GENERAL DISCUSSION Mr. B. T. Pickup (University of Manchester) said I would like to initiate the discussion on Hall’s paper by making a comparison between the new variation principle and the usual Rayleigh-Ritz principle. The Ritz procedure cannot be used for discontinuous wavefunctions but continuous functions with discontinuous derivatives can be used if we utilize the variation principle in its more fundamental form,l involving the square of the gradient of the trial wavefunction 6. In a forthcoming paper Hall Hyslop and Rees use Hall’s principle for a cut-off radial wavefunction of the form where A is the unit step function. This function gives about 90 % of the hydrogen ground state energy (table 1). The present work evolves from a suggestion of W. Kutzelnigg that a quadratic function of the form should give a better energy since it has a more realistic shape.+ = (a-r)A(a-r) 4 = (a - r>(P - r > W - r> r (a.u.) FIG. 1 .-Quadratic wavefunction. The optimized functions are shown in fig. 2. The quadratic function was found to give minimum energy using the Ritz principle when a = p. Only a one para- meter function was examined by the Hall method. An interesting point is the lack of 2s solution in the two parameter Ritz case for unconstrained a and /3. The energies and cut-off distances found using the two variation principles are summarized in table 1. The quadratic function gives about 96 % of the hydrogen atom 1s energy using the Hall principle. ’ J. C. Slater Physic. Rev. 1937 51 846. Hall Hyslop and Rees Int. J. Quantum Chem. in press. 102 1 03 GENERAL DISCUSSION A wavefunction of the form 4 = (a-r)nA(a-r) is easily optimized using the Ritz principle and it is found that the normalized function goes to the true solution in the limit as n+oo.For the Hall principle the integrals can be must be handled evaluated analytically but the resulting transcendental equation numerically . r FIG. Z.-Optimized wavefunctions (normalized) of form (a- r)2A(a- Y ) for atomic hydrogen. Solid lines (i) Hall optimized function 2.09 (1 - 0 - 2 7 ~ ) ~ A ( 3 . 7 5 - ~ ) . (ii) Ritz optimized function 1.28 (1 -0.25r)2A(4-~). Broken line true solution 2e-r ; distance in Bohrs. TABLE 1 .-VARIATION PRINCIPLES APPLIED TO THE HYDROGEN ATOM. THE CUSP CONSTANT IS DEFINED BY y = (- d In $/dr),= 0 cut-off distance energy wavefunction (Bohrs) (Hartrees) Y linear Hall 2.8584 - 0.4522 0.3499 Ritz 4.0000 - 0.3125 0.2500 quadratic Hall 3,7469 - 0.4805 0.5338 Ritz 4.0000 - 0.4375 0.5000 exact 00 - 0.5OOO 1 .oooo With regard to the use of cut-off wavefunctions in large molecular systems the Ritz principle is easy to use but will probably not give very good energies except at larger JZ.The cut-off distance is always greater than n for the hydrogen atom case and is thus too large to allow wavefunctions to disappear at distances greater than the adjacent atom in the molecule. Thus the only hope appears to lie in the use of Hall’s principle which gives more rapid convergence to the true solution. The new principle is more difficult to apply. Prof. G. G. Hall (Uniuersity of Nottingham) said In reply to Buckingham the original motive for using discontinuous trial functions was to have functions which vanished outside atomic cells so that zero differential overlap is exactly true.I 104 GENERAL DISCUSSION believe that we will be able to demonstrate binding using these functions though some adjustments of scale will probably be needed and the equations will not be quite of the usual form. Calculations on H,+ are now proceeding. In reply to Sack the Green function which we use is a classical one and for an n electron system is where G(1,I ; E ) = -2K~(3n-2)(kR)/R~(3n-2) E = - 4k2 and R2 = (r,-ri)2+(r2-r5)3+ . . . +(rn-rr,)2; K denotes the Bessel function which becomes exponentially small as R increases. In reply to Murrell it would indeed be possible to use trial functions which reduced to zero on a cell boundary so that zero differential overlap could be combined with continuity.This involves considerable distortion of the atomic function however especially in polyatomics and would raise the atomic energies considerably. It also has the disadvantage in conjugated systems of leading to a vanishing p and so giving no explanation of n bonding. Prof. R. McWeeny (University of Sheffield) said Hall’s method appears to be closely related to one developed and extensively used by Svartholm in his thesis on the binding energies of light atomic nuclei. Svartholm writes the eigenvalue equation in the form where p is a “ strength parameter ” giving the actual potential when p = 1. If for a certain value of A 3 = A,, the solution gives a smallest eigenvalue ,u = ,uo = 1 then lo is the ground state energy of the original equation.The eigenvalue equation appears to have been first written in this form by Thomas.2 An equivalent form of the equation is where G = (1-T)-’ is a Green’s operator and E = l/,u which is identical with Hall’s eqn. (14). Svartholm however employs the momentum representation in which T = p2/2m (a function of p instead of a Schrodinger differential operator) and G then becomes simply a function of momentum while the potential V becomes an integral operator. The use of the operator GV (whose largest eigenvalue is required) as a means of purifying a trial function by iteration was also explored by Svartholm. If co is a trial function GVco contains an increased amount of the exact ground state eigen- function. This integral-equation equivalent of the well-known “ power method ” of finding the largest eigenvalue of a matrix goes back to Kell~gg.~ In practice the main difficulty lies in the integration necessary to obtain the iterated function.It seems likely that momentum wave functions could be useful in the applications now proposed discontinuities in the usual functions being eliminated by the Fourier transformation to momentum space and I wonder if Hall has given any consideration to this possibility? (1-T)X = PVX E X = GVx Prof. G. G. Hall (University of Nottingham) said I am grateful to McWeeny for providing additional references. We have been aware that in the theory of light N. Svartholm The Binding Energies of the Lightest Atomic Nuclei a thesis published by HAkan Ohlssons Bokryckeri (Lund Sweden 1945). L. H. Thomas Physic. Rev.1937 51 202. 0. D. Kellogg Methem. Ann. 1922 86 14. GENERAL DISCUSSION 105 atomic nuclei ideas such as these often develop though not usually in variational form. We are also aware of the advantage of momentum representation in giving a very simple form to G. Indeed this seems the most hopeful method of computing the various integrals involving G. We believed however that the main interest of the problem was in the use of trial functions with discontinuities in co-ordinate space and consequently that it was not desirable to use momentum wave functions exclusively. Dr. D. B. Cook (University of ShefJield) said Palmieri and I have recently com- pleted some calculations which have some bearing on Klessinger’s paper and on Pauncz’s earlier paper. We have performed ab initio calculations on the hydrides of some second row elements H2S PH3 SiH4.The LCAOSCFMO calculation is elementary and we have used it as a reference for some localized electron pair cal- culations of the type described by Klessinger. In all three cases we used the naive “ chemical ” picture for the choice of hybrids for localized bonds ; 3p-orbitals at right angles for H,S and PH3 and tetrahedral sp3 orbitals for SiH4. Only for SiH4 are the results in reasonable agreement with the MO results. In H2S for example ‘‘ forcing ” the sulphur 3s orbital to be a lone pair (as is possible in the electron pair description) gives an energy 0.83 a.u. higher than the MO calculation (-410.02 compared with -410.85 a.u.). Thus the simple chemical picture of the bonding in this hydrides does not yield quantitative results and the actual hybridization on the heavy atom must be determined by an optimization procedure of the type that Pauncz outlined earlier at this symposium.We are currently investigating this problem. Inclusion of 3d orbitals on the heavy atom improves the MO calculation but makes the problem of hybridization more acute in the self-consistent group function calculation Prof. J. A. Pople (Carnegie-Mellon University Penn.) said I believe that there is some danger in an overemphasis on localized bonding using hybrid orbitals. Certain features of molecular structure may depend intrinsically on delocaljzation even for a-electrons and wave functions based on localized orbitals may lead to a poor description. One such example is the barrier to rotation about carbon-carbon single bonds which has been successfully calculated by most delocalized molecular orbital theories including the extremely simple extended Huckel method.It appears to be directly associated with delocalization of electrons between vicinal C-H bonds and to be fairly independent of such details as integral approximations or semi- empirical parameterization. The molecular orbital method described by Klessinger fails to reproduce this important feature possibly because of the way it is set up in terms of localized orbitals. As a second example I refer to the work of Santry and Segal who showed that a reasonable description of the valence of second row compounds including their equilibrium geometries could be obtained without using 3d atomic orbitals in the basis set. For compounds such as SF6 and ClF3 the molecular oIbitals cannot then be localized in individual bond orbitals with conventional hybrids.While 3dfunctions undoubtedly do play an important part in accurate treatments it is possible that overemphasis on the hybridization concept has misled us to some extent. Prof. R. McWeeny (University of Shefjeeld) said Doubts have been expressed about the accuracy with which a molecule can be described in terms of localized functions for the various electronic groups. The main difficulty however is not with Santry and Segal J . Chem. Physics 1967,47 158. 106 GENERAL DISCUSSION the validity of the approximations (the energy obtained being superior to that of the normal SCF approximation which in principle it may contain as a special case) but with the definition of the localized groups.One way of combining the attractive features of both SCF and group function (GF) methods is to determine a set of localized orbitals as suitable transforms of approximate SCF functions (thereby ensuring that the choice of orbitals will yield a good charge distribution) and then to exploit the GF method as a means of introducing electron correlation effects within each group. Work along these lines confirms that wave functions substantially better than those of one-determinant SCF theory may easily be achieved with a single antisymmetrized product of group functions ; the GF method provides in fact a simple and physically transparent way of introducing the most effective types of configuration interaction. In higher order,2 the method can allow for inter-group effects (e.g.dispersion inter- actions) as well as intra-group correlation and can thus account for long-range couplings. A major advantage is that with suitable integral approximation techniques the GF method is computationally feasible for very large molecules no large secular equations being encountered. Prof. M. Randik (Zugreb) said Klessinger has found in all his calculations that optimum hybrids point along the bond axes. Only when the valence angle is smaller than 90" (cyclopropane) do bent bonds arise. We have calculated hybrids by the maximum overlap method for a number of polycyclic hydrocarbons and have found that bent bonds arise as a rule not only in three-membered rings (0-22~) but also in four-membered rings (ci) - loo) and in five-membered rings (m-2-4") i.e.when the valence angle is larger than 90". Has Klessinger any preliminary results on other cyclic systems besides cyclopropane? Although we are glad to see that SCGF calculations are in good agreement with the maximum overlap results as it gives to the latter additional justification cyclopropane itself is not suitable for comparison being highly strained and thus not very sensitive to the parameters involved in a computation. For example when Slater orbitals and Clementi double zeta orbitals are used in the maximum overlap method they give much the same results in cyclopropane but in other less strained systems the hybridization differs considerably and that based on Clementi functions is s~perior.~ Prof. Dr. W. C. Nieuwpoort (Rijkuniversiteit te Groningen) said With regard to the paper by Murrell minimum basis sets are sometimes not sufficient to describe adequately even the qualitative feature of atomic wave-functions.This is for instance so for negative ions like H- which is not stable in the restricted Hartree-Fock approxi- mation and F- which leads to a stable result only when using at least a double-zeta 2p-function. Do these considerations influence the meaning of the observed changes in the minimum basis exponents ? Prof. G. G. Hall (University of Nottingham) said Would Boys agree that the orbitals which he uses are closer to the exact or Brueckner orbitals than to the conventional molecular orbitals since his conditions on the correlation function and the orbitals implies that when the wavefunction is expanded into configuration interaction form the single-replacements do not occur ? D. B. Cook and R. McWeeny Chem. Physics Letters 1968 1 588. R. McWeeny Proc. Roy. SOC. A 1959 253 242. M. RandiC and D. StefanoviC J. Chem. SOC. B 1968 423 and references therein. L. Klasinc 2. MaksiC and M. RandiC J. Chern. SOC. A 1966,755 ; M. RandiC J. M. Jerkunica. and D. StefanoviC Croat. CJiem. Acta 1966 38 49.