are of the same magnitude as for the second row.Because the A& values for As and Se differ by only 1.40 eV one can speculate that the corresponding activation barrier for AsH2+ and SeH2+ must be quite similar. Experimental and/or theoretical information for these species is not yet available to the best of our knowledge. Finally the simple energy analysis used before can also be extended to the AH;+ family. By analogy with the monohydrides one Fan calculate the relative energy between the attractive and repulsive limits AE(AH;+) = E(AH2++H) -E(AH++ H+).The corresponding AE values for OH:+ NH;+ and PH;+ are given in table 1. With the exception of SHi+ for which Pope et al. have found that the lA ground state correlates with the excited asymptote [alA(SH+) +lS(H+)] for the remaining species one needs only to consider the corresponding AH;+ and AH2+ ground states. From the results of table 1 one again notes a difference between compounds from the first and second row the attractive channel (AH2++H) lies above the lowest (AH+ +H+) asymptote by ca. 15.40 eV (NHg+) or 11.40 eV (OH:+) while in the case of the heavier systems the energy gap is reduced to 5.10 eV (pH;+) or 6.30 eV (SH;+) respectively. This lowering of the relative energies can be correlated with the calculated increase in the activation barrier for the reaction (AH;+ -+ AH++H+) on going from the first to the second row as pointed out in the work of Pope et al.M. Lewerenz P. J. Bruna S. D. Peyerimhoff and R. J. Buenker Mof. Phjx 1983 49 1. P. J. Bruna and S. D. Peyerimhoff XVth Znt. Symp. Free Radicals,published in Bull. SOC.Chim. Belg. 1983 92 525. C.E. Moore Atomic Energy Leoels (National Bureau of Standards Washington D.C. 1949) p. 467. Dr M. Raimondi (University of Milan Italy) said From the paper given by Dr van Lenthe et al. it appears that when computing intermolecular forces in the framework of MO-CI supermolecule approach one has to correct the results for BSSE and size consistency. The number of these corrections is often of the order of magnitude of the well depths even with reasonable basis sets.The different methods of correcting for these errors seem to produce different results and convergence in this respect has not been reached. We want to call attention to the methods based on valence-bond theory which are size consistent and where each fragment can be described within its own basis set and no BSSE is intr0duced.l In addition the contribution of the different components such as induction dispersion coulomb exchange etc. can be sorted out. (This is what the experiment- alists expect from the theoreticians.) Each fragment can be described by means of a molecular-orbital wavefunction computed at different level of accuracy and the procedure turns out to be open ended (according to the definition of Prof.Davidson). In addition the model is flexible intra-atomic correlation energy effects can be included or not and the importance of charge transfer can be determined. See for an application to the system He-HF M. Raimondi Mof. Phys. 1984 53 161 and for an application to the system Ne-HF J. Gerratt and G. Gallup J. Chem. Phys. in press. GENERAL DISCUSSION Dr G. Figari (University of Genoa Italy) said In connection with Dr van Duijneveldt’s paper on weakly bounded systems I would mention the work in progress at the Theoretical Chemistry Laboratory of the University of Genoa on the variation-perturbation theory of molecular interactions including exchange based on an Epstein-Nesbet partition of the hamiltonian. The energy obtained in the first approximation from a one-configuration wavefunction (which can be variationally optimized in first order) is improved in second order of perturbation theory by small terms arising from singly and doubly excited configurations accounting for induction and dispersion including exchange.Preliminary calculations on the ground state of the He dimerl using the antisymmetrized product of atomic Hartree-Fock functions as yo and accounting for dipole-dipole dipole-quadrupole dipole-octupole and quadrupole-quadrupole dispersion contributions give an energy minimum of -10.58 K at R = 5.6 bohr ca. 98.5% of the accurate value resulting the the HFIMD model2 which reproduces a variety of experimental data for He and the best available ah initio He potentials.At large R our potential goes smoothly into the values resulting from the interaction expanded up to C, dispersion coefficients.3 Calculations on the H20 dimer are in progress as well while applications of the method to the study of rotational barriers in single rotor molecules have just been publi~hed.~ * G. Figari and V. Magnasco unpublished results. R. Feltgen H. Kirst K. A. Kohler H. Pauly and F. Torello J. Chem. Phys. 1982 76,2360. G. Figari G. F. Musso and V. Magnasco ,4401. Phys. 1983 50 11 73. G. F. Musso and V. Magnasco Mol. Phys. 1984,53 615. Dr P. R. Surjan (Chinoin Pharmaceutical Works Budapest Hungary) said The numerical comparison of basis-set superposition errors (BSSE) as obtained by different computational schemes is quite interesting especially the fact that the usual Boys-Bernardi scheme overestimates the counterpoinse error by so much.Concerning the question of the method using virtual ghost orbitals only for the calculation of BSSE I do not see any essential difference between SCF and CI calculations since intermolecular SCF corrections can also be incorporated formally in a CI expansion. It is relevant to remark here that there exists an unambiguous definition of BSSE namely that based on the ‘chemical hamiltonian’ method of Mayer.l This is a many-body approach in which appropriate projections onto the subspaces of basis orbitals corresponding to the individual molecules are performed permitting one to introduce explicitly the BSSE operator which has the form1 where X Y Wand 2contain core elements overlap matrix elements and two-electron integrals of the supermolecule problem.The x,* terms are the usual creation operators on basis orbitals while the q5~ are true annihilation operators which however are not the adjoints of the x,’ due to the overlap of the basis orbitals but can be constructed by making use of the reciprocal (biorthogonal) basis set. The actual amount of BSSE can be obtained as the expectation value of this hamiltonian with the supermolecule wavefunction similar to an energy-partitioning scheme so that no ghost-orbital calculation is necessary. The BSSE energy formula can easily be evaluated for arbitrary wavefunctions provided that the first- and second-order density matrices are known.GENERAL DISCUSSION Such a biorthogonal formalism together with the method of moments also permits one to develop an extremely simple and efficient perturbation approach for calculating intermolecular interactions. (A similar method has been proposed2 for the case of intramolecular interbond interactions.) For the intermolecular case the approach can be summarized as follows. The supermolecule hamiltonian is transformed into a mixed reciprocal-direct basis as I H = z <$/IIxv>x,+ 4; +$ z [4jI 4”IxAx,lx; x 4; 47 jIv flv13fJ according to the indices of basis orbitals. The effective intramolecular problem is solved by HA YA+ 1 vac) = EA YJ I vac) where YA+ is the appropriate composite creation operator for the many-electron wavefunction YAof molecule A which is expanded in the subspace of fermion operators x,’ corresponding to the orbitals centred on A.For this reason one has the following commutation rules Yl Yg+&Yg+Yy,+= 0 where the plus sign stands only for the case when both A and B possess an odd number of electrons and This second quantized formalism permits one to define a zeroth-order wavefunction as Y = YA+Yg+Ivac) which is properly antisymmetric. It is easy to show that this zeroth-order wavefunction is an eigenfunction from the right-hand side of the unperturbed hamiltonian HA+ H [HA+ HB]Y = (EA+ EB)Y. This result is a consequence of using the biorthogonal formulation and the above commutation rules. Note that operators HA HB and HA + HB are not Hermitian and their eigenfunctions from the right do not form an orthonormal set.We can however define wavefunctions bKexpanded in reciprocal space forming a biorthogonal set with respect to the direct-space wavefunctions Y, and develop the perturbation theory according to the method of moments.2 For example for the second-order energy we have K The excited states correspond either to intramolecular excitations or to charge transfer. The matrix elements are trivial to evaluate since Wick’s theorem applies. An important feature is that BSSE terms can be dropped entirely. In actual calculations the intramolecular Schrodinger equations cannot be solved exactly in the general case not even in a given basis set. Then HA and H may correspond to certain model hamiltonians (e.g.the Hartree-Fock one) whose eigenfunctions from the right-hand side provide us with the zeroth-order wave-functions YA and YB respectively. Electron correlation then corresponds to additional perturbing terms; it is easy to show however that there is no interference between correlation and intermolecular effects up to second order. Discussions with I. Mayer are gratefully acknowledged. I. Mayer Int. J. Quantum Chem. 1983 23,341. P.R. Surjan. I. Mayer and I. Lukovits to be published. GENERAL DISCUSSION Prof. P. Siegbahn(Stockholm University Sweden) said I address my remarks to Dr van Lenthe. My comment concerns how differently basis-set superposition errors (BSSE) are regarded in the literature and your viewpoints on this problem.Let me first state that I find your very careful analysis of the BSSE very interesting and in line with my own viewpoint of them as highly unphysical effects which one should possibly correct for. However this is not the viewpoint given in a series of recent papers written by Wright and coworkers. Wright uses bond functions which are removed at long distances and are therefore very efficient in generating superposition errors. This fact was pointed out by Bauschlicher several years ago. Wright is aware of this and deliberately optimizes the BSSE to reproduce for example a known experimental dissociation energy. With this optimized basis set he goes on to generate whole potential surfaces and claims extremely high accuracy. The viewpoint is best seen in a recent paper by Wright and Buenker on N,.It is stated there that ‘All CI calculations suffer to a greater or lesser extent from basis-set incompleteness problems. Indeed such incompleteness is often desirable because of the impossibility of calculating larger molecules or potential surfaces using huge basis sets.’ In a comment on this paper Bauschlicher has criticized this viewpoint and he says instead that ‘ . . . the r dependence of the superposition error and the basis-set incompleteness are not expected to be the same’. It is clear that a deeper aspect of this problem concerns the arguments for doing a6 initio calculations at all. I wonder if you have any comments on this ongoing debate? Dr J. H. van Lenthe and Dr F. B. Van Duijneveldt (University of Utrecht The Netherlands) said In reply to the comments made by Dr Raimondi Dr Figari Dr Surjan and Prof.Siegbahn we would like to comment as follows. We agree with Dr Raimondi that intermolecular forces may be computed in a non-orthogonal valence-bond framework in such a way that no BSSE is introduced. However we stress that if charge-transfer states are included in the usual way i.e. using the orbitals of the other fragment the basis-set superposition error is unavoidable. With respect to Dr Figari’s work we feel that omission of intramonomer correlation may lead to serious errors in the calculated potential-energy surface. We observe that his He-He result of -10.58 K at R = 5.6 bohr must be due to a fortuitous cancellation of errors since it is known’ that for example the contribution to the dispersion energy arising from the omitted higher-order terms is ca.1.0 K. We agree with Dr Surjan that intermolecular SCF corrections are incorporated in a CI expansion and that the corresponding BSSE should be corrected for in the same way in both SCF and CI calculations. We do this by employing large-basis occupied orbitals in the CI BSSE calculations. However we feel that the typical CI BSSE in which the monomer uses the ghost in order to be better correlated is a different matter which should be considered separately. We do not think that the BSSE problem is solvable at a formal theoretical level since it does not exist in a complete basis. Finally in agreement with Prof. Siegbahn’s comments we feel that there is little virtue in obtaining the right curve for the wrong reason.In fact it is possible to obtain a potential curve for He showing a ‘van der Waals’ minimum within the Hartree-Fock approximation if no BSSE correction is applied.2 Fitting to experiment is perfectly legal but should preferably be carried out using some simple semiempirical scheme. ’ R.Feltgen J. Chem. Phys. 1981 74 1186. B. J. Ransil J. Chem. Phys. 1961,34 2109. 200 GENERAL DISCUSSION Prof. P. Gray (Uniuersityofleeds)said For a number of years we have been making precise measurements of the diffusivity of hydrogen atoms in gases. This ~rogrammel-~ is largely due to my colleague Dr A. A. Clifford. We have from measurements in nitrogen hydrogen carbon dioxide and the noble gases helium neon and argon.So far the measurements have been made at ca. 300 K although we hope to extend our temperature range upwards eventually. We are now starting to look round for suitable theoretical intermolecular potentials with which to compare our data. I suppose it is mainly the repulsive part of the potential that is relevant to our experimental data the potential for hydrogen atoms and the noble gases in particular ought to be amongst the simplest to obtain. Our experience when comparing transport-property data with theoretically obtained intermolecular potentials has been more than a little disappointing in the following ways. First although calculations were possible in principle the potentials that we needed were not readily available to us in a usable form.Also they did not appear to be nearly as accurate as we needed. For comparison with transport-property data we are looking for an accuracy expressed as the uncertainty in distance for a particular energy of 1 % for the diffusivity data I have just mentioned and much less for the viscosity and thermal conductivity of stable species. Finally a theoretical potential claimed to be the last word that we used was replaced a few years later by a superior product from the same manufacturer. I should therefore like to ask the speaker and the audience what is the current state of the art in calculating ab initio potentials particular for the systems H +Ar H +Ne and H+He and would they expect there to be sufficiently accurate results available yet to compare with our diffusivity measurements.A. A. Clifford R. S. Mason and J. 1. Waddicor Chem Phys. Lett. 1980 76 298. A. A. Clifford P. Gray R. S. Mason and J. I. Waddicor Faraday Symp. Chem. SOC.,1980 15 155. A. A. Clifford P. Gray R. S. Mason and J. 1. Waddicor Proc. R. SOC.London Ser. A 1982 380 241. 4 T. Boddington and A. A. Clifford Proc. R. SOC.London Ser. A 1983 389 179. Prof. A. D. Buckingham (Cambridge University) said Intermolecular potentials that are accurate enough to provide useful predictions of static and dynamic properties of gases may be obtained through the perturbation approach to intermolecular forces. At long range the potential energy is the sum of electrostatic induction and dispersion energies which can be deduced from a knowledge of the charge distribution and polarizabilities of the isolated molecules.1 At short range the repulsion comes from the overlap forces which may be obtained approximately by simple SCF computations.The intermediate region (the vicinity of the minimum) is generally obtained by multiplying the long-range potential by a ‘damping’ function which is unity at large separation and goes smoothly to zero at short ~eparations.~-~ For H +Ar the C, Cs and C, coefficients are known,5 and useful potentials have been deduced2q by combining them with SCF interaction energies at short range. Of course such a potential is not obtained ab initio but it should enable us to gain a useful understanding of the diffusion of H atoms in argon.A. D. Buckingham Adv. Chem. Phys. 1967 12 107. J. Hepburn G. Scoles and R. Penco Chem. Phys. Lett. 1975 36 451. K. T. Tang and J. P. Toennies J. Chem. Phys. 1977,66 1496. R. Ahlrichs R. Penco and G. Scoles Chem. Phys. 1977 19 119. K. T. Tang J. M. Norbeck and P. R. Certain J. Chem. Phys. 1976 64 3063. GENERAL DISCUSSION 20 1 Prof. J. Morrison (University qf Utah U.S.A.)said Since Prof. McCullough and I do work in collaboration perhaps I also could take up Prof. Schaefer’s informally posed question about the prospects of numerical calculations. In our calculation we expanded the pair function in natural orbitals and this enabled us to reduce the pair problem to a set of one-electron equations. These equations can be solved readily in the way we have done here or by employing two-variable programs such as those used for some time in Goteborg and Finland.Our principal difficulty is that the pair natural-orbital expansion itself is not very well suited to very accurate calculations or to evaluating coupled-cluster effects. For this reason it would be desirable to obtain a direct numerical solution of the pair equation as we have done for atoms. One way of approaching this problem is to use an empirical potential of the form as the starting point of the perturbation calculation. Heref(<) and g(v) may be chosen to give a good representation of the Hartree-Fock potential. The underlying one-electron equations are then separable and can be solved numerically to yield functions that are better suited to the diatomic molecule than are the spherical harmonics.Hopefully the partial-wave expansion will be a good deal shorter with these generalized Hi functions. Also by taking advantage of the symmetry associated with the separation constant one can solve for each partial-wave component independently. Finally I would like to say that pair functions provide an intuitive description of the correlation problem at an intermediate level of difficulty between the orbital theory on the one hand and large matrix calculations on the other. The pair function describes the depletion of charge between the two electrons due to their mutual correlation. Prof. E. A. McCullough Jr (Utah State University U.S.A.) said The 0,n 6 incompleteness errors that we found in the basis sets for LiH and FH were to us surprisingly large given the sizes of the basis sets employed.This illustrates the difficulty of choosing adequate basis sets for really high-accuracy correlation-energy calculations. Incompleteness errors in the second-order energy due to neglect of m > 6 are shown by the PNO calculations to be significant. This may be less serious than it appears. Recent work on H and LiH has demonstrated that these errors display a remarkable tendency to cancel in higher orders so that very accurate total energies can be obtained even with neglect of m > 6. Why this cancellation should occur is not apparent to us but if it turns out to be true in general truncation of the mm’ sums may prove not to be a limiting feature of practical calculations.1 A critical need for numerical perturbation calculations on molecules is for a procedure that can accommodate a multiconfiguration zero-order function.Dr Morrison is continuing work on this problem. I would also like to emphasize that all our numerical procedures are restricted to diatomic molecules. We see little hope of extending our methods to any other class of molecule not even to linear triatomics. L. Adamowicz R. J. Bartlett and E. A. McCullough Jr Phys. Rev. Lett. 1985 54 426.

ISSN:0301-5696    

DOI:10.1039/FS9841900175

出版商:RSC    

年代:1984

数据来源: RSC

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ADDITIONAL REMARKS Dr H-J. Werner (J. W. Goethe Uniuersitat Frankfurt West-Germany) (communi- cated). My first comment concerns Prof. Siegbahn's paper. Recently we have also done some calculations on transition-metal atoms and simple molecules containing transition metals. In these calculations we found it extremely difficult to achieve a similar accuracy as we are used to calculations which involve first- or second-row atoms only. The problems start with the choice and size of the basis sets generally one needs diffuse d-functions and at least twof-functions on each first-row transition metal atom. Secondly the results obtained in multireference-CI calculations may sometimes be surprisingly sensitive to the choice of the one-electron orbital basis (e.g. SCF MCSCF or natural orbitals).Thirdly higher than double excitations in particular their unlinked cluster contributions may be very important ;for example their effect increases the dissociation energy of Cu,' from 0.23 eV (CISD) to 1.SO eV (CEPA-1) (exptl 2.05 eV). Finally relativistic effects are usually not negligible as was assumed in many previous calculations. Many of these difficulties arise from the large electron correlation effects in the compact 3d shells. On the other hand the diffuse 4s electrons give only a small contribution to the correlation energy. This leads to very large differential electron correlation effects to the relative energies of the atomic s2dn sldn+land dn+2electron states. For instance the &o -s2d8separation in Ni is reduced from 5.47 eV at the SCF level to ca.2.0 eV at the CEPA-1 level. The relativistic effect increases this value to 2.55 eV which is ca. 0.85 eV larger than the experimental value (1.71 eV). Note that we tried very hard to improve this result employing very large basis sets with up to 4f 3g and lh functions so far without significant success. Since as Prof. Siegbahn noted 3d's hybridization may be important in bond formation it appears to me essential first to understand and solve the atomic problems before doing large-scale calculations on transition-metal complexes. It is my feeling that the results obtained with present methods and basis sets for such large molecules are quite questionable. Also in my opinion the MCSCF or CASSCF method alone is not an appropriate tool for treating transition-metal compounds even though recently much progress has been made in optimizing very long CASSCF expansions2? (we have been able to optimize CASSCF wavefunctions with up to 178916 configurations).The CASSCF method should be used to describe near-degeneracy effects and dissociation processes correctly but in order to account for the important dynamical correlation effects it must be followed by a multireference CI calculation. In conclusion I have the feeling that my view about our ability to do reasonable calculations for transition-metal compounds is more pessimistic than Prof. Siegbahn's. H-J. Werner and R. Martin Chem. Phys. Lett. 1985 113 451. H-J. Werner and P. J. Knowles J. Chem. Phys. in press. P. J. Knowles and H-J.Werner Chem. Phys. Lett. 1985 115 259. My next comment refers to a remark of Prof. Wright concerning the use of bond-centred basis functions printed on p. 187. It is certainly correct that the use of such functions can apparently improve the results in particular if rather small (DZ or DZP) atom-centred basis sets are employed. Since bond functions do not contribute to the energy at large (infinite) internuclear distances it is obvious that the use of such functions will always lead to an increase in the calculated dissociation energies and 202 ADDITIONAL REMARKS usually to better agreement with the experimental values. It appears quite possible however that the addition of several bond functions will lead to binding energies which are too large.Hence the use of bond functions makes it possible to influence (within certain limits) the results arbitrarily. The problem is to find those bond functions which lead to a balanced description of the molecule and the separated atoms. It appears to me that this is possible only by comparison with known experimental data. I admit however that it may eventually be reasonable to optimize the bond functions for a number of molecules for which experimental data are known in order to make predictions for other closely related molecules. With reference to Dr Hunter’s comment on p. 188 I did not say that the use of bond functions would violate the variational principle and such a statement is of course wrong. The problem is that we are always dealing with incomplete basis sets and that certain incomplete basis sets can lead to an unbalanced description of different regions of the potential-energy function.I would now like to make a comment on the question of Dr Davies printed on p. 184. I can confirm Prof. Peyerimhoff s experience that the dipole length operator appears to be preferable to the velocity operator for the calculation of electronic transition probabilities. Generally the transition moments calculated with the length operator appear to bemuchless sensitive to the basis set and electroncorrelationeffects than those obtained with the velocity operator. A particular problem with the velocity form is that the calculated values must be divided by the energy difference to obtain the transition moment.We found that the errors of the values obtained from the velocity operator often become very large as the energy difference between the states under consideration becomes small; e.g. at large internuclear distances when both potentials have the same dissociation limit. We have calculated transition-moment functions for considerable numbers of diatomic molecules and molecular ions. These calculations showed that electron correlation effects on the calculated radiative lifetimes are often very large; for example for the A-X transitions in CH or OH they are of the order of 100%.In order to obtain accurate results it is therefore necessary to use very large and flexible basis sets and highly correlated multireference CI wavefunctions. MCSCF and CASSCF wavefunctions often yield poor results.Typical errors of our calculated radiative lifetimes are 10-1 5% as compared with the most reliable experimental values. In many cases the differences between various measurements are much larger even though the (statistical) error bounds of the individual measurements are mostly fairly small. I therefore believe that the calculation of radiative transition probabilities provides a useful opportunity for quantum chemists to provide experimentalists with much needed quantitative information. I do not think that there is an established basis for the ‘6 electron’ or ‘20 electron’ limit in ab initio calculations of accurate transition moments. The problem seems to be rather the proper vibrational averaging of the transition-moment surfaces of polyatomic molecules.For a review see H-J. Werner and P. Rosmus ‘Ab initio Calculations of Radiative Transition Probabilities in Diatomic Molecules’ in Comparisonof ab initio Calculations with Experiments -The State of the Art ed. R. Bartlett (D. Reidel in press). Finally I turn to Prof. Peyerimhoff s paper :In the section ‘transition probabilities and radiative lifetimes’ and in fig. 5 there is an erroneous reference to our calculations of the dipole-moment function for OH. The MCSCF and MC-CI results in fig. 5 were obviously derived from dipole-moment functions of Stevens et a1.l and Chu et a1.,2 204 ADDITIONAL REMARKS respectively. These functions yield Einstein coefficients which are in error by 50% or more.To my knowledge the ‘experimental’ Einstein coefficients for Av = 1 to v’ = 8 are unknown. There have been a few measurements of A A and A which differ however by several hundred percent. The empirical dipole-moment function of Murphy3 was based on relative intensities and its slope at re is therefore undetermined [arbitrarily set to 1 D A-l in ref. (3)]. We have employed the MCSCF-CI method for the calculation of an accurate dipole-moment function of OH.4The reliability of the dipole-moment functions can be checked indirectly by comparing calculated and measured transition-probability ratios. As has been shown by Mieq5 for such an analysis it is essential to take vibration-rotation and spin-uncoupling effects into account and to average exactly over the same rovibrational lines as in the experiments.With our MCSCF-CI dipole-moment function and an RKR potential-energy function this yields (experimental values in brackets) &/A = 0.38 Ifr 0.06 (0.44+ 0.03,30.41),6 A;/Ai = 1.10k0.37 (1.15+0.05,3 1.25).6 With the MCSCF (17) dipole moment function of ref. (1) one obtains the ratios 0.46 If:0.06 and 1.62 0.28 respectively. The ‘error bounds’ in the theoretical values arise from the averaging over several rotational lines (K = 2-7 in the P branches). W. J. Stevens G. Das A. C. Wahl M. Krauss and D. Neumann J. Chem. Phys. 1974 61 3686. S. I. Chu M. Yoshimine and B. Liu J. Chem. Phys. 1974 61 5389. R. E. Murphy J. Chem. Phys. 1971 54 4852. H-J. Werner P. Rosmus and E. A. Reinsch J. Chem. Ph-vs. 1983 79 905. F. H. Mies J. Mol. Spectrosc. 1974 53 150. F. Roux J. D’Incan and D. Cerny J. Astrophys. 1973 186 1141.

ISSN:0301-5696    

DOI:10.1039/FS9841900202

出版商:RSC    

年代:1984

数据来源: RSC

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LIST OF POSTERS Potential-energy Surface Intersections for Triatomic Molecules I. M. Mills R. N. Dixon and S. Carter Unitw-sity of Reading Distributed Multipoles from partitioning by Zero-flux Surfaces D. L. Cooper Universify qf Ulcj’ord N. C. J. Stutchbury ICI Macclesjield and D. G. Bounds Continuum States of the Hydrogen Molecule J. Tennyson and C. J. Noble Daresbury Luboratory Warrington Analysis of Interbond Interactions Theory and Application in Ab Znitio and Semi-empirical Schemes of Calculation P. R. Surjan G. Naray-Szabo and I. Mayer Chinoin Budapest Hungury A New Potential Surface for Ff H + HFfH J. S. Wright and R. J. Williams Carlcton University Ottawa Canada Hole Localisation in Ionised States of Ordered Solids :Cluster Calculations on Cuprous Halides G.Janssen and W. Nieuwpoort Unicersity of Groningen The Netherlands Hyperfine Splitting in the t-Butyl Radical R. E. Overill King’s College London Computations on the Cluster Photochemistry of Ammonia Water and Ammonia +Water Mixtures E. M. Evleth H. Z. Cao and E. Kassab UniversitP Paris VI France The Exact One-electron Schrodinger Model of Molecular Structure and the Molecular Envelope G. Hunter York University Downsview Ontario Canada Comparison of the Electronic Structures of V(0) Hal and P(0) Hal Analogues (Hal = F C1 Br) by Ultraviolet Photoelectron Spectroscopy and SCC -XaCalculations S. Elbel J. Kudnig G. Riinger and M. Grodzicki Uniziersity of Hamburg WesfGermany The Electronic Structure of the HCO+ Ion M. Hilczer and B.C. Webster University ojGlasgow Properties of the Vibrating Rotating Methane Molecule W. T. Raynes P. W. Fowler P. Lazzeretti and R. Zanasi University of Shefield Quantum-mechanical Approach to Infrared Intensities :The Nuclear Electric Shielding Tensor R. Zanasi and P. Lazzeretti University ofModenu Italy EXAFS Phase Shifts Evaluated from Cluster Calculations J. Paul Chalmers Unioersity oj’ Technology Goteborg Sweden Potential-energy Curves for Various C Il and A Molecular States of some Effective One- and Two-Electron Diatomic Systems Lil LiH+ Na NHe6+ LiH and NHe5+ Mme Frecon and A. Alikacem Universitk Cluudc Bernard Villeurbanne France 205 LIST OF POSTERS Calculation of the C,H Potential-energy Surface Using PAOs G. Chambaud H.Lavendy B. Pouilly J. M. Robbe E. Roueff and B. Levy ENSJF Mon tr ouge France Parity-violating Energy Differences and the Origins of Biomolecular Chirality G. E. Tranter University of Oxford An Ab Initio Study of the Excited States of A1F D. Hirst University of Warwick Valence-electron-only Calculations G. G. Balint-Kurti M. J. Davis T. P. Martin and A. H. Harker University of Bristol Theoretical Characterization of the Trifluoromethoxy Radical I. H. Williams and J. S. Francisco University of Cambridge Potential-energy Curves for Excited States of BeF C. Marian University of Bonn West Germany Investigation of the Reaction Pathway for the Formation of Urea C. Dijkgraaf DSM Geleen The Netherlands and A. D. Buckingham N. C. Handy and J. E. Rice University of Cambridge Structure Dipole-moment Derivatives and Infrared Intensity of Dimers of H,O and H,S R. D. Amos University of Cambridge Intermolecular Perturbation Theory Applications to some Hydrogen-bonded van der Waals Molecules A. J. Stone and G. J. B. Hurst University of Cambridge Reaction Paths Hamiltonian Study of the Isomerisation of the Methoxy Radical S. M. Colwell University of Cambridge The Calculation of CI Gradients and Molecular Properties J. E. Rice University of Cambridge Artefact-free Models of Molecular Structures of Polyatomics P. G. Burton University of Wollongong New South Wales Australia

ISSN:0301-5696    

DOI:10.1039/FS9841900205

出版商:RSC    

年代:1984

数据来源: RSC

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Bernardi F. 137 Bottoni A. 137 Bruna P. J. 193 Buckingham A. D. 176 200 Colvin M. E. 39 Cooper D. L. 149 176 185 Davidson E. R. 7 49 175 176 Davies D. W. 184 Figari G. 197 Gerratt J. 149 190 Gray P. 200 Guest M. F. 109 192 Handy N. C. 17 Hillier I. H. 109 192 Hunter G. 175 188 Jerrgensen P. 85 189 Koga N. 49 Kroon-Batenburg L. M. J. 125 McCullough E. A. 165 201 McDouall J. J. W. 137 Mills I. M. 175 179 Morokuma K. 49 Morrison J. 165 180 201 Obara S. 49 AUTHOR INDEX* Ohta K. 49 Olsen J. 85 189 Overill R. E. 175 Peyerimhoff S. D. 63 179 182 183 184 186 193 Pickup B. T. 188 Pope S. A. 109 192 Rairnondi M. 149 196 Richman K. W. 165 Robb M. A. 137 Saunders V. R. 79 Schaefer H. F. 39 178 180 Schlegel H. B. 137 Siegbahn P. E. M. 97 179 181 182 199 Surjan P. R. 177 183 197 Tennyson J. 181 van Dam T. 125 van Duijneveldt F. B. 125 199 van Lenthe J. H. 125 199 Werner H-J. 202 Williams R. J. 191 Wright J. S. 178 187 190 191 Yeager D. L. 85 189 * The page numbers in heavy type indicate papers submitted for discussion.

ISSN:0301-5696    

DOI:10.1039/FS9841900207

出版商:RSC    

年代:1984

数据来源: RSC