摘要:

Calculations on Some Simple Systems Using a New Variation Principle BY G. G. HALL Dept. of Mathematics The University Nottingham Received 1 1 th October 1968 Calculations of upper and lower bounds for the energies of some one-electron atoms and other systems are discussed. These employ a new form of variation principle which permits the use of discontinuous trial wavefunctions and which is generally superior to the traditional principle. The implications of this principle for future calculations are also discussed. 1. INTRODUCTION In practical applications traditional forms of variation principle such as the Rayleigh-Ritz principle suffer from a number of disadvantages. There are various circumstances for example in which it would be advantageous to use trial functions which are discontinuous.This is not possible because the form of the functional requires the trial function to be differentiable. The variational principle described here uses a functional which does not involve differentiation and so allows for trial functions which are discontinuous. If the trial function is differentiable then this principle leads to lower upper bounds to the energy than the older principle. This improved accuracy is achieved at the expense of a doubling of the dimensions of the integrals which have to be evaluated. The evaluation of these integrals for many-electron systems has proved to be difficult and the examples quoted here are of one-electron systems. 2. A MATRIX INEQUALITY The basic principle behind this variation principle can be illustrated most simply by reference to a similar principle for finite matrices.If A is a positive definite Hermitean matrix the Rayleigh-Ritz principle asserts that for any trial vector u the ratio satisfies where A. is also asserts p = U~AU/U~U (1) > & 7 (2) the smallest eigenvalue. that the ratio Since A-1 is also Hermitean the Rayleigh-Ritz v = u~A-'u/u~u (3) is less than the largest eigenvalue of A-l which is just A; so that v-l > 2 0 . (4) For given u A-lu can be found by solving a set of linear equations and consequently v-l can be calculated readily. 69 70 NEW VARIATION PRINCIPLE These two upper bounds for Lo can be related using the Cauchy-Schwarz in- equality for the positive definite matrix A which states that and consequently Thus the additional effort in calculating v-l is repaid by a more accurate estimate of Lo.This can be illustrated using the simple matrix (utAu)(utA-'u) > (u~u)~ (5) p>v-? 5 3 - 1 * = (-; -; -:> For the trial vector (O,O,l) and for the more accurate vector (0,1,3) 213 (8) p = 1 ; v-1 = (9) p = 315 = 0.6; v-' = 417 = 0.57. Since ib0 = 0.55 the improved estimates obtained from v - I are significantly better than those from p. 3. VARIATION PRINCIPLE The new form of variation principle was introduced in a letter and established formally in a subsequent paper '. The Schrodinger equation for the system which is where T is the kinetic energy operator and V the potential is first generalized by introducing a new parameter E and writing (T+V)+r = Er$r (10) (T+ ~ / E V ) X = AX or Green's operator G is the operator reciprocal to (A-T) VX = &(A - T)x.G(A) = ( A - T)-l (1 3) GVX = E X . (14) where 1. is now a fixed negative number so that eqn. (12) becomes This operator can always be found in the form of an integral operator over a known kernel. This equation in which E appears as the eigenvalue of GV with ;1 fixed is known as the conjugate eigenvalue equation. For boundstates of Coulombic systems both G and V are negative definite so that the E eigenvalues are all positive. Since the product of two Hermitean operators is not generally Hermitean the conjugate eigenvalue equation is not in convenient form until it is made Hermitean by operating with V Thus V on the right of (15) acts as a weighting function in the Hilbert space of the eigenfunctions. The Rayleigh-Ritz principle for eqn. ( 1 5) is the variation principle now proposed.It asserts that the ratio VGVX = EVX. (15) G . G. HALL 71 for any trial function co such that the integral Jw*Vcod.r converges is always less than the largest eigenvalue E~ W E O . If the system is Coulombic eqn. (1 1) can be immediately reduced to the original Schrodinger equation by means of a scale factor such that It follows that the ground state energy Eo satisfies the bound In this expression A may be varied to minimize the bound and this is the equivalent of introducing a scale factor into co and optimizing it. The bound given by eqn. (19) can be compared with that given by the usual ratio. As before the use of the reciprocal operator means that a better result is always obtained. ~ r 2 = Erin; xr(x) = $ r ( x / & r ) * (18) Eo < Lq2. (19) 4. ILLUSTRATIVE C A L C U LA T I 0 N S Since the kinetic energy term does not appear in (16) the ratio q can be given a meaning even for a discontinuous trial function.The only restriction is that co should belong to the function space whose basis vectors are the eigenfunctions of the conjugate eigenvalue equation. This will be so provided that the integral sco* Vcodz (20) is convergent. be inadmissible we may consider the trial function As an illustration of the use of the principle for a function which would otherwise for the hydrogen atom. In this function a is a parameter and the lowest energy is -0.2905 obtained when a = 2.483. Thus despite the discontinuity at r = a and the lack of a cusp at r = 0 about 60 % of the exact energy is found. Much better energies are obtained using more realistic functions even if they still contain dis- continuities.The elimination of the kinetic energy operator has another consequence for the theory. Several of the best known lower bound formulae depend on a calculation of the variance of the energy. For the Schrodinger equation this is not often feasible since the term involving the square of the kinetic energy operator may not be defined or may diverge. For the conjugate eigenvalue equation this difficulty does not arise and in principle the variance can always be calculated although in practice it may be difficult to evaluate all the integrals. The variance for a trial function w becomes c2 = fw*VGVGVcodz/Jw*Vwdz-q2. (22) The Weinstein lower bound is then Eo > A(q + To illustrate this lower bound formulae we consider electron moving in a one-dimensional potential box of function is (23) the simple example of an length 1 Bohr.The tria 1 inside 0 outside 72 NEW VARIATION PRINCIPLE and the zero of potential is adjusted so that the potential inside is 1 Hartree. In this example with the potential everywhere positive the bounds are reversed and the variation principle gives a lower bound to the ground state energy of 574 Hartrees. The Weinstein relation gives an upper bound of 6.29 Hartrees. The exact energy is 5.935 Hartrees. 5. CALCULATION OF MEAN VALUES Many atomic and molecular properties depend essentially on the mean value of some operator. If the ground state wavefunction is not known accurately those mean values can be considerably in error. These errors can be progressively elimi- nated by a repeated use of the conjugate eigenvalue equation.Since the ground state corresponds to the largest eigenvalue of GV the effect of GV on a trial function is to increase the weight of the ground state eigenfunction in the result. This filtering operation can be repeated if necessary until sufficiently accurate results are obtained. The effect is exactly comparable with the power method of finding the largest eigenvalue of a matrix. If the operator in question is F the simplest mean value is Jo*FcodT/i co*codT. jw*FG Vcodz / Jw*GVwdT One stage of filtering gives while two stages gives the more symmetrical result 6. PROSPECTS The illustrations used above are of a simple kind so as to display the advantages of this new formulation. The prime motive for using discontinuous trial functions is that they enable more elaborate systems such as molecules and solids to be considered in termsof their localized parts. If these parts correspond to trial functions which vanish outside non-overlapping cells then a considerable reduction can be made in the number of integrals that require evaluation. Several molecular examples are now being considered. In the molecular as in the atomic examples the evaluation of the integrals has presented difficulties. These are due primarily to their high dimensionality and relative unfamiliarity. The possibilities of Gaussian methods of integration are now being investigated. It may be that these difficulties will prevent the method being applied more generally than to the integral equation analogue of the Hartree- Fock equations. G. G. Hall Chem. Physics Letters 1967 1 495. G. G. Hall J. Hyslop and D. Rees Znt. J. Quantum. Chem. in press.

ISSN:0430-0696    

DOI:10.1039/SF9680200069

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Minimum Basis Self'-consistent Group Function Calculations* BY MARTIN KLESSINGER Organisch-Chemisches Institut 34 Gottingen Germany Received 20th September 1968 The interpretation of complicated wave functions in a way which is illuminating and suggestive to a chemist requires concepts such as localized pairs of electrons. This is realized by writing the wave function as an antisymmetrized product of localized two-electron functions. Thus the separated-pair approximation appears to be of great value in discussing chemical problems parti- cularly in its most simple form which is characterized by the use of a minimum set of basis orbitals. Following a brief review of the self-consistent group function method (SCGF-method) and of the use of contracted gaussians within this approach results of SCGF calculations are reported for methane methyl fluoride ethane ethylene acetylene and cyclopropane.INTRODUCTION The idea of casting the total electronic wave function of a molecule in a form that exhibits localization of pairs of electrons goes back to Hurley Lennard-Jones and Pople who proposed to write the N-electron function "(1 . . . N ) as an anti- symmetrized product of pair functions At the same time they introduced the strong orthogonality condition Y(1 ... N) = d[QA(1,2)@,(3,4) ...I. (1) which leads to the important simplification that only interactions within one pair function iDR or between any two pair functions OR and as enter the energy expecta- tion value. developed a theory of generalized group functions of which the separated pair approximation is a special case.On the basis of this theory a practical method of computing wave functions and energies of polyatomic molecules was developed which takes into account groups of two and more electrons each group (localized or non-localized) being made self-consistent in the field of all others. This self-consistent group function method (SCGF-method) represents a generally applic- able approach strictly defined within the approximation determined by the strong orthogonality condition and leading to localized orbitals which show up a direct dependence on molecular structure. Thus it is hoped that this approach will make possible a genuine understanding and a proper explanation of molecular structure and of chemical regularities and their implications for the chemical behaviour of a molecule. The aim in developing this method was not to calculate absolute energies as accurately as possible but rather to obtain a scheme which is simple enough to be McWeeny * The work reported forms part of the Hubi/itutions.schr$t submitted by the author at the Georg- August-Universitat Gottingen 1968.Financial support by the Deutsche Forschungsgemeinschaft Bad Godesberg is gratefully acknowledged. 73 74 SCGF CALCULATIONS applicable to sufficiently large molecules and which gives results that may be helpful in discussing problems of a chemical nature. At the same time the semi-empirical development of the method was aimed at as it was thought that in spite of the encouraging progress which has been made in computing ab initio wave functions it would for some time be unavoidable to rely on semi-empirical methods in order to achieve what may be called chemical accuracy i.e.to calculate energy differences for large systems accurately to +_ 1 kcal/mole. For semi-empirical methods to be reliable enough to make chemical predictions they have to be put on a firm basis which allows the inherent approximations to be well understood. With this objective in mind non-empirical minimum basis set SCGF calculations were carried out on a sufficiently large number of molecules in order to find out to what extent this method is suited for solving chemical problems and in order to have suitable data to test the necessary approximations for a semi- empirical development. Results of some of these calculations are reported and discussed in the present paper. MINIMUM BASIS SET SCGF CALCULATIONS As a consequence of the strong orthogonality condition eqn.(2) each pair function (D in eqn. (1) may be written in the form @R(1,2) = ~c~’($~(1,2) (3) P as a linear combination of Slater determinants (or suitably coupled sets of Slater determinants) 4; built up from an orthonormal set of orbitals (OAO) rl,r2 . . . sl,sz . . . in such a way that the orbitals ri appear only in the functions 4; of group R the si only in the 4; of group S etc. If the basis orbitals form a minimum set there are just two OAO rl and r2 for each bond pair R of electrons which for a singlet state is therefore conveniently = (ri - r J I J2. ( 5 ) which only one suitably hybridized @dW = 4SCL2) = I s(l)ms(2)P(2) I. A self-consistent ground state is obtained by using an effective Hamiltonian for a two-electron group R which is defined as 2 2 t where the matrix elements of hfff(i) which take into account the interactions with all other groups of the molecule in a self-consistent way are given by (hk)rirj = (hR)rir + C [xRs(rirj) + X,IRxs(rirj) + ~SxFTrirj)] S(#R) M .KLESSINGER 75 with Here X and Ys are a " bond order " and a " polarity " (measuring the amount of charge transferred from orbitals s2 in bond S to orbital sl) respectively and are defined in terms of the first order density matrix by xs = Pi 2 = P; 1 = J2c',s'(c',s' + CS"') (9) Ys = $(PS 1 -E2) = -(cy))2. (All details together with formulae for the calculation of excited states and ionization potentials as well as for the inclusion of many-electron groups are given by Klessinger and McWeeny. 3 The set of OAO r1,r2 .. . s1,s2 . . . is obtained from a set of atomic orbitals (AO) by a transformation with a matrix V which may be written as v= ws,*u. (11) W makes the inner shell orbitals orthogonal to all valence orbitals. Preliminary calculations showed that the polarity of the bonds in a molecule increases if valence orbitals are allowed to mix into the inner shell orbitals. This polarity increase leads to a gain in bond energy which is not sufficient to compensate for the accompanying loss of inner shell energy. Thus the most suitable procedure to construct inner shell orbitals is a Schmidt orthogonalization with respect to all valence orbitals which keeps the inner shell orbitals pure and leads thus to the best value of the total energy.4 of the valence orbitals (which are already orthogonal to the inner shells) i.e.Sb is the overlap matrix of the A 0 after applying the Schmidt orthogonalization. By the Lowdin procedure the orbitals are disturbed as little as possible in a least-squares sense.6 Finally U transforms the OAO into suitable hybrids. U is an unitary matrix so that S 3 represents a symmetrical Lowdin-orthogonalization v = WS,fU = wus,+ (12) (where S is the overlap matrix of the nonorthogonal hybrids) i.e. the hydridized OAO of eqn. (11) are equivalent to orthogonalized hybrids obtained by the same transformation U. W and S;* are entirely determined by the initial choice of A 0 and by the geometry of the molecule whereas U is arbitrary within certain limits and determines how the original basis set is divided up into subsets one for each group of electrons.As a consequence of writing the bond pair functions as linear combinations of Slater determinants i.e. of taking into account configuration interaction within each bond pair the total energy is not invariant with respect to the way in which the original basis is divided up. Therefore in the SCGF method the total energy of a given state of a molecule is a function of hybridization. This leads apart from the deter- mination of the self-consistent group functions to an additional optimization process for the hybridization. At the same time it enables one to determine optimum valence hybrids by applying the variational principle i.e. by an energy criterion. This is of great importance for chemical applications of the method as hybridization is a 76 SCGF CALCULATIONS concept well suited for relating results of quantum mechanical calculations and chemical ideas.Calculations according to the above described method have been performed on a number of small molecules using a minimum basis TABLE COMPARISON OF RESULTS OF SCF AND SCGF BASIS OF THE STO molecule geometry ground-state energy Eo(a.u.1 SCF SCGF HF R = 1.733122 a.u. - 99.4786 - 99.4972 HzO R = 1.8103 - 75.6807 -75.7139 CH R = 2.06742 a.u. - 40.0606 - 40.0980 Nz R = 2.06928 - 108.5736 - 108.6498 CO R = 2.13202 - 1 12.3437 - 1 12.401 9 e = 1050 of Slater A 0 (STO) (table l).? CALCULATIONS USING A MINIMUM hybridization A.Eo(eV) -0.51 A = 0.40" -0.90 a = 102.7'** - 1.02 sp3 hybrid. -2.07 3 = 0.74* - 1 *58 Rc = 0.59* 20 = 0.86* * Bond hybrids bx = N[A(2s)x+(2pz)x] ** cc = angle subtended by bond hybrids -40[ R(;.u.)--r ~ -""I I R (a.u.)+ I 1.0 20 3.0 1.0 2.0 30 FIG.1.-Potential energy curves for HF (a) SCF and SCGF results (b) SCF and SCF-CI results of Karo and Allen. A consequence of taking into account some electron correlation within each bond pair a ground state energy is obtained which is lower than the SCF energy calculated with the same set of basis orbitals by an amount of A& = 0.25-0.50eV per two- electron bond. Potential energy curves for the ground state of the HF molecule as calculated by the SCF and the SCGF r n e t h ~ d ~ are shown in fig. la and may be compared with results of an SCF calculation with and without CI by Karo and Allen in fig. l b (the SCF curves are not identical as Karo and Allen used a basis of atomic Hartree-Fock orbitals). That graph shows that by including CI within the bond pair function aR(1,2) (eqn.(3)) about the same effect is obtained as by extensive CI within the basis of SCF-MO. As the SCGF calculations involve at 1- Atomic units are used unless otherwise stated i.e. 1 a.u. = 0.529 x lO-'O m for the unit of length and 1 a.u. = 27.210 eV for the unit of energy. M. KLESSINGER 77 most 3 x 3 secular problems and convergence is rapid they are considerably faster than conventional SCF calculations even without CI. This demonstrates clearly the great value of the SCGF approach in calculating ground state properties. R(CI u.)- USE OF CONTRACTED GAUSSIANS I N THE SCGF APPROACH R (a u.) -+ In principle SCGF calculations of the type described could be carried out for any kind of molecule. But as with all ab initio calculations the determination of all electron interaction integrals makes the use of Slater A 0 for larger molecules prohibitive.During the last few years gaussian type orbitals (GTO) originally introduced into quantum chemical calculations by Boys and by McWeeny,lO as well as pure gaussian orbitals l 1 (GO) have proved successful in calculations on large molecules as the need to use larger basis sets is far more than compensated by the ease with which molecular integrals over the GTO and the GO can be calculated. I I A way of preserving the simplicity of the minimum basis SCGF method in spite of the slow convergence of the GTO is the use of a minimum basis set of contracted gaussians i.e. linear combinations of GTO (LCGTO). In order to be able to extend the SCGF calculations to larger molecules it was thus decided to expand the minimum STO set in terms of the GTO in the manner described explicitly by Huzinaga.In order to find an appropriate compromise between sufficient accuracy and reasonable computing time extensive calculations with different expansions of the basis set were performed for the methane molecule. It turned out that an expansion of the Is 2s and 2p A 0 in terms of five two and three GTO respectively will be adequate for first row atoms whereas for the H-atom an expansion in terms of three GTO seems to be sufficient. This basis set is well suited to reproduce the results of SCGF calculations with a minimum basis set of STO. This may be seen from fig. 2 which shows for the HF molecules the ground state potential energy curves and the optimum hybridization of the bond orbital as a function of the internuclear distance as determined by the SCGF method.A detailed comparison of the results 4 TABLE 2.-cOMPARISON OF THE RESULTS OF SCGF CALCULATIONS ON HF WITH A MINIMUM BASIS OF THE STO (UPPER PART) AND LCGTO (LOWER PART).* minimum basis of the STO S-C** R = 1.25 R = 1-50 Eo - 99'24802 - 99.43002 HZ -3.51874 - 2.95060 X 0.9514 0.9725 Y 0.2900 0.1763 P 1.4801 1.1317 jZopt 0.7207 0.5165 %s 34.1 8 21.06 R = 1.73312 - 99.4971 7 - 2.54864 0.9741 0-0908 0.8076 0.3962 13.57 R = 2.00 - 99.5 1889 - 2.2092 1 0.9588 0.0184 0.4719 0.3006 8-29 R = 2-50 - 99.50128 - 1.79913 0.8831 0.001 8 0,1814 3.19 - 0.0520 R = 3-99 - 99.47089 - 1.56785 0.7385 - 0.0563 - 0.1763 0.1044 1 *08 R = 3.50 (a.u.) - 99.44982 - 1 ~44075 0.5473 - 0,0341 -0.1396 0.0541 0.3 1 minimum basis of the LCGTO S-"* minimum basis of LCGT0,C.R.-<*** R = 3-50 (a.u.) EO - 99.05520 - 99.30212 - 99030304 - 99.24467 - 99,12027 - 99.34837 - 99.30542 - 99.23 1 10 - 1,43525 R = 1.25 R = 1.73311 R = 2.50 R = 1 -25 R = 1.73312 R = 2.50 R = 3.50 - 3.57046 - 2.59238 - 1.81334 - 1.42471 - 3.64547 - 2.67497 - 1.83335 0,4786 X 0,9566 0.9776 0.8903 0,5605 0.9658 0.9735 0.8799 Y 0.2755 0,0746 - 0.0724 - 0.0468 0.2422 0.1272 0.0172 - 0.0066 0.1 156 I.1 1 5426 0.8523 -0.0129 -0.1886 1 -5290 1.1952 0.6046 Jopt 0.7691 0.4239 0.2006 0.061 9 0-7507 0.4335 0.1936 0.0425 %s 37.17 15-23 3.86 0.38 36-04 15.82 3.61 0.18 *Ground-state energy Eo and effective bond energy HzT in a.u.dipole moment p in D bond order X and polarity parameter Y optimum hybridiza- **S.-C = orbital exponents according to Slater rules CF(~S) = 8.7 C~(2s) = CF(2p) = 2.6 C~(ls) = 1.0.***C.R.-C = orbital exponents according to Clementi and Rairnondi,l3 C~(1s) = 8.6501 C~(2s) = 2.5638 C~(2p) = 2.5500 c ~ ( 1 ~ ) ~1 1-20. tion parameter hopt (cf footnote to table 1) and percentage s-character of the bond orbital %s. M . KLESSINGER 79 obtained with these two kinds of basis functions is made possible by the data in table 2. These data also show that calculated binding energy and optimum hybridiza- tion are little affected by the choice of the basis orbitals whereas calculated charge distributions and dipole moments are sensitive to the choice of the orbital exponents as is well known for SCF calculations Similar results were obtained for the water molecule where binding energy and optimum hybridization appear to be insensitive to the choice of basis orbitals and unexpectedly to the magnitude of the valence angle.The minimum of the total energy is obtained when the valence angle coincides with the hybrid angle (the angle between the bond hybrids) i.e. there is no indication for the existence of bent bonds in the equilibrium ground state of the water molecule. SOME RESULTS OF SCGF CALCULATIONS A fully automatic computer programme MOLCAL has been developed for the IBM 7040 which requires only the following input data the molecular co-ordinates a specification of the basis functions and some data indicating the symmetry of the system. By means of this MOLCAL programme SCF and SCGF calculations were carried out for the following molecules methane methyl fluoride ethane eclipsed and staggered ethylene acetylene and cyclopropane.The results of the ground state total energy Eo for an expansion of the STO with exponents given by Clementi and Raimondi l3 together with 5s = 1.20 for the hydrogen atom in terms of the LCGTO as described in the previous section are collected in table 3 and compared with the best available calculations of other authors. TABLE 3. SCGF GROUND-STATE ENERGIES Eo(a.U.) ENERGY LOWERING A&,(eV) AS COMPARED TO SCF CALCULATIONS WITH SAME BASIS SET AND RESULTS OF OTHER AUTHORS. molecule EoWGF) AEO (eV) Eo (other methods) - 40.0606 (SCF ST0)Q - 40.3 12 (ASPG NO)b methane CH4 -40.1022 - 1.22 methyl ff uoride CH3F - 138.2500 - - ethane stagg. C2H6 - 79.0107 - 1 *48 - 79.1478 (SCF Hartree-Fock-AO)C ethane ecl.C2H6 - 79.01 19 - 1.65 - 79.1438 (SCF Hartree-Fock-AO)C ethylene CzH4 - 77.7814 - 1.47 - 78.0063 (SCF ext. basis of LCGTO)d acetylene C2H2 - 76.5733 - 1-75 - 76.7418 (SCF ext. basis of LCGTO)e cyclopropane C3H6 - 1 16.6797 - 1.05 - 1 16-02 (SCF G0)f * M. Klessinger and R. McWeeny J. Chem. Physics 1965,42,3343. b R. Ahlrichs and W. Kutzelnigg Chem. Physics Letters 1968 1 651. c W. H. Fink and L. C. Allen J. Chem. Physics 1967 46,2261. d J. M. Schulman J. W. Moskowitz and C. Hollister J. Chem. Physics 1967 46 2759 e J. W. Moskowitz J. Chem. Physics 1965 43 60. f H. Preuss and G. Diercksen Ini. J. Quantum Chemistry 1967 1 361. In all calculations optimum hybrids were determined and found to point along the bond axes ; even in cyclopropane the direction of the CH-bond orbitals coincides with the line joining the C and H atoms.Bent bonds consequently do arise only when the valence angle is smaller than 90° as for the CC-bonds in cyclopropane where the deviation of the hybrid direction from the valence direction is calculated to be co = 21.1" in good agreement with maximum overlap considerations which lead to w = 2 1 ~ 4 O . l ~ Energy localized orbitals (LMO) were determined according to the Edmiston- Ruedenberg procedure for ethylene and acetylene by Kaldor.16 As for the triple 80 SCGF CALCULATIONS bond in N2 and CO the LMO for the CC double and triple bond respectively turn out to be equivalent orbitals corresponding to a description of multiple bonds in terms of bent " banana bonds " whereas the SCGF method gives in these cases a lower energy for a description of a multiple bond in terms of a a-bond and one or two n-bonds.17 Whereas the CH-bond orbitals in ethylene determined by the Edmiston-Ruedenberg method form an angle of 11 1 O the SCGF method leads to an hybrid angle of 117" in good agreement with the experimental valence angle.Similar differences between the LMO and SCGF-orbitals are found for acetylene where the SCGF method leads to 45 % s-character of the CH-bond orbitals (corres- ponding to nearly pure sp hybridization) compared to 33 % s-character (sp2 hybridiza- tion) of the LMO. In all cases the SCGF value of the total energy is lower than the SCF value cal- culated with the same basis orbitals by AEo = 0.2-0.4 eV per two-electron bond. For ethane the total energy lowering is 1.48 eV for the staggered conformation and 1.65 eV for the eclipsed conformation.Thus while the SCF calculation gives a barrier of internal rotation of 3 kcal/mole with the staggered conformation being the more stable one the SCGF calculations leads to a value of 0.6 kcal/mole favouring the eclipsed conformation. This seems to suggest that the SCGF method is not well suited for calculating potential energy barriers. The reason for this may be that the localizability of electron pairs is different for the two conformers. For cyclo-propane the SCGF energy is calculated to be only 0.12eV per bond lower than the SCF energy. Thus in open-chain compounds electron pairs appear to be better localized than in cyclic compounds exhibiting ring strain. TABLE 4.-EFFECTIVE INNER SHELL ENERGIES HJ$ OF c- AND F-ATOMS IN DIFFERENT MOLECULES (ENERGIES IN eV).K shell of the C-atom K shell of the F-atom molecule H$ (eV) molecule H: (ev) methane -712.98 HF R = 1-25 a.u. - 1575.96 ethane stagg. - 71 3.09 R = 1.733 - 1577.57 ethane ecl. -713.13 R = 2.50 - 1579.86 methyl fluoride - 71 5.87 R = 3.50 - 1579.59 cyclopropane - 71253 methyl fluoride - 1578.21 ethylene - 714.07 acetylene -715.45 Table 4 gives the effective inner shell energies which lie for the carbon atom between H$f = -712.5 eV and = -715.5 eV; they depend on hybridization and on the nature of the neighbouring groups and are thus slightly different from molecule to molecule. Within an homologous series however they are approximately constant. The values of H:tf for the fluorine atom show that the internuclear distance is of greater influence on the actual value of this quantity than the replacement of the hydrogen in HF by a methyl group.In table 5 the effective framework integrals a, and Priri (cf. eqn. (7)) the effective bond energies HZF and the bond orders X, are collected for CC 0- and n-bonds; the same quantities for the CH bonds together with the polarity parameter YCH are given in table 6. These data clearly reflect different bonding situations in different M . KLESSINGER 81 molecules pCc is for cyclopropane appreciably smaller than for ethane so that the ring strain in the former compound becomes apparent in alower bond order Xc and a smaller absolute value of the bond energy H$$ compared with ethane. The CC a-bonds in ethylene and acetylene are considerably stronger than the o-bond in ethane due to the shorter internuclear distances.The effective bond energy of the n-bonds is roughly half that of a a-bond. TABLE 5.-PROPERTIES OF CC-BONDS AS CALCULATED BY THE SCGF METHOD (ENERGIES IN ev) molecule a-bonds ethane stagg. ethane ecl. cyclopropane ethylene acetylene ethylene acetylene n-bonds 0% - 24.38 - 24.39 - 23.43 - 24.99 - 23.32 - 19.63 - 20.06 Bcc - 11.57 - 11.58 - 10.43 - 16.40 - 21.59 - 3.93 - 4.90 xcc 0.9872 0.9872 0.9822 0.995 1 0.9978 0.891 3 0-93 19 - 54.41 - 54.43 - 50.44 - 64.10 -71.26 - 35.07 - 37.20 TABLE 6.-PROPERTIES OF CH-BONDS AS CALCULATED BY THE SCGF METHOD (ENERGIES IN ev) molecule methane ethane stagg. ethane ecl. methyl fluoride cyclopropane ethylene acetylene 0% - 24.84 - 24.85 - 24.83 - 25.86 - 25.35 - 25.59 - 27.94 MH 'CH xCH =CH H2tY - 23.31 - 11.60 0.9800 0.0980 - 53.58 -23.36 - 11.68 0.9805 0.0918 -53.66 - 23.35 - 11.71 0.9806 0.0914 - 53.68 -23.92 - 11.73 0.9797 0.1045 -55.42 - 23.46 - 11.82 0.9809 0.0950 - 54-52 -23.61 -11.98 0.9812 0.0960 -55.20 - 23.76 - 12.36 0.9768 0.1496 - 58.87 Whereas all CC bonds of the molecules considered in this paper are nonpolar by symmetry the CH bonds show a polarity ranging from a transfer of 0-09 electrons from the hydrogen to the carbon atom in ethane to a transfer of 0.15 electrons also from H to C in acetylene.But as the atomic moment of the bond hybrid by far overcompensates the moment due to charge transfer the calculated bond moment lies for all CH bonds considered between pCH = 1.40 D and pcH = 1.77 D the C atom being the positive end of the dipole in all cases. The change of polarity of the CH bond in going from methane to acetylene as predicted by Cou1son,18 is not observed in these calculations.The influence of hybridization at the carbon atom on the properties of a CH bond is also apparent from table 6 ; in ethylene the calculated polarity parameter is YcH = 0.096 in acetylene Y, = 0.150. The CH-bond orbitals in cyclopropane show the same hybridization as the CH-bond orbitals in ethylene; the calculated bond orders polarity parameters and bond moments of the CH-bonds in these two molecules are nearly identical. The highest value of the polarity parameter was calculated for acetylene in accord with the relative acidity of the CH-bonds in this compound. DISCUSSION From the numerical results of the preceding sections the SCGF approach cannot compete with those methods which were particularly designed for obtaining absolute 82 SCGF CALCULATIONS energies as accurately as possible.But the SCGF approach compares as far as ground-state properties are concerned favourably with the more commonly used approximate SCF method and it offers a number of advantages which may be important. (1) The description emphasizes throughout the structure of the system in terms of chemically significant parts (bonds lone pairs etc.) and the interactions between them. Time-consuming localization procedures are therefore not required. This much facilitates the interpretation of the calculated wavefunctions and makes similarities between different molecules immediately apparent. At the same time it is recognized that a bond (CH for example) is not quite the same in two different molecular situations.The nature of the differences which is related to deviations from chemical " additivity rules " are indicated by the bond order and polarity parameters which take account of the environment in a self-consistent way. (2) As in the SCGF approach the total energy is not invariant with respect to the way in which the basis set is divided up into subsets for individual groups of electrons optimum valence hybrids can be determined using the variation principle i.e. by means of an energy criterion. (3) The individuality of different groups is reflected in the relative weakness of the coupling between them. This results in a rapid convergence of the iterative calculation in contrast to the situation usually encountered in Hartree-Fock theory. In addition there is always a good starting approximation available and the dimension of the secular problems which have to be solved is generally low ; in the minimum basis calculations the secular determinants are never larger than 3 x 3.This again leads to short computing times. As with all ab initio calculations by far the most time-consuming step is the determination of the molecular integrals. (4) The wave-function for any single group in the molecule may be refined indefinitely by adding further basis orbitals in order to include correlation effects. Results of calculations on Hz04 N2 and CO suggest that the localized pair functions obtained by the SCGF method are better suited for including intrapair correlation effects than e.g. the Ruedenberg LMO or other localized orbitals. For the methane molecule it is possible to estimate the amount of correlation energy recovered by the SCGF calculation Ahlrichs and Kutzelnigg l9 calculated the intrapair correlation energy of any single CH-bond in methane by a direct determination of the approximate natural orbitals of a localized pair of electrons in the Hartree-Fock field of all other electrons; they obtained a value of 0.6 eV and estimated the correct value to be 0.76eV.To the calculated value of 0.6 eV orbitals of a-symmetry with respect to the bond-axis contribute 0.48 eV the remaining 0.12 eV come from orbitals of n-symmetry. In the minimum basis SCGF calculations where only a,-orbitals are considered the calculated intra-pair correlation for each CH-bond is 0-26 eV i.e. roughly half the value obtained by direct determination of the NO; the computer time necessary for the SCGF calculation is shorter by a factor of approximately 20.From the same calculation by Alhrichs and Kutzelnigg one may conclude that the best possible separated-pair wave function for methane allows only half of the total correlation energy to be recovered i.e. nearly half of the correlation in CH is interpair correlation. Similar results concerning the ratio of intrapair to interpair correlation were obtained by Sinanoglu.20 This seems to set a limit to the SCGF approach as far as the exact determination of absolute energy values is concerned. ( 5 ) By employing throughout orthogonal basis orbitals which are strongly localized the neglect of all many-centre exchange and hybrid intergrals (the zero differential overlap approximation) can be justified.Thus the SCGF approach lends itself admirably to semi-empirical development as suggested by Klessinger M. KLESSINGER 83 and McWeeny3 and further investigated by Cook Hollis and McWeeny.21 It is hoped that such a semi-empirical method based on the SCGF approach could become most useful in discussing chemical problems. A. C . Hurley J. Lennard-Jones and J. A. Pople Proc. Roy. SOC. A 1953 220,436. R. McWeeny Rev. Mod. Physics 1960 32 335. M. Klessinger and R. McWeeny J. Chem. Physics 1965,42 3343. M. Klessinger J. Chem. Physics 1965 43 Sl17 and references therein. P. 0. Lowdin J. Chem. Physics 1950 18 365. B. C. Carlson and J. M. Keller Physics Rev. 1957 105 102. ' M. Klessinger Chem. Physics Letters 1968 2 562. * A. M. Karo and L. C. Allen J. Chem. Physics 1959 31,968. S. F. Boys Proc. Roy. SOC. A 1950 200 542. lo R. McWeeny Acta Cryst. 1953 6 631. l 1 J. L. Whitten J. Chem. Physics 1966 44 359 H. Preuss Mol. Physics 1964 8 157. *3 E. Clementi and D. L. Raimondi J. Chem. Physics 1963 38,2686. l4 C. A. Coulson and T. H. Goodwin J . Chem. SOC. 1962,2851 ; 1963 3161. l 5 C. Edmiston and K. Ruedenberg Rev. Mod. Physics 1963 35 457; J. Chem. Physics 1965 l 6 U. Kaldor J. Chem. Physics 1967 46 1981. S. Huzinaga J. Chem. Physics 1965 42 1293. 43 s97. M. Klessinger J. Chem. Physics 1967 46 3261. C. A. Coulson Valence (Oxford University Press Oxford 2nd ed. 1961) p. 219. l9 R. Ahlrichs and W. Kutzelnigg Chem. Physics Letters 1968 1 651. 2o 0. Sinanoglu and B. Skutnik Chem. Physics Letters 1968 1 699. 21 D. B. Cook P. C. Hollis and R. McWeeny Mol. Physics 1967 13 553.

ISSN:0430-0696    

DOI:10.1039/SF9680200073

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Optimum Atomic Orbital Exponents for Molecular Wave Functions BY J. N. MURRELL AND C. L. SILK Chemical Laboratory University of Sussex Brighton Received 9th September 1968 Calculated potential energy curves have been obtained for H2 and LiH using valence bond wave functions based on a minimum basis set of Slater type orbitals. The orbital exponents have been optimized at all internuclear distances and the results compared with the curves obtained using best atom and best equilibrium-molecule exponents. The vibration frequencies obtained from the best Re exponents are appreciably larger than those from the calculations in which the exponents are opti- mized at all R. In constructing molecular wave functions from atomic orbitals particularly when a minimum basis set is used it is important to optimize the energy with respect to the orbital exponents 5.Such an optimization not only leads to a considerably improved energy but is also necessary to obtain good bond lengths and to obtain agreement with the virial theorem. In most of the recent a priori calculations based on atomic orbitals (Slater or gaussian type) considerable attention has been paid to this exponent optimization. Its importance decreases as the number of linear variation parameters increases but even for large atomic basis sets such as those used by Cade and his co-workers it has been found to be important. As yet however there appear to be no general rules available for estimating the optimum exponents and few regularities have been noted from one molecule to another. There have been few studies carried out on orbital optimization except with the molecule at its equilibrium geometry.It is in fact the general practice to optimize at this geometry and then use the resulting set of orbitals for calculations at other internuclear distances. However this technique leads to wave functions that are relatively slightly better at the equilibrium geometry than elsewhere and force con- stants calculated from the resulting potential energy curves might be expected to be too large. Thus the problem of exponent optimization is of importance for a priori calcula- tions as well as being a phenomenon of general interest in the theory of the chemical bond. It is hoped that the calculations described in this paper will help to increase our understanding of the subject. Apart from some early calculations of H and Hz the only calculations in which the orbital exponents are optimized at each internuclear distance appear to those of Sahni La Budde and Sawhney.2 These are calculations on CO and Nz and their ions using a minimum basis LCAO MO SCF formulation.However the limitation of this method is the failure of the MO method without configuration interaction to give the correct dissociation products. As a result the orbital exponents found in these calculations do not tend towards those of the ground state atoms at large inter- nuclear distances. To get over this difficulty we have chosen to use valence bond functions as a basis for our calculation. The calculations have been carried out with 84 J . N. MURRELL A N D C. L . SILK 85 a single Heitler-London function and with one or more covalent or ionic structure included.We have however imposed the restriction that an orbital exponent shall be the same in each valence bond wave function. If the complete set of valence bond structures is taken then the calculation is equivalent to the MO method with complete configuration interaction. CALCULATIONS ON H The first such calculation of this type was made by Rosen who varied the expo- nent for the Heitler-London function of H2 as a function of internuclear distance (the earlier calculation by Wang Fig. 1 shows Rosen’s results together with our own using the ionic plus covalent function of H2 with a single 1s exponent. was only for the equilibrium distance). 0 1 2 3 4 5 6 R (a.u.) H-L+ ionic. FIG. 1 .-Variation of 1s orbital exponent with bond length.Broken line H-L only (Rosen) full line The two curves show similar behaviour more so we shall see than in the other molecule we examine. The initial expansion of the 1s orbital as the two atoms come together can be associated with increased overlap that can be gained in this way and the resulting decrease in the potential energy. For small variation in 5 about the optimum value the atom energy varies as The stabilization arising from the orbital overlap at large R is roughly proportional to exp (-2cR) and the variation of this is to a first approximation linear in AC. Hence at large R energy is gained by an orbital expansion. Consider a Heitler-London function formed using two orbitals with exponents c1 and 5,. At large R the term in the Hamiltonian matrix element leading to stabilization will vary roughly as exp [ - (C1 +5,)R].For small changes in 5 from the best atom values this This is clearly a result that can be generalized to heteronuclear systems. 86 OPTIMUM ATOMIC ORBITAL EXPONENTS will vary linearly in AC1 and AC2 hence an expansion of both orbitals at large R is bound to lead to a lower energy. If the calculations on H2 are carried out with best-atom exponents (f = 1) then the dissociation energies are approximately 0.025 a.u. worse for both calculations but more seriously the equilibrium bond length is calculated to be approximately 0.25 a.u. too large and the virial theorem is not satisfied. If the calculations are carried out throughout with the best-molecule exponents (that is the best equilibrium molecule) then the error in the energy of the separated atoms is 0.013 a.u.for the Heitler-London function and 0.018 a.u. for the H-L plus ionic. These values will approximately correspond to the maximum error to be found for values of R greater than the equili- brium distances Ro. The error for R<R, however becomes progressively larger as the optimum exponents get smaller. Coulson showed that the vibration frequency calculated from the H-L curve with best Re exponent was 15 % greater than that calculated using optimum exponents throughout CALCULATIONS ON LiH We turn now to calculations on LiH. An extensive bibliography of calculations on this molecule has been given by Cade and H u o . ~ Valence bond calculations using a minimum basis of STO have been made by Hurley and by Miller Friedman Hurst and Matsen.8 Both use sets of exponents appropriate for the free atoms- J 1.2 - 1.1 - n )s s 1.0 - -9 I r these are roughly what we now know as best atom exponents-and a number of valence bond configurations.Miller and co-workers attempted some variation of their orbital exponents but were unable to find any better than their free atom values. From our calculations it is clear that they made a bad selection. Fig. 2-5 showthevariation of the exponents of the Is 2s and 2pa orbitals of Li and the hydrogen 1s orbital (h) for the followingsetsofcalculations. (Details ofthe numeri- cal work are given in the appendix.) J . N . MURRELL AND C. L . SILK 87 "V 8 1 2 3 4 5 6 7 8 R (a.u.) FIG. 3.-Variation of the lithium 2s exponent. Code as in fig. 2 but (d) apd ( e ) are superimposed. L I 2 3 4 - 5 6 7 8 R (a.u.) 1 I I I 2 3 4 5 6 7 8 R (a.u.) ( e ) - - -.FIG. 4.-Variation of the lithium 1s and 2po exponents. (a) ~ ; (c) - - - * * ; (b) ( d ) and 88 OPTIMUM ATOMIC ORBITAL EXPONENTS (a) the Heitler-London 2s - h function JH I 1sals/Jsahp I - I 1sE1ss2spha I 1. (b) the above plus the ionic function Li+H- I 1salsphahp I (c) the two Heitler-London functions 2s- h plus 2p - h. (d) (c) plus Li+H-. (e) the full six configurations that can be obtained without promoting the lithium Is electrons. 1.2 0.0 a -1-1 -1.2 Is \ \ \ \ \ \ \ '\\ \ \\\ 2Po \ \ \ \ . \ > 1 2 3 4 5 6 7 8 R (a.u.) FIG. 5.-Variation of the exponents relative to best atom values for calculation ( d ) drawn to same scale. The best atom C(2po) is taken as the same as '5(2s). Table 1 shows our full calculated data for the set (d).Comparable data for the other calculations are available on request to the authors. The general trend of the exponents is similar for all the calculations. Except for the hydrogen exponent the results of calculations (d) and (e) were insignificantly different. The main anomaly is in the behaviour of c(2s) for calculation (b); at distances shorter than the equilibrium bond length the 2s orbital begins to expand again. This is probably due to the increasing importance of the ionic structure at these short distances but we have found no obvious correlation between the 2s exponent and the coefficient of the ionic structure in the wave function. The expansion of the 2s orbital is found in all calculations above RE 5 a.u. The expansion of the hydrogen orbital is found for all calculations except (a) and (c) which only use Heitler-London functions.However the region R> 5 a.u. is one for which TABLE 1 Numerical data for calculation (d). C1 Cz and C3 are the coefficients of the H-L(2s-h) H-L(2pa-h) and Li+H- functions respectively in the ground state eigenfunction. 2.0 2.5 3.0 3.25 3.5 4.0 5.0 6.0 7.0 8.0 a0 1.190 1-131 1.083 1 *064 1 -049 1 -026 1 -003 0.998 0.999 1 *ooo 1 -000 2,678 2.682 2.684 2.685 2.685 2.686 2.687 2.687 2.687 2.687 2,6865 0.707 0.704 0.693 0-686 0.678 0.661 0.634 0.624 0.626 0.631 0.6372 0.870 0-807 0.747 0.718 0,691 0.641 0-559 0.495 0.440 0.388 0.61463 0.60888 0.62204 0.63389 0.64844 0.68560 0.78641 0.89757 0.97005 0.99902 1.0 0.45009 0.40039 0.37318 0.36422 0.35673 0.34313 0.30025 0.21511 0.11807 0,0551 1 - 0.10293 0.15046 0.17726 0.18522 0.19053 0.19341 0.17356 0.12779 0.07831 0.04407 - 8.32486 8.1 3285 8.00870 7.96528 7.93023 7-88225 7.84525 7.85735 7.88474 7.90309 - 16.2286 - 16.1 003 - 15.9926 - 15.9493 - 15.91 16 - 15.8540 - 15.7943 - 15.7896 - 15.8083 - 15.8230 90 OPTIMUM ATOMIC ORBITAL EXPONENTS the two-centre electron repulsion are least accurate and we believe that the predicted expansion is outside the accuracy of our calculation.The expansion of the 1s orbital is small. For calculations (b) (c) and (d) the be- haviour of c(ls) was found to be almost the same and the full calculation (e) was therefore carried out using this same set of 1s exponents. The 2p exponent does not extrapolate to any atomic value as the orbital is not occupied in the ground state of the separated atoms.The diffuse 2p orbital at large R is determined by the advantage of having the maximum overlap between the 2p and hydrogen orbitals because the dominant term in the Hamiltonian matrix element between the two Heitler-London functions at large R is the two electron integral < W ) W I e 2 h 2 I2P(2)h(2)). TABLE 2 The quadratic coefficients in the Taylor expansion of the total energy about the best-molecule exponents for R = 3 a.u. FREE ATOMS i\' h Is 2s - - h 1 -00 1s 1 -96 0.06 2s 1.18 CALCULATION (a) h 1s 2s I1 1.14 0.00 - 0.05 1s 1 -94 0-05 2s 1-35 CALCULATION ( d ) j\' h Is 2s 2PO 12 0.77 0.00 0.01 0.09 1s 1 -95 0.04 0.01 2s 0.65 0.01 2PO 0.20 As a measure of the importance of small changes in C and incidentally a measure of the accuracy to which 5 may be determined from a given accuracy in the total energy we show in table 2 the quadratic coefficients of the Taylor expansion of the total energy about the optimum exponent values for calculations (a) and (d) at R = 3.0 a.u.The corresponding free-atom coefficients are also shown The exponents are almost independent variables in the calculation up to the quadratic terms. The quadratic coefficient for the 1s orbital is almost the same for the molecular calculation as for the free atom-as is to be expected-and is larger than the coefficients for the valence orbitals. This alone means that it is energetically unfavourable to have much change in c(1.s) from the free atom value. The change we find at Ro is 0.003 and this would lead to an increase of the atomic energy of only J . N. MURRELL AND C. L.SILK 91 a.u. a number of marginal significance. For the single Heitler-London function the quadratic coefficients of the 2s and hydrogen orbital are fairly close to the free atom values both having become slightly larger. For the three-configuration calcu- lation however the coefficients are almost halved showing that when more linear variation parameters are allowed in the wave function the non-linear parameters become less important. However even for calculation (d) the quadratic coefficients are still quite large. Why is it that the hydrogen exponent is the one to show the greatest change from the free atom value to the optimum value at Ro? From table 2 one can see that less intra-atomic energy is lost for a given change in c(h) than for the same change in the C(2s). However the difference is not large enough to explain why the change in [(IT) is about twice that of ((2s).We have no answer to this question. -7.88- -7.90- -792- T 2 W -7.94. -7.96- - 7.98 1 2 3 4 5 6 7 8 R (a.u.) FIG. 6.-Potential curves calculated using optimum exponent at a R (a)-@) defined in text. Fig. 6 shows the potential curves for calculations (a)-(e) and table 3 shows the spectroscopic constants calculated from these by a five term polynomial fit to seven calculated points in the range 2-6 a.u. We also include in this table the spectroscopic constants obtained from the curves calculated with best-atom and best-equilibrium- molecule exponents and fig. 7 compares the curves for calculations (a) and (d) with the three types of optimized exponent. The differences in energy between the best-atom and best-molecule results at R = 3.0 a.u.are well predicted from the coefficients of table 3. The energy gained by optimization is Zarger in calculation (d) than in calculation (a) even though there are more linear variation parameters in the calculation. The use of best-atom exponent overestimates the equilibrium bond length but not to the large extent found for HZ. The vibration frequencies calculated from the 92 OPTIMUM ATOMIC ORBITAL EXPONENTS curves based on best Re exponents are appreciably larger than those obtained from exponents optimized at all R . The virial theorem is held to the accuracy of our calculated energies for the best molecule exponents. Miller and co-workers using a minimum basis of approximate best atom orbitals and valence bond functions corresponding to our set (e) plus four functions in which ,? ? rrl s.-7.94- - 796- TABLE 3 Spectroscopic constants calculated by a 5 term polynomial fit to the potential energy curves. The virial test for the best molecule zetas was less than l O V . V.B. basis optimum equilibrium minimum dissociation virial test functions zetas distance (Re) energy (a.u.) energy (eV) wet (cm-1) V+2T/E at Re all R Re R all R all R all R Re R all R R 3.397 3.400 3.522 3.263 3.08 1 3.127 3.128 3.258 3.127 3.217 3.015 -790- -7- 92 - 4 -7.98 - 7.95072 - 7.9507 1 - 7.94892 - 7.96632 - 7.97883 - 7.98437 - 7.98437 - 7.98062 - 7.98451 - 7.98 181 - 8.0705 I 0.8924 0.8923 0-8436 1.3170 1 -6574 1 -8082 1 *8080 1.7061 1.81 18 1.7384 2.516 994 1062 972 8 . 6 ~ 1137 1302 1273 1373 1202 1.ox 1272 1406 1210 7 . o ~ 10-3 I V 0 1 2 3 4 5 6 7 8 R (a.u.) FIG.7.-Potential energy curves for calculations (a) and ( d ) showing the effect of using best atom (-.-. -) or best Re (- - -) exponents. The full line is the calculation with exponents opti- mized at all R. J . N. MURRELL AND C. L. SILK 93 the 1s electrons were promoted obtained a total energy of -7.9820 a.u. at R = 3.01 a.u. Our best energy is better than this. The changes in exponent examined by these workers were c(h) = 1.2 which is much too large a contraction at Re and 5(2s) = 5(2p) = 0-55 which is an expansion instead of a contraction. These all give poorer energies than the best atom exponents. Karo and Olsen using valence bond functions based on atomic SCF orbitals for Li and a set of six functions analogous to our set (e) obtained a lower energy than we have.However their calculated dissociation energy defined as the difference between the calculated atomic and molecular energies was poorer than ours and so was their equilibrium bond length and it is these quantities which show the importance of using best-molecule exponents rather than best-atom exponents. Amongst the many LCAO MO SCF calculations that have been made on LiH the one by Ransil is most relevant for comparison with ours. The best molecule exponents he found at R = 3.015 a.u. were These are very different for our best values. In particular he has a contraction in the 1s orbital and an expansion of the hydrogen orbital ; the opposite trend to ours and a much larger contraction of the 2s orbital. The resulting energy is poorer than any of our calculations except (a) and (b).The calculation is much worse than ( c ) which uses the two Heitler-London functions and which can be expressed as a single Heitler- London function between a 2s-2p hybrid of the lithium and the hydrogen orbital. When configuration interaction is added to this SCF calculation Fraga and Ransil '' obtained with 13 configurations (at 3.01 5 a.u.) an energy - 7-9836 a.u. which is about as good as our calculation (d) which only uses three V.B. configurations. ((1s) = 2.6909 c(2~) = 0.7075 5(2p) = 0.8449 [(h) = 0.9766. CONCLUSIONS The programme we have available will carry out calculations of the type described here on the ground states of the molecules BeH BH and CH and it is being extended to a general valence bond diatomic programme with optimized exponents for first row atoms.The calculations made so far are not sufficiently extensive to allow any firm predictions about exponent optimization to be made except for the expansion of the orbitals at large R. It would tentatively appear however that the use of best atom orbitals will over-estimate bond lengths but not as badly as for H2. Using best Re exponents the vibration frequencies will be appreciably larger than those obtained using exponents optimized for all R. It does not appear to be a general rule that the energy gained by orbital optimization is less the greater the number of wave functions in the basis. For these small molecules the V.B. approach with few basis functions gives much better energies than SCF MO calculations and no greater computational effort is required.APPENDIX DETAILS OF THE CALCULATION The matrix elements of one and two-electron operators between determinant wave func- tions formed from non-orthogonal atomic orbitals were evaluated by a corresponding orbital transformation of the orbitals in each pair of determinants to an orthogonal set. The method of King et al. has been used for this.12 The two-centre two-electron integrals were calculated numerically by the method proposed by Roothan et u Z . ~ ~ and checked using integrals from the programme " Midiat " written by Switendick and C0rba1to.l~ The ground state energy was calculated as the minimum root of the equation (H-ES)C=O. This energy was minimized with respect to the orbital exponents using a 94 OPTIMUM ATOMIC ORBITAL EXPONENTS routine written by J. Powell of the Atlas Computing Laboratory Harwell which iterates until the exponents have not altered by more than a specified amount on two succes- sive estimates.The molecular energy was minimized at 10 internuclear distances using a Fortran pro- gramme which requires the following data at each distance. The bond length initial estimates of orbital exponents and convergence criteria and a set of integers specifying the wave function. The complete 3 configuration potential curve (calculation (d)) required about 200 min computing time on an ICT 1905. see e.g. references in R. F. W. Bader W. H. Henneker and P. E. Cade J . Chem. Physics 1967 46 3341. R. C. Sahni and B. C. Sawhney Trans. Faraday Sac. 1967 63 1 . N. Rosen Physic. Rev. 1931 38 2105. ‘S. C. Wang Physic. Rev. 1928 31 579. C. A. Coulson Trans. Furaday SOC. 1937 33 1479. P. E. Cade and W. M. Huo J. Chem. Physics 1967 47,614. A. C. Hurley J. Chem. Physics 1958 28 532. J. Miller R. H. Friedman R. P. Hurst and F. A. Matsen J. Chem. Physics 1957 27 1385. A. M. Karo and A. R. Olsen J. Chem. Physics 1959 30 1232. S. Fraga and B. J. Ransil J. Chem. Physics 1962,36 1127. 1936. lo B. J. Ransil Rev. Mod. Physics 1960 32 235. l2 H. F. King R. E. Stanton H. Kim R. E. Wyatt and R. G. Parr J . Chem. Physics 1967 47 l3 A. C . Wahl P. E. Cade and C. C . J. Roothaan J. Chem. Physics 1964,41 2159. l4 A. C . Switendick and F. J. Corbato Methods in Comp. Physics 1963 vol. 2. M. J. D. Powell Compufer J. 1964 155.

ISSN:0430-0696    

DOI:10.1039/SF9680200084

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Independent Assessments of the Accuracy of Correlated Wave Functions for Many-Electron Systems BY S. F. BOYS Theoretical Chemistry Department University Chemical Laboratory Cambridge Received 1 1 th September 1968 The assessment of the accuracy of fully correlated wavefunctions by means of variance methods requires computation which only varies as N 3 a lower power of N the number of electrons than had been expected. This seems to be dependent on an indirect approach first constructed for a trans- correlated method. This means that various different variance tests could be used for the assessment of the accuracy of wavefunctions calculated by the transcorrelated method developed by Handy and Boys. These would require much less equipment in programmes and computer facilities than the original calculations of such wavefunctions.Supplementary investigations on correlated wave- functions at this level might make possible a whole variety of informative experiments on very exact wavefunctions and energies. 1. INTRODUCTION This investigation was made to try to find the simplest method of testing the accuracy of a correlated wavefunction of the form C@ = nf(ri,rj)@ = cdC#Il&. . . . . . C#IN (1) i > j where CJr is detor of space spins orbitals 4. This would be valuable because Handy and Boys have developed a method applicable to all molecules which has already given an energy value for LiH of -8.063 a.u. compared with the experimental -8.070 a.u. and the SCF value -7.987. It is likely that many such wavefunctions will be obtained but there are some special uncertainties because the convergence is not monotonic.The result of examining the briefest test which can be applied to these wavefunctions by other workers with much less facilities than necessary for the original calculation has turned out to be the variance type of criterion. However in the examination of this from a different approach in which this result was not foreseen a method of evaluation has been developed which appears to increase only as N 3 whereas the shortest direct method which had been found for the variance increased as N4. N here denotes the number of electrons. Hence this method of calculation will be reported although it can be regarded as a special approach to the methods which have already been used by others (Conroy,2 Frost 3 for systems of few electrons. It is hoped that by these techniques it may be used for much larger systems.The guiding principle was to obtain the variance for the transcorrelation wave equation solved by Handy and Boys instead of that for the simple Schrodinger equation (C-lHC-W)@ = 0 (2) ( H - W)$ = 0. 95 13) 96 ACCURACY OF CORRELATED WAVE FUNCTIONS Actually at the final stage of the calculation these can all be changed into each other and into other types of variance. For an equation (R-W)$ = 0 (4) the variance of an approximate solution +’ is d 2 obtained by If d = 0 the $’ must be a true solution of (4) and $’ with small dcan be regarded as a high accuracy approximation to the true solution. we can write either the Schrodinger H or the transcorrelated operator C-lHC and the convergence conditions are the same.It was the use of the latter which was originally found to give a shorter computation and then it was found that this could be changed into the ordinary variance. If the integration is now per- formed by taking the values of the functions at some many-electron space-spin points RI and by the multiplication of these values by suitable weights h, then approxi- mate values of all the above integrals can be obtained. This will be referred to as the method of numerical integration. If the values at these points are represented by EI FI with then the variance may be written as For EI = i w f ( R I h FI = +‘(RI) (6) In the next section the values of EI and FI will be obtained for the transcorrelated operator by a method which gives the N 3 dependence. After that stage it is not necessary to restrict the analysis to the transcorrelated case because the value of E corresponding to the ordinary variance is obtained by E f = C(RI)E:; F f = C(RI)FT (8) where E denotes the Er for the Schrodinger equation and EF the corresponding transcorrelated quantity.These inserted in eqn. (7) give the corresponding variance If any weighting function p 2 is used to form another variance integral such as d2 = (P@- W+’ I P(R- mtww 1 $7 (9) this can be evaluated by using in eqn. (5) The essential property of all these variances is that they can only be zero if the wave equation is satisfied at all (RI). In practice the valuable use for these is to assess two approximate wavefunctions such as CA@A and CB(DB. These could be results obtained either by different workers or in one calculation at different stages of the iteration.The comparison of two such results by an independent method would be very informative. If then it would generally be reasonable to assume C&? and W would be the more accurate solution. E f = PIE,; F,P = P I F I d i > 2d& S . F. BOYS 97 2. THE DETERMINATION OF THE TRANSCORRELATED AND DIRECT VARIANCES FOR MANY ELECTRONS There are three possible definitions of the quantity which it is convenient to call the transcorrelated variance. The first given in definition (A) defines it in effect as the variance of the operator C-lHC in what would be considered the orthodox way for this operator. This is the definition which is simplest in concept and corresponds to the direct approach of sampling at many points. The second definition is formu- lated to give exactly the same numerical value but in a form which is the result of the calculation of the energy by a particular numerical integration grid.The third definition is a transform of the latter which has the valuable property that it can be reduced to three electron sums by the well-known detor integral theorem. It is considered that this latter is important because only a multiple of N 3 basic operations are required in the computation whereas the direct approach appears to lead to N4. It is necessary to define the integration points in electron space-spin variables. If the spin variable is not included at this stage a useful aspect is omitted. These four definitions will be given as compactly as possible and after each the relevant identity will be established.DEFINITION A WEIGHTED POINT SYSTEMS FOR N ELECTRONS. A set of values of (h, XI1 Yzr Zzi V,,) for i = 1,2 . . . . N and I = 1 to M where M can be finite or infinite and where the spin variables V, can only take the values + 3 or - 3 will be called an N-electron point system. Such a set may frequently be obtained by selecting for each value of I a sub-set of N points out of a set (xi yi zi u i ) for a single electron but this does not affect the principle of the way in which the N-space points enter the theory. An example is given in eqn. (27) and (28). It will be assumed that there are no values of i $. j for any value of I where Xri = Xrj Yri = Yrj ZIi = Zrj (10) When V, = Vzj this is not a deliberate exclusion but merely a matter that such a point would give a zero value in any antisymmetric functions and hence zero in all the integrals which occur here.For a case V, + Vzj this is a deliberate exclusion because if this is not made the points with r I 2 = 0 can easily occur with dispro- portionate probability. Since these are the points at which cusps occur this increases the errors due to the use of finite sets of points. For the set of points associated with a given value of I we shall define the repeated delta operation which selects the value of a function at these points. Thus if Qr = ~ ( ~ i X r i > ~ ( ~ i ~ i ) ~ ( ~ i ~ ~ ) s ( ~ v > S ( x ~ ~ r ~ ~ - - Hence for any function F and from this it follows for any G that (Qr I FG) = <Qr I F)<Qz I G). (13) Also if the whole I set is sufficient and has correct weights to perform a numerical integration with an error of order O(p,) C M Q I I F I G) = <F I G > + O ( P ~ > .(14) 98 ACCURACY OF CORRELATED WAVE FUNCTIONS DEFINITION ( A ) Let EI and Fz be defined to be E = ( Q I I C-lHC I a> FI = <QI I @)>- By relations (7) Hence when a method of calculating EI and F are given the integral transcorrelated variance can be directly estimated to within O(pQ). As explained before the ordinary variance can be obtained by using CzEZ and C,F in place of EI and Fz where C = <QI I c> DEFINITION ( B ) If 4; for which &(RIJ) = dij let F ; = <QIJ94;(b; - * * 4; I a> E; = ( Q I d + ; & . . . I C-IHC I a)) where i j Here the (b' would generally be dependent on I but they are only a device to multiply some terms in an expansion by 1 and others by zero and it is simpler to write as 4;.The Q I is the operation which would be obtained if it were decided to do an extremely crude numerical integration over only N points in all dimensions just those of X I i ... f o r i = 1 . . . N . THEOREM 1. If qt is written for the same constant quantity then EI = qtE;; F = 4tFf. (19) Hence the variance formula with E, FI gives the same value as with E; F;. PROOF. Ef can be written as where A denotes the relevant antisymmetric quantity and hence auPQuzA = Q,A where auPu = kPu is any permutation of the electronic variables with the sign according to its parity. It then follows E; = (N!)-3C(auPu4;(b;. . . I QZA) U = (N!)-*E(PU-'P,,+i4;. . . I o,,P,'Q,A) = ( N ! ) - + I ( 4 ; + ; . . . QI [ A ) = ( N ! S ( Q I A ) - U U S . F .BOYS 99 In the second equality Pv permutes all variables in the complete integral and hence does not alter its value. For the last equality 4 cancels all but the term 6(x1,xz1) . . . out of the first factor of Qz and so with the other 4; obtains the result which from the definition (A) is the first statement of the theorem. DEFINITION (C) Let some intermediary orbitals $7 = ZrU:#+ be defined where the coefficients are the solutions of ZrU;4r(RIj) = 6, so that <Qd I 4;) = d i j (21) Let AI = I 4r(Rz,) I = I U; I -l where this determinant and the coefficients can all be evaluated from the original i j coefficients 4r(RIJ with a multiple of N 3 arithmetic operations. Let the final definitions be E;’ = <Q,@’ I C-’HC I @“)A1 F; = (Q@’ I @”}A1 where THEOREM 2 E; = E; F;’ = F;. (25) PROOF.E;’ becomes E‘ by the insertion of = (NU-* I Cuf4r(R,j) I r = I Uf I = A;’@ COMMENT. The value of the definition (C) is that the matrix elements occuring as ex- pressions for the N-space integrals can be immediately reduced by the detor integral theorem. In fact because the @ orbitals only choose the values each at a single point another level of simplicity is introduced. In the most laborious integral for a three- electron interaction which occurs in C-’HC there are only a multiple of N3 terms This simplification is dependent on the fact that the detor integral theorem holds exactly for approximate numerical integrations which are obtained by using the same points of integration for every electron. The proof follows just the same steps as in the theorem for exact integration and the theorem will be stated here without proof for the case when T(rl r2 r3) is an operator dependent on three electrons.where P are the permutations of the three sets of electron variables which follow this operat or. COMMENT. The above analysis has shown that the transcorrelated variance can be evaluated by use of the E;’ by computation which only increases as N 3 for N electron problems. If the direct variance is obtained by using C,E; in place of Ej’ since C only involves computation varying as N 2 the computation again increases as N 3 . Without this analysis the lowest rate of increase of the computation for the direct variance which the author had been able to obtain had been N4. This is due to the 100 ACCURACY OF CORRELATED WAVE FUNCTIONS evaluation of N determinants in each of which one column had been altered from the original wavefunction by a v operator.So although this analysis had been com- menced with a mistaken intention of demonstrating a shorter evaluation for the trans- correlated wave equation it appears to have provided the shortest method for either of these two variances. Any variance with another weighting sayp2 in the form (P2 I ((H- w>w> could also be obtained by this analysis since it is merely necessary to evaluate p(Rzi) and use p(RI,)Ez = E; in the original variance formula. It appears that there is still interesting exploration to be made on the possibility of some p providing a better practical method. The author thinks that the use of p = CD has some promising features. There are many sets of the N-space-spin points previously denoted by R,, and explained in detail in definition (1).But to illustrate the essential properties of these one example will be given. It may have practical value but it appears to be the simplest for a schematic demonstration. Let T for m = 1 2 . . be the set of single electron points whose components are where qnZ3 = (m J3) mod(1) which is the fractional part of (m J3) ; and qm5 similarly for (mJ5) etc. Let Rzk be all selections of N of these points arranged in the order that if the Ith set consists of with m > m12 > m13 and so that if is positive then RIk occurs after RJk. This is just a formal way of defining an order of increasing " seniority ". The weights of these are complicated to write explicitly in algebraic form but are easily obtained from the Jacobian of the transformations in (27) when the initial weights in the q space are all taken to be equal.The dependence on the surds ensures that these tend to a uniform distribution over all q space. DISCUSSION The number of point sets denoted by M for I = 1,2 . . . M has not been discussed because this will generally be completely a matter of expediency. If by some M value the variance of CAQA is persistently greater than for C,@,, for the same progression of points then it will be sufficient to regard the test as complete and W as the better prediction of the energy etc. There are several circumstances in which these assessments will be interesting and there is one which is forward-looking but which is particularly inspiring. It appears that it will be possible to transfer localized orbitals from one molecule to another and also transfer the correlation functions. To a first approximation these latter are n exp (4rij). In this case many estimated wavefunctions may become available. i > j S . F . BOYS 101 If these can be tested directly even inspired empirical estimates of correlated wave- functions may become a valuable subject. At a more realistic level however the resolution of instabilities in predictions of excited states ; the assessment of the claims of different systematic calculations ; and the assessment of rates of convergence with different systems of expansion functions will probably form the major use of the direct assessment of the accuracy of correlated wavefunctions. S. F. Boys and N. C . Handy in course of publication. A. A. Frost R. E. Kellogg and E. C . Curtis Rev. Mod. Physics 1960 32 313. ’ H. Conroy J. Chem. Physics 1967 47 5307.

ISSN:0430-0696    

DOI:10.1039/SF9680200095

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

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.

ISSN:0430-0696    

DOI:10.1039/SF9680200102

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

AUTHOR INDEX * Arrighini G. P. 48. Boys S. F. 95. Briggs M. P. 56. Buckingham A. D. 41. Cook D. B. 55,105. Doggett G. 32 54. Grimaldi F. 59. Hall G. G. 56 58 69 103 104 106. Hehre W. J. 15. Hibbard P. G. 41. Klessinger M. 73. Lecourt A, 59. Maestro M. 48. McKendrick A. 32. McWeeny R. 7 55 104 105. Moccia R. 48. Moser C. 59. Murrell J. N. 84. Nieuwpoort W. C. 54 106. Pauncz R. 23. Pickup B. T. 102. Pople J. A. 15 58 105. Randic M. 54 106. Richards W. G. 64. Sack R. A. 57. Silk C. L. 84 Stewart R. F. 15. Walker T. E. H. 64. Weinstein H. 23. * The references in heavy type indicate papers submitted for discussion.

ISSN:0430-0696    

DOI:10.1039/SF9680200107

出版商:RSC    

年代:1968

数据来源: RSC