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A ”canonical functions” approach to the solution of two coupled differential equations of scattering theory H. Kobeissi, K. Fakhreddine

To cite this version: H. Kobeissi, K. Fakhreddine. A ”canonical functions” approach to the solution of two coupled differential equations of scattering theory. Journal de Physique II, EDP Sciences, 1991, 1 (8), pp.899-908. .

HAL Id: jpa-00247562 https://hal.archives-ouvertes.fr/jpa-00247562 Submitted on 1 Jan 1991

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Phys

J

France

II

(1991)

1

1991,

AOOT

899-908

PAGE

899

Classification

Physics 31

Abstracts 34 40

34 00

15

canonical functions » approach to the differential equations of scattering coupled

A

solution

«

Kobeissi

H

Faculty

(Received

science,

Lebanese

University

Council,

Beirut,

December

3

two

Fakhreddme

Research

of

National

K.

and

of

theory

1990,

and

Group

the

of

and

Molecular

Physics

Atomic

at

the

Lebanon

April 1991,

revised16

accepted 30 April 1991)

differential (CE) arising m the The Abstract. system of two coupled Schrodmger equations functions » descnption of considered A canonical close-coupling electron~atom scattering is « approach is used This approach replaces the integration of CE for the eigenvector with unknown functions initial values canonical » having welLdefined initial values by the integration of CE for « of the then deduced from arbitrary ongin ro (0 < ro < cc The terms reactance matrix at an are canonical functions The results are independent from the choice of ro It is shown that many these difficulties the application are avoided, mainly that of the choice of the conventional numencal m results of the present determined The boundary variable 0 which is automatically starting r~ of with compared those methods and show excellent method other agreement to previous are results

accurate

Introduction.

1.

Scattering

problems

coupled

of

differential

molecules,

ji(d~/d~ where

the

matnx,

r

is

K~ the

radial

etc.

commonly

are

formulated

m

terms

Of

the

I y

matrix square and angular

p*p

wavenumber

electrons, atoms, form [I]

Of

equations

(I

+

is

+

g~

composed

of

I

)/r~)

y(r)

(1)

0 =

solutions

I, U(r)

number

momentum

u(r)i

for

Y~~ is

a

p

*

p

the

matrix,

channel I

is

n

the

with unit

variable

Of (I) Acceptable solutions must Obey the boundary conditions

be

together

continuous

with

their

first

denvatives

and

must

Y(0) Y(r) Here

J

and

N

are

diagonal

0 =

matnces

Jnm

Nnm

N. B

6

J. =

"

r

-

by

given

"

as

K]~~ u nm

&nm Kl~~

in

~~ in

Kn

~

(Kn ~)

oJ

(2) (3)

JOURNAL

900

ji(x)

where

&~

the

is

The

~i(x)

and

constant

6

matnces

6~

B =

Bessel

and

bt

II

respectively

functions

Neumann

8

[2],

delta.

Kronecker

R

matnx

sphencal

are

PHYSIQUE

DE

and

from

formed

B

all

which

(3)

m

scattenng

sufficient

are

determine

to

for

sections

cross

the

problem

the

reactance

calculated

be

can

[3]. In

the

last

decades,

two

methods

numencal

by Thomas integrator (like an boundary condition with As results

r~

sensitive

large

is

clarity,

of this

the

The

For

of R

outlined may be initial point r~

excellent

an

of

review follows

as

the

are

that

show

is to

the

the

avoid

to

one

become

unstable

already

used

method

»

the

to

This

may

difficulty

the

method

problem

channel

solutions

functions

canonical

«

functions

canonical two

small,

too

[15, 16] allows

the

functions

channel

two

I(I

C(r)

=

be

can

r~. if r~ is inaccurate.

point

mentioned

problem

present

method

then

is

~

for

the

above. is

tested

if

;

For

outlined

m

against

a

application.

canonical

where

development

the

m

[4-12])

used

equation of the for

section

conventional

(3)

work

extension

next

2.

the

solutions

Schrodmger

radial the

determination

of

that

to

the

aim

The

example

made

Refs

[14]) is 0 where the Numerov starting at an (2) is satisfied, and stepping on until a sufficiently «large» value the other boundary condition (3) is satisfied. The matching of the computed matrix form (3) allows deduce the R its reactance matrix asymptotic to one straightforward application of most methods mentioned Bayhs [12], by et al gtves

where

ri Y

numencal

The

been

has

progress

of (I) (see for

solution

[13]

al

et

gtven

is

considerable

for the

method.

+

coupled

the

case,

Yi(r)

+

Yi'(r)

+

jKl iK]

1)/~, V(r)

Yi(r)

C

(r)i

u~(r)

c

(r)i Y~(r)

coupling

»

be

can

Ui(r)

the

is

(I)

equations

"

written

V(r) Y2(r)

(4.I)

Yi(r)

(4.2)

v(r) =

potential

boundary

The

conditions

(2)-

wntten

Y~ Y~

(r~)

0

with

r~

=

(ri) =

F~ (ri)

with

ri

0 =

1, 2

(5.I)

1, 2

(5 2)

,

oJ

i

=

,

with

F~ (r) of the

Each

Kj'~rjA~ j I(K~ r)

=

(4)

be

may

Y"(r) and

It

is

Y(r) =

ro

being

an

separately equivalent to the

treated

be

may

equation

Y(ro)

+

arbitrary

(r

B~ ~

i(K~ r)i

i ,

equations

two

+

as

f (r) Y(r) linear

a

(0

+

second




of the

by

l'2(~0) "22(~)

values

"

wntten

Y2(ro) tY12(r)

+

deterrnined

be

can

+

l'((~0) fl21(~)

+

well

"tj(~0)

Y(r)

solutions

Yi(ro) p ii(r)

+

"?1(~)

l~l (~0)

"

functions

the

that

Yi (ro) ail (r)

"

1~2(~) The

finally

find

advantage I'(ro)

an

and p~~ the

other

and «

(pit, p~j),

couples

functions

the

over

completely

not

are

functions

canonical

»

of the

system (4). canonical

( (r) By using

functions

first

the

allow

boundary

the

condition

of

( (ro) and

I'(ro)

0), the

expressions

(10)

determination

(at

r

=

furthermore

and of

((r)

lead

to

of the

relations 0 "

0 "

Yi(ro) tYii(0)

+

Yi(ro) pit(°)

+

Y2(ro) tY12(0)

+

Yi(ro) p12(0)

(13 1)

Yi (ro) tY21(0)

+

Yi(ro) p21(°)

+

Y2(ro) tY22(0)

+

Yi(ro) p22(°)

(13.2)

902

JOURNAL

These

relations

((ro),

and

allow

to

have

to

one

PHYSIQUE

DE

the

I'(ro)

constants

two

N

II

of

tern3s

m

the

two

8

others

write

l'((~0)

"

Yj(ro)

=

l'l(~0) l~ll

+

l'2(~0)1~12

(14.I)

Yi(ro) L~j

+

Y~(ro) L~~

(14.2)

where

~"

ail(0) a~j(0)

jij2(0) ji22(°)

)~~~)

~~~~~

"

~~~

~

~15.1)

ID

(15 2)

~~

(15 3)

/l~

(15,4)

~~

~

a~i(0)

ji21(°)

of (14),

use

the

y~

and

&,

related

are

y~(r) &~

We

notice

since

they

deduced

from

here

too

to

a~i

=

(r) that

the

(according

are

(17)

Yi (~0)

+

yi(r)

+

y~ linear

(17))

Y,(r)

solutions

&j(ro)

I =

+

Y~(ro) &i(r)

(16 1)

+

Y2(ro) &2(r)

(16.2)

+ +

L~i p,~(r), L~~ p,~(r)

i

=

i

=

,

(r) and

&~

(r)

"

y~(ro)

solutions Their

of the

initial

system

values

are

~~

&~(ro)

0,

=

1, 2

p

of a~~

(17 1) (17.2)

1, 2

particular

are

combinations

0,

=

become

by

p~~

Lii p,i (r) Li~ p ii (r)

functions to

fl22(0)

write

we

yj(ro)

and

a~~

a~~(r)

=

fl21(°)

Yi(ro) Y2(r)

"

(r)

fl12(0)

Yi(ro)

=

Y2(r)

fl11(°)

(10) of the

expressions

Yi(r)

where

)~~~)

~

j~

By making

(4)

ji11 (°)

" ~~~~~

~~~

With

(0)

a11

~~'

(181)

=

,

l~

Yi(~0)

~( (~0) "1~12

ii,

"

>

~i(~0)

1~21

"

(18 2)

1~22

>

and

yj(r)

=

Yi'(r) &j(r)

=

=

&I'(r) Thus y, (r) and &~(r) By imposing the second

boundary j

( rf)

F2(ri) given

by (6)

canonical

again

are

F

F,(ri) being

=

These

fj(r) yj(r) f2(r) Y2(r) fj(r) &j(r) f2(r) &2(r)

=

are

Yj

r0

two

+ + +

Y

(3), rf)

Yj(ro) Y2(ri) relations

(19 1) (19.2) (20.i)

V(r)

functions

condition

=

y~(r) Yi(r) v(r) &~(r) V(r) &i(r) v(r)

+

well

1e,

by

(20.2)

deternuned

using

(16)

by equations (18)-(20) (5 2), one gets

and

+

Y2 ~0

(~f)

(21 1)

+

Y2(ro) &2(ri)

(21.2)

between

~

l

Yj(ro), Y~(ro)

and

the

unknowns

N

8

CANONICAL

A

Aj, Et> A~, B~ leading

(Yi(ro),

Y~(ro))

»

reactance

R.

the

to

couples of arbitrary

two

APPROACH

These

unknowns

The

of

computation

and

with

&~

A~, B~ (Eq. (16)) the

Thus

903

by giving

obtained

are

to

values.

According to the theory presented above, the determination coupled equations (4) where the initial values are not known, is functions having well determined initial values arbitrary at an ~U~Cli°~S ("~l)> ("~2)> (fl~l)> (fl~2) With t =1,2, y~

SOLUTION

THE

TO

application.

Numerical

3.

FUNCTIONS

of

2.

effort

for

r

for

ongin

to

ro,

other

of

allows

rmro

((r)

solution

the

reduced

that

the

i-e,

canonical

the

the

of

other

of the

canonical functions

determination

of

of R.

that

the

to

that

allows

ro

«

computation

reduced

is

coupled

the

This

consequently

and

whole

integration

functions

these

=1,

i

of

successive

of

use

single

one

operation,

this

is

the

equations

Yi

Yi

iKl- Ui(r) iKl- U2(r)

+

+

(r)i

C

Yi

V(r)

Y2

(22 1)

V(r)

Yi

(22.2)

=

C(r)1Y2

"

y~ (ro) and and p~~;

y) (ro) are always (it) by (18) when gtven: y, a~j, a~~, stands for and So algorithm single needed for all of the problem the 3~ steps one y~ is y~ In order to test the present method we consider the following problem the calculation of the matrix R for electron-hydrogen scattering m the strong-coupling approxireactance where only the first and the second 0) are included the mation» atomic stages (I m eigenfunction with exchange neglected [18] expansion The potentials Uj, U~ and V are gtven by where

all

coefficients

the

(I) by (11)

known,

are

and

stands

when

where

for

the

values

initial

p,j

for

then

=

Uj

(1+1/r)exp(-2r)

=2-

U~=-2(1/r+3/4+r/4+~/8)exp(-r)

V=41127(2+3r)exp(-1.5r). wavenunibers

The

Table

Computed those by fixed are

I.

compared

fixed

are

to

wavenumbers

Kj

at

of

values

Kj

I

0.5 =

Mohamed

1.138839

0

0.3854490

0

3256349

3855010

0

3256568

0

0.3871720

0

3871420

0

3871970

0

3871970

1530146

0

38719696

0

38719696

Mohamed

method

used

with

a

local

tolerance

s

=

Mohamed

method

used

with

a

local

tolerance

s

=

Apagyi

method

used

with

(~)

Apagyi

method

used

with

PHYSIQLE

It

T

M 8

a

basis

a

basis

ROOT

lwl

3256686

0.3854697

1530147

(C)

B~

3854565

(b)

DE

Bj, A~, B~, by the present method and by Apagyt et al. [20]. The

A~

(a)

JOURNAL

j,

a-u

0.3855010

1.1530019

method

Present

A

[18],

0.3854753 0

138996

Apagyi (C) Apagyi (d)

a-u

Bj

138853

(a) (b)

Mohamed

terms

Mohamed

Aj Chandra

0 5

=

matrix

reactance

=

K~

and

[1, 9], by I and K~

Chandra at

=

size

N

size

N

0.3189450 0

3187691 0.31876919

10~~ 10~~

10 =

=

20 42

PHYSIQUE

DE

JOURNAL

904

N

II

8

it was already treated by since by Mohamed [18] using the same variational vanable method with recently by Apagyi et al. [20] using a steps, and more computed by the present matrix technique. In table I the terms A j, Bj, A~, B~ of the reactance method compared to those obtained by the previous works. are others, we consider this as a first results with the We that note agreement our are m confirmation of the present method. The results by Apagyi given m the fifth line are the limits from N of At, Bj, A~, B~ when the basis size N increases 20 (see Tab. III of 2 to N in those of Apagyi with N 20 excellent with Ref [18]). Our results agreement we are confirrnation of the present method. We note finally that consider this agreement as a further obtain A~ Et which is predicted by the theory [20] The difference A~ Bi decreases we 20. work from 6 x 10~~ for N 2 to 8 x 10~~ for N with N m the Apagyi done on a home Our computations computer (the New Braln AD, gluing 8 significant are the single routine, numencal figures). The quite simple since it uses one i-e, program is differential and gtven equations with given coefficients integration of a system of two coupled request). available from the authors initial values (a listing is upon elsewhere of will published However, we note here details the numencal be The treatment they are obtained and imposed that the boundary points r~ not program m our ri are of a~i(r)/p~i(r)) looking i(r)/p (r) (or when the limit automatically by. (I) constant to at ii of the of looking the obtain and (ii) by the B, 0, to reactance A~, to terms convergence r r~ ; differential obtain ri. We integration of the also that the R when to note matnx co requation may be done, within the present scheme, by using any integrator. That used for the superposition «integrals application is the vanable [21] of the step present version proved be highly [23]. difference [22] accurate equation to This

problem

Chandra

[19]

is

using

well

indicated

the

test

to

method

present

step de Vogelaere's

fixed

a

method,

then

=

=

=

=

=

=

-

4.

Discussion.

In

order

that

study

to

this

the

method

validity

makes

of

use

assumption like that usually used point ri are, here, automatically

m

theoretical any the choice of r~ and

by the

determined

of the

boundary

method,

present

approximation, ri, the starting

of

nore

we

any

a

point r~ and

conditions

as

it

the

notice

priori final

shown

was

in

2.

section

Since

the

integration of

sources

numerical of

the

error

of

those

used

here

the

sensitive

the

«

is

difficulty vicinity of r 0, vicinity (as indicated

coefficients

integration

numerical

is

method

present

given

[22] with

integration

integrator

of the

treatment

system are

numencal

The

The

application

numerical

of the

no

to

integral

the

chosen

single values), system [22]

is

reduced

and

given

of the

integrator

superposition

(IS)

to

one

initial

and

to

the

difference

routine

the

step-size equation

(the unique

strategy.

[22-23]. potentials

here lies in the smgulanty of the hand, and in the necessity of the integration of the the introduction) other hand. the system m this m on did for face this situation, potential m a single channel [24] by using the We as we a singular version of the « integral superposition difference vanable-step equations » [21]. We give m example the quantities (Eqs (15)) deduced from the II Lii, Li~, L~i, L~ canonical table as and computed for functions Ki la-u-, 0.5 K~ starting at ro= I and a-u-, a~~, p~~ The

(Ui

and

main

U~

numerical

in the

~

appeanng

on

one

=

integrating size

m

each

backwards

interval,

simply by looking Another we

give

the

to

illustration terms

towards

r~

0.

We

notice

the

automatic

approach the origin r 0 (to to one 0) of the computed stability (when r

allows the

=

that

of the

=

-

simplicity

At, Bj, A~, B~ of the

determination

imposed

any functions.

of the

step-

accuracy),

method is presented in table III, where present matrix deduced from the R. These reactance tern3s are

of the

bt

8

A

Table

Computed

II.

L~j, L~~ the

CANONICAL

«

deduced

computation

(at

values

from the of the

APPROACH

»

several

of

values

THE

TO

r~l)

of

functions a,~, p~~ (Eqs. (15)),

canonical

SOLUTION

the

and

used

905

Ljj,

quantities initial as

values

l~ll

~12

~21

1~22

0.767 (~) 0,327

4.1582105

0.019421007

0.019421007

0226503

0.071945593

0.071945593

0,127

0.38968764

0,12217359

0,12217359

4.1340494

92288978

0

0.21119275

0.044

0.059900016

0,16729176

0.013

0.11278618

0,19787078

0,19787078

0A1383629

0.003

0.18290949

0.21179547

0.21179547

0.50680657

0.5(- 3) (b) 0.5(- 4) 0.2(- 5)

0.20326702

0.21600709

21600709

0

0.20724818

0.21683973

0.21683973

0

0.20752484

0.21689770

0.21689770

0.53978748

0.6(- 7) 13(- 7) o 12(- 7)

0

(a) The

integration

(b) 0.5(- 3)

Table

stands

0

16729176

0,19134879

53406899 53941574

0.20763839

0.21692150

0.21692150

0.53994007

0.21692970

0.21692970

0.53999262

0.20768748

0.21693178

0.21693178

0.54000603

at

for 0 5

x

The

I

r

=

steps

successively

are

computed by

«

vanable-step

several

values

the

equations [21].

difference

Computed

III

0

0.20767751

started

is

integrals-superposition

Lj~, for

elements.

matrix

reactance

~

r

FUNCTIONS

10~~

elements

A

Bj, A~, B~ of the

j,

reactance

matrix

R at

of

I

>

r

233 (a) 2.206

82

Aj

Bj

A~

0.92512638

0.50674237

0.50674237

6588524

1.0966815

0A1862183

0A1862183

lA204144

5.410

1.1352786

40148657

0.40148657

0.35697847

9.481

1.1527700

0.38793510

o.38793510

o.32347088

14.137

1.1530114

o.38719181

o.38719191

o.31883044

16.415

1.1530141

o.38719946

0.38719946

0.31877888

0

18.856

1.1530149

0.38719700

0.38719700

o.31876926

21.184

1.1530150

0.38719702

o.38719702

o.31876914

23.507

1.1530150

0.38719702

0.38719702

0.31876900

0.38719702

0.38719702

24.652

(a)

1530150

The

is

integration

integrals-superposition

computation

stepping One

on

may

of

the

towards note

that

started

at

r

difference

canonical oJ. riin both

=

I

The

equations

function ri

is

reached

integrations

steps

successively

are

computed by

31876899

0

the

«

vanable-step

[21]

1,2) (Eqs. (16)) started at ro &, (t when Aj, Bj, A~, B~ become constants. (for r < ro, and for r > ro) the number of

y,,

=

I

and

=

intervals

JOURNAL

906

Table

Dependence of computed

IV

h(e),

h

size

=

where

local

the

10-3 10-4 lo-5 10-6 10-? 10-8

effectively The

give the

the

variable

step-

~2

~2

0.38742101

038742101

1533287

0.38723442

0.38723442

1530291

0.38719869

o.38719869

0

31876227

0

31876785

38913309

o

1.1530174

0

38719730

0

38719730

1530146

0

38719696

0

38719696

We

matnx

was

high influence

deduce

to

one

already

tested

accuracy of the

(equivalent

stability of

the

notice

allows

This

local

these

8

vary.

0.38913309

used

reactance

upon

1548922

equation [21] and showed table IV an example of the

m

R

matrix

1.1691259

radial

the

to

~l

limited (~ 20 is quite generated. was present variable-step strategy

error

made

is

e

bt

II

A,, B, of the

elements

tolerance

~l

E

PHYSIQUE

DE

against to

tolerance

values

when

that the

the s s

o.31114486 -0.31787312 0.31861915

0.31876919

no

significant

round-oft

problem for precision) We computer the computed of terms on approaches the computer eigenvalue

precision.

Table

Computed values of compared to those At (N), of the basis size N

V.

method values

Ai(N)

4

1.1490601

0.3820256

0

3851256

8

1.1527882

0.3873718

0

3872978

0.3192054

12

1.1530122

3872067

0

3871988

0.3188105

16

1.1530148

table

obtained V

our

Ladanyi

functions when

N

0

1.1530146

convincing

These eleInents

Bi(N)

1530147 method

Present

and

Aj, Bj, A~, B~ by the present matrix terms A~(N), B~(N) by Apagyi et al [20], for several

reactance

Bj(N),

N

20

m

the

A,,

results here

B~

with matrix

are

those

0

increases

a

this

method

0.3187697

0.3871970

0.3187691

38719696

by

excellent

of the «

3432615

0.3871970

variational

becomes

0

0.3871965

compared

elements

least-squares [20], N being the number using

82(N)

0.3871968

emphasized of the

A2(N)

free

test

0

0.31876919

38719696

comparison of the reactance matnx Apagyi and Ladanyi [20J We give those A, (N), B~(N) obtained by Apagyi to method involving only square integrable test functions. According to Abdel-Raouf [25], the

work by

of

anomalies.

Note

that

our

results

are

the

by Apagyi-Ladanyi when N increases results (and those by Apagyifinally in table I the discrepancy between We notice our that the of Ladanyi) and those by Mohamed [18] (and by Chandra [9]). We assume reasons 0.03, it is only mainly (i) the starting point used by Mohamed is r~ this discrepancy are limits

of

those

N

8

A

CANONICAL

«

FUNCTIONS

APPROACH

»

SOLUTION

THE

TO

907

(u) the final point used by Mohamed is ri 13 7, it is of about 25 m Mohamed de Vogelaere's used by the integrator (a fourth is our fourth-order Runge-Kutta order one), it is equivalent to the integrator [9], much less accurate integrals superposition than [23] integrators our « 0.0000001

work,

work

our

m

,

(iii) the

integrator

Conclusion.

5.

canonical

The

problem [15] differential the

equations

reactance

phase

the

already used with for the diatomic eigenvalues success problem [16], is extended here to the system of coupled determination of scattering problems, and mainly to the some of the electron-atom close-coupling approximation two-state

method,

functions and

shift

arising m R of the

matnx

scattenng It

is

solution

shown

that

(with

difficulties

conventional

initial

values

initial

and of

determination

are

The

of is

that

to

avoided,

combination

boundary

the

generalization scattering problems

values)

thus

linear

the

approach

functions

canonical

the

unknown

of the

the

system

with

namely

mention

we

of

reduces

solutions,

and

of the

determination

initial

known those

related related

those

Many solution

the

to to

system

values. the

numencal

regions.

approach

the

present undertaken

to

the

problem,

N-channel

and

other

to

References

[1] SECREST

D,

chap. II, [2]

ABRAMOWiTz

Atom-molecule p 377 M. and

STEGUN I

collision

A,

«

theory

»

Handbook

(Plenum, of

New

mathematical

York~

1979), chap 8,

functions

»

(Dover,

p

265

New

and

York,

1965) Molecular collision theory » (Academic Press, New York, 1974). [3] CHiLD M S, « (1965) 388 A137 [4] BARNES L. L, LANE N F and LiN c c, Phys Rev [5] LESTER W A, J. Comput Phys 6 (1970) 378 Phys 48 (1968) 4682 [6] JOHNSON B R and SECREST D, J Chem [7] GORDON R G., J Chem Phys sl (1969) 14 [8] SAMS W N and KOURi O J, J Chem Phys 51(1969) 4809 [9] ALLISON A c, J Comput Phys 6 (1970) 378 [10] JOHNSON B R, J. Comput Phys 13 (1973) 445 ill] ANDERSEN P, Chem Phys. Lent 44 (1976) 317 [12] BAYLiS W E and PEEL S J, Comput Phys Commun 25 (1982) 7 MCLEMiTHAN [13] THOMAS L D., ALEXANDER M. H, JOHNSON B R, LESTER W A., LIGHT J c, and WALKER R B, J K D, PARKER G A, REDMAN M J, SCHMALz T J, SECREST D Comput Phys 41(1981) 407 Observatoire Astrophys (Russ ) 2 (1933) 188 [14] NUMEROV B, Pub] central [15] KOBEiSSi H, J Phys 42 (1981) L-lsl, J Phys B IS (1982) 693 ; and KOREK KOBEiSSi H M, Int J Quantum Chem 22 (1982) 23, DAGHER M and H, Comput Phys KOBEiSSi Commun 46 (1987) 445 K FAKHREDDiNE and KOBEiSSi M, Int J. Quantum Chem (in press) [16] KOBEiSSi H. ALAMEDDiNE M. A, J Phys. [17] KOBEiSSi H and France 39 (1978) 43 [18] MOHAMED J. L, J Comput Phys 54 (1984) 457. [19] CHANDRA N, Comput Phys Commun 5 (1973) 417. [20] APAGYi B. and LADANYi K. Phj.s Rev. A33 (1986) 182

JOURNAL

908

[21] KOBEiSSI [22] KOBEiSSI [23] KoBEissi KOBEiSSi

H,

H, H., H,

and

KOBEiSSi

M., J.

DAGHER

M.,

KOBEiSSi

M,

and

KOBEiSSi

M

and

PHYSIQUE

DE

Comput

Phys.

77

(1988)

501.

Comput. Chem. Comput Chem 9 (1988) 844. EL-HAJJ Phys A22 (1989) 287 K., J. Comput Phys (in press).

KOREK

M

EL-HAJJ

and

CHAALAN

A, J. A., J

FAKHREDDINE [24] KOBEISSI H and [25] ABDEL-RAOUF A., Phys Rep. 108 (1984) 1.

A,

N

II

J

4

(1983) 218

8