Vibrational eigenvalues for all levels for the Lennard‐Jones potential

Abstract

In the diatomic eigenvalue problem, and in related topics, physicists often make use of analytic potential functions to illustrate the numerical application of a new method, or to show the improvement of an old one. Among these analytic potential functions, the most widely used are the Morse and the LennardJones functions. The first has the advantage of having the vibrational eigenvalues E,, and the vibrational wavefunctions I),, both given by analytical expressions. The LennardJones function has no such advantage.Yet a persistent trend makes use of the Lennard-Jones function to illustrate numerical applications (to compare accuracy and efficiency) for many problems, namely the centrifugal distortion constants problem,' and the phase-shift problem in elastic scattering. ' Actually, Lennard-Jones eigenvalues E , are known for the first 22 levels (and not for the two highest levels), having been determined by the Cooley shooting m e t h~d .~ In a recent article, Tellinghuisen* considered these eigenvalues as "exact": he mentioned in a still more recent report lC difficulties (already encountered by Hutsonlb) in obtaining the highest eigenvalues (for u = 22 and u = 23). The aim of this note is to give vibrational eigenvalues for all levels of the Lennard-Jones function, reliable up to 15 significant figures.We make use of the numerical canonical functions m e t h~d ,~ which has been shown to be highly accurate for low and high levels, up to the dissociation.6 The numerical application of this method to the Lennard-Jones potential, U(r) = (1 -r-6)2, gives the vibrational eigenvalues E, shown in Table I, along with those computed by Hutson7 using the Cooley method. The present results were obtained in double precision on the computer HP 10008 with a constant mesh-size h = 0.01 A.We notice that the eigenvalues computed by the two methods are in agreement up to 8 significant figures. In order to test these results, we chose to compare the two programs by using the same c~m p u t e r ,~ for a Morse potential function, where the computed eigenvalues can be compared to the Morse theoretical ones.We list in the third column of Table I The canonical function results are equal to the Morse theoretical eigenvalues up to 15 (or 16) significant figures (the computer precision). By examining the results of the shooting method (second entry) and those of the canonical functions method (first entry), we notice that the agreement for each entry between the two results is practically the

Numerical methods for the eigenvalue determination of second-order ordinary differential equations

“…The numerical derivative in Eqs. (8) and (9) is computed by using the highly accurate numerical derivatives in [27]. The numerical integration in Eqs.…”

“…The numerical integration in Eqs. (8) and (9) is carried out by using the highly accurate central-difference integration formula. In [27] we presented formulas for up to n = 8, which are further developed and used for highly accurate integration for a function f (x) with n = 10 as given in Eq.…”

“…Table 8 Comparison The eigenvalues and the MEs of the Hamiltonian for the case x 0 = 2.40873, e = 48.66888, and e x e = 0.977888 [20,8], are of 13-to 15-digit accuracy for the formula 10_4, the whole interval (0.8540425, 6.3227925) and h = 1 128 , which is of comparable accuracy to that using the DM eigenvalue method [27]. The eigenvalues using the SLEIGN2 (seventh column) are incorrect in many cases, but in some cases the accuracy of the eigenvalue ranges from 12 to 14 digits.…”

“…2(a) shows the case of eigenvalue correction according to Eq. (8). In the case of using the eigenvalue obtained via the DM eigenvalue method for the initial guess, the relative errors are initially of order 1D-4, which decrease to the order of 1D-13 in three to four iterations.…”

“…The accuracy of the eigenvalue using the SLEIGN2 (seventh column) ranges from 11 to 15 digits, in general, but in one case the eigenvalue is completely incorrect. The accuracy of the eigenvalue using the SLCPM12 (eighth column) ranges from 10 to 11 digits.Another form of the Morse potential[39,20,8,27] isU(x) = D{1 − exp[− (x − x 0 )]} 2 with D = 2 e /4 e x e , = (k e x e ) 1/2 and k = 1, with the theoretical eigenvalues E = e ( + 1/2) − e x e ( + 1/2) 2 .…”