Generalized KKR-theory for non-Muffin-Tin potentials

Generalized KKR-theory for non-Muffin-Tin potentials

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Molenaar, J. (1988). Generalized KKR-theory for non-Muffin-Tin potentials. (RANA : reports on applied and numerical analysis; Vol. 8818). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1988

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Eindhoven University of Technology

Department of Mathematics and Computing Science

RANA 88-18 October 1988

GENERALIZED KKR-THEORY FOR NON-MUFFIN-TIN POTENTIALS

by

J. Molenaar

Reports on Applied and Nwnerical Analysis

Department of Mathematics and Computing Science Eindhoven University of Technology

P.O. Box 513 5600 MB Eindhoven The Netherlands

ABSTRACT

J. Molenaar

Mathematics Consulting Department Technical University Eindhoven

Den Dolech2 Postbus S13 S600 MB Eindhoven

The Netherlands

We show, that the secular equation in KKR (Koning a, Kohn and Rostaker) theory remains its separable structure also in case of non-Muffin-Tin potentials. This generalization has extensively been discussed last years. During this discussion, in which the possible necessity of so-called Near Field Corrections played an important role, it became more and more clear, that the atten-tion should be concentrated on the basisfunctions used to represent the crystal wave function locally. VIe discuss the construction of reliable basisfunctions, and show that several alternatives can be indicated, which all theoretically satisfy. These different possibilities have quite different implications as far as their numerical evaluation is concerned, and we show, that the generaliza-tion of the construcgeneraliza-tion, which is already in use in "classical" KKR theory, deserves preference. In the literature, it has been claimed, that it is absolutely necessary to take into account the part of the crystal potential between the boundary of the Wigner-Seitz cell and its circumscribing sphere. The present derivations make clear, that basisfunctions, constructed from only the part of the crystal potential inside a Wigner-Seitz cell, may satisfy as well as those constructed from local potentials with larger support.

-2-1. Introduction

In a preceding paper (Molenaar 1988), we discussed the so-called Near Field (NF) error, which is supposed to be frequently made in generalizing the KKR theory of Korringa (1947) and Kohn and Rostoker (1954) for crystal potentials beyond the Muffin Tin (MT) approximation. This paper introduced a new element in the NF error discussion in that it contained a detailed off-shell analysis of Multiple Scattering (MS) theory for crystals. One of the conclusions was, that the well-known KKR equations are valid for both MT and non-MT potentials. This would imply, that the NF error, which Ziesche (1974) claimed to be present in the work of Williams and Van Morgan (1972, 1974), does not exist In the last decade, this issue has often been discussed, e.g. by Van Morgan (1977), Faulkner (1979, 1985, 1986), Gonis (1986), Zeller (1987), and Brown and Ciftan (1983, 1984, 1985, 19800. 1986b). Recent contributions are by Badralexe and Free-man (1987), Brown (1988a, 1988b), Goois, ZIwlg and Nicholson (1988) and Zeller (1988), which appeared after the preceding paper had been written. The approach and conclusions by Badralexe and Freeman differ much from the present contribution and has been commented by other authors (van Ek and Lodder 1988, Brown 1988b). We shall therefore pay no attention to it in the present paper. The approach of Gonis, ZIwlg and Nicholson is quite different from the present one, but their general conclusions agree with ours. They do not discuss the issue of the basisfunctions, used to represent the crystal wave function (CWF). The work of Zeller includes the analytical and numerical analysis of NF corrections for the case of an empty-lattice. He con-cludes that they vanish up to third order. This fully agrees with our work. in which they are shown to vanish up to all orders for general potentials. The contributions by Brown and Oftan have been laid down in a series of papers. Because these papers form a consistent entirety, we shall refer to them by Brown and Ciftan, without specification of a particular article. An extensive and rather complete review of their ideas is given by Brown (1988c). The results of their on-shell approach and the present off-shell approach agree much. They both lead to the conclusion, that the secular equation in KKR theory remains it separability between lattice and potential factors also in case of non-Mf potentials. However, Brown and Ciftan claim, that the greater part of the literature, including the work of the present author, contains another subtle error, which has to do with the basis used to represent the CWF locally. So, the discussion seems to come to an end as far as the separability is concerned but continues at another, equally essential issue.

The main purpose of this paper is to present a detailed discussion of the appropriate choice of basisfunctions. We show that. from a theoretical point of view, a multitude of satisfactory bases exists, which includes the ones used by Wtlliams and Van Morgan (1974), Brown and Cif-tan and the present author. However, if it comes to the question, how these basisfunction could be calculated in practice, the alternatives are not of equal value. We show. that the construction pro-posed by Brown and Ciftan is attractive. but also has some serious disadvantages. We propose an essential modification of this procedure. We also present an alternative derivation of the general-ized KKR equation as given by Molenaar (1988). The present derivation is inspired by the work of Brown and Ciftan and yields similar conclusions. although we start with a different, off-shell analysis.

In Section 2, we deal with general properues of the CWF and pay special attention to the conditions, under which a basis allows for an on-shell, local representation of the CWF. TIle con-struction of appropriate basisfunctions is the subject of Section 3.

In this construction, matrices C i and S i appear, the calculation of which is a central point in the NF error discussion. In Section 4, we show, that these matrices can quite generally be obtained by solving a first order and linear boundary value problem. TIle basisfunctions are uniquely defined except for the choice of a matrix

C

i. TIle consequences of three different choices for this matrix

are discussed in Section 5. Section 6 contains a concise derivation of the secular equation for non-MT potentials, given an appropriate basis. The conclusions are listed in Section 7.

Throughout this paper, atomic units are used with h =2m

=

1, h and m being Planck's con-stant and the electron mass, respectively.

-4-2. Crystal Wave Function

The crystal wave function (CWF) I

'Iff>

is the solution of the time independent SchrOdinger equation (TISE)

(1) We

use

the notation k iii (n, k) with n the band index and k crystal momentum. The crystal

potential Va has lattice symmetry and we write

(2)

Vi is equal to ycr within the i-th Wigner Seitz (WS) cell

,i

and vamshes outside it From transla-tion symmetry, we have

(3) with Ri and R_j lattice positions. The dependence of E on the

quantum

numbers k is the subject of bandstructure calculations, for which we shall derive a secular equation in Section 6. This depen-dence is also referred to as the E - k or dispersion relation and denoted as

E =E,,(k). (4)

From the translation symmetry of ycr, some general properties of I

vf

>

directly follow:

a) Translation over a lattice vector yields a known phase factor for the CWF. The Bloch theorem

(5) assures, that knowledge of the CWF inside one WS cell implies the knowledge of the CWF everywhere.

b) The CWF does not belong to the Hilbert space L2(R3) of square integrable functions on

JR.3, because ya is periodic and extends to infinity. However, in view of a), if suffices to noMlalize and interprete the inner product

<

vf

I

vf

>

as a probability density function only within some WS cell, say

ci.

Potentials yi, which allow for such a nOMlalization, are called regular at Ri .

An essential feature of multiple scattering (MS) theory is the expansion of the CWF within the different WS cells Oi in tenos of sets of basisfunctions, centered around the respective lattice sites Ri . It then remains to calculate the expansion coefficients. In view of the considerations under b) we introduce a function I

'Iff .

i

>

by the definition

<rl'V~r.i>=<rl'Vir> i f r e a (6)

=0 if r t Qi.

continuation would satisfy, provided that it is integrable at infinity. For continuations different from the one given in (6), equation (7) given underneath only holds within Qi and that is all we need in the following. The only important point is, that ,

.r,

i

>

E L₂(R3) and thus can be

rep~nted by each complete set of basisfunctions in this space. Because ,

vf

>

and ,

'l'f ,

i

>

coincide within Oi, the latter still contains all information about the electronic properties of the

crystal.

From (6) it is clear, that , 'f'f,

i

>

satisfies (1) in a reduced sense:

(7) Let us denote the basisfunctions, meant to expand the CWF within

d,

by , E , L , i ,

+

>.

For general Vi, it is not self-evident to label these functions by the quantum numbers

(E, L

=

(I,

m»

of the energy-angular momentum representation. However, for spherically

sym-metric Vi, L is a convenient

quantum

number, and one is therefore used to do it this way. For rea-sons to be explained in the sequel, we assume the basisfunctiODS to satisfy an equation analogous to (7):

(-A

+

Vi) , E, L , i,

+

>

=

E , E, L , i,

+

>.

(8) We further demand, that , E, L, i,

+

>

E L2(R3), which is consistent with the regularity of Vi

at Ri and with its bounded support. In Section 3, we shall deal with the construction of the basis-functions in detail. Here, it suffices to remark, that (_A+Vi) is a Hermitian operator, so that the eigenfunctions can be chosen to form a complete, orthonormal basis in L₂₍ R 3).

We close this section with an explicit derivation of the fact that, for an expansion of I

Vr,

i

>

at energy E

=

E,.(k), only basisfunctions at this particular energy are needed. Although this result is one of the starting points of KKR theory, one seldom realizes that any other expan-sion necessarily contains off-shell components. The essential point is, that the basisfunctions satisfy within Oi the same differential equation as the CWF does. For E '" 0 we have

cr . , E L . t;T . , (-A+ Vi) 'E L .

<Vt,l "I,+>=<'I'k,l E ,,1,+>

(9)

EII(k) c r · 'E L .

=

----e

<

'lit , I " I,

+ >.

This inner product thus vanishes unless E

=

E,.(k), and basisfunctions at other energies do not contribute to the expansion of I

vf,

i

>.

Therefore, we may write

I

vir,

i

>

=

LIE, L, i , + >

dh.t.

(10)

L

If the basis is orthogonal, the coefficients are given by the inner product

di _EL.t

₌

_{< ,}

E L ' _{, I .}

₊

, c r · _{Vt. I}

_>.

₍₁₁₎ In the following we shall not need orthononnality of the basis and only use its completeness.

-6-3. The Construction of Basisfunclions

In the preceding section. we argued that functions IE, L,

i ,

+

>,

which satisfy equation (8) and regularity conditions at Ri and infinity, may fonn a basis in L₂(1R3) and, moreover,

are

appropri-ate to represent the CWF within Oi through an on-shell expansion. In this section, we shall dis-cuss the construction of such basisfunctions. We first remark, that the homogeneous part of equa-tion (8)

-AIE,L,i>=E IE,L,i> (12)

is the well-known TISE for a free particle. The solutions

are

given by

I I

<

r

+

R, IE, L, i

>

=

E;

j,rfi

I r I) YL(r) • E; k(E, r) (13)

1tf 7ti

where the normalization is such, that

<E, L, i I E', L', i

>

=

S(E -E') 3t., L'. (14) The

it

and YL

are

the spherical Bessel function and the (real) spherical harmonic, respectively.

1bese free particle solutions form an orthononnal basis in L2 (R 3), so that we may write

-

IdE'1:. IE,L,i><E,L,il=l (15)

o L

with 1 the unity operator. We remark, that the order of summation and integration may be inter-changed in this projection operator in view of the convergence properties of this kind of expan-sions. To a particular solution of the inhomogeneous equation (8) a linear combination of solu-tions of 'the homogeneous equation (12) may be added. This property is expressed by the Lipmann-Schwinger (LS) equation

IE, L, i,

+>

=

1:

C~'(E) IE,

L', i

>

+G(E) Vi IE, L, i,

+>.

(16)

L'

The free space Green's operator G(E) is defined by G(E)

=

(E

+

i £. - _ft.)-l ii! (E+ - -ft.r1 with £ infinitesimally small and positive.

(17)

If we take the rows of Ch,(E) in (16) linearly independent, we make sure, that the func-tions IE, L , i,

+ >

are also linearly independent in L2(R3). Under the condition

- i

det (Cu,(E» *0 (18)

the functions IE, L, i,

+>

fonn a complete basis in the subspace of L₂(R3) spanned by the

eigenfunctions of the operator (-ft.+ Vi) at eigenvalue E. Although the IE, L, i,

+ >

are orthog-onal with respect to energy, as being eigenfunctions of a Hennitian operator, they are in advance not orthogonal with respect to angular momentum indices. By applying a Gram-Schmidt pro-cedure, they could be orthogonalized, but we do not need this property in the following.

Condition (18) leaves us much freedom in the choice of

C

i. It is important to realize. that

every non-singular matrix

C

i in principle yields a set of basisfunctions. which is formally

appropriate to represent the CWF within

'i.

However. the different alternatives have quite dif-ferent implications. as discussed in Section 5.

Let us evaluate the particular solution G Vi IE, L , i ,

+

>

of equation (8) more in detail. We therefore need the matrix elements of G with respect to the free space basisfunctions

IE, L , i

>.

They are given by

<E', L, i I G(E) I E", L'. i

>

= (E+ _E')-l l)(E' -E'')

au'.

(19)

With the use of the projection operator in (15), we obtain the relation

< r + Ri I G(E) Vi IE, L , i , + >

=

j

dE'

1:

< r + Ri I E', L' • i > + 1 tk-L',EL (20)

o

L' (E -E')

with the t-matrix ti, corresponding to the potential Vi, defined by

tb.,£'.r.:

=

<

E , L, i I Vi I E' , L' , i, +

>

= E;

J

dr' iL(E,r,)Vi(r,)<r' I E',L',

i,+>.

7Ci Ci

In the last line of this equation, we used the fact that for each i

I

dr I r+Ri> <r+Ri 1=1 IR'

(21)

(22)

with I r>, r E R3, the eigenfunctions of the position operator. If we substitute expression (21)

for the t-matrix into equation (20), the energy intergration over E' can be performed analytically by means of the theorem given in the Appendix. To that end. we have to split up the integration over Oi into a subdomain Oi( < r), which is equal to the open sphere with radius r • I r I around Ri and a subdomain Oi(>r), which is the complement of Oi«r) with respect to Oi. In each of these subdomains the theorem in the Appendix applies, and we arrive at

< r + Ri I G(E) Vi IE. L, i, + >

=

1:

{CL:

(E, r)

iL'

(E, r) +

sL:

(E, r) ht, (E, r)} (23)

L'

with h+ the spherical Hankel function of the first kind and the coefficient matrices C i and Si given by

cL:

(E, r)=-i..JE

J

dr' ht, (E, r') Vi (r')

<

r' IE, L, i, +> (24a)

Ci(>r)

sL:

(E, r) =-i..JE

J

dr' iL,(E, r') Vi (ri)

<

r' IE, L, i, +> (24b)

Ci(<r)

-

8-<

r

+

Ri IE, L,

i ,

+ >

=

1:

{(CLl:

+

ciL: )

jL' (E, r)

+

sL"

hL'

(E, r)}. (25)

L'

This expression is completely analogous to the one given by Brown and Oftan. From the defining equations (24a,b) for Ci and Si, some special cases immediately follow:

cL"

(E ,r)

=

0

• 1 I •

Su.: (E, r)

=

-i E4 Ki

t&.a.,

(26a) (26b)

if r is equal to or larger than the radius R~ of the circumscribed sphere of l)i. For r

=

0 we have

sL"

(E, 0)=0. (26c)

Representation (25) is not an expansion in tems of a complete basis in L₂(R3). This is directly seen from the r-dependence of the coefficients Ci(E , r) and Si(E , r). An expansion with respect to a basis would, for example, be obtained by applying the projection operator in (15). The result-ing representation would read

as

-<

r

+

Ri IE, L, i,

+ >

=

r

dE'

1:

<

r

+

Ri I E' , L' ,

i>

<

E' , L' ,

i

IE, L, i fR}

6

L'

-=

J

dE'

1:

k

(E', r) JE'L'.EL

o

L'

with the coefficients d given by

dE'L' FL i

=

- - I (E')t -

< "

E' L' liE L .

"

I ,

+ >

• 11:2 (28)

and independent of r. However, the preference of expansion (25) above an expansion like (28) is clear, if one realizes that (25) is an on-shell, and (28) an off-shell representation.

4. The Coefficient Matrices C j and S i

. . - i

We study the calculation of C' and S' and the choice of C separately. 1bese topics

are

much

related, because it will appear that not all allowed choices for

C

i will lead to a practical algoritJun for the numerical determination of C i and S i .

It has for the first time been pointed out by

Brown

and Ciftan, that a set of coupled, ordinary

differential equations for Ci and Si can be derived by differentiating equations (24a) and (24b)

with respect to the radius r. We shall work out this idea here again, because at this point

we shall

draw conclusions, which are different from theirs. If we write the space variable r in spherical

coordinates with radius r and angle; (both with respect to Ri ). we have for an arbitrary integrand I(r)

=

I(r, ,):

~

I

dr' l(r1

=

~

j

dr' (r12

I

d,' I(r', ,1

dr n'(>r) dr r

(29)

=_r2

I

d,' I(r, '1.

We apply this rule to the r.h.s. of (24a) and (24b) and substitute expression (25) for

<

r' IE, L , i ,

+

> into these expressions. After some rearrangements, we arrive at the following

ordinary differential equation:

d . - i · . . .

dr Ch: (E, r)

= 1:

_L" {(CU " (E)

+

CL,,, (E, r»AL"L' (E, r)

+

SL,,. (E • r) BL"L' (E. r)} (3Oa)

with the matrices A i and B i given by

Ah: (E. r)

=

i

{i

r2

j,({i

r) Vh:(r) hP (..fi r) BL., (E. r)

=

i

.fi

r2

ht (..fi

r) VL.'(r) hP

(VE

r)

and the matrix vL.,.

as

usually, defined by

VL.'(r)

=

I

d ;. YL(r) Vi(r, ;) yL,(r).

(31a) (3Ib)

(32)

Note, that the integration in the r.h.s. of this definition

runs

over those parts parts of the surface of

a sphere of radius r around Rj , which lie within oj. This surface of integration may thus consist

of several unconnected parts.

For Si we have an equation analogous to (30a):

d . - j . . . .

- Su,(E, r)

= -1:

((Cu,,(E)

+

CL,,,(E, r» KL"L,(E. r)

+

Su,,(E. r) LL"L,(E, r)} (3Ob)

dr LN

with Kj and L i given by

-

10-(3Id) Given a potential Vi with support

rt,

the matrices Ai, B i, Ki and L i are easily calculated. For, outside the circumscribing sphere of

n

i, they vanish. To calculate Ci and Si numerically, it remains to specify the values of Ci and Si somewhere in the interval [0, Rb ], with R~ the radius of the circumscribing sphere of

ni.

It is quite clear that, in the first instance, the equations (3Oa)

and (30b) should be considered to establish a boundary value problem with natural boundary con-ditions

CLL, (E, ,)=0 " =R~ SLL' (E , ,)

=

0 "

=

O.

(33a) (33b) With these conditions the problem is well-posed and could be solved by a variety of standard software. Some care is needed in the neighbourllood of ,

=

0, because there the matrices A and B diverge. These divergences are cancelled in the products in the r.h.s. of (3Oa) and (3Ob), but require, from a numerical point of view, subtle treatment. In these considerations, the choice of the matrix

C

i plays an important role, which we deal with in the following section.

- i S. The Matrix C

- i

In this section we present three possible choices for the matrix C and discuss their merits.

a) If we evaluate expression (25) for the single site wave function

<

r

+

Ri IE. L •

i •

+ >

out-side the circumscribing sphere with radius R~. we obtain in view of (33a)

<r + Ri IE, L. i. +>

=

t

{c~,

(E)k,(E. r) -i Et

,d-

tk.FL' hL,(E. r)}.

r~ R~04)

L'

This expression is well-known and commonly used in MS theory in combination with the cooice

- i

CU' (E)

=

au" (35)

The usual interpretation of (34) with (35) is to look at the total wavefunction as the sum of an incoming free wave

iL

and the corresponding scattered waves h+. weighted by the t-matrix. In the MT approximation. one often assumes Vi to be spherically symmetric. In that case. we meet with a diagonal t-matrix and an incoming wave

iL.

which gives rise to only one outgoing wave

ht.

Condition (35) may be called the "classical" choice for Cit because it is commonly used in KKR theory for MT potentials.

b) It is proposed by Brown and Ciftan to choose

- i .

Cu,(E)

+

C

u:

(E. 0)

=

aLL" (36)

This nice idea allows for a different view upon equations (30a) and (3Ob). It is clear that (30a) for Ci also holds for Ci

+

Ci. With condition (36). one has initial conditions for the quantities Ci

+

Ci and Si at the origin. namely (36) together with (33b). This implies that the differential equations could be solved as an initial value problem rather than as a boun-dary value problem. This approach is appealing. because the fonner are numerically easier to solve than the latter. It has. however. three disadvantages:

It is hard to integrate equation (30) away from the origin. because now the r.h.s. cer-tainly diverges.

The proof that choice (36) is consistent with condition (18) fails until now. For the "empty lattice" with Vi == 0 (or constant) this point is trivial. but for less simple Vi the proof is probably hard.

The resulting wave functions and t-matrices. obtained via condition (36). will much differ from the t-matrices usually applied in KKR theory for MT potentials. One should not introduce such a discontinuity. if not absolutely necessary.

-

12-c) To overcome the first disadvantage mentioned under b), one could tty to use the condition

- j .

CU' (E) + ClL: (E, 0) = O. (37)

Although this choice diminishes the numerical problems, the two other disadvantages are still valid, unless one can proof that conditions (37) and (35) are in fact identical. This is a nice subject for further research, but will not pointed out in this paper.

6. The Secular Equation

In Section 2 we formally introduced in each WS cell

oj

a complete set of basisfunctions

IE, L , i , +

>. The defining equation (8), together with the mentioned boundary conditions, do

not uniquely determine this basis. This point has been further discussed in Section 3. Here, we assume that some choice is made and a particular basis set is known. We shall derive the secular equation, from which the expansion coefficients of the CWF and the E - k relation can be calcu-lated, and show that this equation has a separated structure, i.e. the information from the lattice structure and the information from the crystal potential are contained in differem factors. 'The present derivation is much more concise than the one given earlier (Molenaar 1988), though the resulting equation is the same.

The CWF I

yr

>

satisfies the LS equation

In view of the partioning (2), we write this in the form

(1-G Vi) I

'l'Y

>

=

1:

G Vi I

"r

>.

j~

Substitution of the on-shell expansion (10) yields the equation

(38)

(39)

1:

(1 - G Vi) IE, L, i,

+ >

d~A:

=

1: 1:

G Vi IE, L, j ,

+ >

dk,t.

(40)

L i~ L

Note, that in the sequel the expansions of the CWF are always used within the appropriate WS cells.

We take the inner product at both sides with

<

r I and use equation (16). The Bu. of equation (40) then reduces to a sum over free particles wave functions:

. - i

Et .

- j

l:l:da,k

CU ' <r IE, L', i

> =

- I d&.k C_{U ' jL'} (E, r-Rj).

L L' 7ti

If we insert unity operators as given in (15) into the r.h.s. of (40), we obtain the expression

1: 1:

J

dE'

1:

<

r I E' , L', i

>

G~'L',E'L"

(E) Ih'L",FL dh,A:

j~ L 0 L'L"

with the I-matrices Ii given by (21) and the elements of the matrix Gii defined by

G~'L' ,E"L" (E)

=

<

E' , L' , i I G(E) I E" , L" , j

>

=

1 l>(E' - E'') Ji'L" (E,). (E+ -E')

For i

=

j this expression reduces to the one given in (19). The matrix Jii is defined by

(41)

(42)

-

14-JY.,L,(E)=<E,L, i 1 E,L',j>. (44)

Previously (1988) we showed thatJij is explicitly given by

(45)

with CU'L" denoting Gaunt coefficients. For later purposes, we introduce a matrix

G

by replacing the Bessel function in (45) by a Hankel fimction:

Gt;.,

(E)

=

41C

~

;(l-I'-+lj CU'L" h+ (E, Ri -R_{j ).} (46) L"

For i = j we take this matrix vanishing. The Fourier transform G of

G

with respect to lattice sites is given by

- j j

GL,L' (E, k)= ~ GL,L' (E)exp(ik.(Rj-Rj

»

(47)

j(j~)

with j running over lattice sites.

The

energy

integration in (42)

can

be analytically performed. If we isolate the

energy

con-taining terms, we obtain the integral

00

f

dE' (E')Yl +1

k

(~I

r-Rj I)j,-

(~I

Rj-Rj I)k'

(~

1 r'-Rj I) (48)

o

(E -E')

with r' running over

ai.

We restrict the range ofr to an open ball around R j, with its radius such that

I Rj - R_j 1

>

1

r -

Rj 1

+

1

r' -

R_j I. (49)

For the known lattice structures, this open ball is not empty. Under condition (49), we may apply the theorem in the Appendix. Then, we obtain for the integral (48)

xi

VE

k

(Vi

1 r-Rj l)hP

(Vi

1 Rj-Rj I)k

rfi

1 r'-Rj I). If we substitute this product back into expression (42), this simplifies to

~ ~

i..JE

it'

(E, r-Rj )

Gl,.L"

th",EL dh,k'

j~ L.L',L"

(50)

A further reduction is obtained from (3). 'This implies, that all t-matrices tj are the same, so that the position index may be dropped. Further, we have that the expansion coefficients dj are related to each other via the Bloch condition (5):

di;L,k =exp (i k· (Rj-Rj» dh,k' (51)

I',

i

fi

iL'

(E, r-Ri) GL'L" (E, k) tEL".EL dEL,,,· (52) L.L',L "

Expressions (52) and (41) both contain a summation over the particle wave function. If we

equate

them and reorder the indices, we find

I',

iL

(E , r - Ri ) Mu' (E , k) dEL'."

=

O. (53) L,L'

with M given by

Mu ' (E, k)

=

fi

I', [

1.1

~

(Ci)L., aL'L" - i Gu" (E, k) tEL".EL']. (54)

L" E. 7t

By multiplying equation (53) with a particular spherical harmonic and integrating over angles, we project out of the L summation one particular Bessel function. Because Bessel functions are not vanishing in an open neighbourhood of Rj • we may conclude that the secular equation is

I',

Mu' (E , k) dEL'."

=

O. (55)

L'

WerelllllIk that the separation of variables, which is present in the free particle wave functions (13), allows for a projection argument, but is not essential to arrive at the secular equation. Instead of the spherical Bessel functions, each other expansion in terms of linearly independent functions in an open neighbourhood of Rj would suffice to reduce an equation like (53) to the reduced form (55). Equations (54) and (55) have also been derived by Brown and Clftan, except for the constant factors, which have their origin in the normalization used for the free space wave functions'in (13). If we choose C~,

=

au" the matrix M is the usual KKR matrix for MT poten-tials. It thus appears that relaxing this restriction has no influence on the structure of the resulting secular equation. This generalization has only impact on the way the t-matrix is calculated.

-

16-7. Conclusions

Here, we shall list the most important conclusions from the discussions in the preceding sections.

a) Solutions (16) of equation (8) form, under conditions (18) and regularity conditions at the origin and infinity, a complete set ofbasisfunctions in the Hilbert space L2(1l3). Within a WS cell, the CWF can thus be represented by one of these bases.

b) Representation (10) with (25) of the crystal wave function is on-shell, because the CWF and the basisfunctions used satisfy the same differential equation within the WS cell.

c) The choice of the matrix Ci in (16) defines the basis uniquely. It makes quite a difference, which choice is made, because the numerical evaluation of the expansion matrices Ci and

. Si, defined in (24), heavily depends on it The "classical" choice (35) implies that Ci aOO Si are to be determined from a boundary value problem with boundary conditions (33a,b). This problem is well posed and can be solved by stmlard teclmiques, provided that some numer-ical care is taken at the origin. Choice (36) for Ci, proposed by Brown and Oftan, allows in theory the matrices Ci and Si to be determined from an initial value problem with initial conditions (36) and (33b). Although this aspect gives it some preference above the "classi-cal" approach, this idea has some serious drawbacks, both theoretically and numerically.

d) The form of the secular equation (55) for generalized KKR theory shows a separation between structural and potential parts and is in essence the same as the secular equation of KKR theory. The absence of the MT approximation manifests itself only through the 1-matrices used. The form of these I-1-matrices depends on the basis used. However, the band structure equation (55) yields results, which are independent of the used basisfunctions.

e) In the defining equation (8) for the basisfunctions, we take Vi to be equal to the crystal potential within the i-th WS cell and vanishing outside it Brown and Oftan take Vi equal to the crystal potential inside the circumscribed sphere and claim that neglect of the poten-tial outside the WS cell, but inside its circumscribing sphere would introduce an essenpoten-tial error. We have shown above that this is not the case. 'Ibis does not mean that the approach of Brown and Ciftan is wrong. It simply leads to a different basis, which may also be com-plete in L₂(1R3). The only essential requirement in choosing the support of Vi is, that Vi

coincides with Vcr inside the WS cell, because only then an on-shell expansion of the CWF can be obtained.

In general, we may conclude that, after years of discussions, the confusion about the "Near Field" corrections in generalized KKR theory has cleared up both by the on-shell approach of Brown and Ciftan, the wode. of Gonis, and the present off-shell approach. The result is remarkable in

that extending KKR theory beyond the MT approximation lets the secular equation and in partic-ular its separable structure nearly unchanged and only modifies the calculation of the I-matrices in a straightforward manner.

Acknowledgement

I would like to thank A. Lodder from the Free University, Amsterdam, very much for his interest in this wOtX and critical reading of the manuscript.

-18-Appendix

In this appendix we prove the following proposition. Let I be defined by

'"

I

=

S

dk

f(k)(E+i£.-k2)-1~

h,(krj)

o 1=\

with

hi

spherical Bessel functions, E a constant and £. infinitesimally positive.

Under the following restrictions

"

l:

Ij is even j=\ f(k) is an even function of k n-\ l:rj < r" =1= 0 j=1 n-l l:lj+p ~ 1,,+1 j=\ n+2 > t

where p and t are defined by the asymptotic behaviour of

f

i.e.

J(z) - zP and J(z) - Zl

z~o Z-toOC

it holds, in the limit £. ~ 0, that

with hi

=

h+inl a spherical Hankel function and nl a spherical Neumann function. Proof.

We need the following properties of

h

,

nl and hi'

(b.1) (b.2) (b.3) (b.4) (b.5) (b.6) (b.7) (b.B)

z--+oc Z z--+o (21

+

I)!! (b.9)

(b.IO)

(b.ll)

(b.12)

On account of restrictions (b.2) and (b.3) and property (b.ll) we may extend the integration interval to the range -oo<k<+ao. In view of property (b.12) of the spherical Neumann function we may at the same time replace one of the

it

by hi> for which we choose the h(kr) with the largest srgument, i.e.

h(krn). The hln' introduced in this way, diverges at the origin as (krn)-(/.+t) (see (b.lO)) but the product

of the remaining spherical Bessel functions, the function f(k) and hi. does not show this divergent behaviour in view of restriction (b.5) with the properties (b.7) and (b.9).

Let us not close the contour by means of a semicircle in the upperhalf plane. The asymptotic behaviour for k~oo of

it

and hi, given by (b.9) and (b.lO), guarantees that, in view of restrictions (b.4) and (b.6), the contribution of the arc vanishes if its radius goes to infinity. The contour encloses the pole at ko = YE(1 +iE/E)'h. which in the limit E ~ 0 becomes ko

=

YE. The residue of (£ +iE-k2)-t at this pole is (2YE)-1. Application of the integral theorem of Cauchy completes the proof.

-20-References

Badralexe E. and Freeman A.J. 1987, Phys. Rev. B36, 1378. Brown RG. and Ciftan M. 1983, Phys. Rev. B27, 4564.

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