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Analytical expression of the rotational-vibrational eigenvalue for a diatomic RKR potential
H. Kobeissi
To cite this version:
H. Kobeissi. Analytical expression of the rotational-vibrational eigenvalue for a diatomic RKR po- tential. Journal de Physique, 1986, 47 (4), pp.607-615. �10.1051/jphys:01986004704060700�. �jpa- 00210240�
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) eigen-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004704060700
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
write :
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] :
The projection of equations (13) onto 03C8m leads to :
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.
The determination of D is done in the same manner. We find for Dv the expression :
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 figure 1, the second member s(r) of equation (25) is
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 $(2)(r), are obtained by using
all the previous constants and functions already
obtained.
The treatment to obtain DM, D(1), D(2) is similar to
that used to obtain B M, B(1), B(2).
The numerical results are given in tables II and III
reserved to B and D respectively. In each table and for each v, the unperturbed constant C(O) is given along
with the two first corrections C(I) and C(2), and with
the computed constant C = C (0) + C (1) + C(2). This
result is compared to that, C°F, obtained with the
canonical functions method [24], and to that Cs,
obtained with the shooting method [23].
3.3. - One of the interests of the present numerical
application is to avoid the use of the required tests
Table I. - Values of E(0), E(1)v, Ev(2), computed by the present method for the first six vibrational levels of the poten-
tial used by Cashion [23]. In the last three columns Ev = E(0)v + Ev1) + Ev(2) is compared to EcF’ (results of the
Canonical Functions method [24]), and to E$ (results of the Shooting method [23]). All values are in cm-1.
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 analytical expressions of the form : Cy = Cv(’) +
