The Calculation of Potential Energy Curves of Diatomic Molecules: The RKR Method F. Castaio, J. de Juan, and E. Martinez' Departamento de Quimica Fisica, Facultad de Ciencias, Universidad dei Pais Vasco, Apartado 644, Bilbao, Spain The calculation of potential energy curves of diatomic molecules usinr Morse functions has become a common have been shown to he inadequatewhenBccurate potential energy curves are required. In this case the RKR method for obtaining potential energy curves from experimental data is commonly used. The present paper aims to describe this procedure which is not found in standard texthooks for chemistry students. The accuracy of such a method should he reflected in the agreement between observed and calculated (from the Franck-Condon factors; see (1)below) spectral band intensities. In this paper both procedures (Morse and RKR) are compared for the spectroscopically well-known Ie(XIZg+) state. The ready availability of microcomputers and programmable calculators enables students to uerform much more sophisticated analyses of experimental data than has previously been the case. In this context the more laborious and accurate RKR method may be worked out in a quite simple way. Accurate potential energy curves for diatomic molecules are reauired in order to evaluate the Franck-Condon factors ..
radiative lifetimes, vibrational temperatures, predissociations, kinetics of energy transfer, and intensities of vihrational band spectra. Potential energy curves are also important for the interpretation of molecular spectra and chemiluminescent atom recombination processes. The band intensities in emission and absorption between the states (u', J') and ( u " , J") are related to the square of the overlap integral, known as the Franck-Condon factor, i.e. (11: I
-
(S+"T+""J"
dm2
(1)
the #,,J are the rotational-vibrational wavefunctions of the states connected hy the transition. The appropriate wavefunctions may be obtained by solving the radial Schrodinger equation
provided that a good effective potential function, U(R,J) is availahle. Numerical methods for solving this equation have been described elsewhere (2). An Empirical Potential: The Morse Function One procedure for constructing potential curves is to assume an analytical form for the curves involving parameters expressible in terms of the usual spectroscopic constants. One of the most extensively used in the literature is the function proposed by Morse (3). U ( R )- U(R,) = D, 11 - e c P ( R - R *'I
Author to whom correspondence should be addressed.
(3)
where Re is the internuclear separation at equilibrium, Re = (h/8?r2&CB,)~'~, and the parameters /3 and D, are related to the vibrational constants w, and w , ~ ,through the expressions:
The expression (4h) usually gives values for the dissociation energy D, that are too large, so that it is hetter to use the experimental value when available. In this case, one speaks of a quasi-Morse potential function. An advantage of this potential is the feasibility of solving exactly the Schrodinger equation (2) to obtain the vihrational wavefunctions and energies. This potential gives a minimum for R = Re and approaches t~D, for large internuclear distances, as should be expected for a realistic potential. However for R = 0,U does not approach infinity, as i t must do for a correct potential energy function. Nevertheless, the part of the curve in the neighborhood of nil internuclear distance is of no practical importance. The exponential form of the Morse function cannot properly describe the long range dispersion forces between atoms; however, it is generally recognized that this potential function gives a quantitative representation of the energy for internuclear distances near equilibrium. For some states however, (notably BZZ- of CH), aMorse potential has appeared to be seriously in error. Many other empirical potential curves have heen suggested by various authors (see reference (4) for a review). One of the most successful of these curves is the Hulhurt-Hirschfelder function (5). The RKR Method One of the best methods of ohtaining accurate potential curves is by means of the so-called RKR procedure developed by Rydberg, Klein, and Rees (6). This is a semiclassical procedure for determining the two distances which correspond to the classical turning points of the vibrational energy level. The procedure has the great advantage that it makes use of experimental energy levels without reference to any empirical function for representing the potential energy curves. The potential function can he obtained by fitting the turning points to a smooth curve generated hy Lagrangian interpolation. The RKR method is based on the quasi-classical or WKB approximation (see Schiff (7)).In this approximation the eigenvalues for the one-dimensional motion of a particle in a potential well are given by the phase integral condition
where p is the momentum of the particle, x l and xe are the classical turning points, I is the action variable arising from the quantization of the radial momentum and u is bhe vihrational quantum number. The effective potential curve is given in terms of the poVolume 60
Number 2
February 1983
91
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tential curve without rotation U(R) by U,(R) = U(R) d R 2 where K = ( f ~ ~ /J2( ~J ) 1) is the action variable arising from the quantization of the angular momentum for a molecule of reduced mass p. Assuming a Born-Oppenheimer approximation, the total energy of the nuclear motion E, can be expressed as E(u,J) = G(u) + F,(J), where:
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tion. However, if experimental values for D, are taken rather than those obtained from e m . (4h). (i.e.. when a quasi-Morse potential is used), different values for the width of the potential will he obtained. I t must he noted that both procedures, strictly speaking, are to he used only in the case in which the we and w , ~ , values are the coefficients of a two-constant formula, such as in (11). In this case the vihrational levels can he represented in terms and ( V ' 1 ~ without )~ any higher powers of (u of (u Otherwise, in regard to the Morse potential, it is best to take the empirical D, and we in order to calculate P from (4a). However, the coefficient of (u %)2 in ( l l ) , will then fail to agree with the observed w , ~ , . The Rees procedure should he successful only in the case where the ex~ression(11)could be considered valid in the whole range of vibrational levels. However, the vihrational terms for most molecules can rarelv he represented to high u by a quadratic expression in (u '12). ~andersliceand-coworkers (10) improved the Rees treatment by fitting a quadratic expression for each consecutive pair of levels so that an ,optimal set of spectroscopic parameters wi, (wx)i, Bi, cui, were used as the effective vibrational and rotational constants for each vihrational level. Thus, the constants wi and (wx)i were chosen by fitting G(u') and G(u' - 1) to the expression
+
HereB,=B,-a,(u+%)+
... a n d D , = D , - p , ( u + % )
+ . . . ,where Herzberg's notation (8)has been used. E can he hcE(u,J)expressed as afunction of I andx, such a s E ( I , ~ l = In these terms, the phase integral condition (5) becomes
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