Analytical expression of the rotational-vibrational eigenvalue for a diatomic RKR potential

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Analytical expression of the rotational-vibrational

eigenvalue for a diatomic RKR potential

H. Kobeissi

To cite this version:

Analytical

expression

of the rotational-vibrational

_eigenvalue

for a diatomic RKR

_potential

H. Kobeissi

Faculty of Sciences, Lebanese _University, and The _Group of Molecular and Atomic _Physics at the National Research Council, Beirut, Lebanon

(Reçu le 30 _{janvier 1985,} révisé le 3 _{juillet, accepti} le 25 novembre 1985)

Résumé. - Une

expression analytique de la valeur _{propre de} vibration _Ev pour tout _{potentiel diatomique (RKR,}

ou _autres) est _proposée, ainsi que des expressions analytiques de la constante de rotation _Bv, et des constantes de distortion _{Dv, Hv,...} Ces _expressions sont établies en se servant d’une formulation non conventionnelle de la

théorie de perturbation qui se sert des « fonctions _{canoniques »} au lieu de recourir à l’habituel _{usage des} bases de fonctions. Il est montré _{que : i)} les constantes _{Ev, Bv, Dv,} ... sont toutes de la forme : Cv = CMv + C(1)v + C(2)v +

...

et _{ceci pour} tout _potentiel donné U; CMv est la valeur de _Cv pour la fonction de Morse UM « associée à U »

(c’est-à-dire _ayant les mêmes constantes _we et _{we xe que U) ; ii) CMv, C(1)v, C(2)v,} ... décroissent en valeurs absolues ; iii) les

« corrections » C(1)v, C(2)v, ..., C(p)v, ... sont toutes de la forme

$$

, où _03C8Mv est la fonction d’onde de Morse _{(bien connue), ~(1)v} est une fonction liée _{explicitement à 03C8(M)v, CMv} et à U - _{UM, ~(2)v} se déduit de ~(1)v et de _C(1)v, et ainsi de suite... Le _problème des valeurs propres de rotation-vibration pour un _potentiel RKR est ainsi réduit à _{l’intégration} d’une _équation différentielle linéaire non _homogène dont les coefficients et les valeurs initiales de la solution sont connus _(et non un _problème du _{type Sturm-Lionville)} ainsi _qu’au calcul _{d’intégrales simples.} L’appli-cation _numérique montre _{que des} résultats _précis sont obtenus _rapidement et aisément en se servant d’un ordinateur

personnel.

Abstract. -

Analytical expression of the _pure vibrational _{eigenvalue Ev} for an RKR diatomic potential is seeked,

as well as of the rotational constant _Bv and the _centrifugal distortion constants _{Dv, Hv,} ... A new « Canonical

Functions Perturbation _{Approach »} is used for that _{purpose. It is shown} that : i) the constants _{Ev, Bv, Dv, ...}

are _{expressed by : Cv} = CvM ₊ Cv(1) ₊ Cv(2)... for _{any given potential U,} where CvM is the value of _Cv for the Morse

function UM« associated to _{U » (having} the same constants _{we and we xe); ii) CvM, Cv(1), Cv(2),} ... decrease in

magni-tude. iii) The « _{corrections »} Cv(1), Cv(2), ..., Cv(p), ... are all of the form

$$

, where _03C8Mv is the well-known Morse wavefunction, ~(1)v is a function related explicitly to _{03C8Mv, CMv} and U - _{UM, ~(2)v} is related to _~(1)v and

to _C(1)v, and so on... Thus the _problem is reduced to the _integration of an _{unhomogeneous} second order linear differential _equation with _given coefficients and _given initial values _(and not the Sturm-Lionville _problem) and to the _computation of _{simple integrals.} The numerical _application shows that accurate results are _fastly, and _easily

obtained, just by using a _{personal computer.}

Classification

Physics Abstracts 31.00

1. Introduction.

The rotation-vibration energy problem of a diatomic

molecule in a _given electronic state _kept the attention of molecular _physicists since many decades.

The first tendency in _solving this _problem was the

empirical one, i.e. the _{potential U(r) (r} is the

inter-nuclear _{distance) characterizing} the _given electronic

state is _{represented by} an _{analytical expression}

depending on few _{adjustable >> parameters.} The

first _{functions, yet} still _popular, were _{presented by}

Lennard-Jones in 1924 _[1], Morse in 1929 _[2] and Dunham in 1932 _{[3]. Many} other functions are

col-lected in the excellent review articles _by Varshni [4]

and Steel et al. _[5].

For some of these _{potential functions,} the radial

Schrodinger equation is solved _{exactly (at} least for the rotationless _case) and the vibration _eigenvalue is

given by an analytical expression. For other potential

functions the vibration _(or rotation-vibration)

608

value is obtained _by one or another of the _quantum

mechanical _approximated methods _{(the perturbation}

approach, the variation or the WKB methods, ...) [6].

Within the _development of the _spectroscopy

tech-nique and the advent of the _computer, a second (and

still _{actual) tendency appeared.} For this _tendency

we underline two dates : _i) the semi-classical RKR method _[7] for the determination of the _potential

U(r) (which is constructed from the _experimental terms

values) ; the RKR _potential is _given in a « numerical >>

form _(by the coordinates of its _{turning points} with

suitable _{interpolations} and _{extrapolations); ii)} the

Cooley-Numerov scheme _[8] of the numerical

inte-gration of the radial _{Schrodinger equation} for the

realistic RKR _potential, or for _any other potential

function.

Yet, more than two decades’ after the earlier work of _{Cooley (1961), many papers} still appear every year

dealing with the same rotation-vibration _eigenvalue

problem. These works look to _improve the _Cooley’s

« _{Shooting Method >> [9],} or to _present an alternative

(like

the «Iterative Method >> _[10]. The «Matrix methods >> [11], the « _{Log Derivative Method >> [12],}

the « Canonical Functions Method >> _[13], or

others,...).

While the numerical methods _give accurate

_(and

even _highly accurate

_[14])

calculated _eigenvalues Ec

for a realistic _potential, the _empirical methods still

attract _many authors _{[15], mainly} because the eigen-value is _{given by} an _{analytical expression Ea, although}

of _{poor physical} interest.

The aim of this _paper is _precisely to _give the solution of the rotation-vibration _{eigenvalue problem by} an

analytical expression E a for the realistic RKR _potential

(and not _just for the _{empirical potential functions).} However the method is _generalized to _{any type} of

diatomic _{potential (numerical} or analytical). In _theory, this _problem is _already solved. _{By using} the conventional formulation of _{Rayleigh-Shr6dinger}

perturbation approach, one can associate to _{any given}

potential U(r) (of any type) a « _sufficiently close »

potential function U(O) such that the difference

u(r) = U(r) -

U(O)(r)

can be considered as a

pertur-bation

_(U(O)

_being the _{unperturbed potential).} If the

eigenvalue E(O) and the _{eigenfunction}

_41(o)

of U(O)

are _known, then those E _{and 0} of the _{given potential}

U(r) are _{given by} the « analytical expression » : :

with a similar expression for 03C8.

E(O) _being known, the other terms _{E(1), E(2), E(3),} ...

are _{given by} the _{expressions :}

where n stands for v _(pure vibrational _quantum

num-ber) or for the _{couple (v, J),} J _being the rotational

quantum number.

This well-known method was never used to deter-mine the vibrational _{eigenvalue E,} in a concrete

physical application. However it was used to deduce from _E, the _expressions of the rotational constant _Bv,

and the _centrifugal distortion constants _{Dv, H,,} ... [ 16].

In all cases this _{analytical expression} of E is _generally

considered as complicated, and suffers _{(for practical}

applications) from several _{disadvantages [17].}

The _present method makes use of a nonconventional

approach of _{Rayleigh-Schr6dinger perturbation}

theory. This _approach was already used to determine

the rotation effect in the rotation-vibration

eigen-value in terms of the _pure vibrational one _[18]

(and

to deduce _{Bv, Dv, Hv,} ...

[ 19] ).

This approach is extend-ed here in order to obtain _{analytical expressions} of the pure vibration eigenvalue E,, as well as the constants

Bv, Dv, Hv, ... for an RKR potential (or for any other

potential).

This « Canonical Functions Perturbation

Ap-proach » is _presented in section 2, where the _analytical

expressions of Ev, Bv, D,, ... are derived for any given diatomic _potential. A numerical _application is pre-sented in section 3, and _compared to other confirmed

numerical methods.

2. The _theory. 2.1. - Within the

Bom-Oppenheimer approximation [20], the motion of the diatomic _molecule, in a _given

electronic state, is described _by the wavefunction

t/lvÂ.(r) and the energy EvÂ.’ eigenfunction and _eigenvalue

of the radial _{Schrodinger equation :}

where : A = J(J ₊ 1), J being the rotational quantum

number r is the internuclear distance k = 2

ylh2,

p and h having their usual significances [21].

According to the classical formulation of the

Rayleigh-Schr6dinger perturbation theory, one can

and _{replace equation (1) by [18] :}

The _projection of _{equations (5)} onto leads to _{[17] :}

These _equations were _already used _by Huston _[17] and _by the author [18-19] to determine _{Bv, Dv, Hv,} ...

successively, by :

i) The _{previous determination of the pure}

vibra-tional _{eigenvalue E,} and _{eigenfunction ql,(r).}

ii) The successive determination of the so-called

« rotational harmonics » %,(r), 0,(r), ...

The _{present approach replaces,} the conventional

expansions of _{$v(r), 0,(r),} ... on the t/lv(r) basis [6],

by the canonical functions method [18] based on the

direct _integration of the «rotational _Schrodinger

equations >>

(Eqs. (5)).

2.2. - The

same procedure _may be used to determine the pure vibrational _{eigenvalue E,} for the _given

potential U(r).

To the _{given potential,} one can associate the Morse

function _{[2] :}

where re is the value of r at the _{equilibrium, Dd} and a

are _{disposable parameters} defined by :

The constants _{we and we} xe are those _appearing in the

well-known Morse _{eigenvalue :}

The function

_UM(r)

is said to be the « Morse function

associated to _{U(r) » when we and We Xe} are close

(or equal) to those _appearing in the _{eigenvalue E,}

related to the _{given potential U(r)} and _classically

represented by [21] :

where _{We, We xe, We Ye,} ... are known to be decreasing.

According to this _choice, one can notice that

Ef’

is « close » to _Ev, and can deduce that

UM(r)

is « close »

to _U(r). We write :

where _y is a scaling _parameter, u(r) is the difference

between U and UM when _y = 1.

The _application of the classical

Rayleigh-Schr6-dinger formulation to the vibrational wave

equa-tion _(4), leads to the set of _{equation [6] :}

610

The « _{corrections »}

E (1), E(2),

... are deduced from these equations, where the functions

o(l), qf(2),

... are obtained from the _{equations (13) by using} the canonical functions method. The vibrational _{eigenvalue E.} is thus _given

(when y = 1) by the analytical expression :

with :

2. 3. - This last

procedure is extended to the determination _{of By, Dy, Hy,} ...

To determine _By, we _apply the _{perturbation approach} to the first rotational _{Schrodinger equation}

_{(Eq. (5.1)).}

The formulation used above leads to the set of _{equations :}

with

The terms

_{BM, B (’), B,(2),}

... are obtained from equations (19) by projection onto

V/m.

We get :

The rotational constant _Bv is _{given (for y = 1) by} the _{analytical expression :}

We notice that :

_{i) B,’}

is well defined in terms of the Morse wavefunction

_{03C8Mv :}

_ii)

_B,(’)

_depends on

03C8(1)v)

solution

of _{equation (13.1)} and

_1St,

solution of _{equation (19.1).} These functions are determined by the canonical functions

method _(see the _{Appendix). iii)}

_B(’)

_depends, furthermore, on

ql(2)

and

%(’)

solutions of equation (13.2) and

equation (19.2) respectively. These two functions are determined _by the same canonical functions method.

where

The terms

_{D m, D (’), D (2),}

_...depend on the

functions Om, q,(’), 0(2),

...,

1S§l, %(1), %(2),

... previously determined,

and on the functions

om, 9)(’), 0(2), ...

_{respectively.} These last functions are the _respective solutions of the

equa-tions :

with

The solutions of these _equations are also determined _by the canonical functions method (see Appendix).

3. The numerical _application.

3.1. - The formulation _presented in the _previous

section can be _simplified, for a _given level v, by taking :

C1 stands for E (and

Cia)

for

_Em)

C2 stands for B _(and

_{C 2 (0)}

for

_{B M)}

C3 stands for D _(and

_{C 3 (0)}

for

_{D M) .}

The _computation of _any term

_C.1p)

makes use of

cia)

= Em (which is given by the choice of the

unper-turbed _potential

_UM)

and _implies the determination

of all the terms

_C(P’)

with 1 _{q’ q} and 0 p’ p. This must be done in « _{good »} order which is shown in _figure 1. In this _figure the arrows indicate the

succession of _operations marked _by the number of the

equation relative to each _{operation. (The} dashed

arrows _represent an _integration of a differential

equation, the other arrows are relative to an _integral

computation).

All these _operations are in fact reduced to two :

i) The _integration of a second order linear

diffe-rential _equation of the _{type :}

where _{only s(r) changes} from one _equation to another.

ii) The _computation of a _{simple integral} of the _{type :}

where/(r) is either _equal to

1/kr2

or to a _{constant, y(r)}

being the solution of the differential _{equation (which}

stands for

_t/J(p)

or ,%(P) or 1)(p), _...).

According to the order of _{operations imposed by} the

612

Fig. 1. -

Diagram of the successive _operations for the

computation of _{E, B, D,...} for a _given vibrational level.

Each arrow _represents the _equation used to _get the _quantity marked in the « _pointed » _{square. Each} arrow is numbered

by the _{corresponding equation} number. The dashed arrows

correspond to a differential _equation, the others _correspond

to an _integral calculation.

always well determined. The initial values _y(ro) and y’(ro) at the _{arbitrary origin ro} are well known

accord-ing to the canonical functions method _{(see Appendix).}

The _integration of _{equation (25)} becomes therefore

of the direct _{type (the} coefficients are known as well as the initial _values) and not of the _{eigenvalue type.} For _this, one has a large variety of difference

equa-tions _[22].

3 . 2. - The _example of the numerical _application

presented here, is that _already used _by Cashion _[23]

to test the _shooting method and _by the author for the canonical functions method _[24]. This same _example was also used with success to test the _application of the

« canonical _{functions-perturbation approach »} to the determination the _pure vibrational _{eigenvalues E, [25].} The _{potential U(r)} used _by Cashion is a Morse

function characterized by : we = 48.668 _{883 264 and}

we xe = 0.977 881 676 cm-1 (with k = 1).

The _{unperturbed potential}

_UM(r)

associated to the

given potential U(r) is another Morse function

cha-racterized _{by :}

_wt

= 48.6

and we xM

= 0.977 cm-1.

These constants are close to _we and _{we xe respectively,} and the difference _u(r) = U(r) -

UM(r)

can, therefore,

be considered as a _{perturbation.}

By using E M, for a given level v, and the initial values

4(m(ro)

= 1 and

ql’m(ro)

given in the Appendix,

4/m(r)

is

deduced from _{equation (12) (see Fig. 1). E}

_1/1(1),

_E(2),

t/f(2),...

are deduced successively by using

equa-tions _{(16.1), (13.1), (16.2), (13.2),...} The values of the

corrections E(I), E (2) _(for the first six vibrational

levels used _{by Cashion)} are _reproduced in table I for

comparison.

The value of BM is deduced from

_{t/fM(r) (Eq. (20.1)) ;}

that of

_BM(r)

is then found

_{(Eq. (19.1)).}

The first

correction B(I) is _{given by equation (20.2)} where _E(I),

t/f(1)(r),

B M,

BM(r)

are _{already found;}

B(1)(r)

results as

the solution of _(19.2). The second correction _{B (2)9}

along with the function

_$(2)(r),

are obtained _{by using}

all the _previous constants and functions _already obtained.

The treatment to obtain

_DM,

_D(1), D(2) is similar to

that used to obtain

_{B M,}

B(1), B(2).

The numerical results are _given in tables II and III reserved to B and D _{respectively.} In each table and for each v, the unperturbed constant C(O) is _{given along} with the two first corrections C(I) and _C(2), and with

the _computed constant C = C (0) ₊ C (1) ₊ C(2). This

result is _compared to _{that, C°F,} obtained with the

canonical functions method _[24], and to that _Cs,

obtained with the _shooting method _[23].

3.3. - One of the interests of the

present numerical

application is to avoid the use of the required tests

Table I. -

Values

_{of E(0),}

_{E(1)v, Ev(2),}

_{computed by the present method for the first} six vibrational levels _{of the}

poten-tial _{used by} Cashion _[23]. In the last three columns _Ev =

E(0)v

₊

Ev1)

₊

Ev(2)

is compared to EcF’

(results

of the

Table II. - Values

of B(O), B(1),

computed by the present method for the first six vibrational levels _{of the potential}

used by Cashion _[23]. In the last three columns _Bv =

B(0)

₊

B(1)

_is

compared to

B:F (results

of the Canonical Functions method

_[24]),

and to

_{Bsv (results}

_{of the Shooting} method

_[23]).

All values are in cm-1.

Table III. -

Values

_{of D(0), D(1),}

_{computed by the present method for the first} six vibrational levels _{of the potential}

used _by Cashion _[23]. In the last three columns _Dv =

Dv(0)

₊

D(1)

is compared to

DcF (results

of the Canonical

Functions method

_[24]),

and to

_{D$ (results}

_{of the Shooting} method

_[23]).

All values are in 10-6 cm-1.

of _accuracy, since the _shooting method results and

those of the canonical functions method were _already

tested in reference _{[24]. According} to that test, we

consi-der the canonical functions method results as « exact ».

The _agreement between the results of the _present method and those of the canonical functions method is considered here as a confirmation of the _validity

of the _present method.

All the _computations are done on the _personal

computer New-Brain AD. We notice that :

i) The _agreement between the _{computed E,} and the

« exact » one

EcF

is good to _{eight significant figures}

(which is the limit of the used _computer). The discre-pancy AE, is averaged to 10-’ cm-’ for the _present

method and to 1.35 x 10-4 cm-1 for the shooting

method.

ii) The _agreement between the _computed B and the

« exact » one BcF

_{is good}

to seven _{significant figures.}

The mean value _ABv of the difference ! I Bv -

B:F

_[

is of 10- 8 cin-’ for the _{present method} and of 6 x 10- 6 CM- I for the _shooting method.

iii) The _computed constant _Dv is less _precise

(accord-ing to the same adopted criterion). The _average

ADv

of the _{discrepancies} for the considered levels is of

10-9 cm-1 for the _{present method,} which is to be

compared to _that, 0.8 x 10-’ cm-1 of the _shooting

method.

iv) As we mentioned before, one algorithm is used to

compute all the terms _{C M, e(1), C(2),...} for all the

constants _{Ey, By, Dy, H,,,} ... (integration of the diffe-rential _{equation (25)} and _computation of the _simple

integral

(26)).

Each _computer run >> is therefore

equivalent to one run in the _Shooting Method _(with a trial value of _E). The total _computer effort can be

evaluated from the number of « dashed » arrows in

figure 1 _(two runs to _{compute Ey,} two others for _{B,, ...).} The whole effort in _{computing Ey, By, Dy, By,...} is

rou-ghly equivalent (or slightly exceeds) that of the neces-sary iterations for the determination of Ey in the

con-ventional methods.

In _{the present paper,} we intended to _present the main

approach and to avoid what we considered as details

in the _theory as well as in the application.

A detailed _study of the sources of error along with some theoretical properties of the functions

$(1)(r),

Jc3(21(r),

...,

g)(11(r), 1)(2)(r), ...

will appear in a

forth-coming paper; other examples of the numerical

application, namely for the RKR _potentials, will also

be _presented. 4. Conclusion.

The Canonical Functions-Perturbation _Approach was

applied to the determination of the _pure

vibra-tional _{eigenvalue Ey} related to _{any given} diatomic

potential, as well as to that of the rotational constants

By and the _centrifugal distortion constants _{Dy, Hv,} ...

It was _proved that all these constants are. _{given by}

614

C(l)

+

C,(2)

+... where

_C,(P)

is the _p-th correction

to the zeroth - order value

C(’). Cv(’)

is the value of

Cv for a Morse function UM associated to the _given

potential U. The corrections _Cv(P) are all of the form

where _f(r) is _equal either to

_1/r2

or to a _constant,

y(r) is the solution of a linear second order

(inhomo-geneous) differential _{equation (with given} coefficients

and _given initial values of the _solution),

_#Qfl(r)

is the Morse wavefunction related to UM.

The numerical _application is _simple in _comparison

with the conventional _methods; its accuracy is as _high as the best of the published results.

Appendix.

Morse canonical functions and initial values.

1. All the differential _equations used in this _paper

are of the form :

where the solution is _subject to the _boundary condi-tions _{[2] :}

The solution of this equation is known to be of the

form [15] :

where a and P are the Morse canonical functions

(particular solutions of _{homogeneous equations),} with :

The function a is the _particular solution of the

inhomogeneous equation given by [15] :

with

By using the _boundary condition _(r - _0), we find : where

with similar definitions _{for P-} and Q-. This _{equation gives :}

By using the other _boundary condition _(r ---> oo), we

find _{similarly :}

where _a+,

_fl+

and Q+ have definitions _(at r - oo)

similar to that of a-.

2. When _y(r) stands for the Morse wavefunction

t/JM(r),

we have _s(r) = 0 (i.e. _a = 0) and _we take

arbitrarily y(ro) = 1; _we find :

When _y(r) stands _{for any one of} the functions

",(1),

0(2), ..., %M, %(1),

%(2), ... I)M, 1)(1), a)(2), ..., we have s 0 0 and _Y(ro) = 0 [27], _we find :

where

We deduce for _{y’(ro) :}

or _similarly

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[2] MORSE, P. M., Phys. Rev. 34 ₍₁₉₂₉₎ 57.

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