Solution of self-consistent field equations by the recursion method

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Solution of self-consistent field equations by the recursion method

J. Julien, D. Mayou

To cite this version:

J. Julien, D. Mayou. Solution of self-consistent field equations by the recursion method. Journal de

Physique I, EDP Sciences, 1993, 3 (8), pp.1861-1872. �10.1051/jp1:1993103�. �jpa-00246835�

Classification Ph_vsics Abstracts

71.20A 71.20C 71.25C

Solution of self-consistent field equations by the recursion method

J. P. Julien and D.

Mayou

Laboratoire d'Etudes des

propr16tds

Electroniques des Solides (*), CNRS, BP 166, 38042 Grenoble Cedex 9, France

(Receii>ed 18 January 1993, revised 8 March 1993, accepted 9 April 1993)

Rksumk. Nous montrons que les

Equations

de

l'approximation

du

potentiel

cohdrent (C.P.A.) peuvent ^{due r6solues dons}

l'espace

r6el par la mdthode de rdcursion. La fonction de Green est

reprdsentde _{par une} fraction continue _et la mdthode foumit _une solution

rapide

et

prdcise.

Une premidre application est prdsentde.

Abstract. We show that the equations of the Coherent Potential Approximation (C.P.A.) _can be solved in real space

using

the recursion method. The Green's function, from which the

density

of

states is calculated, is then represented by _a continued fraction, and the method provides _a

quick

and

precise

solution. A first

application

is

presented.

1. Introduction.

The Coherent Potential

Approximation [I]

is _one of the _most

widely

used _mean field theories of disordered substitutional

alloys.

Even in

partially

ordered systems it is the basis for _a

determination of interaction

potentials

that can lead to

prediction

of

phase stability [2].

This

approximation

leads to self-consistent

equations

for Green's function which _are

usually

solved for each energy z. The aim of this paper is _to demonstrate _a way of

solving

the C-P-A-

equations by

_a

completely

different

approach.

We show that the Green's function is that of _an effective Hamiltonian H~~~ where the matrix elements _are calculated step

by

step. ^{Then the} recursion method

[3]

is

applied

to that effective Hamiltonian and allows _one to express the Green's function _{as a} continued fraction. This method is

quick

and is also

interesting

since continued fractions have proven very useful _to calculate densities of states _or

integrated quantities

like energy or

charge

transfer.

(*) Associated with Universitd Joseph Fourier.

2.

Description

^{of the method.}

For

simplicity

_we consider _a

binary alloy

with _two components ^{A and B}

having

_on site

(matricial) energies

_e~ and e~. The

generalization

for _more than two components ^{will be}

discussed elsewhere

[4].

The Hamiltonian for _a

given configuration

is _:

H=Ho+£E, (1)

Ho

does _not

depend

_on the

configuration (diagonal disorder)

and I denotes the atomic sites with E, = E~ ^or e, = e~

depending

_on the

occupation

of site I

by

A _or B atom. If p~ and p~ are the

probabilities

of

occupation by

_an atom A _or

B,

_one has p~ + pB = I and _{one can}

define the average energy

~

" PA ~A + PB ~B

(2)

In this paper we are interested in densities of states which _are

calculated,

_as

usual,

with the formalism of Green's functions. For _a Hamiltonian H the Green's operator ^G

(z

is defined

by

_:

G(z)

=

~

/

~

(3)

and the

partial density

of _states

n~(E)

on a state

/$r)

is

given by

n~(E)

₌ Im

($r/G(E

₊

ie)/$r) (4)

where Im

~f)

is the

imaginary

_part of the

complex

number

f,

and

e is _a

positive

real number

which tends _{to zero.} More

generally

if _we consider _a

subspace

E

generated by

a set of

N orthonormal _states

/$r,)

_we define _a

projector

_on that

subspace P~ by

FE

=

( /~, ) l~,/ (5)

and the restriction of the Green's operator

G(z)

to that

subspace

^{E is} :

G~(zi

₌

P~G(ziP~. (6)

If E is the

subspace generated by

the orbitals of _an A _or B _{atom we note} the restriction of the Green's operator to this

subspace G~(z)

or

GB(z).

Within the C.P.A. _{an atom} A _or B is embedded in _an effective medium

consisting

of

identical atoms with _on site

energies

> ₊

3(z) [1, 2].

Then the

self-consistency

condition states that the average Green's operator on an A _or B atom in the effective medium is

equal

to the Green's operator G~~~(z

)

_{on an} effective atom with on-site energy ^{> +}

3(z), placed

in the

same effective medium _:

PA

~A(Z)

_{+ PB}

~B(Z)

~

Geff(Z) (7)

G~(z)

=

~ ~

'~

~ ~~~

G~(z)

=

~ ~

~

~ ~~~

(8)

~~~~~~

_~

z I

3(z) ~r(z)

^~~~

«(z)

_represents the

(matricial) self-energy

due to the

coupling

of _an atom with the effective medium. The

self-energy 3(z),

which is

Herglotz [5],

_can

always

be considered _as the self-

energy due _to the

coupling

_{to a} chain C with the Hamiltonian in

figure

I where

A~,

B~ are square matrices with the _same dimension _{as e~} and e~, If there is _one orbital per site

A~, B~

are

just

numbers and

3(z)

is _a scalar

quantity

which _can be

expressed

as a continued

fraction

~2

3(z)

₌ ^°

(10)

B(

z-Aj

B~

~

z

-A~

z

-A~

Z(z)

= AI Az Aj C

B° Bi Bz 83

Fig.

I.

Representation

of the semi-infinite chain C. The

coupling

_to the chain C is

equivalent

to

adding

the

self-energy

3(z). The circles represent

subspaces

of states whose dimension is

equal

to the number of orbitals of the _atoms A _or B. For each circle A~ ^{is the on-site matricial} energy of the

corresponding

subspace. B~ ^is the coupling between the _two adjacent subspaces of the chain.

If there is _more than _one orbital per site

3(z

is _a matricial

quantity

which _can be

expressed

as a matricial continued fraction

3(z)

=

Bj _Bo 3~(z )

₌

Bj

~

B~

ⁱⁱ

)

Z-Ai-~i(Z) Z-An+i- n+i(Z)

As shown in

Appendix A,

this allows _us to

replace

the usual effective Hamiltonian of the C.P.A. which is energy

dependent, by

_a Hamiltonian H~~~ which is

independent

^of energy, but is defined in _a

higher

dimensional space since each _atom with _a

self-energy 3(z)

is _now

coupled

to a chain C which simulates

exactly

the

self-energy 3(z).

Then the

self-energy

«(z)

of the central atom

(A

_or

B)

in the energy

dependent

effective Hamiltonian is also that of the central _atom in the energy

independent

Hamiltonian H~~~

(Fig. 2).

In this

representation

«

(z)

can be calculated

by

a classical recursion

procedure starting

from A _or B. If there is _one orbital per site

«(z)

is _a scalar

quantity

which _can be

expressed

_{as a} continued fraction _:

~2

«(z)

=

°

(12)

hi

z aj

b~

~

~ ~~

z a~

If there is _more than _one orbital per site _«

(z

is _a matricial

quantify

which _can be

expressed

as a matricial continued fraction _:

«(z)

=

hi

z ~~

'

«~~z~ ho ~rn(z)

₌

hi

z ~~~

'

~r~~ ~~z~

bn (13)

where a~, b~ are square matrices with the _same dimension _as e~ and eB.

«(z)

_can also be

interpreted

_as the

self-energy

due to the

coupling

with _any chain C'

(Fig. 3).

E+Z(z) E+Z(z) E+Z(z) E+Z(z)

ffi ---~ O ~ 0~~2a

EAorEB

83 83 83 83

82 Bz Bz Bz

B> Bi Bi Bi

Bo Bo Bo Bo

I ) I I

2b

8A Or 88

Fig. 2. a) ^Schematic

representation

of the energy dependent effective Hamiltonian of the C-P-A- The

central atom A _or B has the on-site energy e, or F~. The other _atoms have on-site energy

7 plus _a

self-energy

3(z). The atoms _are coupled

by

the Hamiltonian Ho ^{which is} independent of the atomic

configurations.

hi Schematic

representation

of the energy

independent

effective Hamiltonian

H~j

used in the recursion

procedure.

This Hamiltonian is obtained from the

previous

one

by attaching

_a

chain C to each _atom. This chain C simulates exactly the effect of the

self-energy

3(2 j.

°~~~* b~8

~~

~

~

~

~

ic,1

Fig.

3.

Representation

of the semi-infinite chain C'. The

coupling

to the chain C' is equivalent to the

self-energy

«(z). The circles represent

subspaces

of states whose dimension is

equal

to the number of orbitals of the atoms A or-B- For each circle _a~ is the on-site matricial energy of the

corresponding

subspace. _b~ is the

coupling

between the _two

adjacent

subspaces of the chain.

Let _us consider first the _case of _one orbital per atom. We _note that if _we

apply

the recursion method

starting

from _an orbital

/$ro)

which is located _on the central A _or B atom, then the vector [fl~~) ^{of the recursion method has} non zero components

only

_up to sites

(n

I

)

in the chain

representing 3(z) (I.e.

the

subspace

with on-site energy

A~_ j).

^{In order} to calculate b~ and _a~

~ i with _:

~n "

I

~eff'i'n)

_~n

i'n)

_~n

_i'n II

^I

(14)

i'fi

+

)

"

) _{(lieff' i'n)}

~fi

i'n) ~n i'n II (15)

~

a~

~ ⁼

j*n

+

Heft 4'n

+

1)

~~

we need _to know the matrix elements of the Hamiltonian of the chain C

only

_{up to}

A~, B~_

_i ^with pm n.

Thus,

_even if all

A~

^and B~ ^are

unknown,

which is the _{case at} the

beginning

of the

calculation,

_{we can} still calculate the first terms

ho

^and _aj. ^{If there is} more than

one orbital per ^atom we can use the _vector recursion which is

presented by Haydock [3]. Again

it is easy to see that in order to calculate b~ ^and a~

~ we need to know the matrix elements of the Hamiltonian of the chain C

only

_up to

A~,

B~ ^{with p w n. This fact is essential in our}

method _{as we} show below.

In order _to go further and calculate the

following

terms _a~~

~ j, b~~

(with

n _~

0)

the next step ^is to use

equations (7), (8), (9),

and express

3(z

_{as a} function of _«

(z).

We get

(see Appendix B)

_:

3(z)

= ~ A

(17)

Z PA E~ p~ EA tT (Z

A is _a Hermitian matrix

/~

"

fi(EA _{~B) (18)}

From the continued fraction

representation

of

3(z)

and «(z) (see

Eq. (I

I) and

(13))

this relation is

equivalent

_to

Bo

₌ A A

i ⁼ PA EB ⁺ PB EA

('9)

B~ ₌ b~

A~

~ i ⁼ a~ for _{n ~} 0

(20)

Using equations (19)

and

(20)

and the terms

bo

and _ai, which _can

already

be calculated _as

explained

above, we get

Bo,

A and B

j,

A~.

^{From the} above remark about the extension of the

vectors

[w~)

we see that the recursion

procedure

allows _us then to calculate

hi,

a~, and

b~,

a~. Then

using equation (20)

this

gives

_us

B~,

A~ ^and

B~, A~.

^{The recursion}

procedure

then allows _us to calculate the next terms

b~,

_a~, and

b~,

_a~.

Clearly,

^with this

procedure

we can calculate

progressively

all the terms of the continued fraction

expansion

of

3(z)

and

«(;),

from which _we deduce the Green operators

G~(z)

_or

GB(z).

In

appendix

C _we estimate the _extent of memory

M(n)

and the

computing

time

T(n

needed for the recursion

procedure

with H~~~ up ^to stage n. We find that these

quantities

are greater ^{than the}

corresponding quantities M~(n)

and

T~(n

^for a Hamiltonian in real space I-e- without chains attached to the atoms of the average medium. For the scalar _case we get ⁱⁿ the limit of

large

_n :

)~(

_~~~

ii~)~l'

_~~~~

This shows that for _a classical value of _n

= 20 the

procedure presented

here is

applicable

to many systems.

3.

Application

and discussion.

Here _we

apply

the method described above _to

study

the electronic structure of _a disordered

alloy Ax

B

x within the

simple

s-band model _{on a} cubic lattice without

off-diagonal

disorder.

In _a

forthcoming publication,

_we will show how the method _can be

directly

extended _to multi- orbital _cases (matricial

self-energy).

The _s orbitals _on the cubic lattice have on-site energy e~

(respectively e~)

with

probability

_p~

(respectively p~)

^and are

coupled

to their six _nearest

neighbours

via _a

hopping integral

t.

In

figure

4, we present ^the coefficients of the continued fraction _versus the number of recursion steps ^{for various} cases of parameters. ^The

gapless

case

(Fig. 4a)

can be

easily recognized

when the coefficients present

damped

oscillations and converge

rapidly

to their

single asymptotic

value. The continued fraction is terminated

by

the

simple square-root

terminator and the

corresponding density

of _states is

pictured

in

figure

5a. When the

coefficients

present undamped

oscillations

(Fig. 4b),

_{we use} the terminator

proposed by

Turchi et al.

[6]

to get ^the

density

^of states, which presents a gap

(Fig. 5b).

As _can be _seen in this

simple

illustrative

example,

the method

provides

_an efficient _way for

solving

the CPA

equations

and presents ^several attractive features. In

particular,

in contrast to

usual

methods,

the Green's functions do not need _to be calculated for

complex energies having

a small but finite

imaginary

_part and then deconvoluted up to real axis.

Furthermore,

ⁱⁿ _many

situations,

_one needs

only

to know

integrated quantities

like band

energies

_or

charge

transfer

(in particular during

the calculation of the self-consistent

potential)

and it is well-known

[6]

that _a very

good precision

for these

quantities

is obtained

by calculating exactly only

the first few coefficients of the continued fraction. Our method is

well-adapted

to these calculations.

Finally,

our method avoids

integration

in the

reciprocal

_space ^which _may present some

advantages.

We also _note that

despite

^its

simplicity,

the one-band-model treated here is of interest in several

approximate

treatments of realistic systems. ^{Indeed it is}

formally equivalent

to the

o 9

,~ ,

-l ^{~ ~}

-i. i

-1 2

-1.4

-1.5 i~

f~ -1.6 +

b

~

~

3.3

°

u

~ ~

3.2 °

3,1

3

2

2.8

~

5 lo

n

a)

Fig.

oefficients are hi.

a)

_{For e, = - eB} " 2 t pA " 0. I, pB "

p~ ₌ o.9.

o

,~ ,

-O.5

~

~~

+ + + + + + + +

-l

-1.5

+ + +

+ +

+ +

+ + +

+

~

-2 +

+ +

~ + ~

~ ~ +

m ~

~ 3

~

~

+

'~n'

u

~

~

o

~

4.4

O ~

O O

O O

o o O

O

O ~

~

~ O

4.2

~ o

o o

3.8

3.6 _o

~ o o o o o o o

o 3.4

5 lo 15 20 25 30 35

hi

Fig.

4 (continued).

unhybridized

multiband _case and the method _can be

straightforwardly applied

to the so-called

« canonical _»

picture [7] developed

in the L.M.T.O. scheme. In that _case, the effect of off-

diagonal

disorder _{can even} be taken into _account

by setting

the

hopping integral

t of

tie

_{reference medium}

equal

to the average p~t~ +p~t~ ^{of the}

hopping integrals

t~ ^and t~ of pure materials and

solving equations (7), (8), (9)

where p~

(respectively, pB)

has to be

replaced by

_p~

(t~/t ) (respectively,

_pB

(tB/t [8].

In _a

hybridized

bands model with

several

inequivalent

orbitals per site it is still

possible

to use a scalar version of the C.P.A. _as done

by Papaconstantopoulos

et al.

[9].

Each orbital of type ^{I has} a scalar

self-energy 3~(z).

For _a

given

I the relations

(7), (8)

and

(9)

_are still valid with all

quantities

in it

corresponding

to the _same I. Then «~

(z)

^is calculated

by applying

the recursion method _{to an}

effective medium where each orbital of type ^{I is}

coupled

to a chain C~

simulating

3, (?).

One has _to

perform parallel

recursions

starting

^{from all}

inequivalent

^orbitals of type

j

and this

procedure

is

particularly

suited _to

parallel

_processor

computing.

To

conclude,

_we

point

out that the present

approach

_can be

generalized

when there _{are more} than _two components ^and that the calculation _can be done within the

reciprocal

_space. We also note that the recursion method _can be

applied

to other self-consistent theories of disordered systems ^{in the} same

spirit [10].

In each _case, the effective medium is

represented by

some

effective Hamiltonian

(the

chains C in this

paper)

where the matrix elements _are calculated step

by

step

using

the recursion method. This _new

approach

to mean field theories of

alloys

^{is based}

on the _use of the

configuration

_{sum space} which is

presented

in _a

previous

_paper

[I I].

It is

O.14

fs~f O.12

o,i

O.08

O.06

O.04

O.02

g

°

,5b,

O.12

o,i

O.08

O.06

O.04

O.02

-lo -5 5 lo

ENERGY (E/t)

Fig. ^5. al Total

density

of _states reconstructed by the continued fraction without gap (same _parameters

as Fig. 4a) hi Total

density

of _states reconstructed

by

the continued fraction with gap (same parameters

as

Fig.

4b).

interesting

_{to note} that this

configuration sum'"I)ace

is

equivalent

to the

augmented

_space of

Mookerjee

^and collaborators

[12]

in the _case of substitutional disorder. However it is _more

general

and allows _one to treat the _case of chemical order. These aspects ^{will be}

developed

in _a

forthcoming

_paper

[4].

Acknowledgments.

We are

grateful

to J.

Kudrnovsky

for fruitful discussions

concerning

the

application

of the C.P.A. method in the L.M.T.O. formalism.

Appendix

A.

Our aim is to show that the Green's function

G~(z)

_or

_GB(z)

of A or B _atom in the effective medium _can be calculated

by attaching

chains

C,

to each atom I in the medium. These chains

C,

simulate

exactly

the self energy

3(z ).

Thus _we consider the system

represented

in

figure

2b.

We define the

projector P,

in the

subspace

of the orbitals of _an atom I and also _:

P

=

£P,. (Al)

We define the

projector Q,

in the

subspace

of all _states of the chains C~ and also

Q

₌

£ Qi (A2)

Thus

P ₊

Q

=

(A3)

Using

the Schur

formulae,

_{we get}

PGP

=

~

(A4)

z P HP

PHQ QHP

z

QHQ

However the chain

C,

is

coupled only

to the _atom I and thus

PHQ

₌

iP, HQ, QHP

=

£ Q, HP, (A5)

Further _{more, as} the chains _{are not}

directly coupled,

it is

straightforward

to show that _:

z

$HQ

^"

(

z

Q,'HQ, ^~~~~

Thus _one has

P GP

=

~

(A7)

: P HP

£ P, 3, (z P,

with

~

~"

~'

~

z

Q, HQ, ~~' ~~~~

Thus

coupling

^each atom I to a chain

C,

is identical to

adding

the

self-energy 3,

(=

(A8)

to the

atom I _as shown

by expression

(A7).

Appendix

B.

We consider _two matrices ~r~

and,<B

and two

probabilities

_p~ and pB with p~ + pB =

I. Our aim is first to show that _:

~~

+

~~

=

(Bl)

'~ _{'~ ~}

p_A IA + pB XB p_A pB (XA .KB (X~ _.KB)

U

with

U

= PA ~B + PB 'A

(~2)

or

equivalently

~~ ~~ ^~ ~~ ~~ ~~

~~~~~ ~~~

~~~

~~~

~

PA PB

~~~~

>A X~

Indeed _one has

~

+

~~

^{" ~I'A XB + PB XA " ~PAXB + PB XA}

⁾

~ (~~)

A B A B B

thus _:

~~+~~

=

U

= U

(85)

XA XB XA XB XB XA

We then write

PA xA + PB xB = U

)

~PA xA + PB .<B ~ ~PB xA ⁺ PA xB

I

_°~A

xA + PB xB

(86)

Developing (86)

_{we get}

PA XA ⁺ PB XB " PA PB XA

j

XA + PA PB XB

j

.~B +

P~

_XA

j

XB +

P~

_XB

)

XA

(~7)

Inverting (85),

we deduce

xA

I

xB = xB

I

xA

(88)

Developing ~p~

+

pB)~

₌ l and

using (88),

_{we get}

Pi

_xA

I

xB +

Pi

_xB

I

xA = xA

I

xB 2 PA PB xA

I

xB

~B9)

applying (85), (88)

^and

(89)

to

(87)

we get

(82)

_or

equivalently (Bl).

In _{our case, we} can choose

x~ = z e~ «

(z) (B lo)

xB = z eB «

(zl' (Bl1)

and _we deduce

z

e)~

«

(z

) ^~ z

e)~

«

(z)

_z

~(z _z(z) ^~~~~~

with _:

2

" PA EA + PB EB

(B13)

and :

~~~~

~

Z PB ~A

A

~B

~'(Z)

^~

~~~~~

A

"

fi

_(EA

_EB). (B15)

Appendix

^C.

It is

interesting

to compare the memory

place

and

computation

time needed for

calculating

_one stage of the continued fraction in the effective medium and in real space

(I.e.

without any

coupling

of _atoms to

chains).

Let _us first define the index _m of _an orbital _as the minimum number _m for which

fl~~)

^has a non-zero component on that orbital. It is easy to see that if _an

orbital is located _at the level _n and is

coupled

from _a chain to an atom with index

m, then the index of that orbital is _{m + n} that is all fl~~) ^with p < m + n have _zero component

on that orbital.

Quite generally,

if _{we want to} calculate the recursion _vectors up to level

m, we need _to take into _account

only

the orbitals with indices I _mm- The number of these

orbitals _can be estimated in the

following

_way. Let _us call

N~(m)

the number of orbitals with

index _m. Thus in order _to calculate

[fl~~)

^{in real} space, we must take into account

ii~(m)

_{orbitals with}

&~(m)

=

f N~~p). (Cl)

p=o

In the effective

medium,

the number of orbitals with indices less _or

equal

to m is

& _(m

=

~o jj NR~P)

_x

(m

_{p +} I

j (c2)

In the limit of

large

_m,

N~(m)

_grows like _m square and _{we can}

approximate

ii~~p)

as

ii~~p)

m

ii~~p)

x

~.

_If

we take _p

=

20 _{as a}

typical

value _{we see} that the memory 4

place

needed for the effective Hamiltonian is about 5 times that for the real space Hamiltonian which is still reasonable for many systems.

We _now compare the number of

operations

needed at step p ^to calculate

a~

^and

b~ ^{in the effective medium and in}

real-space (I.e.

without any

coupling

of atoms to

chains).

If [fl~~_

j),

_[fl~~) and a~, b~ n w p I have

already

been

calculated,

the _next steps are

I)

^calculate

H[fl~~)

;

2)

calculate

a~

=

(fl~~[H[

_fl~~)

3 calcul ate b~ fl~~_~ i

)

= H fl~~)

a~

_fl~~ b~ _fl~~

)

from which

b~

^and _fl~~~

)

_are

deduced ( [fl~~~

i)

is normalized

by definition).

One _sees

easily

that the number of

operations

_per orbital in steps ² or 3 does _not

depend

on

the Hamiltonian. On the contrary, ^{the number of}

operations

per orbital in step I is

proportional

to the number of orbitals

coupled

to one

given

orbital and this number is

larger

for orbitals in real space than for orbitals in the chains. Thus, if _we call

Z~ (respectively Z~)

the number of

operations

_per orbitals

(for

_steps 1, 2 and

3)

in real space

(respectively

in the

chains),

_we

always

have Z~ _~

Z~.

If the number of orbitals

coupled

to one

given

orbital is

sufficiently large

then step ^{I is}

large

and Z~ can be

only

_a fraction of

Z~.

The ratio R between the numerical work for stage _p ^{of the recursion in the effective medium and in real} space is thus

Z~

+ Z~

pm

_Z~

R p

m m +

-.

(C3)

ZR ZR

⁴

If _we take _p

- 20 _{as a}

typical

value, we see that R _S 5 since Z~ _<

Z~.

We conclude that in many systems ^{the solution of C-P-A-}

equations by

the recursion method _can be done in _a

reasonable

computing

time.

References

[Ii Soven P., Phys. Rev. B 156 (1967) 809

Velicky B.,

Kirkpatrick

S., Ehrenreich H., Phys. ^{Ret,. B 175} (1968) 747.

[2] Ducastelle F., Gautier F., J.

Phys.

F 6 (1976) 2039 _;

Treglia

G., Ducastelle F., Gautier F., J. Phys. F 8 (1978) 1437.

[3]

Haydock

R., ^Solid State

Physics,

H. Ehrenreich, F. Seitz, D. Turnbull Eds., Vol. 35 (Academic Press, New York) (1980) _p. 215.

[4] Mayou D., Julien J. P., to be published.

[5] Muller-Hartmann E., Solid State Commun. 12 (1973) 1269.

[6] Turchi P., Duscatelle F., Treglia G., J.

Phys.

C IS (1982) 2891;

Turchi P., Thesis (Paris, 1984)

unpublished.

[7]

Kudmovsky

J., Drchal V., Masek J.,

Phys.

Rev. B 35 (1987) 2487.

[8] Sigli C., Thesis (Columbia University, 1986).

[9]

Papaconstantopoulos

D. A., Pasturel A., Julien J. P., Cyrot-Lackmann F., Phys. Rev. B 40 (1989) 8844.

[10]

Mayou

D., Thesis (Grenoble, 1987)

unpublished.

[I Ii Mayou D., Pasturel A., Nguyen ^Manh D., J. Phys. C 19 (1986) 719.

[12]

Mookerjee

A., J. Phy.I. C 6 (1973) L 205 and J. Phys. ^C 6 (1973) 1340.