Studies in the accuracy of some lower-bound formulae-a simple one-dimensional problem

Abstract

A comparative study is made of several lower-bound procedures for calculating the eigenvalues of a simple exactly soluble one-dimensional potential, when this potential V(x) is regarded as the sum of two harmonic-oscillator potentials 1/2(R-x)2 and 1/2(R+x)2 centred at x=+or-R. This potential resembles a molecular potential, the wavefunctions for which are to be represented as combinations of the 'atomic' wavefunctions of the two separate potentials 1/2(R+or-x)2. The resulting LCAO-type wavefunctions are chosen to give lower bounds according to the classical methods of Weinstein and Stevenson (1938) and the truncated hamiltonian method of Bazley and Fox (1963); upper bounds are also obtained according to the Rayleigh-Ritz principle. The calculations are made for different values of R. Acceptable energies can be calculated by all methods. Of the methods considered the Rayleigh-Ritz upper-bound method converges most rapidly; the Stevenson lower bound is best if the optimum value of a certain parameter is known; the Weinstein method gives the poorest convergence of all.