FFC-a program for calculating Franck-Condon factors and R-centroids for transitions between the vibrational-rotational levels of two electronic states of a diatomic molecule

Computer Physics Communications 47 (1987) 305—309 North-Holland, Amsterdam

305

FFC A PROGRAM FOR CALCULATING FRANCK-CONDON FACTORS AND R-CENTROIDS FOR TRANSITIONS BETWEEN THE VIBRATIONAL-ROTATIONAL LEVELS OF TWO ELECTRONIC STATES OF A DIATOMIC MOLECULE -

Mounzer DAGHER and Hafez KOBEISSI Faculty of Science, Lebanese University, Beyrouth, Lebanon and Molecular and Atomic Physics group, National Research Council, Beyrouth, Lebanon Received 5 December 1986

PROGRAM SUMMARY Title of program: FFC

Franck—Condon factors and the R-centroids for a diatomic molecule [1—3].

Catalogue number: ABBE Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue) Computer: IBM 3090; Installation: 1N2P3, Lyon, France Operating system: VM/SP HPO Programming language used: FORTRAN IV High speed storage required: 70 Kbytes Number of bits in a word: 64 bits Peripherals used: magnetic tape, terminal, line printer Keywords: diatomic molecule, vibration—rotation, Franck— Condon factors, R-centroids, wave function, overlap integral, Franck—Condon integral, canonical method, analytic expression, electronic states, coupling coefficients, normalization integral Nature of physical problem The program presents a new method for computing the

OO1O-4655/87/$03.50

Method of solution The canonical ~function method is used to calculate the wave function for each interval (r,,,, r,,,÷ 1)of the r axis, in the form of a uniformly and absolutely convergent series. The computation of the Franck—Condon factor is then reduced to that of the “Franck—Condon integral” which can be also written in form of a uniformly and absolutely convergent series in (r,,,, r,,,~1).The coefficients of this series are called “coupling constants” and depend uniquely on the eigenvalues E~and Eb of the considered transition and of the potentials U~and Ub. Restrictions on the complexity of the problem Provided that the Born—Oppenheimer approximation is valid, there are no known restrictions. Typical running time: Dependent upon the number of transitions to be calculated. 73 s for the first test run, 3 s for the second and 43 s for the third. References [1] M. Dagher and H. Kobeissi, J. Comput. Chem. 6 (1985) 360. [2] H. Kobeissi and M. Dagher, Mol. Phys. 44 (1981) 1419. [3] H. Kobeissi, M. Dagber and M. Alameddine, Intern. J. Quant. Chem 1980 (Q.C. Symposium 14).

© Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

306

M. Dagher, H. Kobeissi

/

FFC

LONG WRITE-UP 1. Introduction FFC calculates Franck—Condon factors and R-centroids for transitions between the vibrational—rotational levels of two electronic states of a diatomic molecule. The method used reduces the overlap integral

Q a.b

jI ~ a

=

I

\

b

I

\

.

dr

to that of the Franck—Condon integral Ta.b(O, x)

=

f

and 4’b are the wave functions of states (a) and (b), respectively. The potentials Ua(r) and U”(r) of the two electronic states (a) and (b) are given in numerical form. One can devide the r-axis into intervals delimited by the abscissas of well-known points Pm (turning points and/or others) of the potentials. It is convenient, for practical reasons, to take one of these abscissas to be the ongin r0 and to designate the others by r~ r,~,... (for r> r0), and r~,..., r,~,... (for r Qab and Q~a~b can then be written by using the canonical method [1—3]: 1pa

~

dt.

For any potential, this integral is given by a simple analytic expression in terms of the two potentials for each interval (Tm, rm+i) of the r axis. The Franck—Condon factors are well determined by “coupling constants” related uniquely to the coordinates of a set of points ~m on the potentials (turning points and/or others) and the energy levels of the considered transition.

~ and

Q

,a,b =

~ T

2. Theory and numerical method

and

The theory and the numerical method used in this program have been described in many papers [1—3].The principle of the method can be outlined as follows: A transition between two electronic states (a) and (b) may be characterized by the Franck—Condon factor

T’~(m)

Qa.b

=

2/Qa.aQb~b

Q~a.b/Qa.b

Qa.b

f~pa~pb dr

=

andQ’~ QIa.b

=

=

=

f

dr

“‘~‘,~pa~pb

J

‘q,a~pb,.dr.

The summation is taken over all the intervals (Tm, r ~) of the r axis. In any one of these intervals (Tm, rm+ ~), the two considered potentials may be represented by

(1)

U(x)

(2)

where U( x) is the2rotationless to which in the formpotential of a Taylor series expansion. we add J(J + 1)/r The wave function i/i (T) for a given level can be

or the R-centroid where ~ a,b is the overlap integral

(m)

with T’~’(m)

qa.b

,a,b

=

.

.

written for a given point

T(T,,
isdefinedby

j~pa~pbr dr

the coefficients X~(m) are defined by the recur-

M. Dagher, H. Kobeissi

sion formula:

and

(n+2)(n+1)X~~2(m) = —EA~(m)+ ~ X 1(m)y~1(m)

T

307

~ t~b’(m)I [

M

(m)= ~ ô,

m rmn+1 X~~+1~

il~

~n+2

n=O

with

8”(m)

depend

uniquely on E a E b and on the coefficients y,~( m) and y~’(m)of the potentials ua and U~’in the considered interval (rm, Tm+i).

=~!l(Tm),

X1(m) =

,a,b

FFC

The coupling coefficients

J0

X0(m)

/

1Ii’(Tm),

Qa.b

and for the first interval

(T0,

and

become respectively: 1/n + 1 ,b m) X,~

Q~a.b

M

r1):

~a

~

.b =

(

m n—O

and where E is the eigenvalue for the considered level. We notice that E, ~~(T 0) and I~ti’(T0)are calculated by the program DIRIGE [4]. When numerical values of ~pa and ~pb are introduced for each interval we can write: Ta.b(m)

=

f”+’ fx,,,

M

r,,,

=

m ,~+m ~

n + 2

~

~a.a(m)Xn+~/fl+1

m n=O

Qb.b..~

dx

+

M

,~.Ø

~~,a.b(m)xfl

j.

r ~

~,n+2

b’(~m)i ‘I “~m

The normalization integrals become: Qa.a~

M

~‘

m

6(~)(~~Tm) dr

~

F

M

~

Q~a.b =

M ~ ~~bJ1’(m)Xn+l/n+1 m n—O

Thus qa.b and R defined by eqs. (1) and (2) are already calculated.

and M

T ,a.b (m)

frm+1 =

~

n

0

~:‘m)(r

—

Tm)T

dr

3. Program structure

M

=

f

The calling sequence of the various subroutines

X

~:‘F~(m)x~~(x+

~ =

rm)

dx is as follows:

0

where

FFC

Xm=rm+i~rm,

I

x=r~Tm

SPLIN

and o:.b(m)

I

INTER

The functions of the different program segments are as follows. =

X~(m)X~~(m).

~

3.1. FFC Finally M

Ta.b(m)

=

~ n=0

~:‘b(m)x:~/(n

+ 1)

Master segment containing input for details of the potentials of the two states (the coordinates of a set of points ~m’ turning points and/or others), the eigenvalues and the log derivative eigenfunc-

308

M. Dagher, H. Kobeissi / FFC

tions. It calculates Franck—Condon factors and R-centroids for transitions between the vibrational—rotational levels of two electronic states of a diatomic molecule.

N1G

—

N1D

—

3.2. SPUN

NDM2

—

N2G

—

Cubic spline procedure called by FFC. This subroutine is used to interpolate between the points 1~mof the R—K—R potential curve by assuring the continuity of the first and second derivatives in

*

The number of given points to the left of the potential minimum in state (a). The number of given points to the right of the potential minimum in state (a). The total number of points to be supplied for the potential corresponding to the second electronic state (b). The number of points to the left of the potential minimum in state (b). FORMAT (7110)

Card 3

MI,MJ

—

MR,MS

—

JR,JS

—

LM

—

* Card 4 AK

—

RE1

—

RE2

—

3.3. INTER Potentials-generating subroutine; it calculates, at each interval of the pointwise potential, the four coefficients of a third order Hermite Polynomial obeying the conditions given by SPLIN.

4. Input and output channels This version of the program reads data from channels 2 3 and 4. It writes the results to channel 6. 5. Description of the input data The input data are devided into three tables. 5.1. The first array, called DFFC, contains: *

Card 1

FORMAT (7110) *

LA, , LG The program can accommodate different values of LMAX: LA, LB, LC, LD, LE, LF, LG according to the width of the intervals . . .

Card 5

FORMAT (3D20.10) AK = 21s/h2 = ~s/16.857630, where ~s is the reduced mass expressed in atomic mass units. Abscissa of the potential minimum corresponding to the first electronic state (a). Abscissa of the potential minimum corresponding to the second electronic state (b). FORMAT (D23.16)

—

LMAX is the number of terms in the series representing the eigenfunction and the Franck—Condon integral. * Card 2 FORMAT (6110) (Tm, Tm+i).

RI

*

Card 6

—

The total number of points to be supplied for the potential corresponding to the first electronic state (a).

—

The starting point on the r axis (RI = r 0). FORMAT (2D23.16)

(in fact(MJ

—

MI) cards, M .

T1(J) NMD1

These are the first and the last vibrational levels corresponding to the first electronic state (a). The first and the last vibrational levels corresponding to the second electronic state (b). The first and the last rotational quantum numbers to be considered if calculations are to be performed at J ~: 0. Maximum number of points on radial mesh used in solving the Schrodinger equation.

—

=

MI, Mi) .

Eigenvalues corresponding to the different levels of the first electronic state (a) calculated by the program DIRIGE [4].

M. Dagher, H. Kobeissi

Y1(J)

—

T2(J)

—

Y2(J)

—

Log derivative eigenfunctions at the starting point RI calculated simultaneously with T1(J) by “DIRIGE”. Eigenvalues corresponding to the different levels of the second electronic state (b) calculated by “DIRIGE”. Log derivative eigenfunctions at the starting point RI calculated simultaneously with T2(J) by “DIRIGE”.

b

*

Card 1

FORMAT (2D2o.lo)

309

V” ~ 95 of the X state and the level V’ = 43 of the B0~state of iodine. transitions between the vibrational levels 0 ~ ~ 7 of the X ~ + state and the vibrational levels 0 ~ V’ ~ 5 of the A0 + state of the PbO.

Both calculations were performed on rotationless potentials. c

5.2. The second array, called COORDJ, contains:

—

/ FFC

—

transitions between the vibrational—rotational levels (0 ~ V” ~ 20, J” = 70) of the X’~+ state and the vibrational—rotational levels (0 ~ ~ 5, J’ = 70) of the B ill state of NaK.

(in fact NDM1 cards, I = 1, NDM1) (RDO1(I)

—

YDO1(I)

—

The abscissa of the different points of the potential corresponding to the first electronic state (a). Ordinates of these points.

5.3. The third array, called COORD2, contains: RDO2(I)

—

YDO2(I)

—

The abscissa of the different points of the potential corresponding to the second electronic state (b). Ordinates of these points.

6. Summary of output The program output contains a tabulation of the levels of the different transitions between the two electronic states, Franck—Condon factors and R-Centroids.

Acknowledgements This work was supported by the C.N.R.S (Centre national de Recherche Scientifique France). It was written in the laboratory of ionic and molecular spectrometry, M. Dagher wishes to thank Dr. J. Desesquelles, Director of the Laboratory, for his hospitality. We thank Professor J. D’Incan for his kind support and continued interest in this work and for valuable discussions, Mme F. Martin for kindly supplying the iodine and the PbO R—K—R potentials. We are grateful to Miss A. Ross for supplying the NaK potentials and for reviewing the manuscript. —

References [11 M. Dagher and H. Kobeissi, J. Comput. Chem. 6 (1985) 360.

7 Sample ou~’ut

a

Sample data are given for: transitions between the vibrational levels 0

—

~

[21 Dagher, 44 (1981)Intern. 1419. J. [31H. Kobeissi Kobeissi,and M. M. Dagher andMol. M. Phys. Alameddine, Quant. Chem. 1980 (Q.C. Symposium 14). [4] M. Dagher and H. Kobeissi, Comput. Phys. Commun. 46 (1987) 445.