Determination of the radial coupling between molecular states

JoumalofiUolecr&rStmcture,12O(1985)28~289 Tlt&OCHEM Else&r SciencePublishers B.V., Amsterdam - printed in The Netherbds

DETERMINATION OF ME

RADIAL COUPLING BmEEN

MOLECULARSTATES

M.C. BACCHUS-MONTABONEL’*, R. CIMIRAGLIA* and M. PERSICO’ 1 Laboratoire de Chimie Theorique de l’ENSJF, 1 rue Maurice r2ouge (France) Istituto

di

Chimica

Fisica,

Via

Risorgimento,

35,

Arnoux,

56100 Pisa

92120 Mont-

(Italy)

ABSTRACT The radial coupling between the first ‘z+ states of HeNe2+ has been calculated in the adiabatic representation by means of the finite differences technique and in a non-adiabatic representation using a projection method previously developped. The behaviour of the radial coupling matrix elements is discussed in view of using them in the collision equations. In the quasi-diabatic basis, radial couplings are shown to be negligibly small and the non-adiabatic transformation provides a rapid and accurate determination of these couplings between adiabatic wavefunctions. INTRODUCTION An amount of experimental tum-chemistry suitable sings

for

calculations. dynamical

of potential

ween adiabatic

batic

states

finite

studies,

which

differences

technique near

dial

coup1 ing necessitates

great

density

these

ab-initio

results

To get rid

of these

during

gested

(ref.*);

radial

couplings

ance in order functions In the first

the

calculation

all

we have developped are

intended

A great

their

first

part

of this

by Levy (ref.4)

numerical peaked func-

shape of the

matrix

elements

difficult

ra-

for

(ref.3)

dependant

values

have been have been sug-

in which eiectronic

and

of the

dist-

by means for

internuclear

example of spline

equations.

paper we give

a

to interpolate

representations

number of definitions

to be smoothly easily

is of

distances.

an approach

and use them in the collision

proposed

is then

bet-

when

states

case of a sharply

, non-adiabatic

years.

to interpolate

it

cros-

occur

‘a between adia-

by means of the

of the

not

of avoided

particularly

operator

of the gKL(R)

internuclear

difficulties last

are often

between molecular coupling

the evaluation

As a consequence, for

calculations

but in the

crossings,

functions

deep interactions

eL > may be calculated

the

of points.

couplings

by means of Quan-

neighbourhood

regions,

cumbersome

(ref.l),

avoided

in the

of the radial

I+, g,L(R)=c@KI~Ri

as appears

introduced

to

nowardays

the adiabatic

In these

lead

elements

tion

interpreted

specially

curves.

of non-adiabatic

The matrix

are

Nevertheless

energy

states

the determination interest.

results

the great

and in a second part

statements

we discuss

of the

method

the numerical

226 results

obtained

crossing

2+ for the HeNe system which presents

between the two first

‘.E+ excited

an interesting

avoided

states.

M!?ll-iODOLOGY This is a projection tonian H an effective reference

functions

eigenvalues

method which introduces one Fi defined

{Xi)_

Fi is defined

EN of the electronic

on S, JJ,, of its lision.

so that it gives

hamiltonian

n eigenfunctions

L

where

$

of the electronic

hamil-

by diagonalization

and the orthogonalized

I+ describing

The matrix elements iiij of the effective given by: Fiij = ’
instead

on a smal! subspace 3 spanned by a set of the

projections

the channels implied in the col-

hamiltonian

in the subspace 3 are

GL>EL <$,I Xj' -l/2

=

and

; (PJIN) SNL

SK,_ =<$NlPIeL=-

the projector on the subspace 5). An analogous definition can be applied

(P is

such as the transition

The expression radial

coupling

dipole

is less

straightforward

operator.

to define

other effective

operators

moment:

for derivation

operators

such as the

{Xi},

the require-

For a set of independent functions

of H ment similar to the statement used in the definition leads to the following expression for the matrix elements of the radial coupling in the reference basis (ref.5):

Ix J.>with

i,

basis,

projection

calculation easily

of the

calculated

The possible the collision

after

of the neglect

equations.

operators

(ref.i).

basis

coupling

matrix

interpolation

internuclear of the

radial

A numerical

matrix

elements

defined

above are direc-

For an appropriate

, aRxj’
hand of the equation

radial

of the adiabatic

smooth functions

the effective

the matrix elements

two terms of the right

to the is

that all

dependent on the reference

reference the

(1)

= c EiK Xi i

It should be noticed tly

ackL c; IX_> a L 3

’

k,L

choice

of the

should be small with respect (1).

The first

elements

of the

term corresponds

and necessitates

&p operator,

of R whose matrix

to

the

elements

the

second one should

be

distance. coup1 ing test

should

was performed

improve on the

the

r;yl

HeNe

ution of system.

287

RESULTS The potential puted the

energy

by an ab-initio

6-311Gxx

tations

basis

with

curves

calculation set

respect

to the

into

-3-

+ Ne2+( %,2p4)

-l-

The evaluation states chose

is

expression

respectively

the

states

of HeNeil+ were com-

interaction

and including

all

describing , -2-

single

the

states

\He(‘S)

(ref.6)

using

and double

exci-

dissociating

+ Ne2*(1D,2p4)]

,

) .

determined the

‘z+

configuration

+ Ne+(2P,2p5)j

of the matrix

A=O.O012 au),

This

with

first

configurations

{He+(2S)

JIR has been performed

wavefunction

three

of Pople(ref.7)

at infinity [He(%)

of the

elements

gKL(R)

at two close coupling

breaks

=

by means of the finite

up into

differentiation

is

internuclear

then

given

two parts

between adiabatic

differences

technique.

distances

The

R and R+A (we

by:

gKL_ (R)

of the molecular

= g!:(R)

+ g::(R)

orbitals

involving

and of the

CI coef-

ficients. The radial (Table excited

R

1.500

1)in

coupling

calculated

correspondence

states.

This

912 -0.62021

with

the

to the avoided

peak is

913 0.11131

approximately

SCF-CI

functions

crossing

shows a sharp peak

exhibited

9 au high

by the

and 0.1

wide at half

0.11881

0 _07837

0.04044 0.10774

1.750

-0.53281

0.04039

0.87418

1.850

-0.46572

-0.06312

4.31784

4.62565

-0.30781

1.875

-0.43493

-0.12253

6.83474

7.13013

-0.29539

1.900

-0.38369

-0.19926

8.95335

8.51828

0.43507

1.950

-0.25161

-0.29813

5.30032

4.65813

0.64219

2.000

-0.17272

-0.29804

1.99460

1.90338

0.09112

-0.22295

0.26906

0.14294

0.12612

-0.9681

height.

coup1 age residue1

g23

0.76644

2.100

two first

2.500

0.09434

0.07315

-0.50027

-0.52498

0.02471

4.000

0.05952

0.02882

-0.01363

-0.01277

-0.00086

Table 1 : Radial coupling matrix elements calculated by the technique and results obtained by the projection method. (All values are in atomic units.)

finite

differences

288

R(au) 5

Figure 1: Results obtained wavefunctions:

(---)

applying the projection method to the variational de gR5, (-) dx, (....) residual coupling.

CI

The projection method was applied to the adiabatic wavefunctions; as reference basis we chose the appropriate products of atomic species. In the case of two states it is possible to express the coefficients ciK in terms of trigonometric functions: G1 = case x1 - sine x2 and z2 = sine x1 + case x2 the equation (1) reduces then to: de cx1t$RIx27 = $2 - 8 The mixing angle is determined by interpolation tions.

The shape obtained

&@I for the dR

function

the radial coupling calculated by the finite value for the residual coup1 ing was obtained (fig. i).

These results

of R by means of spline

is in very good agreement with

are in good agreement with those obtained

of the shape of the radial

coupling

func-

differences technique; the maximal for R=l. 95 au and equals 0.64 au

diabatization procedure (ref. 6). The projection method thus allows an accurate details,

;

determination,

by a numerical even in fine

without heavy numerical calcula-

283 tions.

The matrix

which

is expected

problems the

elements

and provides

precise

of &

to be a suitable the

CI wavefunctions

are negligibly basis

same dynamics

for taking

in the adiabatic

small the

in the reference

resolution

the effective

of the

set

{xi>

collisional

hamiltonian

fi than

representation.

REFERENCES C. Galley and J.C. Lorquet, J. Chem. Phys., 67 (1977) 4672-80 J.B. Delos and W.R. Thorson, J. Chem. Phys., 70 (1979) 1774-90 R. Cimiraglia and M. Persico, Mol. Phys., 38 (i979) 1707-10 M.C. Bacchus-Montabonel. G. Chambaud, B. Levy, Ph. Millie, J. Chim. Phys., 80 (1983) 425-39 M.C. Bacchus-Montabonel and P. Vermeulin, Comp. Phys. Corn., 30 (1983) 163-67 8. Levy, Spectral Line Shapes, Walter de Gruyter and Co, Berlin. Rew York, 1980, pp 615-29 B. Levy, Current Aspects of Quantum Chemistry, Elsevier, Amsterdam, 1982, DD 127-37. k:C. Bacchus-Montabonel, R. Cimiraglia and M. Persico, J. Phys. B: Atom. Molec: Phys., 17 (1984) 1931-42. ROK;;‘hman. J.S. Binkley, R. Seeger, J.A. Pople, J. Chem. Phys., 72 (1980) .

*

Present

address:

Claude Bernard

Laboratoire Lyon I,

de Catalyse

Organique,

LA 231,

Universite

43 bd du 11 Novembre 1918, 69622 VILLEURBANNE Cedex.