INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XVI, 1265-1277 (1 979)

Transferable Integrals in a Deformation-Density Approach to Crystal Orbital Calculations. I

JOHN AVERY Department of Physical Chemistry, H. C. 0rsted Instifute, University of Copenhagen, Copenhagen,

Denmark

Abstracts

In the usual ab initio method of calculating molecular orbitals, the number of integrals to be evaluated increases as M4, where M is the number of basis functions. In this paper, an alternative method is discussed, where the computation time increases much less violently with the number of basis functions. Matrix elements of the deformation potential are evaluated by Fourier transform methods, while matrix elements of the neutral-atom potential are evaluated by means of transferable integrals. The transferable integrals (moments of the neutral-atom potentials) can be evaluated once and for all and incorporated as input data in computer programs. In an appendix to the paper, a general expansion theorem is discussed. This theorem allows an arbitrary spherically symmetric function to be expanded about another center.

Dans la mtthode ab inifio ordinaire pour calculer des orbitalesmolhlaires le nombre d’integrales B Cvaluer augmente comme M4, ou M est le nombre de fonctions de base. Dans le prtsent article une methode alternative est discutte ou le temps de calcui augmente beaucoup moins rapidement. Les tlkments de matrice du potentiel de deformation sont cafcul6s par des mtthodes utilisant des transformies de Fourier, tandis que les 6ltments de matrice du potentiel des atomes neutres sont calcults par des integrates transmissibles. Ces inttgrales transmissibles (moments des potentiels des atomes neutres) peuvent &re Cvalutes une fois pour toutes et incorporkes comme input dans des programmes d’ordinateur. Dans un appendice on discute un thtorkme de dtveloppement g6nkral. Ce thCorbme permet un dtveloppement d’une fonction radiale arbitraire autour d’un autre centre.

In der ubiichen ab-initio-Methode fur Molekulorbitalberechnungen wachst die Anzahl von Integralen, die berechnet werden miissen, wie &, wo M die Anzahl von Basisfunktionen ist. In diesem Artikel wird ein alternatives Verfahren diskutiert, wo die Rechnungszeit vie1 weniger schnell mit M wachst. Matrixelemente des Deformationspotentials werden mit Fourier-transformmethoden berechnet, wahrend Matrixelemente des Potentials der neutralen Atome mit Hilfe von iibertragbaren Integralen berechnet werden. Diese ubertragbaren Integrale (Momente der Potentiale fur neutrale Atome) konnen ein fur allemal berechnet und als Inputdaten in Computerprogramme eingegtiedert werden. In einem Appendix wird ein allgemeiner Entwicklungssrtz diskutiert, der eine Entwicklung einer wilkiirlichen kugelsymmetrischen Funktion um ein anderes Zentrum erlaubt.

Introduction

In the usual method for calculating molecular electronic structure, the mole- cular spin-orbitals are expanded as linear combinations of atomic spin-orbitals:

@ 1979 John Wiley & Sons, Inc. 0020-7608/79/0016-1265 $01.00

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Substitution of this expansion into the Hartree-Fock equations leads to Roothaan’s equations:

where

and

In the usual ub initio method, the supermatrix of Coulomb and exchange integrals is calculated exactly. This means that roughly M4 integrals have to be calculated exactly, where M is the number of basis functions. In order to achieve good accuracy, one must use a basis set that contains a number of functions several times as great as the number of filled orbitals in the system. Serious errors can result if the basis set is truncated. Thus, for example, if a molecule contains 100 electrons, at least 100 basis functions must be used, and the number of elements in the supermatrix reaches lo8. This violent dependence on the number of basis functions makes it difficult to apply the usual ab initio method to molecules that contain a large number of electrons. In this paper we shall discuss an alternative method, where the number of integrals to be evaluated depends much less violently on the number of basis functions. In this alternative method, the potential is divided into a large part, which would be produced by neutral atoms located at the various atomic sites, and a small correction (the deformation potential) due to the effects of chemical bonding. Matrix elements of the neutral- atom potentials are evaluated by means of transferable integrals, while matrix elements of the deformation potential are evaluated by a Fourier transform method. In this Fourier transform method, the matrix elements are converted into reciprocal lattice sums, a procedure that can only be applied in the case of crystals. If one is interested in isolated molecules, rather than in crystals, the transferable integral method can still be used for the neutral-atom potentials, but matrix elements of the deformation potential must be evaluated by some other means. In general, the deformation potential will be a small correction to the total potential, and therefore, in the case of isolated molecules, matrix elements of the defor- mation potential could be evaluated using methods of lower accuracy, such as the Mulliken approximation. In the discussion that follows, however, we shall confine

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ourselves to the case of crystals, where the lattice-sum method can be used, and where crystallographic measurements of charge density distributions are avail- able. We shall also. confine our attention to nonconducting crystals.

Deformation Density

Much progress has recently been made in the x-ray analysis of charge densities [ 1-51. In experimental charge density analysis, one usually begins by determining the nuclear positions in a crystal by means of neutron diffraction experiments. This is followed by the collection of high-quality x-ray diffraction data, preferably at low temperatures. From the x-ray data, one constructs the measured total charge density p ( x ) in the crystal. One then calculates the neutral-atom charge density po(x ) by superimposing the spherically averaged densities which would be produced, in the absence of any chemical bonding, by neutral atoms located at the positions determined by neutron diffraction. The difference, A p ( x ) , between the measured total density and the neutral-atom density is called the “deformation density,” and it can be ascribed to the effects of chemical bonding. The defor- mation density contains zero total charge, and, at most points in the crystal, its magnitude is very much less than that of the total charge density. This is particularly true in the neighborhood of the nucleus of a heavy element, where the charge density of the inner electrons is very little deformed by chemical bonding. One can also define a “deformation potential” AV as the potential produced by the deformation density Ap. Thus, if we express the total density p as the sum of the neutral atom density po and the deformation density A p

P = P O + A P , (7)

then the total potential will be the sum of the neutral atom potential and the deformation potential:

V = Vo+AV, (8) where

and d 3x ’ A p ( X I )

AV(x) = Ix - X‘I

and where po includes the nuclear charge.

Matrix Elements of the Potential

We shall begin to calculate the Fock matrix Fab of Eq. ( 3 ) by calculating the matrix elements of the potential V which can be divided into two parts, Vo and A V :

v a b = d3x X S VXb = (VO)ab ( A V)nb, (11)

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where

and

(A V)ab = I d3x xg A vxb. (13)

In general, the largest contribution to vab will come from (VO)ab, while (A will be a small correction. We shall use Fourier transform methods for the evaluation of (AV)ob, while (V&b will be evaluated by means of transferable integrals.

Fourier Transform Evaluation of (A Vh, Since we are dealing with the potential in a crystal, AV must have the

periodicity of the direct lattice. Hence A V can be expressed as a Fourier series of the form:

A V(x) = 1 (A V)K exp(iK - x), K

where (A V)K are constant Fourier series coefficients and the sum is taken over the reciprocal lattice vectors of the crystal. Then

(AV)ab = C ( A W K J d3x exP (iK ’ XI XaXb (Av)Kxab(K)? (15) K K

where

Xab(K) = d3x eXp (iK * X) xzxb (16)

is a “generalized scattering factor [6-121”. The reciprocal lattice sum in Eq. (15) converges rapidly because AV is a diffuse function in direct space, and therefore its Fourier transform is concentrated near the origin in reciprocal space. In order to evaluate the sum in Eq. (15), we need to know the Fourier series coefficients (A V)K. Here we have the choice between two alternative methods: Either we can find the coefficients (AV), from the Fourier series coefficients (Ap)K of the deformation density as measured in x-ray experiments using the relationship:

(A V)K = (471./K2)(Ap)K, (17) or else we can calculate the coefficients iteratively from the charge and bond-order matrix using the relationship:

Here Pab and x a b are given by Eqs. (5) and (16), respectively, while the coefficients ( A p ) K are defined by the relationship:

Ap = 1 ( A p ) K exp (iK x). (19) K

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Transferable Integrals

Let us turn next to the problem of evaluating Eq. (12). We can write the neutral-atom potential in the form:

where the sum is taken over all the atoms in the crystal. In Eq. (2) , c is an atomic position, and V, is the spherically averaged potential due to the atom located at x = c . Then

(21 )

Our method of evaluating this term will be to expand xx (x - a) Xb(x - b) about the point x = c. Let us consider first the case where x : (x - a) and X b ( X - b) have spherical symmetry about their respective centers. Then we can write ,ya* (x -a) in the form:

X : ( X - a) = A (ra 1, (22 )

where A(r,) is some function of r, = Ix-a\. We now expand A ( r , ) about the point x = c in terms of Legendre polynomials and in powers of r = Ix - cl.

where PI is a Legendre polynomial and

R, = a - c .

As we shall show in the Appendix, the function al(r) will then be given by

where

Similarly, if X b ( X - b) is spherically symmetric about the center x = b, we can write:

X b ( X -b) = B (rb) (27)

and expand B(rb) in the series

where Rb = b - c

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and (-2)'(21+ l)(n + l ) ! R' 1 a a a n (- -)l(A - R 2 -) B ( R ) . (30) n!(21+1+2n)! R 8R R 8R 8R Bl,n ( R 1 =

Substituting these expansions in to Eq. (21), and carrying out the angular integrations with the help of the relationship:

we obtain:

(32) where

m m m I0 dr r2 al(r) bl(r) Vc(r) = c Al,,(Ra) Br,,,(Rb) I dr r2'r+n+n'+1) Vc ( r ) . n,n'=O 0

(33)

In this way, the matrix elements (VO)ab can be expressed in terms of the moments of the neutral-atom potentials. The important thing to note about these moments is that they are transferable. They can be evaluated once and for all, and incorporated as input data of computer programs.

Let us next consider a case where the atomic orbitals xa and Xb are not spherically symmetric about their respective centers. The simplest example of this type is the case where X b is an s-orbital and can still be represented by Eq. (27), but where xa is a p,-orbital. In that case, ,ya can be written in the form:

X X (x-a) = (2 - Raz)A(ra). (34)

Then, still using the expansions [Eqs. (23) and (28)], we have: I d 3 ~ ~ X ( ~ - a ) ~ b ( ~ - b ) V c ( x - c ) = dx3 ( t - R a , ) A ( r a ) B ( r b ) V,(r) I

We are now faced with the angular integral

(36) Using the addition theorem for spherical harmonics

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We can write this integral in the form:

where the coefficients

are Condon-Shortley coefficients, and where 0, and 4. are the polar coordinates associated with R,, while Ob and 4b are those associated with Rb. The matrix element then becomes

I d 3 x x n * ( x - a ) x b ( x - - b ) V c ( x - c )

We can see from this example that the more general cases, where the atomic orbitals do not have spherical symmetry about their centers, lead to somewhat more complicated expressions. But even in these more general cases, the angular integrations can still be performed with ease by means of the addition theorem for spherical harmonics, and the matrix elements can still be expressed in terms of the moments of the neutral-atom potentials.

Separation of the Neutral-Atom Potentials into “Hard” and “Soft” Parts

The neutral-atom potential V c ( r ) can be chosen to include both the contribu- tion of the nucleus at x = c , and the contribution of the electrons. When the potential of a neutral atom includes both these contributions, it has an asymptotic behavior of the form:

-0

V C ( I ) - c-, (41)

where f is a screening parameter that is typically of the order of 1 or 2 r.a.u. (reciprocal atomic units). However, in cases where xa and X b fall off still more rapidly than V c ( r ) with increasing values of r, and rb, large contributions to the matrix element may come from the neighborhoods near x = a and x = b, and in these neighborhoods the expansions [Eqs. (23) and (28)] are inappropriate. In order to avoid this difficulty, and to ensure the convergence of the series [Eq.

,+oo r

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(33)], we shall split the neutral-atom potential into a “hard” part:

v h c ( r ) = exp [ - (r/rd21 vC ( r ) (42)

Vsc(r) ={I -exp [-(r/ro)21}~c(r). (43)

and a “soft” part:

The “hard” part of the potential is localized in the near neighborhood of the atom c, while the “soft” part is diffuse. In reciprocal space, the opposite relationship holds: The Fourier transform of the “hard” part of the potential is diffuse in k-space, while the Fourier transform of the “soft” part is localized in the neighborhood of k = 0. Since the “soft” part of the potential is localized in k-space, we can include it, together with the deformation potential, in the reciprocal lattice sum of Eq. (15). The procedure used here is similar to that used by Ewald in his method for evaluating lattice sums.

In our numerical work, we have chosen a value of ro equal to 0.5 a.u. We have calculated moments of vhc(r) for all elements up to 2 = 36 using potentials derived from Clementi’s Hartree-Fock atomic wave functions. The tabulated moments will be given in part I1 of this paper.

Fourier Transforms of V,(r)

Having split the neutral-atom potential into a “hard” part and a “soft” part [Eqs. (42) and (43)], and having decided to add the Fourier series coefficients of the “soft” part to those of the deformation potential, we now need to calculate the Fourier coefficients of V,. We can do this in the following way: Let {X} be the set of direct lattice vectors of the crystal, while {K} is the set of reciprocal lattice vectors. Then the “soft” part of the neutral-atom potential can be written in the form:

where cc is a sum over the atoms of a unit cell, while 6 , is the position of atom c with respect to a standard point in the unit cell. If we write Vs(x) as a Fourier series of the form:

then the Fourier series coefficients will be given by

( v ~ ) K = ~ ~ J d3x exp(-iK. x)Vsc(x-&) v c

which can be rewritten in the form:

DEFORMATION-DENSITY APPROACH. I 1273

where Y is the volume of the unit cell and

is the Fourier transform of the soft part of the potential of a neutral atom of type c. We have calculated these Fourier transforms for all the elements up to 2 = 36, and tables of the transform will be given in part I1 of this paper.

Exchange and Correlation

In the previous sections we have discussed a method for calculating matrix elements of the Coulomb potential. We have now to find some method of dealing with exchange and correlation. The simplest way of doing this is to make use of an exchange-correlation potential based on a local density approximation such as the one used by Gunnarson and Johansen [15,16]. We hope to discuss the matrix elements of the exchange-correlation potential in another paper.

Appendix: Theorem for the Expansion of a Spherically Symmetric Function about Another Center 117-261

Theorem

Let F(lx-RI) be an arbitrary analytic function of Ix-RI. Then, within the region of convergence,

where the Pi's are Legendre polynomials, r = 1x1, R = IRI 00

f d r ) = C Fi,n(R)r'f2" n=O

and

(-- -)'(A (-2)'(2/ + l)(Z+n)! R' 1 a n!(21+ 1 +2n) ! R ~ I R R aR aR F1.n ( R ) =

Proof

transform of F(lx-RI) is Let us take the Fourier transform of both sides of Eq. (49). 'The Fourier

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where

F'(k) - [ ( 2 ~ ) ~ ] - ' / ~ I d3k exp(ik x) F(r). (53) Expanding exp(ik R) in terms of spherical Bessel functions and Legendre polynomials, we obtain:

k * R m

IF(Ix-Rl)l' = exp(ik. R)F'(k) = C (21 + l)iiPl(--) jl(kR) F'(k) . (54) r=o kR

The Fourier transform of the right-hand side of Eq. (49) is

DEFORMATION-DENSITY APPROACH. I 1215

where, in the last step, we have used the usual expansion of exp( - i k x), together with a relation similar to Eq. (56). Finally, cancelling B[ (x R ) / r R ] from both sides, we have an integral representation of f i ( r ) :

The next step is to expand j l (kr ) in powers of r:

(- l)"(n + I)! (kr)'+'" jl(kr)=2' c "=O n!(21+1+2n)! *

Substituting the expansion (61) into (60), we obtain:

(:)"' lorn dk k'+2n+2jl(kR) F' (k) . O0 (- l)"(n + 1) ! rlc2"

f r (r ) = 2'(21+ 1) c "=O n ! (22 + 1 +2n)!

We know that

F ( R ) = [ ( 2 ~ ) ~ ] - ~ / ' d3k exp( - ik . R)F'(k)

= ($)"' lom dk k 2 jo(kR) F'(k)

and -V@(R)=[ (27r ) 3 3 - 1 / 2 [ d 3 k e x p ( - i k . R ) k 2 F ' ( k )

= (:)I" JOm dk k4 jo(kR) F' (k) (64)

and in general,

where

V ? = ( z g R 1 a 2 a -) " aR

We also know, from the recursion relations for spherical Bessel functions, that

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Therefore we can write:

1 a 1 2 ' / 2 R ~ R T

= (- 1)"+'R1(- -) (-) I dk k2n+2 jo(kR)F'(k)

Finally, substituting this result into Eq. (62 ) , we obtain Eqs. (60 ) and (61) .

Acknowledgments

It is a pleasure to thank Professors A. J. Freeman, D. Ellis, C. J. Ballhausen, J. P. Dahl, R. W. Stewart, T. Maslen, and F. Harris and Lektors A. E. Hansen, H. Johansen, K. J. Watson, and B. Voigt for many helpful conversations. I would also like to thank the North European University Computing Center at Lundtofte Denmark, for the use of their facilities. Finally, I am grateful to Dr. A. Zugner for sending me a preprint of one of his papers.

Bibliography

[l] P. Coppens and M. S. Lehman, Acta Crystallogr. B 32, 1777 (1976). [2] F. L. Hirshfeld, Acta Crystallogr. A 31, S224 (1975). [3] P. Coppens and E. D. Stephens, Adoances in Quantum Chemistry (Academic, New York, 1976). [4] E. N. Maslen, Acta Crystallogr. B 24, 1172 (1968). [5] P. Becker, Phys. Scripta 15, 119 (1977). [6] J. S. Avery, Theor. Chim. Acta 39, 281 (1975). [7] F. E. Harris and H. H. Michels, Adv. Chem. Phys. 13, 205 (1967). [8] J. S. Avery and K. J. Watson, Acta Crystallogr. A 33, 679 (1977). [9] A. Graovak, H. J. Monkhorst, and T. Zivkovik, Int. J. Quantum Chem. 7,233 (1973).

[lo] R. F. Stewart, J. Chem. Phys. 51,4569 (1969). [ l l ] G. S. Chandler and M. A. Spackman, Acta Crystallogr. A 34, 341 (1978). [12] H. J. Monkhorst and F. E. Harris, Int. J. Quantum Chem. 6, 601 (1972). [13] J. S. Avery and M. Cook, Theor. Chim. Acta 35,99 (1974). [14] F. Harris, Computation Methods in Quantum Chemistry, preprint (Quantum Chemistry Group,

[15] 0. Gunnarsson and P. Johansson, Int. J. Quantum Chem. 10,307 (1976). [16] A. Zunger and A. J. Freeman, Int. J. Quantum Chem., Quantum Chem. Symp. 10,383 (1976). [17] P. 0. Lowdin, Adv. Phys. 5, 96 (1956). [18] M. P. Barnett and C. A. Coulson, Phil. Trans. R. SOC. (Lond.) A 243,221 (1951). [19] M. P. Barnett, in Methods in Computational Physics, B. Alder, S. Fernbach, and M. Rotenberg,

[20] R. A. Sack, J. Math. Phys. 5, 245 (1964). [21] F. E. Harris and H. H. Michels, Adv. Chem. Phys. 13, 205 (1967). [22] F. E. Harris and H. H. Michels, J. Chem. Phys. 43, S165 (1965). [23] H. J. Silverstone, J. Chem. Phys. 47, 537 (1967).

University of Uppsala, Sweden, 1973).

Eds. (Academic, New York, 1963), Vol. 2, p. 95.

DEFORMATION-DENSITY APPROACH. I 1277

1241 E. 0. Steinborn and E. Filter, Theor. Chim. Acta (Berlin) 38, 273 (1975). [25] R. R. Sharma, Phys. Rev. A 13, 517 (1976). [26] M. Abramowitz and I. A. Stegun, Handbook ofMafhernafical Functions (Dover, New York,

1972).

Received March 7, 1978 Revised February 12, 1979 Accepted for publication February 20, 1979