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. 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/SF96802FX001

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Multi-Configuration SCF Calculations BY R. MCWEENY Dept. of Chemistry The University Sheffield S3 7HF Received 13th September 1968 In multi-configuration self-consistent field (MC SCF) calculations a many-configuration approxi- mation to the wave function is optimized by variation of the orbitals. The equations satisfied by the optimum orbitals are formulated along with a practicable method of numerical solution. Incidental problems arising in the computer implementation of MC SCF calculations are also discussed. 1. INTRODUCTION In the configuration interaction (CI) approach to the construction of electronic wave functions the wave function is expanded in the form Y = CKCKQK (1-1) where the QK are antisymmetric many-electron functions each referring to a con- figuration of occupied orbitals (i.e to an orbital configuration).The orbitals used are drawn from say the first M members of any convenient complete set and the expansion (1. l) in principle infinite is therefore truncated accordingly ; nevertheless even with only a small number of electrons the expansion may be too long to be manageable. Thus with five electrons and ten orbitals we have twenty spin-orbitals and can set up 15,504 spin-orbital configurations if the OK were taken to be single determinants this number would be the dimension of the secular problem to determine the coefficients. It is therefore preferable to take as the mK suitable linear combinations of determinants chosen to conform to symmetry requirements in order to reduce the dimensions of the secular problem. The only type of symmetry that is always available is the spin symmetry if the QK are chosen to be spin eigenfunctions (quantum numbers S M ) we need consider only different orbital configurations (a relatively small number) the different deter- minants within each orbital conguration being combined to yield a small number of spin eigenfunctions of each (S M).When the Hamiltonian is spin free (the usual first approximation) only QK of the same (S M ) occur in the expansion which is greatly reduced in length. For five electrons and ten orbitals we can find 3,300 independent spin eigenfunctions with S = A4 = 3. In a complete CI calculation with the given one-electron basis all these functions would be admitted. No further reduction is possible unless spatial symmetry is present and although the construction of spin eigenfunctions is a necessary step in the practical application of CI methods it is not sufficient to reduce the expansion to a manageable length unless the orbital basis is quite small.In order to make progress possible it is usually necessary to truncate further the expansion (l.l) e.g. by admitting only certain types of configuration ; one might for instance take a one-determinant SCF function as the leading term and then admit all configurations formed by promoting one or two electrons into the higher-energy " virtual " orbitals obtained from the SCF calculation. Unfortunately however 7 8 MULTI-CONFIGURATION SFC CALCULATIONS the orbitals which optimise a one-determinant wave function may not be well-suited to its further improvement by CI ; they do not give the best function of given form (unless complete CI is used) and a truncated expansion can be improved greatly in the usual variational sense by variation of the orbitals.In using a basis of m orbitals (with m large) it is therefore natural to turn attention from the extremely large but linear equation system for determining the CI expansion coefficients to the m-dimen- sional but non-linear problem of determining optimum orbitals in a given truncated expansion. This latter problem is in some ways analogous to that encountered in SCF theory and several types of multi-configuration (MC) SCF theory have now been successfully used. These go back originally to Hartree et al. and later to Yutsis et aZ.,2 for atoms but calculations on molecules are relatively r e ~ e n t . ~ - ~ Usually restrictions are imposed on the types of configuration admitted but general equations for the optimum orbitals can be given and the purpose of this note is to outline a practicable method of solution and to consider some of the incidental difficulties common to all MC calculations.2.-CHOI CE OF LINEARLY INDEPENDENT S P I N EI GENFUN CTIONS For a given orbital configuration there are many methods of setting up a complete linearly independent set of spin eigengunctions of given (S M ) . The functions we shall use are of the form a = M,4Q,@,l (2.1) where R is an orbital product 0 a spin function A the antisymmetrizer and M a normalizing factor. We choose 0 to be a spin eigenfunction with M = S since this is the simplest case to consider and matrix elements between functions with M # S can always be derived easily from this case (e.g.by the Wigner-Eckart theorem) and take @,(S1 s2 . . . s,) = q S i sj)e(sk s,) . . . a(s,)a(s,) . . . (2.2) where there are g paired spins each with a factor of the form and N-2g a factors. The resultant spin function has S = M = 3(N-2g) and the prescription used is essentially the one employed in valence bond (VB) theory. FIG. 1 . Branching diagram showing pos- sible resultant spin (S) for N electrons. 0 1 2 3 4 5 6 N If i2 contains a factor u(m)u(n) (doubly occupied orbital) then 0 must contain the factor Q(s s,J and spin factors associated with doubly-occupied orbitals need not be considered further. Different spin functions correspond to different pairing schemes but the number of possibilities is much larger than the number of independent functions.One method of eliminating linear dependencies is to set down the numbers 1,2 . . . N in a ring representing each 8(si s j ) factor by an arrow i-+j and each a(sJ R . MCWEENY 9 factor by a point p . Any functions with crossed arrows are then expressible in terms of those with no crossings and may therefore be discarded. When S # 0 however more limited dependencies still remain and it seems preferable to use a set which is in one-to-one correspondence with the standard set used in the branching-diagram and group-theoretical constr~ctions.~ In the branching diagram approach the spins are coupled one at a time,? the rrth electron spin being coupled to the spin eigenfunctions for 12- 1 electrons in order to obtain those for n electrons. The resultant functions are conveniently classified using the branching diagram (fig.l) in which the routes leading from the origin to any intersection represent the coupling schemes leading to linearly independent eigenfunctions of the corresponding S value. Thus for five electrons the coupling schemes leading to S = 3 are The functions so constructed are identical with those that carry the standard irreducible representations of the spin permutation group,8 associated with standard tableaux (respectively) (the numbers in the top row correspond to up-spin couplings (/) in the branching diagram thosein the bottom row to down-spin (\)) and the five functions are in one-to-one correspondence with (though not identical with) the VB-type functions indicated by In other words the VB function associated with any given branching-diagram function is obtained by coupling the first up-spin (/) with the first down-spin (\) the second up-spin with the second down-spin and so on any remaining spins being given a fix t or s .The final result is that the branching diagram functions (which are orthogonal not only independent) can be obtained from the VB functions by Schmidt ortho- gonalization starting at the left when the functions are listed (as above) in “last letter sequence ’,.lo The whole construction is now so straightforward and easily adapted for fully automatic computation that attention may be confined to functions of VB type which turn out to be extremely easy to handle. t Spins associated with doubly occupied orbitals need not be considered. 10 MULTI-CONFIGURATION SCF CALCULATIONS The evaluation of' matrix elements betwecn spin-paired functions for one- and two-electron operators of all kinds has been discussed by Cooper and McWeeny who give general rules based upon superposition patterns.Here we summarize the rules in their simplest form for spinless operators. Let us take two spin eigenfunctions 0 and OK. draw the associated diagrams showing the paired spins and superimpose these diagrams to form a superposition pattern. For example two 6-electron triplet functions ( S = 1) and their super- position pattern would be * 6 1 6 Such patterns contain islands each formed by a closed sequence of arrows andchains each formed by an open sequence the latter are of two types 0-chains containing an odd number of centres E-chains containing an even number. The pattern above contains one island and two 0-chains.The spin-factor dependence of any matrix element is characterized by such a pattern and in particular by the quantities. g = number of paired spins in each function n K K ' = number of islands in the superposition pattern v,,) = number of arrow reversals (if any) needed to achieve head-to-head tail-to-tail matching dE = 1 no E-chains; = 0 otherwise. The dependence on orbital factors which we denote by (where 4, is simply the orbital in the nth position whatever it might be) is characterized by the following quantities CT,,. = parity of permutation needed to achieve maximum matching of the orbital products inK,mKt = numbers of doubly occupied orbitals in R, QKt mKKt = number of doubly occupied orbitals which after matching S,,. = overlap of orbital products llK,QK) (= 1 products identical; coincide,? = 0 otherwise) SKK#(i)= overlap with orbitals in the ith position removed SKK#(i,j) = overlap with orbitals in the ith andjth positions removed qij(qij) = 1 if 4i t$j (& f#$) are different orbitals = 0 if 4i + j (& 4;) are identical.t For example AABBCDEEF and AABBCDCEF each contain three doubly occupied orbitals but can give only two coincidences ( I T z K K ~ = 2) ; but AABBCCDEF and AABBKKDEF with the same numbers of doubly occupied orbitals give three. R . MCWEENY 11 In t e r m of these quantities the non-orthogonality integral and the only non-zero niatrix elements of the usual spinless Hamiltonian take the forms (3.2) and (<Dtir I <Dk) = hE~,,tA,,tS,,~ <@,PI H I @ti) = &utiK*AKKr C sKtif(N4i I 11 I 4i) + 3SC'SKKf(i,j)(+:4i I I 4i4j) N 1 i = 1 i j = 1 The matrix element vanishes identically if there are more than two E-chains.The factors CO,,. and AKKI are defined by (3.4) COKK = (- lyKK'2(mK +inK'- 2 m K K ' ) / 2 The coefficients x i j depend only on the positions of i a n d j in the superposition pattern and are listed in table 1. TABLE 1.-COEFFICIENT Xc FOR POSITIONS i j I i xij(= X j i ) no E-chains island I island I %3Plj+ 1) 0-chain J 0-chain J W P i l + 1) island I island I' (#I) - i 1 0-chain J 0-chain J' (# J) + ( p ~ + 1) or 0-chain J two E-chains E-chain K E-chain K' (#K) p~ pq = pipj where the " parity factor " pi = -+1 is assigned by giving + 1 to an arbitrary position in an island or to an end point in a chain and then proceeding along the sequence giving rtl to alternate positions.For E-chains the end point chosen is the one where the arrow points into the chain. Normally the expression (3.3) contains only a few terms vanishing altogether when QK and R,. differ in more than two orbitals; thus with one orbital difference in the pth position SKKI = 0 SKKp (i) = 0 except for i = p and SKK#(i,j) = 0 unless i = p o r i = p. Examples of the application of these results are available.12 They are readily programmed for fully automatic computation the only input data required being a specification of orbitals and coupling scheme for each <DK. 4.-THE VARIATION PROBLEM When the orbitals are selected from a complete set A 23 C . . . each matrix element takes the form Optimum values of the expansion coefficients cK for a given choice of orbitals are determined by solving the secular equation HC = EMc (4.2) 12 MULTI-CONFIGURATION S C F CALCULATIONS with matrix elements The energy may then be written The coefficients of the integrals are elements of the one- and two-electron density matrices and depend on the expansion coefficients cK thus and these coefficients determine the orbital forms of the electron density and the pair function.The basic problem of MC SCF theory is to optimize the forms of the orbitals A B C . . . which appear in (4.4) so as to obtain an improved energy. For this purpose the coefficients of the one- and two-electron integrals are regarded as numerical constants and we consider an independent variation A+A+6A B+B+6B . . . of the orbitals. A formal solution of this problem in the form of equations defining the optimum orbitals was given many years ago but no practical application of these equations appears to have been made.To obtain a practicable method we introduce an LCAO-type approximation assuming that the n orbitals referred to in the CI expansion are written as linear combinations of nz basis orbitals $1 $ 2 . . . & Thus writing A = c $ r K A (4.7) r where the coefficients TrA ( A denoting a typical orbital) may be collected into an nz x JZ matrix T the energy expression (4.4) takes the form in which the one- and two-electron integrals are now defined over basis functions and the variational problem now involves the elements of the unknown matrix T. This matrix must be chosen so as to minimize E subject to an orthonormality con- straint which (assuming an orthonormal basis) may be written in the form TfT = 1 (4.9) (the n x y1 unit matrix).The equations that determine the optimum orbitals follow on making a first-order variation T+T+GT and requiring that E be stationary the constraints being in- corporated by the method of Lagrangian multipliers. The result is hTPl +Z = TE (4.10) where E is a Hermitian matrix of mutlipliers which may be determined from the condition (4.9) on multiplying (4.10) from the left by Tt. Here h is the usual matrix of one-electron integrals with elements hrs = (r I h I s) and Z is anelectron interaction matrix whose elements are zrA = C TJs<rs I 9 I tLL)TCT~DP2CD,AB* (4.11) B,C,D s.t,u R . MCWEENY 13 Equation (4.10) reduces to the Roothaan equation l4 hFT = (h+ G)T = TL (4.12) in the one-determinant approximation in which P1 is twice the unit matrix and Z may be factorized in the form Z = 2GT; but in general Z does not factorize and the equation does not reduce to eigenvalue form.5.-METHOD OF SOLU'I'ION Instead of trying to solve (4.10) directly it appears to be simpler to start froni the energy formula (4.8) and to minimize E by a descent procedure analogous to that used in some forniulations 1 3 9 of the one-determinant SCF method. To incorporate the orthonormality constraint (4.9) we suppose T-+T+6T = UT = (l+V)T (5- 1) where U is an HZ x IIZ unitary matrix and V must therefore be to first order anti- Hermitian Vt = -v (5.2) The corresponding first-order change in energy is where 6E(') = trVt(X-Xt) X = (hTP1 + Z)Tt (5.3) (5.4) The reqiiircinent 6E(' = 0 (all V) yields incidentally an alternative stationary condition namely that the matrix X be Hermitian.Away from the stationary point however X is non-Hermitian and we may regard rs as the Hermitian scalar product of two vectors in a space of dimension 172. For a V of given magnitude this quantity takes its greatest negative value corresponding to steepest descent of the energy surface when the two vectors are anti-parallel v = -A(X-Xf) (5.5) where A is a positive number measuring the magnitude of the change. The method of reaching an energy minimum is now clear. We start from a T matrix corresponding to the orbitals of a first approximation calculate the correction matrix V with some suitable estimate * of ;1 and revise T accordingly. The process may be repeated recalculation of Z X and V leading to a new descent direction and iteration may be continued until (X-Xt) vanishes to any desired accuracy.In order to preserve orthonormality accurately (not just to first order) T may be corrected between iterations by the process T,+T,+ = +T,(31 -T,?T,) which is rapidly convergent when T (used as first approximation) has columns which are almost orthonormal. * e.g. A may be chosen so that aE/aA = 0 corresponding to the achievement of the lowest point accessible in the given descent direction.l39 l 5 14 MULTI-CONFIGURAT ION SCF CALCULATIONS 6.-CONCLU S I ON A programme has been outlined for the fully automatic calculation of mutli- configuration molecular wave functions with optimization of the orbitals and with no restrictions on the types or configuraton admitted.The main steps are (i) construc- tion of a linearly independent set of spin eigenfunctions (ii) evaluation of matrix elements and solution of the secular equations and (iii) iterative adjustment of orbitals to obtain the best CI expansion of given form. Algorithms have been devised for the computer implementation of each step and pilot calculations on small molecules are now in progress. Convergence has been found satisfactory and detailed results will be published elsewhere. D. R. Hartree W. Hartree and B. Swirles Phil. Trans. A 1939 238 229. Proc. Roy. SOC. A 1933 139 311. A. P. Yutsis Zhur. Eksp. Teor. Fiz. (Soviet Phys. JETP) 1952 23 129. G. Das and A. C. Wahl J. Chem. Physics 1966,44,87. E. Clementi and A. Veillard J. Chern. Physics 1966 44 3050. J. Hinze and C. C. J. Roothaan Suppl. Prog. Theor. Physics 1967 40 37. ti R. McWeeny Proc. Roy. SOC. A 1955,232 114. ’ M. Kotani A. Amemiya E. Ishiguro and T. Kimura Tables of Molecular Zntegrals (Maruzen. Tokyo 1966). T. Yamanouchi Proc. Phys. Math. SOC. Japan 1937 19,436. R. McWeeny 1968 unpublished. D. E. Rutherford Substitutional Analysis (Edinburgh University Press 1948). I1 I. L. Cooper and R. McWeeny J. Chem. Physics 1966,45,226. I. L. Cooper and R. McWeeny J. Chem. Physics 1968,49,3223. I3 R. McWeeny Rev. Mod. Physics 1960 32 335. I4 C. C . J. Roothaan Rev. Mod. Physics 1951 23 69. R. McWeeny Proc. Roy. Soc. A 1966 235 496.

ISSN:0430-0696    

DOI:10.1039/SF9680200007

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Atomic Electron Populations by Molecular Orbital Theory W. J. HEHRE R. F. STEWART AND J. A. POPLE Dept. of Chemistry Carnegie-Mellon University Pittsburgh Pennsylvania 15213 U.S.A. Received 20th September 1968 The detailed distribution of electrons in molecules can bz broken down into atoiiiic orbital populations if an atomic orbital basis is used for a molecular orbital wave function (LCAO). A total assignment of all electrons in a molecule to atomic orbitals (and hence to particular atoms) can be made by finding gross populations as suggested by Mulliken. These are of some interest in discussing various molecular properties associated with charge density particularly if a minimal basis is used consisting of only atomic orbitals in the valence and inner shells. It has now become possible to evaluate atomic orbital populations for a range of molecules both by semi-empirical and ab initio methods.The semi-empirical methods (based on zero differential overlap) utilize some experimental atomic data but can be applied to large systems with little difficulty. Ab initio methods generally require the evaluation of a large number of difficult integrals but they are becoming available for many small systems. The main aim of this paper is to compare atomic electron populations obtained by the semi-empirical INDO molecular orbital theory with some values from full SCF calculations using a minimal basis set of exponential-type functions. Agreement is moderately good but only if the exponents in the full calculations are chosen carefully to minimize the total energy. INTRODUCTION Molecular orbital wave functions and associated electronic charge distributions in linear combination of atomic orbital (LCAO) theory may be obtained either by approximate semi-empirical methods or by full ab initio computations.Semi- empirical methods frequently make use of experimental atomic data in determining parameters and attempt to determine properties of molecules in terms of those of the constituent atoms. Such theories by virtue of their approximate nature have the advantage of being relatively easy to apply to quite large molecules of chemical interest. The full quantum-mechanical calculations however involve much more computation and are only readily available for small molecules but they do lead to well-defined upper bounds for the total energy. Some of the approximate methods make use of the ab initio methods for calibration purposes.Clearly a comparative study of the prediction of these various kinds of wave functions for some simple molecules will be of value in illuminating their deficiencies and pointing the way to more satisfactory theories. This paper is primarily concerned with a comparison of the electron distributions given by the semi-empirical INDO scheme and by full LCAOSCF calculations using a minimal basis set of Slater-type atomic orbitals. 2.-C 0 M P UT A T I 0 NA L D ETA1 L S In all the wave functions discussed here molecular orbitals $i are written as linear combinations of atomic orbitals @/, 15 16 ATOMIC ELECTRON POPULATIONS where 4!t are Slater-type exponential functions. If we deal with a minimal basis set and no atoms heavier than fluorine these have the form 41s(Cl,O = (c;/4* exp (-Cd 42S(C2,0 = (C337+ r exp (-C2r) 42pK2,r) = cts; In>* y exp ( - 123) cos 0 (2.2) and similar expressions for the other 2p functions.For closed-shell ground-states of diamagnetic molecules the electron density (corresponding to a single-determinant wave function using the molecular orbitals (2.1)) is where Ppv is the density matrix occ The overlap integral is the total magnitude of the overlap charge distribution $P+v and if this is " assigned " equally to the orbitals p and v the Mulliken gross atomic orbital population is then a measure of the total number of electrons associated with the orbital 4 in the molecule.2 A gross atomic population QA may be obtained by summing the q,L for all orbitals on a particular atom A.These populations then give a theoretical distribution of electron density throughout the molecule. and full details will not be given here. Only valence electrons are considered explicitly so that two electrons are assigned to inner shell 1s orbitals and these are included in an effective core. The scheme is formally based on a set of Slater-type atomic orbitals with values for the C-exponents chosen by Slater's rules for atoms (except hydrogen for which 5 = 1-2). However the atomic electro-negativities are obtained from experimental atomic energy levels rather than by computation so that some of the deficiencies of a calculation using a limited basis set are avoided. The ab initio calculations obtain the LCAO coefficients in (2.1) by minimizing the calculated total energy.This leads to the Roothaan self-consistent equations involving a set of one- and two-electron integrals. To simplify the evaluation of these integrals the exponential functions (2.2) are replaced by linear combinations of K gaussian functions a procedure first suggested by Foster and Boys.' Thus The semi-empirical INDO method has been specified elsewhere W. J. HEHRE R. F. STEWART AND J . A . POPLE 17 Here yls and g2p are the gaussian-type functions sls(a,r) = (2a/.Y exp (-ar2) gZp(a,r) = (128a5/n3)4r exp (- ar2) cos 0. (2.9) (The 2s exponential function is written as a linear combination of 1s gaussians). The constants d and a are chosen to minimize the integral (2.10) subject to it normalization constraint and are independent of the c-exponents. This procedure was suggested by 0-ohata Taketa and Huzinaga6 Calculations have been carried out for K = 3 4 5 and 6.Values of d and a for K = 3 4 and 5 were obtained by methods described elsewhere and are given in table 1. Values for K = 6 were taken from ref. (6). (Some preliminary calculations have already been published for K = 4 using the exponents and coefficients proposed by 0-ohata Taketa and Huzinaga. However the values given in table 1 lead to lower &-values. Also the E value for the 2s function with K = 5 is less than the value with K = 6 given in ref. (6)). Using these expansions the various integrals can be reduced to integrals involving gaussian functions and these can be evaluated by methods originally introduced by The self-consistent equations were solved in an iterative manner.Sub- routines from programmes QCPE 47 and QCPE 92 were used in the computation.1°9 '' Calculations have been carried out both for a standard set of [-exponents (using Slater's rules but with = 1.2 for hydrogen) and by optimizing the valence 5's to give a minimum calculated energy. For the latter inner shell cl were chosen equal to optimum atomic values l 2 (rounded to two decimals) and the valence i2 (c for hydrogen) were varied in steps of 0.01 until the lowest energy was found. 3.-RESULTS AND DISCUSSION The first point to study is the efficiency with which a gaussian expansion of the type used in this paper reproduces the results of a calculation based directly on Slater exponential orbitals. Table 2 gives results for H F and NH with K = 3 4 5 and 6 using the same nuclear geometry and orbital c-values as the original exponential-type calculations reported in the literature.' 3 9 l4 Convergence to the exponential result with increasing K is fairly rapid for energies dipole moments and populations.The differences between the gaussian and exponential results are much less for the binding energy (difference between molecular and atomic values using the same exponents) than for the total energy. This suggests that much of the difference arises from the atomic inner shell description which is largely unaltered in the mole- cule. Atomic populations with the 3-gaussian functions differ from the exponential values by only about 0.02. This suggests that this simplest level of calculation is sufficient to study the relation between the ab initio and semi-empirical populations.Results for a series of simple molecules and ions with K = 3 are listed in tables 3-9. For the neutral molecules the nuclear geometry is chosen according to the standard model A defined previo~s1y.l~ For the ions model B is used l6 except for CH; where the HCH angle is taken as tetrahedral. The following points may be noted about these results. I . The optimum values of the c-exponents vary considerably from one molecule to another. This is most marked for hydrogen where the range is 1.14-1.48. For carbon in all neutral molecules and positive ions the optimum exponent is substantially K als 3 0.109818 0.405771 2,227660 4 0.088019 0.265204 0.954620 5.21 6850 5 0,074453 0.197573 0.578651 2.07 1740 11.30570 TABLE 1 .-COEFFICIENTS AND EXPONENTS FOR GAUSSIAN FIT OF SLATER ORBITALS dlS &Is 012s d2s E2s U2P d2P 0.444635 0.0601 83 0,458 179 0*080098 0.422307 0.535328 3.31 x 0.156762 0.596039 6.87 x lo-’ 0.235919 0.566171 0.1 54329 2.581 580 - 0.059945 0.919238 0.1 62395 0.291 626 0.061257 0,477008 0.065439 0.26323 1 0.532846 4.38 x lo-’ 0.1 60728 0.580559 2 .7 0 ~ lo-’ 0.164372 0.551787 0.2601 41 2*000240 - 0.054721 0.466262 0,285746 0.056752 11.615300 -0.01 1984 1.798260 0.0571 32 0.193573 0.04249 1 0.1 47266 0.056063 0.1 65345 0.482571 0.088064 0.54805 1 0.127331 0.483493 0.331815 6.88 x 0-194473 0.369824 2.70 x 0.307982 0.366774 0.113540 1,673710 - 0.056859 0.864327 0.1 23547 0.0221 40 8,984940 - 0.01 5964 3.320390 0.020790 TABLE 4.-cOMPARISON OF ATOMIC POPULATIONS Q FOR NEUTRAL MOLECULES H,X,Y QH Qx QY standard optimized INDO standard optimized INDO standard optimized 5.909 5.903 5.971 8 *09 1 8.097 0.866 0.755 0.725 9.1 34 9.245 9.275 0.850 0.781 0.829 8.301 8.437 8.342 0.849 0.832 0.912 7.452 7-503 7.264 0-87 1 0.978 1 -009 6.51 8 6.088 5.963 0.877 0.979 1.018 6.370 6.063 5.946 5.997 0-866 0.917 1.001 6-267 0.816 0.813 0.947 6.184 6.1 87 6.053 0.886 1.001 1.031 6.299 5.838 5.676 9.042 0.890 0.973 1 a045 6-155 5.823 5-672 8.065 0.790 0.792 0.950 6.145 6.073 5.91 1 7.064 0.848 0.829 0.902 7.302 7.277 7.027 9.001 8.022 0.858 0.824 0.948 7.120 7-093 6-928 6.1 66 HOF 0.840 0.750 0.789 8.172 8.227 8.094 8.987 2P 2 .6 9 ~ 10-4 2.90 x 10-5 3 . 7 2 ~ INDO 8.029 9-1 59 9.23 1 8.230 8-238 7.1 34 7.1 39 9.065 9.169 8.082 8.124 9-023 9.1 17 W . J. HEHRE R. F . STEWART AND J. A . POPLE 19 greater than the standard Slater value for the free atom (1.625).The same applies to nitrogen except for the negative ion NH;. 2. The atomic populations after optimization of exponents differ considerably from those using standard exponents. For the neutral molecules the optimized calculations lead to a wider variation of hydrogen populations corresponding to a greater variation in polarity. For both neutral molecules and ions the values of Q H for the optimized set agree better with the INDO charges than do the standard TABLE 2 total energy binding energy dipole moment atomic populati2,ns molecule basis set (hartrees) (hartrees) (D) H 3 gaussians - 98.5072 0.0484 0.815 0.866 4 gaussians - 99.1922 0.0476 0.870 0.851 5 gaussians - 99.3893 0.0468 0.818 0.847 6 gaussians - 99.4656 0.0468 0.88 1 0.846 Slater b - 99.4785 0.0468 0.878 0.846 3 gaussians - 55.4453 0.3055 1-729 0.854 4 gaussians - 55.8372 0.3013 1 -763 0.846 5 gaussians - 55.9522 0.3006 1-767 0-845 6 gaussians - 55.9974 0.3001 1 a765 0.845 Slater - 56.0052 0.3001 1.764 0.845 a ref.(6); b ref. (13); c ref. (14). A 9.1 34 9.149 9.153 9.1 54 9.154 7.439 7.462 7.465 7.466 7.467 TABLE 3.-oPTIMIZED EXPONENTS FOR NEUTRAL MOLECULES HmXnY 5iW) 1.19 1.33 1 a28 1 *25 1.18 1.18 1 -23 1.31 1-19 1.21 1.37 1.28 1-31 1.33 12(C) 1 -65 1.75 1 -74 1.69 1.67 1 -78 1.75 1 -69 1 -97 2.26 2.28 2.22 2.55 2.53 1.94 2.5 5 2-23 1.95 1.96 2.56 1 -96 2-25 2.23 2.55 set. For the entries listed in tables 4 and 8 the root mean square values of QH- QH(IND0) are 0.109 for the standard set and 0.084 for the optimized set. 3. Dipole moments calculated with the optimized exponents are in better agreement with experimental values than are those with standard exponents.However calculated values still show insufficient polarity in most cases and the INDO results are superior. 4. The optimum exponents for hydrogen are linearly correlated with the electron population of the atom. As the atom becomes more positive the orbital becomes more contracted. This is illustrated in fig. 1. 20 ATOMIC ELECTRON POPULATIONS 5. Many detailed features of INDO charge densities are reproduced by the optimized calculations but less satisfactorily by the standard set. In the paraffins CH4 and C2H6 the optimized results and INDO give a nearly neutral hydrogen whereas the standard exponents lead to considerable positive character. Both the TABLE 5.-cOMPARISON OF DIPOLE MOMENTS FOR NEUTRAL MOLECULES HmXnY HmXnY co HF H20 H3N HjCF H2CO HCN H2NF HNO HOF standard 0.8 15 0-985 1.520 1,602 0-651 0-922 2.065 1353 1 -343 1 -275 optimized 0.730 1 -465 1.841 1 -677 0.981 1 ~626 2.457 1-783 1.601 1-714 INDO 0.941 1.966 2.1 34 1-876 1.692 1 -999 2.455 2.433 1.756 2.244 expt.0-13 a 1.8195 1.846 C 1.468 d 1-855 2.339 f 2.986 g QA. L. McClellan TabZes of Experimental Dipole Moments (W. H. Freeman and Co. San Fransisco b R. Weiss Physic Rev. 1963 131 659. C G. Birnbaum and S. K. Chatterjie J. Appl. Physics 1952 23 220. d D. K. Coles W. E. Good J. K. Bragg and A. H. Sharbaugh Physic. Rev. 1951 82,877. e M. Larkin and W. Gordy J. Chem. Physics 1963 38,2329. f J. M. Shollery and A. H. Sharbaugh Physic. Reu. 1951 82,95. 9 B. N. Bhahacharya and W. Gordy Physic.Rev. 1960,119 144. Calif. 1963) p. 48. TABLE 6.-TOTAL ENERGIES FOR NEUTRAL MOLECULES H,X,,Y total cnergy standard optimized - 1.1191 - 1.1 192 - 107.4868 - 107.4884 - 147.6193 - 147.6197 - 195.9269 - 195.941 6 - 111.2186 - 111.2176 - 98.5432 - 98.5588 - 74.9437 - 74.95 18 - 55443 1 - 55.4441 - 39.7102 - 39.721 6 - 78.2692 - 78.2947 - 77.0444 - 77.0584 - 75.831 6 - 75.8459 - 137.1213 - 137.1514 - 1 12.3205 - 112-3379 - 91.6521 - 91.6620 - 152.841 1 - 152.8480 - 128.0387 - 128,0426 - 172.3290 - 172.3430 optimized set and INDO lead to a marked increase of positive charge on hydrogen along the series CH4 NH3 H20 and HF. The same is true for the hydrocarbon series C2H6 C2H4 C2H2. In methyl fluoride the hydrogen is more negative than in the isoelectronic hydrocarbon ethane for all three calculations.The same applies to the formaldehyde-ethylene pair. This provides some further backing W. J. HEHRE R. F. STEWART AND J . A . POPLE 21 for the alternating inductive charge displacements in a-systems suggested previously on the basis of CND0/2 calc~lations.~~ Among the ions the large negative charge on N in NH; predicted by the standard calculation is much reduced when optimized exponents are used. INDO predicts the nitrogen to be approximately neutral. TABLE 7.-OPTIMIZED EXPONENTS FOR IONS (XH,) * XHni i 1 (HI M C ) 4XN) MO) 52(F) H 1 -40 CHf 1 *41 1 072 CHZ 1-33 1.77 CH 1.15 1.53 NH+ 1 -44 2-04 NH; 1.36 2.05 NHZ 1.33 2-06 NH 1.16 1.80 OH+ 1 -45 2.35 OH 1.41 2.33 OH 1-37 2-31 OH- 1-14 2.09 FH+ 1 *48 2.66 TABLE 8.-cOMPARISON OF ATOMIC POPULATIONS Q FOR IONS (XH,)' ion CH+ CH CH; NHf NH; NH NH OH+ OH; OH,+ OH- FH+ standard 0.695 0-674 1 so87 0.594 0.621 0.637 1-124 0-536 0.572 0.602 1-239 0.488 QH optimized 0-705 0.756 1.012 0.637 0.689 0-71 5 0-967 0.583 0.604 0.621 1.017 0.529 INDO 0.848 0.849 1.131 0.697 0-734 0.752 1.108 0.562 0.601 0.638 1.113 0-454 standard 5.305 5.978 6-739 6.406 7,136 7.454 7-75 1 7.464 7.856 8.195 8.761 8.512 ex optlm1zed 5.295 5.731 6.965 6.363 6.934 7.140 8.066 7.417 7.793 8.138 8-983 8.471 TABLE TO TOTAL ENERGIES FOR IONS (XH,)* ion XHm Hi- CH+ CH CH; NHf NHZ NH NH OH+ OH; OH$ OH- FH+ electronic state "$ IZ+ 1A 1 ' A 1 2rI 2A1 1A 1 l A 1 3c- 3B1 total energy standard optimized - 0.5307 -0.5510 - 37.41 75 - 374476 - 38.7270 - 38.7821 - 38.8748 - 38.8957 - 53.8128 - 53.8452 - 55.1 626 - 55.2008 - 55.8451 - 55.8825 - 54.5625 - 54.621 1 - 74.0397 - 74.071 1 - 74.6549 - 74.6823 - 75.3162 - 75.3369 - 74.01 89 - 74.1 276 - 98.1 934 - 98.221 8 INDO 5.1 53 5.454 6.606 6.303 6.799 6.994 7.784 7.43 8 7.797 8.085 8.887 8.546 22 ATOMIC ELECTRON POPULATIONS Overall we may conclude that the ab iizitio calculations reported here do provide some support for the charge distributions predicted by the simpler semi-empirical method.However it is most important to optimize exponents if such a comparison is to be made. The correlation between optimum exponents and electron populations suggests that exponent variation is necessary to give an adequate account of the + 1.30 0 * 1.20 11h3 i ~ l . l ~ l ~ l ~ l ~ - 4 0 *SO * 6 0 - 7 0 -80 .90 1.00 1-10 Atomic population FIG. 1.-Atomic Population of Hydrogen in Various Molecular Environments as a function of Cls(H).basic electronegativities of the atomic orbitals. No exponent variation is used in the semi-empirical approach but this kind of effect is partly allowed for by the use of empirical electronegativity parameters. Valuable discussions with Dr. M. D. Newton are acknowledged. This research was supported by U.S. Army Research Office-Durham Grant DA-ARO-D-3 1-124- G722 and National Science Foundation Grant GP-8472. J. A. Pople D. L. Beveridge and P. A. Dobosh J. Chem. Physics 1967 47 2026. R. S. Mulliken J. Chem. Physics 1955,23 1833 1841 2338,2343. J. C. Slater Physic Reu. 1930 36 57. C. C. J. Roothaan Rev. Mod. Physics 1951 23 69. J. M. Foster and S. F. Boys Rev. Mod. Physics 1960 32 303. K. 0-ohata H. Taketa and S. Huzinaga J.Physics SOC. (Japan) 1966 21 2306. R. F. Stewart Small Gaussian Expansions of Atomic Orbitals J. Chem. Physics to be published. W. J. Hehre and J. A. Pople Chem. Physics Letters. 1968 2 379. S. F. Boys Proc. Roy. SOC. A 1950,200 542. lo I. G. Csizmadia M. C. Harrison J. W. Moskowitz S. Seung B. T. Sutcliffe and M. P. Barnett Quantum Chemistry Program Exchange (Indiana University Bloomington Indiana U.S.A.) l 1 D. R. Davis and E. Clementi Quantum Chemistry Program Exchange (Indiana University Bloomington Indiana U.S.A.) l2 E. Clementi and D. L. Raimondi J. Chem. Physics 1963 38 2686. l 3 B. J. Ransil Rev. Mod. Physics 1960 32 239 245. l4 W. E. Palke and W. N. Lipscomb J. Amer. Chem. Soc. 1966 88 2384. Correction given in l5 J. A. Pople and M. Gordon J. Amer. Chem. Soc. 1967,89,4253. M. D. Newton and W. N. Lipscomb J. Amer. Chem. Soc. 1967,89,4261. J. A. Pople D. L. Beveridge and P. A. Dobosh J. Amer. Chem. SOC. 1968 90,4201.

ISSN:0430-0696    

DOI:10.1039/SF9680200015

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Molecular Orbital Set Determined by a Localization Procedure B Y HAREL WEINSTEIN * AND RUBEN PAUNCZ Department of Chemistry Technion Israel Institute of Technology Haifa Israel Received 12th September 1968 A starting set of molecular maximum overlap orbitals is localized by means of an external procedure using local density maximization as transformation criterion. For a group of hydride molecules (LiH BH BH3) improved localization of the starting orbitals leads to a set having increased overlap with LCAO SCF calculated molecular wave functions and improved molecular energies. The possible use of the easily obtained localized orbitals as a starting set for more elaborate calcula- tions is considered. The interest in localized molecular orbitals (L.M.O.) originally arose from the early observation that wave functions obtained from molecular orbital calculations do not exhibit direct connection to well-established chemical concepts.LMO describ- ing electronic densities localized in well-defined regions of a molecular system are however found to be closer to the chemical picture and therefore represent an important factor in the conceptual analysis of the results obtained from the molecular orbital method. Moreover since they also seem to be useful for different refinements of existing calculation methods various attempts have been made to find effective procedures for calculating LMO. Using the definition of Edmiston and Ruedenberg localization procedures can be classified as being intrinsic or external in character according to the localization criteria proposed.Orbitals obtained by both procedures exhibit particular properties which might favourably be used when dealing with correlation effects in many- electron systems. Thus localized orbitals are expected to have maximal intra- orbital correlation and divide the total electronic distribution into groups which can be later dealt with by means of special methods.3* 4 9 The localization method proposed by Edmiston and Ruedenberg being intrinsic in character uses the maxi- mization of the sum of orbital self-repulsion energies as localization criterion. The idea is based on the suggestion of Lennard-Jones and Pople6 and has been used successfully in localizing SCF starting orbital^.^ The method however necessitates the calculation of a large number of two-electron integrals. Another method proposed by Boys * is simpler computationwise but more limited in its range of application.Recently Magnasco and Perico suggested an even simpler procedure using localiza- tion criteria obtained by imposing an extremum principle on the sum of local electron populations within the molecule. Nevertheless their results are in fairly close agreement to the ones obtained by Edmiston and Ruedenberg although the localiza- tion criterion is an external one. The closeness of these results indicates that the * This work is part of a thesis submitted to the Senate of the Technion Israel Institute of Tech- nology in partial fulfillment of the requirements for the degree of M.Sc. 23 24 MOLECULAR ORBITAL SET BY A LOCALIZATION PROCEDURE connection between properties of localized molecular orbitals and the chemical concepts might be independent of the intrinsic or external features of the localization criteria emphasizing the importance of the easier external procedures.The aim of the present work is to investigate the possibility of using only an external localization criterion similar to the " localization functions " of Magnasco and Perico in order to obtain a set of molecular orbitals which can serve as good starting wave functions for more elaborate calculations. Since by the use of such a procedure an orbital set is obtained without cumbersome calculations of two-electron integrals results being close enough to those obtained from an SCF procedure would motivate the use of the resulting localized functions for these purposes. LOCALIZATION CRITERIA AND STARTING MOLECULAR SET The molecular model which has been successfully used in chemistry considers the electron pair as being the fundamental structural unit in the construction of the local electron densities known as inner-shells chemical bonds and lone pairs.Since the partition of the molecular density used here will be based on chemical evidence it will use doubly occupied molecular orbitals to describe the local densities known in chemistry. The set of orbitals will thus consist of b molecular functions describing bonds between pairs of atoms i molecular orbitals describing inner shells and I lone pairs the total number of molecular orbitals being b+i+l= n where n = +N = half the number of electrons in the molecule. (2.2) The choice of localization criteria is based on the partition of the total electronic population into subtotal electronic densities and local densities described by defined orbitals.According to Mulliken,lo an orbital @(j) defined by an LCAO-method where the summations include all atoniic orbitals Y and s on atoms k and I respectively. The total orbital density can be further decomposed into n(j,rksl) = partial overlap populatioii = 2N(j)cjrkcjSlSrkSI (2.5) (2.6) and n(j,rk) = net atomic population = ~ ( j ) I cjrk I ' Unlike populations representing sums over all MO the n(j,rk) and iz(j,rm) popula- tions are non-invariant quantities with respect to orthogonal transformations. The localization procedure we present here being based on a series of such orthogonal transformations will be expected to increase the numerical value of certain local electron densities defined through the n( j r k q ) and n(j,r,t) partial densities.The confinement of the electronic densities to certain previously defined regions of the molecule is achieved by restricting the summations in the total density eqn. (2.4) to certain groups of atomic orbitals only. H . WEINSTEIN AND R. PAUNCZ 25 The definitions of the resulting localization functions are similar to those given by Magnasco and Perico where to yield the electronic density described by orbital @(j). shells the localization function will take the form and Aj2) represent indices of certain groups of atomic orbitals considered If the molecular orbital @(j) belongs to the first i wave functions describing inner where Aj = I(A,) represents the inner shell atomic orbital of atom A. For @(j) representing a bond orbital which describes the interatomic charge density in the region between the bonded atoms B and C the localization function becomes where Ajl) = V(Bj) and Aj2) = V(Cj) are indices of the valence atomic orbitals of atoms B and C respectively.An orbital @(j) considered to belong to the group of I lone-pair wave functions yields a localization function given as (2.10) where Aj G V(Dj) is the group of valence atomic orbitals on atom D. So far there is a strong correspondence between the localization procedure derived for LCAO-SCF-MO given by Magnasco and Perico and the one presented here. The crucial difference lies in the fact that we do not start with an SCF solution but with a conveniently chosen basis set obtained from a procedure being much simpler than SCF.Given a group of such molecular wave functions it may be considered as belonging to a subspace of the general Hilbert space X . If the molecular orbitals are obtained by an LCAO-procedure then they are said to be elements of the subspace dcYf spanned by the atomic wave functions. Different molecular orbitals formed by linear combinations of the same set of atomic basis functions form different subspaces of the same space d. Let A and 9 represent two such subspaces of d. According to the basic projection theorem,l' each element fc%' can be uniquely decomposed into two components with where is the orthogonal complement of the subspace 4 and f o is called the projection off on A. Consider now a case in which the subspace .Acd is spanned by molecular wave functions obtained by an LCAO-SCF procedure and 9 is also a subspace of a? defined by some arbitrary wave-functions.The decomposition of every vector in 9 according to eqn. (2.1 1) being unique and the norm off being split according to the same decomposition procedure I f 1 = i f 0 I 2 + If1 I 2 (2.13) f = f o +fl (2.11) f 0 c 4 and flc.Al (2.12) 26 MOLECULAR ORBITAL SET BY A LOCALIZATION PROCEDURE the overlap between the two spaces 9' and d e a n be defined as 09"4 = c I .fk I (2.14) wherefk are the molecular basis vectors of the 9 space and ft is their orthogonal projection on the A space. If a certain subspace Y f is now defined in 9 orthogonal transformations between elements of 9' and 9 can define a new subspace 9. The orthogonal transformations can be chosen so as always to cause an increase in the sum (2.15) 1 The resulting subspace 9 will thus have an increased overlap with the 4 space.Consequently the aim of this work will be achieved if the transformations needed in the localization of a certain starting set of molecular orbitals (Y) will also lead to an increase of the defined overlap between the new space of localized functions @ and the LCAO-SCF functional space. THE LOCALIZATION PROCEDURE The localization method described has been applied to a group of hydride mole- cules LiH BH and BH3. As a starting point in each calculation use was made of a basis set of molecular orbitals obtained as eigenvectors in the diagonalization procedure of the atomic overlap matrix. These orbitals defined by Lykos and Schmeising,12 represent a set of maximum overlap orbitals.These authors showed that for homonuclear systems the Hiickel MO are identical to maximum overlap molecular orbitals. Moreover MOO have also been shown l2 to be in better agree- ment with the SCF-MO without overlap than the Huckel MO in a general case merely because nonclosest neighbour interactions are included in their construction. Taking also into consideration the readiness with which they can be obtained MOO have been considered as offering a good starting set. The number of molecular maximum overlap orbitals coincides with the number of basis atomic orbitals which can usually be taken to be larger than half the number of electrons in the molecule. The first step in the proposed procedure therefore consists in the partition of the starting set 9 into 9' and (9-9'). For diatomic molecules the subset 9' was chosen to contain functions yielding maximal numerical values of the respective localization functions Lj.Each element in the defined subset 9' of size n represented a molecular orbital describing the electronic charge distribution in a well-defined molecular region. The molecular charge concentrations were defined according to a certain chemical picture of the molecule. However the total charge partition is by no means unique and several empirical " charge maps " can be investigated as to which of them yields the most accurate results. Once the set 9' was defined orthogonal transformations between pairs of orbitals were performed each pair being composed of an element fc9" and an element g c (9 - 9'). Every new f obtained by the transformation changed 9'' into a new subset which was further treated in the same way.The transformations were thus completed for all possible pairs subject to the restriction that the new f obtained yield a higher numerical value for the corresponding localization function Lj. Since the transformation matrix T cos 0 -sin 6 T = ( sin 6 cos 6 H . WEINSTEIN AND R . PAUNCZ 27 was made dependent 011 the parameter 0 it was possiblc to construct it function of 0 from which an optimal value for the rotational angle in each 2 x 2 transformation was found l 3 by solving the equation dLj(0)/dO = 0. For each diatomic molecule considered an atomic basis set of the same magnitude was defined consisting of STO with energetically optimized exponents. The basis functions were chosen to be of Is 2s and 2pa type on the heavy atom and included only one 1s orbital on the hydrogen.The LiH molecule was considered to be described by two doubly occupied mole- cular orbitals one of them describing an inner shell around the Li atom and the other generating the bonding charge distribution in the region between the two atoms. The exponents of the atomic STO as well as the value for the interatomic distance were taken from Ransil’s calculations of heteronuclear diatomic molecules,14 in order to make possible an ulterior comparison of the results. The molecular energies calculated for the set of orbitals obtained in several stages of the localization pro- cedure numerical values for the overlap function OgA defined in (2.15) where d represents the LCAO-SCF functions reported by Ransil and numerical values of the localization functions,* are listed in table 1.The BH molecular orbital set was considered to consist of three doubly-occupied molecular orbitals forming an inner shell centered on the B nucleus a bond molecular orbital and an orbital describing a lone pair charge distribution. Table 2 contains results obtained at various stages of the localization procedure. For the polyatomic boron hydride the choice of the 9’-set was first made accord- ing to a maximum-population criterion made possible by the cumbersome calculation of the charge density matrix for a multiconfigurational wave function using maximum overlap orbitals as basis functions. The 9’-set turned out to be identical with the one obtained from a choice using maximal numerical values of corresponding localiza- tion functions as a criterion.This was encouraging and the possibility was con- sidered to rely completely on maximum values of localization functions as the criterion to be used for the definition of the 9’-set. The atomic basis STO were chosen to be of Is 2s 2p, 2p and 2p,-type on the B atom and to contain one 1s type function for every H atom in the molecule. Of the eight functions contained in the 9-space four were considered to form the 9 ’ - subspace. They included one molecular orbital describing each 2-electronic bond in the molecule and one doubly occupied orbital forming an inner shell around the B nucleus. The exponents of the basis atomic orbitals and the inter-nuclear distance were taken to be identical to the constants used by Pipano l5 in an LCAO-SCF and a full C.I.calculation of BH3 to facilitate comparison of the obtained results. Table 3 contains the results of the localization procedure applied to the BH3 molecule. DISCUSS I ON An observation common to all the investigated example is that the starting molecular set yields energies which are in poor agreement with the LCAO-SCF calculated energies and have the smallest overlap with the SCF orbital set. In each case by performing the localization of the orbitals one obtains an improvement of the calculated energies and overlap with the LCAO-SCF space which accompanies the requested increase in the values of the localization functions. One therefore observes the gratifying relation between localization and improvement of the molecular orbital set. However the required improvement in molecular energy as compared * Ptotd represents values of localization functions Pi defined by Magnasco and Perico ; Pspeciec are the numerical values of the localization functions Lj defined in this work.28 MOLECULAR ORBITAL SET BY A LOCALIZATION PROCEDURE 00 2 W 00 00 0 9 0 4 - 0 0 0 3 8 3 0 E x I4 .r( 1. H f r; I b I I * m e H c\1 00 m W 6\ m ? I w h d 8 Q\ 00 9 I H 2 c .- x kind of transform Ekin Epot M.O.O. 24.853762 - 59.515638 inner shell and lone p. optim 24.907302 - 57.140541 lone p. optim. 24.082540 - 57.603794 SCF 25.1246 - 58.01482 Function exponent TABLE 2.-BH MOLECULE ‘total i ; 1.24875204 I ; 0.41 5760702 overlap with SCF Epo t 2 el. Etotal 8.530937 - 23.984939 0.97533 16 b ; 1,88720029 i ; 1*99701091 5.910709 - 24.17653 0-978223 b ; 1.88720029 I; 1-00865878 i ; 1 -24875204 7.1 25723 - 24.249531 0.985320 b ; 1 ~88720029 1 ; 1.88691773 5.6686 -25.0756 * 1 SH 1 S B 2SB 2PB ; 1.1860 4.6805 1 -2955 1.3168 ; ‘specific i ; 1.24875204 b ; 0.792066 163 1 ; 0.41 5760702 i ; 1.99701091 b ; 0.7920661 63 I ; 140865878 i ; 1 -24875204 b ; 0.792066163 I ; 1.88691773 internuclear distance 2-329 a.u.; internuclear repulsion 2.146 am. * Rand l4 TABLE 3 .-BH3 MOLECULE Epot 2e1. Etotal 16.641091 -23.0930 overlap with:SCF 0.7464549 0.789704 0.920575 0.920899 ‘total i ; 1 -9554295 1 b ; 0.869304179 b ; 2-8 1892274 b ; 1.45779479 ‘specific i ; 1.95542951 b ; 0.267928363 b ; 048800799 b ; -0’858374978 Ekin Epot 27’222153 - 74.104605 kind of transform M.O.O. i ; 1 -9554295 1 b ; 0.956055994 b ; 2.77342052 b ; 1.37954702 i ; 1 m9554295 1 b ; 0.271768867 b ; 0.27563364 b ; 0.508858310 14.594344 - 24.736825 27’080747 - 73.560276 bond opt.I i ; 1 -9554295 1 b ; 0.272450727 b ; 0.535184892 b ; 0.513980190 i ; 1 -9554295 1 b ; 0,985927778 b ; 1.76734181 b ; 1.37230965 14.143330 - 25.639230 25.851089 - 72.782009 bond. opt. I1 i ; 1.96536577 b ; 0.985927778 b ; 1.767341 84 b ; 1.37230965 i ; 1 *96536577 b ; 0.543528790 b ; 0.535184892 b ; 0.508858314 inner shell and bond opt. 14.163857 - 25.654446 25.977 1 18 - 72.94378 1 i ; 1.98581803 b ; 1.16591427 b ; 1.33254770 b ; 0.841003964 14.197128 - 26.3 17308 25.887759 - 73.586555 SCF* 2PzB 1-30 ; 2SB 2PXB 2PyB 1.30 1.30 1-30 Function exponent internuclear distance 2.383 a.u. ; internuclear repulsion 7.14836 a.u. * A. Pipano l 5 H. WEINSTEIN AND R. PAUNCZ 31 to the initial values strongly depends upon the possibility of performing a satisfactory number of orthogonal transformations-which implies a requirement of a large 9’- set.For the BH molecule only a small improvement of the starting results is achieved mainly because the (9-9’) set contained only one molecular orbital because of the very small atomic basis used. For the LiH and BH3 molecules the improvement is sensibly larger as compared to BH since the two transformed sets 9’ and (9-9‘) were equal in size. The result of the localization procedure is also found to be encouraging as to the connection existing between chemical concepts and the results of molecular orbital calculations in view of the possible construction of a good starting molecular orbital set by the simple use of chemical concepts. Since the handling of the starting functions and of the results in the presented procedure is simple and the procedure showed up to be dependent on the magnitude of the basis set used fairly large sets of functions can be used probably causing a further improvement in the results and determining a final minimal set which can then be used in more accurate calculations.In view of the fact that determination of the best minimal set of functions when starting with an arbitrarily chosen basis is not always easily achieved with more accurate calculations the readiness with which the determination is achieved with the presented procedure only emphasizes its usefulness. Also since accurate calculation methods based on localized electron pair functions can be sensibly simplified if the molecular system is localizable and such information being quite difficult to obtain for some systems,16 the presented localization procedure seems to be adequate to use in localizability investigations and subsequent starting functional-basis construction.C. Edmiston and K. Ruedenberg Rev. Mod. Physics 1963 35 457. J. M. Parks and R. G. Parr J. Chem. Physics 1958 28 335. E. Kapuy Acta Physica Acad. Sci. Hung. 1958 9 237 ; Acta Physica Acad. Sci. Hung. 1961 13,461. R. McWeeny Proc. Roy. SOC. A 1959,253 242. J. E. Lennard-Jones and J. A. Pople Proc. Roy. SOC. A 1950 202 166. C. Edmiston and K. Ruedenberg in Quantum Theory of Atoms Molecules arid the Solid State ed. P. 0. Lowdin (Academic Press Inc. New York 1966) p. 263. S. F. Boys in Quantum Theory of Atoms Molecules and the Solid State ed. P. 0. Lowdin (Academic Press Inc. New York 1966) p. 253. V. Magnasco and A. Perico J. Chem. Physics 1967 47,971. lo R. S. Mulliken J. Chem. Physics 1955 23 1833. B. A. Lengyel in Adv. in Quantum Chemistry ed. P. 0. Lowdin (Academic Press Inc. New York 1968) p. 14. P. G. Lykos and H. N. Schmeising J. Chem. Physics 1961 35 288. * J. E. Lennard-Jones and J. A. Pople Proc. Roy. SOC. A 1951 210 190. l 3 M. Rossi J. Chem. Physics 1965 43 2918. 14B. J. Ransil Rev. Mod. Physics 1960 32 239. * l 6 E. Kapuy J. Chem. Physics 1966 44 956. A. Pipano DSc. Thesis (Technion Israel Inst. of Technology).

ISSN:0430-0696    

DOI:10.1039/SF9680200023

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Use of Integral Approximations in Non-Empirical LCAO MO SCF Calculations on XCN Systems BY G. DOGGETT AND A. MCKENDRICK Chemistry Department University of Glasgow Glasgow W.2. Received 30th October 1968 The accuracy of three multicentre integral approximations has been evaluated for a number of LCAO MO SCF calculations on the ground states of HCN and FCN. Mcban’s results for HCN using a basis set of best atom atomic orbitals are best reproduced by two integral approximations suggested by Lowdin while the Mulliken approximation is poor. All three integral approximations yield total energies lying below the ab initiu value. But double-zeta calculations on HCN and FCN using a fixed CN internuclear separation both yield minimum total energies which straddle the ab initiu values. The calculated equilibrium CH and FC internuclear separations straddle the experi- mental values of 0.1063 nm and 0.1262 nm respectively RCH = 0-1041 nm and RFC = 0.1 196 nm when using the Mulliken approximation and RCH = 0.1 104 nm and RFC = 0-1 34 nm when using the best Lowdin approximation.Dipole moments and one-electron energies are also calculated for each approximation. The charge distributions are discussed in terms of effective atomic charges and one-electron difference density functions. 1. INTRODUCTION There have been few detailed investigations into the accuracy of invoking integral approximations in calculations on polyatomic molecules. Too often integral approximations are made in molecular calculations without first evaluating their accuracy. This could be carried out for e.g.by attempting to reproduce the results of an ab initio calculation. Unless such a procedure is adopted the results obtained from the approximate calculation may be of little value. But detailed evaluation of approximate methods has only become possible recently as a consequence of the rapidly increasing number of ab iizitio calculations. The possibility of finding a reliable approximate method offers some hope for making reasonably accurate calculations on molecules which are at present beyond the scope of a completely ab initio calculation. The linear nitrile systems XCN are of interest and the present work was initiated in an attempt to elucidate the ground state electronic structures of HCN and FCN. The effect of X on the electron distribution in the CN fragment is of particular interest.The most convenient method of calculation is the single configuration LCAO MO SCF approach using a basis of either simple Slater or double-zeta atomic orbitals. The basic problem in this kind of calculation is in the evaluation of the three-centre one- and two-electron integrals particularly when dealing with extended basis sets. The search for a suitable integral approximation is made by first calculating the ground state electronic structure of HCN using the same molecular parameters as in the accurate ab initio calculation of McLean.’ It is then possible to assess the accur- acy of a particular approximate ab iizitio calculation by a direct comparison of the calculated total energy dipole moment one-electron energies and LCAO coefficients with the corresponding values as determined by McLean.l 32 G.DOGGETT AND A . MCKENDRICK 33 The basic problem in approximating a multicentre integral is in finding the opti- mum expansion for a given overlap density in terms of suitably chosen atomic densities. The most widely used approximation is the one suggested by Mulliken,’ hereafter referred to as the MA d ) a ( l ) d ) b ( l ) = *sab{d)a(l)d)a(l) f d ) b ( l ) d ) b ( l ) ] * (1 * 1) A slightly modified version of the MA which is made invariant to orthogonal trans- formations of the atomic orbital basis set-the IMA has also been investigated in view of the work by Pople Santry and SegaL3 Although (1.1) preserves charge it is basically unsatisfactory because the centroid of the overlap density is not maintained. Lowdin suggested that an asymmetric partitioning of the overlap density was more satisfactory d ) a ( l ) + b ( l ) = sab{Al#a(l)d)a(l) + A2d)b(1)d)b(1)} (1.2) where A + A2 = 1.This approximation is referred to as the partial Lowdin approxi- mation or PLA and A2 is chosen so that the dipole moments of the densities of the left- and right-hand sides of (1.2) are identical. A more complete approximation has also been suggested by L o ~ d i n ~ hereafter called the full Lowdin approximation or FLA + a ( l ) + b ( l ) = ~ ~ ~ d ) a ( l ) + a ~ ( l ) s a ~ b + A ~ ~ d ) b ( l ) ~ b ~ ( l ) s b ~ a a’ b’ where again A + A2 = 1 and A2 is chosen to reproduce the dipole moment of the overlap density. The summations over a’ and 6’ include all orbitals on a or b having principal quantum number equal to or less than that of The calculations are performed within the framework of SCF theory as outlined by McWeeny.6 This analysis assumes the basis functions are mutually orthogonal.Thus the f and G matrices initially calculated using a basis of ordinary atomic orbitals must be first transformed by the matrix S-* which symmetrically ortho- gonalizes the set of basis functions or +b respectively. - x = s-+ x s-4 where X stands for f or G. No difficulties were experienced with the convergence of the iterative procedure and the occasional oscillatory behaviour could always be traced to errors in the data. All one- and two-centre molecular integrals are evaluated numerically except for the one-centre two electron integrals which are evaluated analytically. The remaining three-centre integrals are evaluated according to the MA PLA or FLA as required.2. THE ELECTRONIC STRUCTURE OF HCN The molecular geometry for the pilot calculation on HCN is taken from the work of McLean RCH = 0.1058 nm (2-000 a.u.) RCN = 0.1157 nm (2.187 a.u.). The calculated energies and dipole moments are compared with McLean’s results in table 1 for the integral approximations under discussion. The first point to be noted from table 1 is that the use of a multi-centre integral approximation in this study of HCN lowers the total energy relative to the ab initio value. There is also little difference between the MA and IMA calculations. There is no reason to expect the approximate energy to lie above or below the ab initio value. This follows since the use of an integral approximation induces an arbitrary change in the effective Hamiltonian matrix thereby causing the calculated energy to suffer an arbitrary displacement from the ab initio value.If the total 2 34 I N T E G R A L APPROXIMATIONS IN SCF C A L C U L A T I O N S energy is used as a criterion of accuracy then the FLA is the best integral approxima- tion as it gives a total energy within 0.05 % of the McLean value. However the PLA calculation reproduces the dipole moment more satisfactorily. The accuracy of each calculation incorporating an integral approximation is further checked by calculating the one-electron energies. These energies listed in table 2 again show the superiority of the Lowdin approximations over the Mulliken approximation. The occupied molecular orbital energies are apparently better reproduced by the PLA calculation.TABLE 1 .-CALCULATED MOLECULAR PROPERTIES FOR HCN USING BEST ATOM ATOMIC ORBITALS IMA MA PLA FLA McLean 1 electronic energy - 116,6159 - 116.6079 - 11 6.5314 - 116.4692 - 116.4236 total energy E - 92.7397 - 92.7317 - 92.6552 - 92.5930 - 923474 kinetic energy T 91.2547 91.2412 91.4292 91.4905 91.3209 potential energy Y - 183.9944 - 183.9728 - 184.0843 - 184-0834 - 183.8683 -2T/V 0-9919 0.99 19 0.9933 0.9940 0.9933 dipole moment 1.309 1.304 2.033 1.803 2.100 All energies in a.u. (1 a.u. = 27.20963 eV). Dipole moments are in Debyes (1 D = 0.33356 x lo-'' C m) TABLE 2.-cOMPARISON OF ONE-ELECTRON ENERGIES FOR HCN (BEST ATOM ATOMIC ORBITAL BASIS SET) IMA MA PLA FLA McLean * 10 - 15.7427 - 15.7469 - 15.7287 - 15.7479 - 15.7402 20 -11.3972 -11.3918 -11.4035 -11.4142 -11.4277 30 - 1.2404 - 1.2430 - 1.2526 - 1.2575 - 1.2522 40 - 0.8263 - 0.8158 -0.7934 - 0.781 1 - 0.7965 50 -0.5691 - 0.5646 - 0.5644 - 0.5735 - 0.5582 In - 0.4960 - 0.4974 - 0.5030 - 0.51 37 - 0.5074 60 0.8400 0.6860 0.5215 0.3535 0.3648 7a 1.0923 1.1502 1.0799 0.961 3 1.0860 2rG 0.2592 0-2603 0.2557 0.2453 0-25 16 5o-ln gap 0.073 1 0.0672 0.0615 0.0598 0.0508 1n-271 gap 0.7551 0.7577 0.7586 0.7590 0.7590 1st I.P.(eV) 13.49 13.53 13.69 13.98 13.81 All energies are in a.u. But overall since the 60 level is particularly sensitive to the choice of integral ap- proximation the FLA is considered as the best choice. Two orbital energy gaps are also given in table 2 as well as the estimated value for the molecular ionization potential deduced by application of Koopmans' theorem. A Mulliken population analysis gives the results listed in table 3.The Lowdin approximations again yield results which are more in accord with those of McLean. The comparison between the dipole moments or effective atomic charges although useful does not reveal in which regions of space the wave function is inaccurate. G . DOGGETT AND A . MCKENDRICK 35 Some information on this problem is forthcoming from an examination of the differ- ence density function AP obtained by subtracting the approximate density function from the one given by McLean.l This Ahp function is shown in fig. 1 for the FLA calculation. The inaccuracies are mainly confined to regions close to the nuclei. A similar situation obtains for the other approximations but as expected the regions of inaccuracy are more extended in the MA calculation.Fig. 1 also shows that the overall effect of the multicentre integral approximation is to remove charge from the bonding regions and deposit it in the vicinity of each nucleus this behaviour is also apparent in the AP plots for the other integral approximations. The PLA or FLA gives results which are very close to the accurate ab irzitio values the MA on the other hand is poor. TABLE 3.-EFFECTIVE ATOMIC CHARGES DEDUCED FROM A MULLIKEN POPULATION ANALYSIS OF THE HCN MOLECULAR WAVE FUNCTION (BEST ATOMIC ORBITAL BASIS SET) N C H IMA + 0-083 - 0.261 +0*178 MA + 0.09 1 - 0.300 + 0.209 PLA - 0.100 -0.181 + 0.281 FLA -0.113 - 0.1 57 + 0.270 McLean - 0.08 1 -0.155 + 0.236 \ '\O.O i0.0 \ 0.0 N C H - 0.6k FIG. 1.-The upper part of the figure shows contours of the difference density function AP obtained by subtracting the FLA one-electron density function for HCN from that of McLean the basis functions for both the FLA and McLean calculations are best atom atomic orbitals.The lower part of the figure shows the variation of AP along the internuclear axis. The units of AP are electrons (a. u .)-" The success of the FLA and PLA indicates that a further study of the cyanide systems would be profitable. The results of some IMA and FLA calculations on HCN using a double-zeta basis a CN internuclear separation of 0.1159 nm (2.190 a.u.) and five different CH internuclear separations are shown in fig. 2. The IMA and FLA calculations yield minimum total energies of - 92.8970 and - 92.7449 a.u. respectively at CH internuclear separations of 0.1041 nm (1.968 a.u.) and 0.1 104 nm(2.086 a.u.) respectively so straddling theexperimental value of 0.1063 nm (2.009 a.u.).The molecular dipole moments are 1.75 and 2.59D respectively as against the experimental value * of 3.00D. But the assumed CN internuclear 36 INTEGRAL APPROXIMATIONS I N SCF CALCULATIONS separation is 0.0004 nm longer than the experimental value,8 to facilitate comparison with the FCN calculations in the next section. have given the results of a double-zeta calculation on HCN using RCH = 0.1066 nm (2.014 a.u.) and RCN = 0.1153 nm (2.179 a.u.). Since this molecular configuration is close to the IMA and FLA minimum energy configurations it is interesting to find the IMA and FLA total energies straddling the McLean and Yoshimine value of -92.8369 a.u. McLean and Yoshimine -92.710- -92.730- - 92.750 ? ? - I G - 9 2 .8 6 0 - -92.880. -92.900- 1 I . I 1-6 1-8 2.0 2.2 2'4 RCH(a.u-) FIG. 2.-A plot of the total energy E(a.u.) against RcH(a.U.) for the IMA (lower curve) and FLA (upper curve) calculations on HCN using a basis of best atom atomic orbitals. The one-electron energies follow the same general trends as before. In both approximate ab initio calculations all energy levels except 60 and 7a become more strongly binding as RCH decreases. Also the 1n-27-c gap is virtually constant over the range of RCH values. The predicted ionization potentials are now 15.43 and 15.67 eV respectively for the IMA and FLA calculations. A Mulliken population analysis of the molecular wave function gives rather similar results to those found earlier. In the IMA calculation the effective charge on H remains approximately constant at about +0-24 and as RCH decreases n charge is transferred preferentially to C.This 7-c transfer is accompanied by a smaller transfer of a charge from C to N. The results for the FLA calculation cannot be generalized so readily. As RCH decreases charge is transferred from H and redistri- buted predominantly on C while N increases its share of charge to a smaller extent. A more satisfactory representation of the way in which the electronic charge is redistributed on molecule formation is obtained by plotting contours of the difference density function : G . DOGGETT AND A. MCKENDRICK 37 The neutral atoms placed at the appropriate nuclear sites in the molecule are assumed to have the following configurations H Is C ls2 2s2 2px 2py N ls2 2s2 2px 2py 2pz where the 2p atomic orbitals are directed along the internuclear axis.Contours of the AP function are shown in fig. 3 for the FLA calculation at the experimental value of RCH. Electronic charge is displaced from the vicinity of the nuclei and accumu- lated in the bonding regions as well as the region usually ascribed to the N lone pair. Furthermore as the CH internuclear separation is decreased less charge is removed from the vicinity of the proton in contradiction to the results obtained from the Mulliken population analysis. N C H -0.6 FIG. 3.-The upper part of the figure shows contours of the difference density function AP for HCN obtained from the difference between the FLA and free atom one-electron density functions situated at their respective nuclei.The basis functions for the calculation are double-zeta atomic orbitals. The lower part of the figure shows the variation of AP along the internuclear axis. The units of AP are electrons (a.~.)-~. 3. THE ELECTRONIC STRUCTURE OF FCN The calculations on FCN are performed in a similar manner to those described for HCN. A double-zeta ' basis set is chosen and RCN is again fixed at 0.1 159 nm which is now the experimental value.' IMA and FLA calculations are performed using five different FC internuclear separations. In addition one PLA calculation is performed at the experimental geometry (see later). The IMA calculations yield a minimum energy of - 192-2624 a.u. at RFC = 0.1 196 nm (2.26 a.u.). The corresponding value for the dipole moment is 1-66 D (expt.,' 2.17 D).The apparent anomaly in the FLA calculation at the experimental value of RFC requires further investigation. The results of a PLA calculation at the same molecular geometry yield a total energy of - 191.6657 a.u. which is reasonable in view of the earlier cal- culations on HCN. McLean and Yoshimine find a total energy of - 191-6275 a.u. (no other molecular properties are quoted) for a double-zeta calculation on FCN using a slightly different molecular geometry RFC = 0.1260 nm (2.381 a.u.) and RCN = 0.1 165 nm (2.202 a.u.). If the anomalous FLA calculation is disregarded the estimated total energy and equilibrium value for RFC are - 191.566 a.u. and 0.134 nm (2.54 a.u.) respectively. Hence the IMA and FLA energies probably straddle the ab initio energy as was found in the earlier double-zeta calculation on The variation of total energy with RFC is shown in fig.4. 38 INTEGRAL APPROXIMATIONS I N SCF CALCULATIONS HCN. The IMA and FLA calculations also predict equilibrium values for RFC which straddle the experimental value of 0-1262 nm (2.385 a.u.). However the PLA may in fact be superior to the FLA as not only is the total energy better but so is the molecular dipole moment 1-68D as against an estimated 1.54D for the FLA calculation. The question as to why the FLA calculation breaks down should be answered. The determination of A2 involves the calculation of a ratio of two numbers of which the denominator is given as the difference of two summations. This difference is usually not small but for the particular choice of molecular parameters under con- sideration it is small enough to cause appreciable instability in the value of A2 e.g.A2 = 5.97 for the 2pz 2p carbon-fluorine overlap density. That this instabilityis the I 191.650 I I 1 - 192.1 5 0 - 192.250 ia I I 0 1 I I I I -T- I I I I . 2.0 2.2 2.4 2.6 2.8 RFC(a.u.) FIG. &A. plot of the total energy E(a.u.) against RFc(a.U.) for the IMA (lower curve) and FLA (upper curve) calculations on FCN using a basis set of double-zeta atomic orbitals. exception rather than the rule is confirmed to some degree by the reasonable values obtained for A2 at the other FC internuclear separations. So far it has been impossible to find any rules to predict when instability will occur. The best method of avoiding the instability is to examine the values of A at various molecular geometries before performing the SCF calculations.By this means any configuration leading to instability can be avoided. A comparison of the one-electron energies for the IMA and PLA calculations at the experimental molecular configuration is given in table 4. Also included are the results of McLean and Yoshimine lo for an extended basis set calculation incor- porating 3d and 4fatomic orbitals on each centre. Although their molecular con- figuration differs slightly from the one used here it is satisfying to find the PLA results for the occupied molecular orbital energies are in such good agreement with those of McLean and Yoshimine. However the IMA calculation predicts a different sequence of occupied energy levels with the 70 and 2n transposed with respect to the other two calculations. G .DOGGETT AND A . MCKENDRICK 39 The charge distribution in FCN is examined by means of Mulliken population analyses and maps of appropriate difference density functions. The population analyses give the following results for the effective atomic charges -0.183 + 0.358 -0.175 F C N -0.123 + 0.263 - 0.140 where the IMA and PLA values are given above and below each atom respectively. These charges are conventionally interpreted in terms of a displacement of charge TABLE 4.-A COMPARISON OF ONE-ELECTRON ENERGIES FOR THE IMA AND PLA CALCULATIONS OF FCN USING A BASIS SET OF DOUBLE-ZETA-ATOMIC ORBITALS 1 0 2 0 30 40 50 60 70 171 271 IMA - 26.413 1 - 15.7950 - 11 *6723 - 1.7721 - 1.41 12 - 0.9192 -0.6610 - 0.971 6 - 0.723 1 PLA - 26.4593 - 15.7749 - 11.6230 - 1.7679 - 1.3652 - 0.9391 - 0.6581 -0.8158 - 0.5618 McLean and Yoshimine 10 - 26.4375 - 15.6115 - 11.4018 - 1.7673 - 1.2523 - 0.9308 -0.6012 -0.8141 - 0.4984 80 0.5473 0.473 1 0,1922 90 1.4127 1.3190 - 3lT 0.5089 0,1619 0.2359 All energies are in a.u.The McLean and Yoshimine results have been obtained from a calculation using an extended basis set and a slightly different molecular geometry (see text). f 0.0 FIG. 5.-The upper part of the figure shows contours of the difference density function AP for FCN obtained from the difference between the FLA and free atom one-electron density functions situated at their respective nuclei. The basis functions for the calculation are double-zeta atomic orbitals. The lower part of the figure shows the variation of AP along the internuclear axis. The units of AP are electrons (a.~.)-~.40 INTEGRAL APPROXIMATIONS I N SCF CALCULATIONS from C to each end of the molecule. The difference density function obtained by subtracting the appropriate atomic densities from the FLA molecular density function is shown in fig. 5. Again the pattern is similar to that found for HCN charge is displaced from the regions around the nuclei and accumulated in the bonds or in regions normally considered as being occupied by N or F lone pairs. The plot of AP along the molecular axis shows this effect very clearly. 4. CONCLUSION The systematic use of three integral approximations in LCAO MO SCF cal- culations on HCN and FCN has been examined. Preliminary calculations on HCN using the same molecular parameters as McLean show that the two Lowdin approxi- mations are superior to the Mulliken approximation.Further calculations on HCN and FCN using a double-zeta basis set of atomic orbitals yield total energies which straddle the ab initio value. The FLA appears to be breaking down in one of the calculations on FCN. The PLA devoid of such instability provides a simpler yet reliable approximation for reproducing the results of an ab initio calculation with a surprising degree of accuracy. One of us A. McK. thanks the S.R.C. for the award of a Research Studentship. A. D. McLean J. Chem. Physics 1962,37 627. R. S. Mulliken J. Chim. Physique 1949 46,497 675. J. A. Pople D. P. Santry and G. A. Segal J. Chem. Physics 1965,43 S129. P. 0. Lowdin J. Chem. Physics 1953 21 374. P. 0. Lowdin Adv. Physics 1956 5 1. R. McWeeny Rev. Mod. Physics 1960 32 335. E. Clementi Tables ofAtomic Wave Functions (I.B.M. J. Res. Develop. Suppl. 1965). J. Sheridan and J. K. Tyler Trans. Faraduy Soc. 1963 59 2661. A. D. McLean and M. Yoshimine Int. J. Quantum Chem. 1967 1% 313. Develop. Suppl. 1968). lo A. D. McLen and M. Yoshimine Tables of Linear Molecule Wave Functions (I.N.M. J . Res.a

ISSN:0430-0696    

DOI:10.1039/SF9680200032

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Polarizability and Hyperpolarizability of the Helium Atom BY A. D. BUCKINGHAM AND P. G. HIBBARD School of Chemistry The University of Bristol Received 18th October 1968 The zeroth fist and second order perturbed Schrodinger equations for the helium atom in an external electric field have been solved to high accuracy through the variation principle. A ground- state energy differing by only about 1 part in lo9 from Pekeris's extrapolated figure was obtained with a wave function do containing 181 adjustable parameters (compared to Pekeris's 1078). The first-order wave function d1 and hence the second-order energy and polarizability of the atom was obtained with up to 84 adjustable parameters in dl. The polarizability a is 1.38319 a.u. = 0.204956 x cm3 = 0.228044~ C2m2 J -'. The dipole shielding factor differs from its exact value of unity by a few parts in lo5.The second-order wave function d2 and hence the fourth-order energy and hyperpolarizabilityy were obtained for various wave functions do and dl and with up to 106 adjustable parameters in d2. Smooth convergence was obtained yielding y = 43.10 a.u. = 2-171 x C4m4J-3. However an extension to d1 may be needed before an accurate value ofy can be computed. Accurate values have also been obtained for the %values of the unperturbed atom and for the quadrupole and octopole polarizabilities. e.s.u. = 2.688 x Polarizabilities and hyperpolarizabilities describe the changes in the charge distributions of atoms and molecules due to external electric fields. Polarizabilities have been of interest for a long time and their calculation has been reviewed by Dalgarno hyperpolarizabilities describing nonlinear distortions by strong fields are of special importance because of their relationship to " non-linear optics ".2 If an atom is in a uniform static electric field F its energy E and dipole moment YM may be written as power series in Fz E = Eo+E2Fz+E4F:+ ...m = - (aE/aF,) = aF + &F + . . . where the polarizability a = - 2E and the " second hyperpolarizability " y = - 24&. The differential polarizability n = am/dF is x = a + 3 ' y F 2 + ... (3) The Kerr effect is due to anisotropy in E viz. Zzz-EXx = ~(yzZ,,,-y,,,,,)F,2 ; for static fields this reduces to 3yF2 but dispersion may be significant. Measurements of the Kerr constant of helium have been r e p ~ r t e d ~ . ~ and yield y = 2.7+0.2 x e.s.u.= 3.3 f 0 - 2 x 10-63C4 J-3 m for an optical wave-length of 632.8 nm. Various attempts have been made to calculate y for helium; 4 9 6-8 the present work was undertaken in the hope of providing a definitive non-relativistic quantum-mechanical description of a helium atom in a strong static electric field. The total electric field at a stationary nucleus is zero ; thus the field due to the other charges in a molecule in a field precisely compensates F,. If this total field at the nucleus FiN) = F,( 1 - ol) then o1 is the " dipole shielding factor " and is unity in a neutral atom. 41 42 HYPERPOLARIZABILITY OF THE HELIUM ATOM Similarly if the atom is in a potential V which is symmetric about the z-axis the multipole moment of order I (i.e. the expectation value of xeirfPl (cos 0,)) is in the linear approximation where a is the appropriate polarizability* and Fzzz .. . = - (8 V/azz). The corres- ponding shielding factor ol is defined by and is the expectation value of zZ!eir,('+ l)PI(cos Bi)/Fzzz . . . Shielding factors are also equal to minus the ratio of the Zth order moment that would be induced in the molecule by a nuclear moment of order Z to the nuclear moment itself. i (4) r ( 0 s z z z . . . = UlFzzz. . . 3 Fir;. . . = Fzzz . . .(I - 0,) ( 5 ) I THE PERTURBATION EQUATIONS Schrodinger's equation for a helium atom in an electric field Fz is ( 2 0 +XlF,)Y = EY where is the Hamiltonian for the free atom and the perturbation Since F is a variable it is convenient to write the ground-state wavefunction Y as a power series in Fz %IFz = [r,Pl(cos 01) + ~ ~ P ~ ( c o s '82)]Fz.(7) Y = Y o + Y l F z + Y 2 F ~ + Y 3 F ~ + ... The identity (6) and eqn. (1) and (8) lead to the equations Eo = (Yo I *o I ' u o > / ( ~ o I 'yo> E2 = ( Y o I X l I Yl>/(YO I y o > E.4 = [(Wl I 2 1 I ' u 2 > -E2((Yo I ' u 2 > +(Yl I ~ l > ) I / ( ' Y O I y o > * (9) (10) (1 1) No analytic solution to the zeroth-order equation #oYo = EoYo exists and it is usual to obtain an approximation 4o to Yo through the variational condition A suitable 4o may be constructed from a power series in the three interparticle distances rl r2 and r12 with a factor exp [ - +k(rl + r2)] the best set of coefficients for any particular expansion being chosen by minimizing c0. The introduction of a scaling parameter k for all interparticle distances and minimization of c0 with respect to k ensures that the virial theorem is ~atisfied.~ Schwartz lo has demonstrated that functions of r l r2 and r12 are particularly appropriate.Expansions involving only positive powers of distances (Hylleraas-type expansions) give rather slow convergence to Eo when high accuracy is sought ; the rate of convergence is governed then by the ability of 4o to approximate Yo in the regions of the singularities in the potential (rl = 0 r2 = 0 rI2 = 0). Kinoshita l 1 has shown that the inclusion of negative powers of the distances (subject to restrictions l 2 placed on all satisfactory $o) improves 4o near these singularities. E o e o = <40 I 2 0 I40>/(+0 1 4 0 ) . (12) * The polarizability ccz is I ! times that defined in ref.(l). A . D. BUCKINGHAM A N D P .G . HIBBARD 43 The perturbed wave function is obtained in a similar way expansions in powers of r l r2 and r12 having the appropriate angular symmetry being used. The computa- tion of an approximate ith order wave function 4i is simplified if it is written as 40& and the variation theorem used to find the coefficients in 4;. This product form for 4i is valid provided 4o is nodeless. While Yo is nodeless there could be " accidental " nodes in q50 but as the expansion length of 4o increases these should occur in ener- getically less important regions and convergence to EZi should be satisfactory. Whereas E~ obtained from eqn. (12) is an upper bound to Eo the value of c l i would only necessarily be if all 4 j ( j < i ) were exact. The variational condition is that E< E for all F,.In practice it was found that E and E~ converge from above as 40 $1 and 4 are made more accurate. THE UNPERTURBED WAVEFUNCTION A N D ENERGY OF HELIUM Apart from the wavefunctions of Pekeris,13 no sets are available for the purpose of obtaining smooth convergence of properties. However Pekeris's coordinate system has not been found convenient for the atom in a field. Preliminary studies of published wavefunctions showed that considerable variations in computed properties exist between wavefunctions of similar length and energy; some of these variations are due to insufficient minimization. In this work new sets of wavefunctions for the helium atom in a field have been calculated (following Pekeris,13 using Kinoshita- type Since a and y depend on the long-range behaviour of Y no attempt was made to obtain the best possible c0 for any particular expansion length but rather to choose the functions so that E and E~ were gradually improved.expansions) and the convergence of atomic properties studied. The functions used may be written 40(n) = exp (-is)[ f cpQrsPtqu' + Kinoshita-type terms where s = k(r + r 2 ) t = k( -rl +rz) u = kr, and all n>p,q,r>O (because the spatial part of the singlet ground state wavefunction must p + q + r = O ( 4 even) -1 with q even be symmetric with respect to electron exchange) are included in the sum ; the Kinoshita-type func- tions are 20 of the most important terms with negative powers of s introduced by Kinoshita 1 1 in his 80 parameter 40. All 40(n) for JZ = 2-10 (27-181 adjustable parameters in all) were prepared using the fast direct minimization procedure of Fletcher and Powell,14 with double length arithmetic throughout.Full convergence for all +o was not achieved accumulation of round-off errors causing premature termination of the iterations for y1 = 5 and 8. The other $o(yl) probably have energies less than a.u. from the true minimum for the particular expansion. The energies and k values for these wavefunctions are in table 1 with the best four results of Pekeris l 3 (who used no negative exponents). The energies for our comparatively simple wavefunctions compare well with the accurate results though there is evidence from the rate of convergence that the number of Kinoshita-type terms is becoming disproportionately small. Values for are in table 2. Our &values are more accurate and converge more strongly than Pekeris's because of the inclusion of Kinoshita-type terms.44 HYPERPOLARIZABILITY OF THE HELIUM ATOM The original and re-minimized energies and functions for the six parameter 4o of Hylleraas l5 (40 = exp (- +s)[l+ clu + c2t2 + c3s+ c4s2 + c,u2]) are in table 3. This re-minimization gave considerable changes in calculated properties. TABLE EN ENERGIES co (IN A.u.) AND SCALING PARAMETERS k FOR TRIAL UNPERTURBED WAVE- n 2 3 4 5 6 7 8 9 10 12 15 18 21 original reminimized THE number of parameters 27 33 42 54 70 90 115 145 181 252 444 71 5 1078 FUNCTIONS ($0 (n) EO - 2.903714647 - 2.90371 9257 23329 23988 4236 4329 4354 4367 4372 4290 4356 4370 4375 k 3.6612 3.6352 3.8362 3.8461 3.8639 3,9432 3.9287 3.9500 3.9493 (3.4081) (Pekeris 13) (3.4081) (Pekeris 13) (3,4081) (Pekeris 13) (3.4081) (Pekeris 13) TABLE 2.-&VALUES FOR VARIOUS UNPERTURBED WAVEFUNCTIONS n 7 8 9 10 15 18 21 0 1) 1 -8 10447 1.810403 1 *8 1 0430 1 08 10433 1*810389* 1 *810410* 1.810419" * from Pekeris.13 d(r 12) 0,106369 0,106359 0.106355 0.106352 0.1 06377" 0*106362* 0.1 063 55 * TABLE 3 .-THE SIX-PARAMETER HYLLERAAS WAVEFUNCTION k c1 c 2 c3 c 4 C5 EO 3.63586 0,0972 0,0097 -0.0277 0.0025 -0.0024 - 2.90324 3.51 352 0.0961 0.0092 -0.0409 0.0019 - 0.0030 - 2,90333 FIRST ORDER WAVEFUNCTION AND POLARIZABILITY The first order wavefunction Y satisfies the equation and E2 is given by eqn.(10). Because of the presence of rl in Z0 there may be terms in Y1 with angular symmetries of the form of Yll,m(Ol,41) Yz,m(82,42). Those contributingto c2 can be generated from Yl,o(81 4,) Y ,0(02,42) through multiplica- tion by powers of u in the trial function.A suitable function l6 for obtaining g2 is (Xo-Eo)Y,+X1Yo = 0 (14) 41 = 404Lwhere 4;(n> = f c;qrsPtqur(rlP1(cos el) + r,~,(cos e,)> + p+q+r=O (4 even) 5 ckqrsPtqur(rlP1(cos 0,) - r,~,(cos 0,)). (15) p+q+r= 1 (4 odd) A . D. BUCKINGHAM AND P . G . HIBBARD 45 Results for E~ and crl for IZ = 0-6 are in table 4. Since 1 a.u. of polarizability is (b2/lme2)>" = 0.148176 x C2 m2 J- l the polarizability of the helium atomis a = 1.383193 = 0.204956 x C2 m2 J-l. The dipole shielding factor o1 provides a valuable check on 41 since ol is 1 ; owing to computational problems only about five figure accuracy was achieved for crl. Our result for a is to be compared with the value 1.384 obtained by Chung and Hurst." cm3 = 0.164863 x cm3 = 0.228044 x TABLE 4.-THE SECOND ORDER ENERGY Ez FOR THE 181 PARAMETER 40(10) AND THE DIPOLE SHIELDING FACTOR FOR THE 90 PARAMETER $0(7) FOR VARIOUS 4 l ( n ) with no terms in r12 n parameters ep (a.u.) 01 0 1 -0,5662528 1-53649 1 3 -0'6911853 0.98333 2 6 -0.6912874 0.99 195 3 10 -0.6914775 0-99735 4 15 -0.6915125 0.99948 5 21 - 0.691 5279 1 -00009 6 28 -0.6915349 1 -0002 1 with terms in r12 parameters 62 (a.u.) 0 1 1 -0,5662528 1 *5 3649 4 -0.6911886 0.98291 0.99490 10 -0.6914819 20 -0.6915716 0.99899 35 - 0.691 5897 0.99993 56 - 0.691 5944 1.00005 84 - 0.691 5959 1 -00003 TABLE 5.-THE FOURTH-ORDER ENERGY E4 FOR THE 181 PARAMETER +0(10) AND VARIOUS 4dn1) AND 42(n2) \nl 0 1 2 3 4 5 6 m\ O* 1* 2* 3* 4* 5* 0 1 2 3 4 5 - 10727567 - .07508832 - -091 5776 1 - -09212278 - '0921 3712 - e09214761 el0727567 -*37157956 - .43 5 8 8434 - -43743006 -*43751062 -125991 54 - 056828633 - *744 1 1092 - '74988604 - -749921 16 - '74994987 - 12599154 - 1'2587623 - 1.7514343 - 1'7723120 - 1.7724772 -126091 87 -12655203 *I2660158 -a57746122 - e57528826 - '57528369 7'76045337 -*75556923 - e75576099 - a76701894 - a76154030 - '761 84665 - -76706149 - '76160061 - .76190417 -76709204 -.76163687 -.76194189 *12609 1 87 -1 2655203 -1 2660 1 58 -1.2731227 - 1.2705417 - 1.2705538 - 1.7820801 - 1.7736862 - 1.7740656 - 1.8053507 - 1.7950649 - 1.7957793 - 1.8055247 - 1.7953062 - 1.79601 17 * with diqr = e;qr = 0.* 1266 1534 - -57527346 -*75577051 - -761 85908 - '7619 1710 - -76 195674 *12661534 - 1.2705490 - 1.7740688 - 1.7957708 - 1.7960018 * 12661 944 - -57526823 - -75576609 - -761 85153 - -76190986 - -76 194821 * 1266 1944 - 1.2705448 - 1'7740633 - 1.7957630 - 1'7959926 THE SECOND ORDER WAVE FUNCTION AND THE HYPERPOLARIZABILITY The second-order wave function Y satisfies the equation ( 2 0 - WY2 + 2 1 % = E2Y0 (16) and E4 is given by eqn.(11). zeroth- and first-order equations + 2 = $04i may be found where Using the functions q50 and q51 determined from the n +i(n) = 1 [c;qrsPtqur + d ~ q r ~ P t 4 ~ r ( r ~ P 2 ( ~ ~ ~ 0,) + r2P2(cos 0,)> + p + q + r = 1 e~qrsPt4urrlr2P1(cos 0,)P1(cos O,)] + n 1 &~qrsPtqur(r~P2(~os 0,) - riP2(cos 62)>. (17) o + a + r = 1 - (<odd) The calculation of z4 and 4; is similar to that of c2 and 41 and no difficulties were found beyond the need for care with accumulation of round-off errors.Values of c4 for &(n) for n = 0-5 are in table 5 for the 181 parameter 40(10) and various $,(n) both, 46 HYPERPOLARIZABILITY OF THE HELIUM ATOM for the full expansion (17) and also for restricted expansions involving the spherical terms only (i.e. (17) with d& = edbr = 0). The final result for c4 is - 1.796 a.u. equivalent to y = 43.10 a.u. = 2.171 x e.s.u. = 2.688 x The scaling parameter k has so far been considered to retain its field-free value ; greater flexibility is permitted by lettering k = ko + k F + k4F$ + . . . . This has no effect on #1 or E, but c4 is reduced by minimization with respect to k2.4 The im- provement is valuable for simple functions #$(n) but is insignificant with the larger expansions which have sufficient flexibility without this additional nonlinear variation- al parameter.Our hyperpolarizability y is in good agreement with Grasso Chung and Hurst’s value of 42.8 1 a.u. (these authors obtained E~ = - 2.903721 a.u. with a 25 parameter 40 and a = 1.3830 a.u.) and with Sitz and Yaris’s 42.6 a.u. However the experi- mental value from the Kerr effect is 53.6 +4 a.u. for a wave length of 632.8 nm ; a “ constant excitation energy ” model indicates that this experimental optical y is equivalent to a static value of 52.8 a.u. There is therefore a significant discrepancy between theory and experiment. This may be due to insufficient flexibility in q51 and 4,; it is possible that there could be additional terms in 41 contributing to E ~ though not to E or to the dipole shielding constant. We are investigating this possibility.We have found that it is particularly important to use a very accurate unperturbed wavefunction 40. C4 m4 J-3. THE QUADRUPOLE A N D OCTOPOLE POLARIZABILITIES In an axial field gradient Fzz = -2F, = -2F,,, the perturbation is where a, = - [r?P,(cos 61) + Y P (cos 6,)] is the quadrupole moment operator. The energy is given by the equivalent to eqn. (10) and the quadrupole polarizability C defined in ref. (4) is C = a2 = - 4 ~ ~ . Values for both a and the octopole polar- izability a3 = - 12c2 are shown in table 6 for the 181 parameter unperturbed wave- function 40(10) and for various perturbed functions 41(n). Using Kinoshita’s 80 parameter 40 Davison l 7 obtained a = 1-2202 a5 for helium. TABLE 6.-THE QUADRUPOLE AND OCTOPOLE POLARIZABILITIES FOR 40(10) AND VARIOUS (bl(n) with terms in r-12 n with no terms in r12 az(aS) a3(a7) a2(a5) a3(a7> 0 1 *05 1206 1.531 1 *05 1206 1.531 1 1.221 389 1.561 1.222038 1-561 2 1 -221 841 1 *567 1 -222388 1 -568 3 1 ~ 2 2 191 7 1 -570 1 -222499 1-571 4 1.221995 1 *222533 A.Dalgarno Adv. Physics 1962 11 281. N. Bloembergen Nonlinear Optics (Benjamin New York 1965) ; see also A. D. Buckingham and B. J. Orr Quart. Rev. 1967 21 195. A. D. Buckingham and J. A. Pople Proc. Physic. SOC. A 1955 68,905 ; A. D. Buckingham Proc. Roy. SOC. A 1962 267 271. L. L. Boyle A. D. Buckingham R. L. Disch and D. A. Dunmur J. Chem. Physics 1966 45 1318. A. D. Buckingham and D. A. Dunmur Trans. Faraday Soc. 1968 64 1776. G. W. F. Drake and M. Cohen J. Chem. Physics 1968 48 1168 (and references therein). ’ M. N. Grasso K. T. Chung and R. P. Hurst Physic. Reu. 1968,167,l (and references therein). A . D. BUCKINGHAM AND P . G . HIBBARD P. Sitz and R. Yaris J. Chem. Physics 1968 49 3546. E. A. Hylleraas Adv. Quantum Chem. 1964 1 1. lo C. Schwartz Physic. Rev. 1962 126 1015. l 1 T. Kinoshita Physic. Rev. 1957,105,1490; 1959,115 366. l 2 T. Kato Trans. Amer. Math. Soc. 1951 70 195 and 212. l3 C. L. Pekeris Physic. Rev. 1958 112 1649; 1959 115 1216. l4 R. Fletcher and M. J. D. Powell Cump. J . 1963 6 163. l5 E. A. Hylleraas Z. Physik 1929 54 347. l6 K. T. Chung and R. P. Hurst Physic. Rev. 1966 152 35. l7 W. D. Davison Proc. Physic. SOC. 1966 87 133. 47

ISSN:0430-0696    

DOI:10.1039/SF9680200041

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Calculation of Dipole Hyperpolarizabilities of H20 NH3 CH,and CH,F BY G. P. ARRIGHINI M. MAESTRO AND R. MOCCIA Centro di Chimica Teorica del C.N.R. Istituto di Chimica Fisica Via Risorgimento 35 56100 Pisa Italy Received 29th August 1968 From the first-order corrections of the SCF MO induced by a static homogeneous electric field the third-order corrections to the SCF molecular energies have been evaluated. These SCF hyper- polarizabilities were determined by using basis sets of STF of various sizes to express the unperturbed and the perturbed orbitals for HzO NH3 CH4 and CH3F. In some cases the calculated value compare badly with the experimental estimates. The possible reasons for these disagreement are briefly discussed. INTRODUCTION The distorsion of the electronic charge of a molecule under the influence of an electric field is important in determining several phenomena of interest.The evalua- tion of this distorsion for general situations represents a formidable problem which cannot be solved accurately not even for molecules of moderate complexity. Fortu- nately the computations can be carried out with reasonable accuracy for a particular situation which is not so restrictive as to diminish their importance. This situation is realized when a molecule is subjected to a static electric field whose sources can be considered external. For this situation the theory leads to the definition of several quantities through which the interaction energy between the molecule and the field is expressed. While some of these quantities like the permanent dipole moment and the polarizability can be determined experimentally with ease most of them like the higher permanent multipole moments and the higher polarizabilities are much more difficult to measure '9 both for experimental reasons and because of the complexities involved in the interpretation of the quantities experimentally determined.2* It seems therefore of interest to take recourse to the theory in order to evaluate them although the actual computation must be of an approximate nature.In this paper we report the results for the dipole hyperpolarizabilities of H,O NH3 CH4 and CH3F calculated by the rigorous perturbed H-F scheme where both perturbed and unperturbed orbitals were expressed by linear combinations of STF. The theory of the molecular polarizabilities has been developed by Bucking- ham.lV 3 9 McLean and Yoshimine have published a complete review always referring their treatment to Cartesian coordinates.Here we will limit ourself to a brief sketch using spherical coordinates which present some advantages over the car- tesian ones. If the electric field is charge of the molecule static and its sources are completely external to the electronic the perturbing potential V(r) may be written as 48 G . P . ARRIGHINI M. MAESTRO AND R . MOCCIA 49 where VL,M are the parameters which characterize the field and Y L M are the usual spherical harmonics. The transformation properties under rotations of the quantities VL,M will be those of the irreducible h a n k tensors while there the transformation properties under translations can be easily obtained from those of the YL,M.6 In presence of the field (2.1) the molecular energy E can be expanded as (ll3!)C C (a3E/avL,MavLi,M~avL~,M~)OvL,MvL~,M~vL2,M~+* * * (2'2) L M L i M i L 2 3 2 The coefficients appearing in the successive summations in eqn.(2.2) may be ideiitifiedwith the permanent multipole moments the general polarizabilities and so on. The use of the spherical coordinates and of (2.1) automatically eliminates from the expansion of the energy E all terms which must be zero because of Y(r) must obey the Laplace equation AV = 0. The fact that the VL,M behave as irreducible tensors can be exploited to express their product as linear combinations of new irreducible tensors,'. i.e.? VL M vL. 1 M2 = Cc( L 1 9 L' ; M ) v,&:2 + M 1 (2.3) L' where the C(L,L1,L';M,M1) are Clebsch-Gordan coefficient^.^ By standard procedure 7,8 it is thus possible to bring eqn.(2.2) in a form which contain only contractions of irreducible tensors. At this stage it may be expedient to return to Cartesian axes where only real quantities appear. This can be readily accomplished by using suitable unitary transformations. Once this last form of eqn. (2.2) is obtained the trans- formation properties of the various coefficients appearing in it are evident and it will be easy to find a particularly convenient set of axes. A suitable arrangement will be that coincident with those used to represent the symmetry operations of the molecular point group. In this case the only tensors not vanishing will be those which transform according to the total symmetric representation. As an example we consider an homogeneous field characterized only by the vector 8 = VV(r).In this case only the VI,M coefficients of eqn. (2.1) will be different from zero. Going through the steps previously described it is possible to write E = E0+47cN1 x 3)-1CY,,m(')[Y,,m(tl)E]o +(I x 3 x 5)-1Cy,,,,(b)Cyz,,(tI)Elo+ in m (1 x 3 x 5 x 7)- 'Cy,,m('>[y3,m(q1)E10 + (1/6)~2~0,0(')[r12~0,0(~)EI~ + m (1/30>~~2~1,,('>[~z~~,m~~)EIo + * * * * (2.4) m where the vector operator q stands for ( d j d 8 ) and the cSPl,m(r) are real harmonic polynomials obtainable by a suitable unitary transformation from the usual solid harmonics gI,m(r) = r Z Yl,m(0,q5).8 In eq. (2.4) the quantities [cSPl,m(q)E]O are proportional to the permanent dipole moments the [Y2,m(q)E]0 and [y2cSPo,o(q)E]o to the polarizabilities and so on.The evaluation of the above quantities requires the computation of the derivatives (dE/o'iffg)O (d2E/i%'gd8'gr)o (a3E/a8'gaiffg,a8'gll) etc. (9 g' 9" = X y 2). Because of the impossibility of their rigorous evaluation several approximate methods have been p r o p ~ s e d . ~ - ' ~ It seems that among them the perturbed H-F method without further approximation the so-called coupled H-F scheme (CHF) is capable of giving reliable results for the atomic polarizabilities 1 4 9 as well as for the molecular polarizabilities 9* '* l6 and magnetic sus~eptibilities.~~ This method permits one to obtain the rigorous derivatives of the H-F energies and hyperpolarizabilities ' 50 CALCULATION OF HYPERPOLARIZABILITIES EHF with respect to a parameter which give the intensity of a one-electron perturbation.Thus the coupled H-F scheme has been employed here to calculate the third derivatives of the molecular energy with respect to the three components of 8. As it has been shown,18J9 it is necessary to calculate only the first-order corrections 4:) of the occupied orbitals 4 j ) in order to evaluate the energy derivatives up to the third order with respect to the parameter p of a one-electron perturbation phP. Here only closed shell cases will be considered. These corrections 4$) must satisfy the equation wheref" is the unperturbed H and F hamiltonian YP indicates the correction upon the H-F electronic potential 2o which depend upon the &)'s while the other symbols are self-evident. In the above equation the off-diagonal Lagrange multiplier corrections appear.Without lack of generality the 4;)'s can be projected upon the unoccupied orbitals only. Thus the following formula is obtained which can be solved iteratively.16* 2 o also the following formula for the energy derivatives are obtained. With a little manipulation by exploiting the higher order perturbation equations FIRST DERIVATIVE * O SECOND DERIVATIVE occup. EgF = C [($; I hY I 4;)+(4; I hP I4S)+C.C.]. (2.8) i THIRD DERIVATIVE occup. occup. j i . j EaBy = 2c [(44 I hP+ GP I 4:) + C.C.] - 2 c [~gi(@$ I (by) + C.C.] +(pya)+(y~p). (2-9) In the last equation E:~ = (4; I ha+ga I 4:) and (pya)+(yap) means that the contributions corresponding to the others aPy arrangements must be added to the first one. In the present computations the orbitals 4;) and +$) were approximated by the Roothaan method i.e.4:) = { x>C. ;) 4:) = ( x } C . $. Once a given basis set { x ) is employed it can be shown 21 that thefollowingexpres- sion is obtained in place of eqn. (2.6) unoccup. C.\+(WP + GP)CPj 0 &g-&E,O Ctj = c CP (2.6a) Subsequently it is easy to derive the analogues of the eqn. (2.7) (2.8) and (2.9). Thus (2.7a) G . P . ARRIGHINI M . MAESTRO A N D R . MOCCIA 51 occup. occup. j L j Igg. = 2 c [cajt(Hfl+Gfl)c:j+c*c.]+ -2 c [C:J(Wa+Ga) cp,x C!JSC’ii+C.C.]+ (PY4 +(Yap>. (2.94 where q l y = (x I hP I x,> q = (x I gfl I x,> and = (x 1 x,>. The symbol EsCF has been used to distinguish it from the true H-F quantity EHF. If the basis set employed ( x ] is complete they will be coincident. For a static homogeneous electric field the perturbation matrices Ha WP and HI’ are those of the dipole moment operator.RESULTS AND DISCUSSION As reported elsewhere the first-order corrections to the SCF MO have been evaluated with eqn. (2.6a) for H20,16 NH3,16 CH4 16 2 2 and CH3F 2 3 using several basis sets of STF some of them of such a size as to obtain an unperturbed energy EiCF which are estimated to be close to the H-F limit. These corrections were used to evaluate through eqn. (2.8a) the dipole polarizabilities 16* 22* 2 3 and have presently been used to evaluate by eqn. (2.9a) the dipole hyperpolarizabilities. In table 1 are reported the basis sets employed and the SCF molecular energies E& obtained by the standard Roothaan procedure. The calculations were performed for fixed molecular geometries coincident with the experimental ones.Only for CH3F due to reasons given el~ewhere,~ the geometry employed is negligibly different from the experimental one. The figures reported in table 1 indicate that the largest bases give energies close to the estimated H-F Table 2 reports the results of our calculation for the dipole hyperpolarizabilities of HzO. For a reference frame which has the z-axis coincident with the two-fold symmetry axis and the zx plane coincident with the molecular plane only the [q29’,,,(q)E], [93,2(q)E]0 and [9,,o(q)E]o tensor components are not vanishing therefore table 2 reports only three quantities which are proportional to them. These quantities are expressible by the p coefficients as defined by Buckingham.’ For these molecules with a reference frame which has the z axis coincident with the three-fold symmetry axis and the y z plane coincident with one of the reflection planes only the [q29,,,(q)E], [ 9 3 0 ( q ) E ] 0 and [93,-3(q)E]0 tensor components are different from zero.As for H20 table 3 reports three quantities proportional to them expressed as linear combination of the p coefficients. Finally table 4 reports the CH4 results for the only not vanishing tensor component proportional to [ 9 3 - 2 ( q ) E ] 0 when the orienta- tion of the reference frame is such to have the x y z-axes coincident with the three twofold symmetry axes. The experimental results listed in table 3 and 4 (there are no available data for H20) were taken from ref. (3). Values in parentheses are approximate only. Tables 2 3 and 4 show that with the exception of CH4 the calculated values compare poorly with the available experimental data.The disagreement is so great as to make apparently negligible the importance of the size of the basis sets employed. At first sight this would seriously undermine the accuracy of the computed values but it seems to us that it would be a hasty conclusion. We examine e.g. if the fault might lie in the computing scheme we adopted i.e. the CHF technique. That this seenis not the case may be conjectured by the following arguments. First we observe that the CHF technique has proved to be satisfactory for the atomic polarizabilities I4 and hyperpolarizabilities I 5 and certainly superior to other methods even for the 24 Table 3 shows the results obtained for NH and CH3F. 52 HzO I H20 I1 NH3 I NH3 I1 CH4 I CH4 I1 CHSF CALCULATION OF HYPERPOLARIZABILITIES TABLE 1 .-BASIS SETS EMPLOYED basis functions no.of basis functions 7 27 8 32 9 39 47 lSN(6.67457) ; 2S~(1.9426) ; 2P~(1*9426) ; lS~(1.19545) EicF = -56.0058 a.u. lSc(5.716) ; 2Sc(l.625) ; 2Pc(1*625) ; lS~(1.28) E;& = - 40.1 129 a.u. (a) The 2P components perpendicular to the molecular plane are omitted. (b) 0. Salvetti to be published. (c) Located at a distance of 0.7 a.u. from the nitrogen approximate position of the centroid of the ( d ) R. Moccia and C . Vergani to be published. ( e ) ref. (22). (f) ref. (23). lone pair (CLP). TABLE DIPOLE HYPERPOLARIZABILITY RESULTS FOR H20 t2.S.U.) (Pxxz + by y z f P z zz) (Pxxz - PYYZ) (282 z z - ~ P X X Z - ~ P Y Y Z ) H20 I + 0.1723 + 0.1087 -0.1983 H20 I1 + 0.1598 +0~1000 - 0.0778 expt. I - - TABLE 3.-DIPOLE HYPERPOLARIZABILITY RESULTS FOR NH3 AND CHjF ( t3.S.U.) ( P x x z - t P y y z - k P z z z ) (2Pz2z-313xxz-3317,Yz) U P X X Y - P Y Y Y ) NH3 I - 0.05 197 + 0.1 451 4 -0.1884 NH3 I1 -0.1363 + 0-2739 - 0.3051 expt.(- 11.7) I - expt. (-3.33) - - CH3F - 0.2667 - 0.021 1 + 0.0944 TABLE 4.-DIPOLE HYPERPOLARIZABILITY RESULTS FOR CH4 ( t3.S.U.) ( P X Y Z ) CH4 I + 0.0400 CH4 I1 +0.0191 expt. - f0*01 G . P. ARRIGHINI M. MAESTRO AND R. MOCCIA 53 molecular polari~abilities.~* l 1 On the other hand this result had to be expected because as it could be easily shown the CHF technique give the derivative of an approximate energy which is as much corrected with respect to the fluctuation potential 2 5 as the unperturbed H-F energy. It appears rather improbable that the higher corrections due to the neglected part of the electronic correlation should drastically change the values of these derivatives.Thus the CHF technique is expected to give results of satisfactory accuracy for quantities like the electrical polarizabjlity hyperpolarizability paramagnetic susceptibility etc. If this reasoning is correct one is led to conclude that the cause of the observed discrepancies between calculated and experimental hyperpolarizability is due to the fact that our calculations were performed even in the best cases with basis sets of large but limited extension. To this criticism a definite answer cannot be given at the moment ; however some consideration may be helpful. The differences among the values calculated by using basis sets of different sizes are percentage wise not negligible and of the right magnitude as it may be inferred from the differences encountered for other quantities like the dipole polarizabilities 6 22 and the paramagnetic su~ceptibilities.~~ As for the hyperpolarizabilities these quantities depend only on the first-order corrections of the orbitals.In addition the results obtained with basis sets of increasing size (which are not reported here) show a regular behaviour and their values tend to those corresponding to the largest bases. It is rather improbable then that by increasing the size of the bases which is tantamount to approaching the H and F limit the results would change drastically. It is our opinion therefore that the computed values cannot be that wrong and that the experimental estimates should be reviewed. This conclusion is also supported by the fact that the only reliable experimental result which refers to CH4 is in satisfactory agreement with that calculated by the largest basis.Accordingly it would establish a + sign for this quantity. A. D. Buckingham Quart. Rev. 1959 13 183. A. D. Buckingham R. L. Disch and D. A. Dunmer J . Amer. Chem. Soc. 1968,90,3104. A. D. Buckingham and B. J. Orr Quart. Rev. 1967 21 195. A. D. Buckingham Adv. Chem. Physics 1967 12 107. A. D. McLean and M. Yoshimine J. Chem. Physics 1967,47 1927. Y. N. Chiu J. Math. Physics 1964 5 283. ’ M. E. Rose Elementary Theory of Angular Momentum (J. Wiley and Sons 1957 New York). see e.g. A. R. Edmonds Angular Momentum in Quantum Mechanics (Princeton University Press 1960) p. 124. R. M. Stevens and W. N. Lipscomb J. Chem. Physics 1964 40,2238.R. E. Wyatt and R. G. Parr J. Chem. Physics 1964 41 514. l 1 H. J. Kolker and M. Karplus J. Chem. Physics 1964,41 1259 ; and ibid. 1964 41,2011. l 2 H. D. Cohen and C. C. J. Roothaan J. Chem. Physics 1965,43 S34. I 3 J. M. O’Hare and R. P. Hurst J. Chem. Physics 1967 46 2357. l4 H. D. Cohen J. Chem. Physics 1965 43 3558 ; ibid. 1966 45 10. l 5 G. F. W. Drake and H. D. Cohen J. Chem. Physics 1968,48 1168. l 6 G. P. Arrighini M. Maestro and R. Moccia Chem. Physics Letters 1967 1 242. G. P. Arrighini M. Maestro and R. Moccia J. Chem. Physics 1968 49 882. l 8 R. McWeeny Physic. Rev. 1962 126 1028. l 9 G. Diercksen and R. McWeeny J. Chem. Physics 1966,44 3554. ’,’ R. Moccia Theor. chim. Acta 1967 9 192. 22 G. P. Arrighini C . Guidotti M. Maestro R. Moccia and 0. Salvetti J. Chem. Physics in press. ’!3 G. P. Arrighini C. Guidotti M. Maestro R. Moccia and 0. Salvetti to be published. 24 C. D. Ritchie and H. F. King J. Chem. Physics 1967 47 564. 2 5 0. Sinanoglu Proc. Nat. Acad. Sci. 1961 47 1217. R. McWeeny Chem. Physics Letters 1968 1 567.

ISSN:0430-0696    

DOI:10.1039/SF9680200048

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

GENERAL DISCUSSION Prof. Dr. W. C. Nieuwpoort (Rijksuniversiteit te Groningen) said With regard to Pople’s paper even an optimized minimum basis set offers a rather poor description of atomic charge densities. This is true for the inner parts as well as for the outer parts of the atomic wave-functions. Extension of the basis with diffuse functions to describe the outer parts better will generally lead to changes in the exponents of the functions already present. Precisely such an extension takes place in a molecular calculation where in a sense every atomic minimum basis set is extended by the functions on other centres. The question then is how much of the exponent changes observed must be attributed to these purely atomic effects and how much of it has to do with molecule formation.This can be investigated by repeating one or some of the calculations with just the functions but no charges present at the various centres but one. Dr. G. Doggett (Uniuersity of Glasgow) said I would question the reliability of using Mulliken gross atom charges for discussing bond ionicities. The partitioning of the overlap density in a population analysis is not unique. In general the Mulliken partitioning does not even maintain the centroid of the overlap density. In calcula- tions on HCN McKendrick and myself found that the variation in the effective atomic charge on H with changing CH internuclear separation is sensitive to the method used for partitioning the overlap densities for some molecular configurations different methods of partitioning can yield effective atomic charges of opposite sign.Prof. M. Randid (Zagreb) said In Pople’s paper a comparison is made of atomic electron populations obtained by the semi-empirical INDO molecular orbital theory with some values from full SCF calculations using a minimal basis set of exponential- type functions. A total assignment of all electrons in a molecule to atomic orbitals is made by finding gross populations as suggested by Mu1liken.l We in Zagreb became interested in the question of the definition of the population analysis and would like first to stress that although Mulliken definition is universally accepted it contains an artibrary element in assigning the overlap density 2ciacibSab equally between atoms a and b. Other choices are possible and we proposed recently that the overlap density on the bond a-b to be so divided that the centre of the charge does not change.2 Such a definition will lead to different results when the atomic orbitals 4a and 4b are considerably different in their exponential (screening) part.With wider use of double zeta and other limited bases sets the situations with con- siderably different exponents are more common and Mulliken population analysis may lead to a biased distribution of electrons in molecules. The modified definition might provide an essential and important improvement. It would be of interest to know if the moderately good agreement of the comparison of atomic electron populations reported by Pople between the semi-empirical and the full SCF calculations is affected by the limitations of the Mulliken population analysis. R. S. Mulliken J .Chem. Physics 1955 23 1833. T. Zivkovii and N. Trinajstib Chem. Physics Letters 1968 2 369. 54 GENERAL DISCUSSION 55 Dr. D. B. Cook (University of ShefJieZd) said Since the question of the convergence of Gaussian expansions for STO’s has been raised I would like to describe my own experience in this field. If one looks at an SCF calculation using this Gaussian expansion method then in order to reproduce the total energy one-body density matrix etc. of the full STO calculation a Gauss STO ratio of at least 6 1 is needed as Pople points out. Thus a calculation on a fairly large molecular system is soon swamped by the large number of two electron integrals which arise. However if one examines the convergence of the integrals to the STO values some important conclusions emerge.It is found that the one electron integrals converge very slowly but the two electron repulsion integrals converge rapidly to the correct values an expansion of two Gaussians per STO yielding results in good agreement with the true STO values. These conclusions are easily understood in terms of the functional form of the Gaussians-in particular their behaviour near the origin. Thus the rather long expansions necessary are due to the poor convergence of the one electron integrals and the large numbers of repulsion integrals which arise are largely redundant since these integrals have converged after two or three terms. We have used these conclusions to develop a “ Mixed Basis ” method in which we use one-electron integrals calculated over STO and a short Gaussian expansion repulsion integrals thus combining the advantages of STO and Gaussian bases.Prof. R. McWeeny (University of ShefJieZd) said I agree with Pople’s remark that a mixed-basis approach cannot be viewed as a satisfactory ab initio procedure. It is put forward as method of closely reproducing the results of or “ simulating ” an ab initio calculation by evaluating all one-electron quantities with high accuracy and estimating all electron interaction effects with somewhat lower accuracy. Such “ simulated ” ab initio calculations 2* retain all the features of a complete non- empirical calculation and attempt to include all integrals except those that are negligible in a fairly strict numerical sense they give results in better accord with those of ab initio work than do the more empirical approximations-unless of course the latter are actually Jitted to the results by purely empirical adjustment of parameter values.The work referred to by Cook4 appears to us t o be a good compromise between theoretical rigour and computational simplicity ; the use of only two Gaussians per Slater orbital appears to be satisfactory and will probably be essential in dealing with larger molecules. Prof. R. McWeeny (University of ShefJieZd) said One major disadvantage of localization criteria based on orbital overlaps or populations is their disregard of the energy which implies that localization may lead to a wave function which is energeti- cally quite poor. In particular one finds that contamination of the high energy inner shell orbitals by admixture with valence orbitals can lead to very inferior wave f ~ n c t i o n s .~ I suggest that such criteria should be applied with caution and then only to valence electrons the valence orbitals being Schmidt-orthogonal to those of the inner shells. A similar difficulty arises with lone-pair orbitals when the corresponding orbital energies are not taken into account and it is for this reason that in recent work we have used an energy criterion. In some cases ( e g when D. B. Cook and P. Palmieri to’be published. * D. B. Cook P. Hollis and R. McWeeny Mol. Physics 1967 13 553. D. B. Cook and R. McWeeny Chern. Physics Letters 1968 1 588. D. B. Cook and P. Palmieri to be published. R. McWeeny and K. Ohno Proc. Roy. SOC. A 1960,255 367. R. McWeeny and G. Del Re Theor. chim. Acta 1968,10 13. 56 GENERAL DISCUSSION there are no lone pairs) the results are not very different from those obtained by overlap or population criteria but in general there are significant differences and the energy criterion appears to be more reliable.The main advantage of the method of Weinstein and Pauncz seems to be that it involves an intrinsic criterion and can therefore be applied even when an approximate wave function is not available. Prof. G. G. Hall (University of Nottingham) said There are two rather different motives for using localized orbitals and it seems to me that they may sometimes lead to different results. The wish for a convenient physical interpretation leads naturally to a minimization of the exchange energy and similar criteria for localized orbitals. On the other hand the desire for a compact treatment of correlation also leads to localization since its main effects are short range.I would like to ask in particular whether any specific criterion for localization arises from the alternant molecular orbital theory. Mr. M. P. Briggs (University of Sussex) said In connection with the point raised by Hall concerning the choice of localized orbitals such that they are a suitable starting point for the improvement of the wavefunction by the introduction of correlation effects I would mention some calculations as yet unpublished by J. G. Stamper and myself. For the two four-electron systems Be and LiH we have taken a limited configuration interaction wavefunction and examined the effect on the total energy of the function of a unitary transformation among the occupied orbitals E (a.u) - 14.62 5 i C F.-14640 0.1 8 (radians) FIG. l.-Plot of E against 8 for Be based on Clementi double-zeta S.C.F. function. of the ground-state configuration. The C.I. function consists of pair-correlations chosen to account for the main features of the correlation error. For example for Be the function is - I,!/ = N( I ls22s2 I +Al I lS22p2 I + A 2 I 2d22F1 +A2 I 2~’~2?1 } where the correlating orbitals 2p 2s’ and 2p’ are single Slater-type orbitals ortho- gonalized to all previous orbitals. Such a function with optimized correlating GENERAL DISCUSSION 57 orbitals gives about 70 % of the difference between the energy of the exact wave- function and that of the first determinant in the C.I. expansion. For a four-electron system the unitary transformation is a single 2 x 2 rotation ; hence the transformed starting orbitals are given by Is = cos e ls+sin e 2s; 2s = -sin 8 1s+cos o 2s.The correlating orbitals are optimized for each value of 8 taken and the energy of the C.I. function evaluated. A typical plot of energy as a function of 8 is shown in the figure for Be based on the double-zeta S.C.F. function of Clementi;' EsCF = 12.56868 (double-zeta). Although such a method does in itself provide a criterion for localization it was our intention to see how far existing criteria give good starting orbitals for this particular sort of improvement to the wavefunction. Results to date show that as expected the best transformations are rather small i.e. that the initial orbitals are well localized. They also show that the orbitals obtained by the method of Edmiston and Ruedenberg are not very good starting functions and in particular that they form a worse starting point than the S.C.F.orbitals (see fig. 1). Dr. R. A. Sack (University of Salford) (partly communicated) The product $A$B of two Slater orbitals based mainly on centres A and B lies between the nuclei and hence any expansion in functions centred on one nucleus at a time requires a relatively large number of terms to approximate the higher moments of the distribution. It is known 3 9 that can be expressed as an integral of charge distributions with centres P lying along AB resembling extended Gaussians when P lies between the nuclei and becoming weaker and at the same time more concentrated as P approaches A or B. Hence as a logical first approximation $A$B can be replaced by one or two Gaussians with centres along AB supplemented by point charges at A and B (for orbitals with Z>O the Gaussians have to be multiplied by appropriate angular factors and the point charges replaced by multipoles).Exchange integrals evaluated by such an approximation will be the more accurate the further the two charge products are separated. When one of the nuclei is common to both orbital products the Coulomb interaction of the point charge in the approximation to the one product with the whole of the other product can be evaluated in one operation as a l-electron 2-centre integral. So far I have only been able to derive explicit formulas and results for products of 1s orbitals with the same exponents. Taking the abscissas of A and B as * a the approximations are (i) point charges AS each at A and B a Gaussian integral approx.(i) approx. (ii) accurate 5 (hl,hZ ; h3,h4) 0.03023 0.03054 0.030682 (hlh2 ; hlh3) 0.03534 0603517 0-03 5 694 of total charge 4 S a t the midpoint ; (ii) point charges & S each at A and B Gaussians of total charge -& S each at x = &(a/ J7). Here S is the overlap integral and the width of the Gaussians is chosen so as to yield the same second moments as for $A$n. for two of the exchange integrals occurring in the methane molecule I obtained the results given Using the same data and notation as Shavitt and Karplus E. Clementi Z.B.M. J. Res. Dev. 1965 9 2. C. Edmiston and K. Ruedenberg Rev. Mod. Physics 1963 35 457. P. J. Roberts Proc. Physic. SOC. 1966 89 63. I. Shavitt and M. Karplus J. Chem. Physics 1965 43 398.4R. A. Sack Int. J. Quantum Chem. 1967 IS 369. 58 GENERAL DISCUSSION below. The values calculated with the aid of a Curta calculating machine and 4-figure tables are accurate to within 1-2 %; the relatively poorer result for the approximation (ii) in the 3-centre integral is probably due to the greater interpenetra- tion of the charge clouds. Further work along these lines is in progress. Prof. J. A. Pople (Carnegie-Mellon University Penn.) said I would emphasize the importance of rotational invariance in LCAOSCF schemes involving integral approximations. To be useful in chemical applications such methods must be appropriate for general three-dimensional molecules with no particular spatial symmetry where no unique choice of Cartesian axes is possible. Methods leading to results which depend on the choice of axes should in my opinion be avoided.I would also mention some work related to that of Doggett and McKendrick which has been developed at Carnegie-Mellon University principally by Dr. M. D. Newton. This is a general method of integral approximation in which two-centre differential overlap products are projected onto the space defined by a set of one- centre functions AA centred on the atoms A. Thus if +; is an atomic orbital on A and 4; is another on B the product +f+ is approximated as a sum over the sets of functions AA and AB. The coefficients dpvu and dLvu are chosen so that the integral of the square of the difference between the two sides of this equation is minimized. This method is thus described as projection of diatomic diferential overlap.It is rotationally invariant and all four-centre integrals are reduced to two-centre coulomb integrals involving the A’ functions. The set AA is taken to include the one-centre products +?#:. If the basis set q!$ consists of a minimal set of Slater-type orbitals (1s for H and 2s 2s 2p for Li to F) the products +?+; are spanned by a larger set of Slater-type functions (1s for H and Is 2s 2p 3s 3p 3d for Li to F). If AA is limited to this then each diatomic differential overlap product in (1) is written as a sum of up to 28 terms. Calculations at this level of approximation lead to total energies within 0.1 hartree of accurate values. In a more refined version of the theory the set A* is supplemented by additional 2s’ 2p’ functions for H and additional 2s‘ 2p’ 2d’ functions for heavier atoms.The expression (1) may then contain up to 46 terms. This reduces the error of calculated total energies to about 0.01 hartree. Prof. G. G. Hall (University of Nottingham) said The nub of the discussion is that the variation expression for the energy is accurate to second order in the wave- function. When the Hamiltonian and the trial functions depend on a perturbation parameter A then = wo+Aw1+A’W,+ ... being an identity in A has each term accurate to second order. This gives e.g. Wl = IJ$o*W,dz + j$T(H- Wo)$od.c + j3x(H- ~o)$ldt)/p:$od~ which is accurate to second order whereas the first term generally is accurate only to first order. belongs to the domain of the trial functions used in determining $o then the remaining terms vanish and the first term which is often assumed to be the only term becomes accurate to second order. On the other hand if

ISSN:0430-0696    

DOI:10.1039/SF9680200054

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Calculation of the Electric Dipole Moment of some Diatomic Molecule Hydrides BY F. GRIMALDI," A. LECOURT * AND C. MOSER -/- Received 6th Septetnber 1968 The electric dipole moments for six diatomic molecule hydrides LiH BH AlH FH CIH and NaH have been calculated using functions built from the molecular Hartree-Fock function plus about 200 singly and doubly substituted configurations. The same procedure proved to be quantitatively accurate for CO. In three of the six molecules the electric dipole moment has been measured and the agreement between calculated and observed moments is good. The electric dipole moment of diatomic molecules has been calculated from molecular Hartree-Fock (MHF) functions for several molecules with an error of only a few per cent.'. However when there is near cancellation between the electronic and nuclear contributions to the dipole moment a MHF can give erroneus results.The molecule carbon monoxide is a particular example of this situation. The electronic contribution given by the MHF function is (at Re calc.) - 12.425 a . ~ . ~ and the nuclear contribution is 12.486 a.u. If the MHF function predicts an electric dipole moment (+0-15 D) which is fortuitously in good agreement with the observed moment (0.12 D) the calculated sign (PO-) is contrary to the " observed " sign This coincidence led Nesbet ' to re-examine carefully the hypotheses which were used to deduce the sign of the dipole moment from a microwave e~periment.~ He showed there was an error in the reasoning of Rosenblum et al. who presumed that a positive g factor would lead to completely unreasonable mass ratios.Nesbet has shown that either sign of the g factor would give internally consistent results for the mass ratios. The final conclusion as to the sign of the dipole moment of CO could still be correct. The doubt raised of the interpretation of the microwave experiment has been resolved by Ozier et aL6 who determined the sign of the g factor (and consequently the sign of the dipole moment) from a beam experiment. The sign of the dipole moment in CO seems to be firmly established as C-O+. For molecules with a closed shell Hartree-Fock ground-state doubly-substituted configurations (dsc) give an important contribution to correlation energy but only a very small contribution to the average value of single electron operators. Singly substituted configurations (ssc) contribute much less than do dsc to the correlation energy but make relatively larger contributions to the mean value of the one electron operators.In general only a few out of all possible ssc make significant contributions. These are the ssc which arise by substituting the highest occupied orbital by the lowest unoccupied orbital.8 For the molecule CO adding the 200 most important dsc to the MHF function changes the value of the dipole moment from the MHF value of +0.15 D for +Ow07 (at R = Re ~ a l c ) . ~ Addition of more dsc would change this value of the dipole (c-0'). 5 6 * Commissariat 1'Energie Atomique 29 rue de la FCderation Paris 15e. t Centre de MCcanique Ondulatoire AppliquCe 23 rue du Maroc Paris 19e. 59 60 D I P O L E MOMENT OF DIATOMIC H Y D R I D E S moment by a negligible amount.By contrast a function which is MHF plus the I38 most important dsc plus all 62 ssc (in the basis set used) gives a dipole moment of -0.19 D which certainly confirms the sign which had been deduced from the microwave experiment. The most important contributions which are given by the ssc are those given by the substitution 1z+mn and amongst these the most important are ln+2n and 1n+3z.' We thought it would be of interest to repeat the configura- tion interaction calculation of the electric dipole moment for six diatomic molecule hydrides viz. LiH BH AIH FH CIH as good MHF functions are available from the work of Cade and Huo." We limit our list to these as our configuration inter- action programme is restricted to the use of a closed shell reference state.RESULTS AND DISCUSSION The method of calculation has been described elsewhere 8 r and so it will be necessary only to give a brief r6sumC here. The normalized configuration interaction function has the form ab d 4o is the Hartree-Fock function &{ and 4 are respectively the dcs and SCC ; a b d are occupied orbitals which are replaced by virtual orbitals a j? and y. The c are the coefficients. In the expectation value of the electronic part (z), it is useful to divide up the various contributions to the change A(Z),~ in the H.F. value. I (non diagonal b ) ] (diagonal terms dsc-dsc) terms dsc-dsc) (non diagonal terms dsc-ssc) +C(c:>"(a I z I I z I 4 1 (diagonal terms ssc-ssc) a a a # b \ (non diagonal terms ssc-ssc) (non diagonal terms 4b0-ssc). (2) The largest term in eqn.(2) is generally the last one which is the matrix element of Z between the HF function and a ssc. The reasons for this are (i) As co is nearly equal to one the element is of the order of C," (second order in the present case). (ii) In fact C may be as large as some of the largest C:f (due to summation over inter- mediate states in perturbation series s) when la) and la) are close in energy. F . GRIMALDI A . LECOURT A N D C . MOSER 61 (iii) The integrals ( a I z I a) may be large when the same condition holds which makes C; large. This is due to the fact that I a) and I a) must be orthogonal even with spatial degeneracy (cf. ref. (8)). The other contributions from ssc are negligible (i) The diagonal terms are small since when the difference ( a I z I a)-(a I z I a) is large the probability I C; I is small and vice versa.(ii) Nondiagonal terms are small as the integrals ( a I z I b) ( a I z I p ) are not large and the product of the C is of fourth order. Much the same holds for the contribution of dsc (the coefficients of all terms are of 2nd order). (i) When the difference ( a I z 1 a)-(a I z I a ) is large in the diagonal terms the coefficients C;l are small and conversely. (ii) In the nondiagonal elements the integrals (a I z I a') and ( a I z I a') are never large. Matrix elements of z between ssc and dsc are small as the coefficient CiC is quite small even if the integral ( b I z I p ) may be large. The basis sets for each molecule are those given by Cade and Hu0.l' The R values are those which correspond to the observed equilibrium distance.We have used a programme written by R. K. Nesbet and R. Stevens to calculate the H F functions and orbitals. A large number of dsc can be constructed. The most important of these are chosen amongst all possible dsc by calculating the 2nd order contribution to the energy of each configuration using the Raleigh-Schroedinger method (the sum of single particle Hartree-Fock Hamiltonians is chosen as the unperturbed Hamiltonian). The number of ssc is much smaller. We can choose the most important of these by selecting the single substitutions 4; for which ( a I z I a) is large compared to ( a I z I a) . I a) and I a) are then close in energy-the highest occupied and lowest unoccupied orbitals of the same symmetry type. We limited the number of ssc only for NaH ClH and AlH. For LiH BH and FH all possible ssc were used.For all molecules there are a total-occupied and unoccupied-of 16 (T and 8 rc orbitals. In table 1 we collect the results for (i) the electronic contribution obtained from respectively (a) the H F function ; (b) HF plus 200 dsc ; and (c) HF plus 200 dsc and ssc. Similar results are given for (ii) the total dipole moment i.e. nuclear minus electronic contributions. It is only necessary to use enough dsc to interact with the ssc and it was convenient to limit the sum of both to 200 configurations. For the three molecules for which the dipole moment has been measured LiH FH and ClH the agreement with experiment is good. The error is about 1 1 and 7 % respectively. As the calculated Re values will probably not coincide with the observed R values the larger error for ClH may not be significant.The change in the electronic part in going from function (a) to (c) is 1 4 and 3 % for LiH FH and ClH respec- tively. No experimental measurements are available as far as we know for BH AlH and NaH. AlH will be a particularly sensitive test of the value of our calculations as the function (c) gives a dipole moment which has a different sign from that given by the HF function. In table 2 we give the percentage of the contribution of 2 JTCoECz(a I z I a) to A(z)elect. and show that these terms are by Far the most important. In table 3 we give evidence that the largest part of the sum in table 2 is given by at most two configurations of the type which we have previously mentioned. As the contributions may have both positive and negative signs it is possible to have the contributions total more than 100 %.BH is an exception which merits special mention. The largest coefficients Cz in this molecule are of the order and are at least 3 times smaller than the smallest coefficients quoted in table 3. These coefficients are smaller than those of LiH as the 62 DIPOLE MOMENT OF DIATOMIC HYDRIDES TABLE 1 .-CALCULATION OF THE DIPOLE MOMENTS OF DIATOMIC MOLECULE HYDRIDES molecule LiH BH AIH FH ClH NaH nuclear part (a.u.) 3.015 2.336 3.1 14 1.7328 2.4087 3.566 a (a.u.) 5.3763 1.6541 3.0478 0.9689 1-9381 6.3054 b (a.u.) 5-3658 1.6660 3.044 0.9864 1-9983 6.2717 electronic contribution c (a.u.) 5.3028 1.6663 3.1175 1.0116 1-9984 6-1228 % change (ato c) 1.3 0.73 2.3 4.3 3.0 2.85 a (a.u.) -2-3613 +0-6819 +0.0662 +0.7639 +O-4706 -2.7393 Debye -6.0005 + 1.7338 +0.1682 + 1.9412 + 1.1958 -6.9611 total b (a.u.) - 2-3508 + 0.6700 + 0.0736 + 0.7464 + 0.4503 + 2.7057 c (a.u.) - 2.2878 + 0.6697 - 0.0035 + 0.721 1 + 0.4103 - 2.5567 Debye -5.8137 + 1-7018 -0.0089 + 1.832 1 -0426 - 6.497 1 experimental (Debye) - 5.882 + 1.8195 + 1-12 % error C - 1.19 a + 3.15 0.7 - 6.5 6.6 + 6.9 (a) Hartree-Fock function (ref.(1)) ; (6) H.F.+200 doubly substituted configuration ; (c) H.F.+200 doubly and singly substituted configurations. L. Wharton L. P. Gold and W. Klemperer J. Chenz. Physics 1960 33 1255. R. Weiss Physic. Rev. 1963 131 659. C. A. Burrus J. Chem. Physics 1959 31 1270. TABLE 2.-RELATIVE CONTRIBUTION TO A(Z)elect OF 22/2CoCC,"(a I Z ] a) (3 molecule LiH BH AIH FH CIH NaH % 80 2-5 120 71 77 78 A1H 0.972 42; 0.0183 1.513 0.075 107 * FH 0.982 44; - 0.010 - 0.780 0.0217 51 4:: - 0.022 - 0.047 0.0029 6 4; 0.001 5 0.549 0.0029 6 ClH 0979 $2; - 0.0069 - 1.327 0-0253 42 42: 0.0078 0.7687 0.0166 26 NaH 0.984 42; 0.0409 - 0.9858 -0.185 101 * * As other contributions niay have different signs one niay have more than 100 %.F. GRIMALDI A . LECOURT AND C . MOSER 63 energy difference between the highest occupied and lowest unoccupied orbital is 0.7 a.u. in BH as compared to 0.32 in LiH. The value of A ( Z ) ~ ~ ~ ~ . is small and princi- pally comes from the diagonal dss-dss terms. Our conclusion is that calculation of the important second order contributions to the dipole moment will likely make it possible to produce quantitative estimates of the dipole moment of diatomic molecules even when there is near cancellation between the nuclear and electronic parts. P. E. Cade and W. H. Huo J. Chem. Physics 1966,45 1063. S. L. Kahalas and R. K. Nesbet J. Chem. Physics 39 1963 529. W. H. Huo J. Chem. Physics 1965 43 624. C. A. Burrus J. Chem. Physics 1959 31 1270. B. Rosenblum A. H. Nethercott and C . H. Townes Physic Rev. 1958 109 400. I. Ozier P. Yi A. Khosla and N. F. Ranisey J. Chem. Physics 1967 46 1530. R. K. Nesbet J. Chem. Physics 1964 40 3619. F. Grimaldi Adv. Chern. Physics 1968 14 in press. F. Grimaldi A. Lecourt and C. Moser Int. J. Quantum Chem. 1967 IS 153. P. E. Cade and W. H. Huo J. Chem. Physics 1967 47 614 649.

ISSN:0430-0696    

DOI:10.1039/SF9680200059

出版商:RSC    

年代:1968

数据来源: RSC

摘要:

Ab Initio Computation of Spin-orbit Coupling Constants in Diatomic Molecules BY T. E. H. WALKER AND W. G. RICHARDS Physical Chemistry Laboratory South Parks Road Oxford Receiued 1st October 1968 Use of the Dirac equation leads to an expression for the spin-orbit coupling constant which involves both one and two-electron integrals the former being due to nuclear charge and the latter providing screening of this charge. In this work both types of molecular integrals are considered and expanded into atomic integrals. Application to diatomic hydrides produces results which are in excellent agreement with experi- ment for molecules including a first row atom and further demonstrates that the two-electron two centre integrals are very small. Calculations on non-hydrides indicate that this is not always the case.1 . INTRODUCTION Experimental spectroscopists often use the measured value of the spin-orbit coupling constant A of a molecule to assign a molecular configuration to an observed electronic state. The properties of a Hartree-Fock wavefunction which are most often used to correlate it with an observed state are the energy and those constants related to the change of energy with internuclear distance such as the vibration frequency. Hartree-Fock wavefunctions give values of these constants that are incorrect and it seems reasonable to use a value of A obtained from the wavefunction to obtain a more unambiguous correlation with the observed state. Moreover the value of A obtained would not necessarily have the large errors inherent in a calculation of the total energy neglected the two-centre integrals and assumed that the one-centre integrals were identical with those in the isolated atom.The second did include some two-centre integrals but neglected exchange terms. This work will show that these assumptions are not always justified. The Dirac Hamiltonian has been used for the calculation of spin-orbit coupling in and has given good results when the spin-orbit coupling between the outer electrons and the closed shells is included. This interaction is proportional to the spin-orbit coupling and reduces the value of A from that given by simple Z / r 3 terms. Only a few accurate expectation values of molecular two electron properties (other than the energy) have been carried O U ~ ~ - ~ and in one of these the wavefunction was based entirely on one centre.6 A calculation of A will also serve as a test of Hartree-Fock molecular wavefunctions for two electron properties.There have been previous calculations of A the first of which 2. METHOD The Pauli approximation of the Dirac equation gives H, = (a2/2)o(grad U x y) 64 T. E . H . WALKER A N D W. G . RICHARDS 65 where CI is the fine structure constant Q is the Pauli matrix U is the potential and p the momentum of the electron. in their calculations on NO. However when a Hartree-Fock wavefunction is used it is not clear what should be employed for the potential U and particularly how exchange terms are to be included. The previous workers did not include exchange terms and this could account for their calculated value being higher than experiment. We follow Blume and Watson in using the Dirac Hamiltonian which was given by Kayama and Baird * for a diatomic molecule as This was used by Hellmann and Ballhausen where capital letters refer to nuclei and small ones to electrons and r, = rf -rj.The Hamiltonian in this form is not suitable for the calculation of matrix elements ; Blume and Watson rewrote it in tensor operator form and were then able to calculate the matrix elements between atomic orbitals. Because I is not a good quantum number in a diatomic molecule a molecular orbital is composed in general of atomic functions of s p d andfcharacter all with the same value of ml (= A). Thus the matrix elements between molecular orbitals cannot be given by simple expressions analogous to those for atomic orbitals. However since the molecular orbitals are expanded in terms of atomic Slater type functions the molecular matrix elements reduce to sums of individual atomic matrix elements which can be calculated using Blume and Watson's expression for the Hamiltonian.Another problem arises when the matrix elements involve atomic orbitals centred on different atoms. In this case the integration cannot be carried out analytically with ease and numerical techniques must be used. These are relatively trivial for the one-electron integrals but are lengthy for the two electron integrals. The resulting expressions contain matrix elements of both one and two electron operators. The former are the simple Z/r3 terms and the latter having the opposite sign reduce the value of A and may therefore be interpreted as that part of the potential which shields the outer electron from the nuclear charge.3. RESULTS Table 1 gives the results for 211 states of a number of diatomic hydrides containing both first- and second-row elements and also the two non-hydride molecules BO andCO+. All integrals except for the two-electron two centre ones areincluded. This method gives good results for hydrides containing first row elements while the values obtained when second row elements are involved are all too low. The same divergence has been obtained for calculations performed on atoms.4 It has been attributed to the effects of spin and orbital " polarization ". In the usual Hartree-Fock approximation p orbitals have the same spatial wavefunction whatever the value of rn and m,. If the atom has an open shell configuration this degeneracy is removed.In unrestricted Hartree-Fock calculations the p orbitals of different m and m are allowed to have different spatial wavefunctions and this results in closed p shells having a resultant contribution to the ( Z / r 3 ) terms. This will therefore raise the final value of the spin-orbit coupling constant. All semi-empirical methods for calculating A involve the approximation that the one-centre integrals remain unchanged in going from the atom to the molecule. 5 TABLE 1 .-CALCULATIONS OF MOLECULAR SPIN-ORBIT COUPLING CONSTANTS molecule (AH) BeH(C) BH+ CH@) NH- OH(C) FH+ MgH AIH+ SiH PH- SH BO co+ zAlr2 (ii) 0) 5.90 5.96 29.8 30.1 56.1 56.5 88.96 93.7 216.8 2 17-9 444.7 445.1 35.86 36.2 119.7 121.2 165.0 166.5 219.3 222.3 437.0 443.2 1.00 3.3 6.25 10.2 0.003 0.0008 0*0006 0.012 0.0007 0-004 o.Oo01 0.002 0.001 0.004 0.005 195.32 186.29 (ii) 0.20 0.42 0.60 0.75 0.96 0-77 0.10 0.18 0.22 0.26 0.35 198.32 192.80 - 3-82 - 14.96 - 26.8 - 35.95 - 77.5 - 137.2 -9.1 - 26.9 - 35.5 - 35.5 - 80.98 - 0.52 - 2.82 -0*001 - 0.0021 -0402 - 0.003 - 0,006 -0.0003 - O*OOo7 - O*OoO7 -0.0013 - 0.0024 0 - 70.54 - 64.82 calc.2-3 15-9 30.4 57.8 141.4 305.8 27.2 94.6 139.2 178.5 362.0 133.4 134.3 A obs.10 2.14 14.0 28.0 139-7 35 108 142 382.4 116.7 1175 I - - (i) one centre contribution ; (ii) total contribution ; (a) contribution from atom A ; (b) contribution from hydrogen ; (c) the values for BeH CH OH are taken from a previous paper by the authors.' TABLE 2.-cOMPARISON OF ATOMIC AND MOLECULAR ONE-CENTRE INTEGRALS CONTRIBUTING TO SPIN-ORBIT COUPLING exchange {atom- Amol AH state atom 4 mol atom 4 mol atom 4 mol calc.expt.4 10 Zit3 direct BeH A211r 6.9 5 -9 - 3.0 - 2.3 - 1.79 - 1.48 0 0.12 MgH A211r 42.7 35.9 - 8.09 - 6.76 - 3.12 - 2.34 4.3 5.5 CH X 2 n r 58.5 56.1 - 15.5 - 15.7 -11.4 -1194 2 4.6 OH X2ni 230 21 6.8 - 44.0 - 45.1 - 3.4 - 32.3 4.6 6.0 SiH X21Tr 168.2 165.0 - 26.56 - 25.84 - 10.73 - 9.66 0 6.0 SH X2lli 452.2 437.0 - 60.3 1 - 58.54 -21.81 - 22.44 4.98 1.6 T. E . H . WALKER AND W. G . RICHARDS 67 Table 2 compares some of these values and shows that this is not always the case particularly for the Z / r 3 terms. The Hartree-Fock wavefunctions reproduce the change from atom to molecule quite accurately both for the first and second row hydrides. The contribution from the exchange terms is of the same order as the direct terms emphasizing their importance and providing an explanation for the high values previously calculated.2 Work on the two-centre two-electron integrals has not yet been completed but these results show that they make onlya small contribution in the hydrides of first row elements whose calculated values of A are only slightly higher than experimental.Table 3 gives the numerical values for some of the two electron integrals involved in the direct term (M" in the notation of ref. (3)). The two-centre integrals are often orders of magnitude smaller than the one centre and this together with the small contribution of the hydrogen basis orbitals to the wavefunctions makes their neglect reasonable. However when the charge density is more evenly spread over the molecule the two-centre integrals would be expected to make a larger contribution to the value of A and the higher calculated values of BO and COf may be ascribed to their neglect.TABLE 3 .-SELECTED TWO ELECTRON TWO-CENTRE INTEGRALS a = 2p centre A orbital exponent Z = 10.0 a' = 2p centre A 7 9 9 9 9 2.0 b = 2p centre B 9 9 7 , 10.0 b' = 2p centre B 9 9 7 99 2.0 The integrals considered are electron 1 electron 2 integral electron 1 electron 2 integral aa aa 106.5 a'a' 0.64 bb 1*3glO - 12 a'a' b'b' 6.521 0 - 4 ab' 0.18 a'b' 1-9610 -2 4. CONCLUSIONS Provided that spin-other orbit interaction between the outer electrons and the closed shells is included good agreement is obtained for calculations of the spin- orbit coupling constants of first-row diatomic hydrides. For molecules involving nuclei of more equal charge the two-centre integrals become important but even neglecting them the computed values of the coupling constant are accurate enough to resolve some of the experimentally difficult problems concerning the nature of excited states.The authors thank Dr. P. E. Cade for communicating his wavefunctions to them before publication. T. E. H. W. thanks Brasenose College Oxford and the Science Research Council for research grants. 68 SPIN-ORBIT COUPLING CONSTANTS E. Ishiguro and M. Kobori J. Physic. Soc. Japan 1967 22,263. * J. Hellmann and C. J. Ballhausen Theor. Chim. Acta 1965 3 159. M. Blume and R. E. Watson Proc. Roy. SOC. A 1962 270 127. M. Blume and R. E. Watson Proc. Roy. Soc. A 1963,271,565. J. W. McIver and H. F. Hameka J. Chem. Physics 1966,45767. J. B. Lounsbury J. Chem. Physics 1967,46,2193. ’ K. Kayama J . Chem. Physics 1965,42,622. a K. Kayama and J. C. Baird J. Chem. Physics 1967,46,2604. H. Lefebvre-Brion J. Wajsbaum and N. Bessis La Structure hyperfine magnetique des atomes et des molicules (C.N.R.S. Paris 1967) p. 299. G. Herzberg Spectra of Diatomic Molecules Van Nostrand New York 1950. l 1 T. E. H. Walker and W. G. Richards Physic Rev. 1969 177 100. l2 T. E. H. Walker and W. G. Richards Proc. Physic. SOC 1967,92,285.

ISSN:0430-0696    

DOI:10.1039/SF9680200064

出版商:RSC    

年代:1968

数据来源: RSC