Analytic vibrational matrix elements for diatomic molecules

Computer Physics Communications 39 (1986) 439—446 North-Holland, Amsterdam

439

ANALYTIC VIBRATIONAL MATRIX ELEMENTS FOR DIATOMIC MOLECULES J.P. BOUANICH, J.F. OGILVIE* Laboratoire d’Infrarouge, Associé au CNRS, Université de Paris -Sud, Bâtiment 350, F-91405 Orsay Cedex, France

and R.H. TIPPING Department of Physics and Astronomy, University of Alabama, P.O. Box 1921, University, AL 35486-1 921, USA

Received 15 May 1985; in final form 30 December 1985

PROGRAM SUMMARY Title of program: VIBMATEL

No. of lines in combined program and test deck: 5915

Catalogue number: AAFQ

Keywords: molecular, diatomic, vibrational, matrix elements,

Program obtainable from: CPC Program Library, Queen’s Uni-

expectation values, Dunham potential—energy function, rotational dependence

versity of Belfast, N. Ireland (see application form in this issue)

Nature of the physical problem

and sufficient core

The vibrational matrix elements and expectation values for a diatomic molecule, including the rotational dependence, are calculated for powers of the reduced displacement in terms of the parameters of the Dunham potential—energy function [1].

Computer: Sperry 1100/91; Installation: P.S.!. Université de

Method of solution

Pans-Sud, F-91405 Orsay, France

Explicit expressions for the matrix elements in terms of the vibrational quantum numbers v and v’, or ~v, are given for x’, 0 1 8 or 7, respectively, in two different groups: in one group, 0 v v’ 7 and 0 1 8, up to the eighth order in the potential—energy function; in the other group, 0 z~v= v’ — v 7 and 0 1 7, up to the sixth order in the potential—energy function.

Computer for which the program is designed and others on which it is operable: any computer having a FORTRAN-77 compiler

Operating system: SPERRY OS1100 Exec-8, Level 39R2 Programming language used: FORTRAN-77 with nonstandard

IMPLICIT statement High-speed storage required: 169 Kwords

Typical running time No. of bits in a word: 36

1.3 s.

Overlay structure: optional

Reference

Peripherals used: card reader or input console, line printer

El] J.F. Ogilvie and R.H. Tipping, Intern. Rev. Phys. Chem 3 (1983) 3.

*

Present address: Department of Physics and Astronomy, The University of Alabama, P.O. Box 1921, University, AL 35486-1921, USA.

OO1O-4655/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

440

J.P. Bouanich et al.

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Vibrational matrix elements for diatomic molecules

LONG WRITE-UP 1. Introduction According to quantum mechanics a matrix element of the corresponding operator is associated with any physically observable quantity. For the comparison between theoretically and experimentally determined properties, one thus requires accurate expectation values or matrix elements. For the evaluation of these quantities, one can use either numerical methods, requiring numerical integration over an appropriate range, or analytic expressions, if the theory to produce the latter has generated an adequate supply of expressions over the necessary range of quantum numbers of the discrete states. The numerical method is rather general, but requires a relatively long calculation for each value, whereas once the analytic expressions have been formed, probably as the result of long calculations, the results are fairly simple expressions that require only brief times for evaluation.

Here x is the reduced internuclear displacement in terms of the instantaneous R and equilibrium Re distances, x (R Re )/R e Be is the equilibrium having rotational parameter, Be h/(8~i2c~tR~), the same (wavenumber) units as V; y is the appropriate dimensionless expansion parameter, the ratio y 2 Be/We of limiting rotational and vibrational spectral intervals, that governs the convergence of the analytic expressions to follow. The a 1 coefficients are usually determined from the frequencies of lines in the observed vibration—rotational spectra, whether in absorption, emission or Raman scattering. It is convenient [2] to express various molecular properties, for instance rotational parameters Be,, shielding factors or spectral line intensities, in the form of power-series expansions of the corresponding functions in terms of the reduced displacement x; e.g. =

—

=

=

M(x)

=

~ M1x’.

(3)

1=0

2. Summary of the theory and its implementation In the program VIBMATEL, we use a series of analytic expressions for the vibration—rotational expectation values and vibrational matrix elements of a diatomic molecule (in a ~ state) in terms of the parameters of the potential—energy function due to Dunham [1], commonly used in spectroscopic investigations of these molecules: V(x)

=

BeY_2X2(1

~ a1xi).

+ j

=

(1)

I

The vibration—rotational wavefunctions ~ implicit in the calculations are the solutions of the dimensionless Schrodinger equation

d

2 X

+

1

-~-

,~

E~,—

\

V~x

—

BCJ(J + 21) (1 + x)

‘

~

—

0 ~ \

In this case, any vibrational matrix element of the function M(x) can easily be expressed in terms of the sum of the corresponding matrix elements of x: KM(x)k))

=

~ MiKv~x’Iv’)

(4)

/0

in which the (v I and v’) correspond to the vibrational eigenfunctions ‘4i~0(x)and i~0(x).If the values of v and v’ are identical, then we have the expectation values of the function M(x) in a particular vibrational state v v’. Two groups of analytic expressions are used in the program VIBMATEL. In the first group, the expressions are given for explicit values of v and v’, in the range 0 v v’ 7, useful for powers of x in the range 0 I 8. (Higher powers can be =

generated from exact recurrence relations [3].) The expressions contain all terms in the potential-energy parameters up to a8 inclusive. In the second difference vibrational quantum numbers, group, the between expressions are given in terms of the ~v=v’—v,intherange0z~v7forpowersof x up to x7 and contain all terms up to a 6 inclu-

J.P. Bouanich et al.

/

Vibrational matrix elements for diatomic molecules

sive. These two groups of expressions have been derived by different methods, that we summarize as follows, The matrix elements and expectation values in terms of v and v’ have been generated [21entirely by computer algebra [41,through the use of the vibrational wavefunctions [5], q~~0(x)IV~,g1,(x) tJ.~ 0(x).The derivation of the wavefunctions t?11~,O(x) with the aid of quantum-mechanical relations [3] has been described in detail elsewhere [2]. The preexponential functions g~,(x)are polynomials with coefficients involving y and a1. 1Thus, the I v’) can general vibrational matrix element (v I x be written as =

1

(v~xIv)

/

=

N,N,.K0 g (x)xg~(x) 0),

(5)

so all that is required is a method of generating the ground-state (v 0) expectation values of xt. For the latter purpose, a simple two-term recurrence relation has been previously devised [4]. By =

this means, all the expectation values and matrix elements in the first group have been evaluated. The vibrational matrix elements with a functional dependence on v, in the second group, have been evaluated by the application of perturbation theory [6] using the harmonic-oscillator basis set and including the anharmonic-oscillator terms as successive perturbations; these matrix elements have been generated by numerical computer programs. The method has been previously described [7] and some expressions have also been published; the present set of terms extends the treatment. It is important to confirm that the given expressions are correct. For this purpose some procedures are available. First of all, if all the values of the coefficients a 1 are set to zero, the expressions reduce to the harmonic-oscillator results that can be written in closed form. Secondly, the two groups of expressions can be used to check each other, by analytic evaluation of the appropriate differences. Tests have confirmed the correctness of the expressions by this errors). method However, (and in fact located few transcription some expres-a sions for which overlap between the two groups does not occur could not be checked in this way. The expressions in the first group are in this category. However, because of the nature

441

of the generation of these expressions, that in no way deviates from the method of generating the expressions for 1 < 8 and that is in fact a trivial extension of the algorithm for which all necessary terms were confirmed to be correct, it is believed that these expressions are also correct. The expres1 v + 7) in the second group could be sions (v I x checked against those in the first group for only the special case v 0. As an alternative method of checking all the matrix elements, one can use a well known potential-energy function, such as that due Kratzerand [2],calculable for which form. the matrix elements take to a simple It may be desirable to know the rotational dependence of these matrix elements and expectation values, thus in effect to generate the matrix =

elements
=

‘~‘~

—

~

~‘

—

+

7$3( ‘1’

a1(/3)

=

. . . ~

+

•..

/

a1 + y2$[_3a1(1 + a1)

+(j+ 3)(a1+ (—1)’)] 4$2(...)+y6$3(...)+ (7) +yalmost all known diatomic molecules, Because for the range of y is 102 > y> iOn, there is a rapid convergence of these series provided that the ~

values of J are not too large (J 301forv’J), y 10 a 2) All the expressions for KvJ I x 1(f~) and y(/3) have been automatically converted into FORTRAN statements and have then been assembled into ten subroutines. The first eight subroutines contain the statements corresponding 1 v’), 0 tov thev’expressions the first value groupof(vvI xin each sub7, with aindifferent routine. The ninth subroutine contains the expressions for (v I x1 I v + ~v) belonging to the second group. The tenth subroutine produces the J-dependent parameters a 1($) and y(/3) if necessary,

442

J.P. Bouanich et al.

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Vibrational matrix elements for diatomic molecules

Table 1 Values of contributions of terms containing successive powers of y to vibrational matrix elements of HCI with ~v = I or 2, as v increases 3/2 ~S/2 ~S/2 Sum Matrix element ~ <10_1 0.6393x102 0.8596x10~ <151x3116) 0.4047xl0~ 0.1029 0.5993x10~ O.344SxlO’ 0.2377 Matrix element <01x212)

y 0.5009>< lO_2 0.2295x10~ 0.4069<10~ 0.5841 x101


~,4

—0.2250x103 —0.4467x102 —0.l40lx10~ —0.2886x 10_i

and is executed before the previous subroutines are called as required. The entire program is thus readily adaptable to an overlay or segmented structure for use in a small core, The limitations of the validity of the results can be easily ascertained because the analytic expressions are series in powers of y. As already mentioned, the f-dependence is found by a formal substitution that depends on a rapid convergence of the expressions forconvergence a1(fl) and y($); explicit requirement for this is thattheJ(f + 1) 2. Another limitation of the use of the ex<
Table 2 Description of subroutines of program VIBMATEL Subroutine Input name parameters

Output parameters

MELO MELI MEL2

)~,a1 ~8’

<01 x’I v’), 00 t~v’),

MEL3 MEL4 MEL5 MEL6 MEL7 MELDV ROTDEP

y, y, y, y, y, y, y,

v’

y, y, aa1 —a8, v’

/1 8,8, 10 v’v’ 77 <11 x <21 xt I v’), 0 1 8, 2 v’ 7 <31 x’l v’), 0 I 8, 3 v’ 7 <41 xt I v’), 0 1 8, 4 v’ < 7 <51x’I v’), 0 / 8,5 v’ 7 <61 x’I v’), 0 I 8, 6 v’ 7 <71x’17), t I v + 0dv), l 08 / <7

1—a8, v’ a1—a8, v’ c11 — a8, v’ a1—a8, v’ ai — a8, v’ ai—as, v’ a1 — a8, J ai—as, v, LXv y(f3),
Note that if J * 0 then all the quantities
—0.1100x104 —0.7645x103 —0.4203x102 —0.1239x i0~

—0.7830x106 —0.2213x103 —0.2146x102 —0.9070x102

Sum 0.4772x102 0.1750x101 0.2033x102 0.8092x 10_2

a 1 merely set equal to zero. If the set of known a1 terminates at ak, then all terms (coefficients of y to some power) that contain any potential-energy coefficients a~with j> k should be removed from the expressions. An alternative procedure would be to extend the set of known a1 with values appropriate to a particular model potential—energy function (such as that of Morse or LennardJones [2]). The tsame injunction applies the I v + ~v) for which in mosttocases aexpressions set of a Ky I x 1, I j 6, is required. Also in the

Table 3 Description of COMMON variables and arrays Variable or array name

Description

G Al A2 A3 A4 AS A6

expansion parameter potential energy coefficient potential energy coefficient potential energy coefficient potential energy coefficient potential energy coefficient potential energy coefficient

y a1

A7 A8

potential energy coefficient

a7 O~

MEL (V, L, VP) matrix element V vibrational quantum number L power of the reduced displacement x VP vibrational quantum number R(L, DV) L matrixof power element the reduced displacement x DV difference of vibrational quantum numbers

a2 a3 a4 a5 a6


/ v’

/(clx &

/

Ic +

~v)

J.P. Bouanich et al.

/

443

Vibrational matrix elements for diatomic molecules

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J.P. Bouanich et al.

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Vibrational matrix elements for diatomic molecules

second group in which the v-dependence appears as a parameter to be set, it is important to recognize that for values of v greater than some value (that has to be determined empirically in each particular application) the series in powers of y will fail to converge. For instance, the data in table 1 illustrate the values of the various contributions to
and ten subroutines. In the routine MAIN, the input data are read, the results are printed and the subroutines are called as required. The functions of the subroutines are stated in table 2. The meanings of the common variables and arrays are explained in table 3. The input data are described in table 4.

=

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References [1] J.L. Dunham, Phys. Rev. 41(1932) 721. [21 J.F. Ogilvie and RH. Tipping, Intern. Rev. Phys. Chem. 3 (1983) 3. 131 R.H. Tipping, J. Chem. Phys. 59 (1973) 6433, 6443. 14] J.F. Ogilvie, Comput. Chem. 6 (1982) 172. IS] R.M. Herman, RH. Tipping and S. Short. J. Chem. Phys. 53 (1970) 595. 16] EU. Condon and G.H. Shortley, Theory of Atomic Spectra (Cambridge Univ. Press, Cambridge, England, 1957) p. 34. 17] J.P. Bouanich and C. Brodbeck, J. Quant. Spectrosc. Radiat. Transfer 15 (1975) 873. 181 J.P. Bouanich. J. Quant. Spectrosc. Radiat. Transfer 16 (1976) 1119. 191 RH. Tipping and J.F. Ogilvie, J. Mol. Spectrosc. 96 (1982) 442. [10] J.F. Ogilvie and RH. Tipping, J. Quant. Spectrosc. Radiat. Transfer 33 (1985) 145.

3. Program description Written in the language FORTRAN-77, the program VIBMATEL comprises a MAIN routine

111] J.P. Bouanich, Nguyen-Van-Thanh and I. Rossi, J. Quant. Spectrosc. Radiat. Transfer 30 (1983) 9.

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Vibrational matrix elements for diatomic molecules

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