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 Towards the end of the present decade silicon-based technology will start to be superseded by that based on gallium arsenide with the resulting machines (such as the Cray 3) having projected speeds of up to 10 gigaflops.By the turn of the century the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.