Transferable Integrals in a Deformation Density Approach to Crystal Orbital Calculations. V. Coupling Coefficients for Crystal Symmetry

Functions

JOHN AVERY AND PER-JOHAN (BRMEN Department of Physical Chemistry, H . C. (9rsted Institute, University of

Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

Abstract

Coupling coefficients for symmetrized sinusoidal functions in crystals are discussed. These coupling coefficients, which are analogous to Condon-Shortley coefficients, can be used to calculate generalized scattering factors.

Introduction

In parts I-IV of this series [ 1-41, we discussed a proposed new method for calculating crystal orbitals, making use of transferable integrals. In this method, the Coulomb potential experienced by an electron in a crystal is divided into two parts: a large part VO due to the neutral atom potentials, and a small correction AVdue to the effects of chemical bonding. In our method, matrix elements of the neutral atom potential VO are evaluated by means of transferable integrals, while matrix elements of the deformation potential AV are converted into re- ciprocal lattice sums.

If the Fourier coefficients of the deformation density Ap are known from crystallographic experiments [ 5-81, then the Fourier coefficients of the defor- mation potential can easily be found by means of the relationship:

Here ( A U K and (Ap), are coefficients in the Fourier series expansions:

and

Ap(X) = (Ap)KeiKeX (3) K

and K represents a reciprocal lattice vector. Equation (1) follows from (2) and (3) because AVand Ap are related by Poisson's equation.

International Journal of Quantum Chemistry: Quantum Chemistry Symposium 14,629-636 (1980) 0 1980 by John Wiley & Sons, Inc. 01 6 I -3642/80/0014-0629S.01 .OO

630 AVERY AND 9 R M E N

Let us now consider the matrix representation of AV based on Bloch functions of the form:

&,a ( x ) = e iqXvu(x ) (4) where the periodic part of the Bloch function is given by

In Eq. (9, xu is an atomic orbital, X is a direct lattice vector, and 6, is a vector giving the position of the atom on which xa is located with reference to a standard point in the unit cell. Then, from Eqs. (2) and (4), the matrix elements of A V will be given by

.fd3X dJi.aAV4q.b = . f d3xvu*(x )AV(x)vb (x )

= C ( A v ) , . fd3xe iK*xvz (x ) r ]b (x ) (6) K

The sum in Eq. (6) can be expected to converge rapidly since the deformation potential is delocalized in direct space, and hence localized in reciprocal space.

A number of methods are available for evaluating the generalized scattering factors [3,9,10]:

. fd3xeiK*"v",x)q , (x) (7)

which appear in Eq. (6), and some of these methods have been discussed in Parts I-IV. In this article, we would like to discuss an alternative method which makes use of the point group symmetry of the crystal.

Evaluation of Generalized Scattering Factors by Means of Group Theoretical Coupling Coefficients

Let R1, R2, R3, . . . , R, be the elements of the point group of a crystal, so that

is a projection operator corresponding to the mth basis function of the 7th ir- reducible representation of the group. If we act on the function e iKoX with this operator, we will obtain a symmetrized function,.

S & . K ( ~ ) = Pi,eiK*X (9)

which has the translational symmetry of the crystal (since K is a reciprocal lattice vector) and which transforms like the mth basis function of the 7th irreducible representation of the crystal's point group. (The contour diagram of a function of this type is shown in Fig. 1 .)

If we multiply two such functions together, we will obtain a product which

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632 AVERY A N D P R M E N

s g ( x ) = ' [cos E (x + y )] + cos [F 0, + z)] + cos E (x + z)] 6

["," ] (x - .)I + cos [? ( x - 4} + cos -(y - z) + cos

etc. (13) where

S,(x) = PAlgei2~/d(hx+ky+1z), h,k,l = O , f l , f 2 , . . . , j = h 2 + k 2 + l 2 (14)

From Eq. ( 1 2) we obtain the following multiplication table for these func- tions:

S&j = sj

1 1 2 2

s3s4 = -s3 +-s11

etc. (15)

Now let us expand the periodic function v , ( x ) of Eq. (5) in terms of the functions S,(x) . For example, suppose that

v a ( x ) = c e - l l x - X l (16) X

where the direct lattice vectors of the fcc lattice are

TRANSFERABLE INTEGRALS. V 633

X = ~ I U I + m2u2 + m3u3. m1.m2,m3 = 0, f l , f 2 , . . . u1 = (- d d - ,o)

2 ' 2

d d

Using the relationship [ 121

we can show that if

then

where u is the volume of the unit cell. From Eq. (20) it follows that va(x) can be written in the form:

where the expansion coefficients a, are given by

(22) 329nj (

d3 [(F)) + r2l2 aj = , j = h 2 + k 2 + I 2

Here nj-I is the coefficient of SO in the product S,S;. From Eq. (1 5) it can be seen that no = 1, n3 = 8, n4 = 6, ns = 12,. . . , etc. Note that the functions S,(x) are normalized in such a way that if Nu is the total volume of the crystal, then

Now suppose, for example, that qb(x) is given by

Then, from Eqs. (1 9) and (20),

where = b&o(x) + b3S3(x) + b&4(X) + bsSs(x> + * - (25)

634 AVERY A N D P R M E N

From Eq. (1 5) we can see that the product of the two functions is given by

VI,(X)Vb(X) = CosO(X) + C 3 S 3 ( X ) + C 4 S 4 ( X ) + ~ s S s ( x ) + * * (27)

where

etc. (28)

Since the deformation potential is invariant under the elements of the crystal point group (i.e., since it transforms like the Al , representation), we can rep- resent it by a series of the form:

AV(X) = doSo(x) + d 3 S 3 ( ~ ) + d 4 S 4 ( ~ ) + d g S g ( x ) + * (29)

Thus, finally, combining Eqs. (6), (23), (27), and (29) we have for the matrix elements of the deformation potential in our simple example:

where the coefficients Cj are given by Eq. (28).

Effects of Truncating the Fourier Series

If great accuracy is desired in evaluating the matrix elements of the defor- mation potential, it may be necessary to take into account the effects of trun- cating the Fourier series. Figure 2 shows, for example, a function of the form given in Eq. (16) together with its Fourier series representation [Eqs. (21) and (22)] truncated after the first 13 symmetrized functions. This truncation cor- responds to excluding all the regions of the reciprocal lattice outside the sphere with h2 + k2 + l2 = 32. It can be seen from Figure 2 that the truncated Fourier series gives a very accurate representation of va(x) in almost all regions of space, but the truncated series fails to represent accurately the cusp at the nucleus. Of course the cusp could be more accurately represented by extending the series.

TRANSFERABLE INTEGRALS. V 635

Figure 2. The function Z,e-clx-x I is shown tagether with its representation by a truncated Fourier series.

However, since the region of direct space where the Fourier series fails is confined to the near neighborhood of the nucleus, it may be more convenient to divide qla into a “soft” part qa,s and a “hard” part 7a .h = qa - the soft part being the part represented by the truncated Fourier series, and the hard part being the remainder. Then the product q a ( x ) q b ( x ) can be represented to a high degree of accuracy by

q a ( X ) q b ( X ) 1 q a , s ( x ) q b , s ( X ) + ar ] la ,h (x ) + P q b , h ( X ) (31) where, in our simple example, the constants a and are given by

The contribution to the matrix elements from the product of the two soft parts has already been discussed, while the correction due to q a , h ( x ) + PT1b.h ( x ) may conveniently be evaluated by integration in direct space.

Acknowledgments

The authors are grateful to Lektor Helge Johansen for the use of his contour plotting program. We would also like to thank the Northern European University Computing Center of Lundtofte, Denmark, and the Computing Center of the University of Lund, Sweden, for the use of their facilities.

636 AVERY AND P R M E N

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Received March 17, 1980 Revised May 23, 1980