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Journal of Physics A Mathematical and General

Optimisation of parameterised perturbative approximations to wavefunctions

Steven Kenneth Kauffmann, S. M. Perez

Published Jun 21, 1987• 0 citations
Publisher

Abstract

The authors have justified a point-by-point optimisation procedure for parametrised perturbative approximations to wavefunctions by appealing to a principle of minimal sensitivity. Here it is shown that, if the diagonal matrix elements of the perturbation vanish (which can always be arranged via a simple redefinition of the unperturbed Hamiltonian), then the optimisation procedure is also justified by an underlying stationary principle, with the nth-order perturbative approximations being stationary to that order.

DOI
10.1088/0305-4470/20/9/045

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The authors have justified a point-by-point optimisation procedure for parametrised perturbative approximations to wavefunctions by appealing to a principle of minimal sensitivity. Here it is shown that, if the diagonal matrix elements of the perturbation vanish (which can always be arranged via a simple redefinition of the unperturbed Hamiltonian), then the optimisation procedure is also justified by an underlying stationary principle, with the nth-order perturbative approximations being stationary to that order.
Stevenson's principle (1981) of minimal sensitivity is applied directly to configuration space wavefunctions calculated in first-order perturbation theory. The redundant parameters, which comprise the essential elements of minimal sensitivity calculations, actually become functions of configuration space, giving the method considerable flexibility and potential accuracy. It is shown that the method is exact in an illustrative class of simple cases and, for the more realistic case of the ground state wavefunction of the quartic oscillator, it is considerably more accurate than the closely related perturbative variational approach, particularly in the asymptotic region.
The direct point-by-point application of minimal sensitivity optimisation to the first-order perturbative approximation to the ground-state wavefunction of the quartic oscillator, carried out previously, is extended to low-lying excited states. An analytic formula for the derivative with respect to the redundant parameter is developed and used to obtain the minimally sensitive value for this parameter as a function of configuration space. Generally, more than one such minimally sensitive value is found at each configuration space point-these values are categorised into disjoint continuous curves. For all the cases studied, it has proved possible to piece together sections of such curves in an unambiguous manner so as to obtain optimised wavefunction approximations which are both renormalisable and continuously differentiable. Comparison of these optimised wavefunction approximations with the exact results revealed satisfactory agreement, particularly in the large mod x mod asymptotic region where orthodox perturbation methods fail completely due to the strong coupling.
Abstract As a preliminary to a wide range of applications, conventional perturbation theory is re-examined and a number of useful properties emphasized. It is shown in particular that the total, kinetic and potential energies can be obtained to the (2s + 1)th order from a knowledge of only the sth-order wave function and that similar though less powerful theorems hold for other diagonal matrix elements. A combination of variational and perturbation techniques is suggested as the best method of calculating small disturbances of a stationary system.
The use of the shifted inverse Hamiltonian in the Lanczos algorithm is examined. We present a perturbative inverse and demonstrate that the Krylov subspace formed from it is better suited to approximating the ground state wavefunction compared with the Hamiltonian itself. We further demonstrate that the perturbative inverse provides an adequately accurate approximation to the inverse Hamiltonian at second order. Naturally, the quality of the results depend on the quality of the initial guess, here we use the Hartree-Fock, CISD, MP-PT1, and EN-PT1 wavefunctions. Lastly, we examine the scaling of the method where we find that accurate results can be found for similar cost as PT5.
The bound states of a weak attractive potential never come out of doing perturbation theory on free particle states. This is a quite general rule: if the perturbation commutes with a constant of the motion, A, of the unperturbed Hamiltonian, then matrix elements of the perturbation between eigenstates of A with different eigenvalues must vanish. This rule, an example of a selection rule, often simplifies the problem of selecting the correct linear combination of degenerate states with which to do the perturbation expansion. The Rayleigh-Schrodinger perturbation theory – yields rather complicated expressions for the change in the states in second order and beyond. There is a somewhat different form of the perturbation theory, due to Brillouin and Wigner that enables to see more clearly the structure of the perturbed states.
The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.