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CRC Press eBooks

Hypervirial-perturbation methods

John P. Killingbeck

Published Jan 18, 2018• 0 citations
Publisher

Abstract

This chapter gives a brief sketch of some parts of Rayleigh-Schrödinger perturbation theory, the author have written in detail about perturbation theory elsewhere and concentrate here on a few ideas which are directly useful in numerical work on a microcomputer. The intellectual puzzle of how to reduce everything to an energy calculation gets really interesting if we want to calculate a local quantity, e.g. the value of the (normalised) wavefunction at the origin. The renormalised series trick seems to give the results to greater accuracy and more easily. The program makes an essential use of arrays, but since the array subscripts on the Pet cannot be negative the subscripts have to be changed from their values in the theoretical equations. In many-body theory, then, the sum-overstates formalism can be useful for giving a semi-quantitative account of various important effects, and the so-called diagrammatic perturbation theory used in atomic theory leans heavily on that formalism.

DOI
10.1201/9781351074582-9

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Chapman & Hall/CRC eBooks•Dec 12, 2008•0 citations
This chapter gives a brief sketch of some parts of Rayleigh-Schrödinger perturbation theory, the author have written in detail about perturbation theory elsewhere and concentrate here on a few ideas which are directly useful in numerical work on a microcomputer. The intellectual puzzle of how to reduce everything to an energy calculation gets really interesting if we want to calculate a local quantity, e.g. the value of the (normalised) wavefunction at the origin. The renormalised series trick seems to give the results to greater accuracy and more easily. The program makes an essential use of arrays, but since the array subscripts on the Pet cannot be negative the subscripts have to be changed from their values in the theoretical equations. In many-body theory, then, the sum-overstates formalism can be useful for giving a semi-quantitative account of various important effects, and the so-called diagrammatic perturbation theory used in atomic theory leans heavily on that formalism.
Background There are two stages in the study of perturbation theory and related techniques (although they are mixed intimately in most derivations in the literature). The first is the formal development , carried out in terms of the total Hamiltonian and total wave function (and total zero-order wave function), without attempt to express anything in terms of one- and two-body quantities (components of Ĥ, orbitals, integrals over orbitals etc.). We can make a considerable amount of progress in this way before considering the detailed form of Ĥ. The second is the many-body development , where all expressions are obtained in terms of orbitals (one-electron states) and oneand two-electron integrals. We shall try to keep these separate for a while and begin with a consideration of formal perturbation theory. Another aspect of the study of many-body techniques is the large variety of approaches, notations and derivations that have been used. Each different approach has contributed to the lore and the language of many-body theory, and each tends to illuminate some aspects better than the other approaches. If we want to be able to read the literature in this field, we should be familiar with several alternative formulations. Therefore, we shall occasionally derive some results in more than one way and, in particular, we shall derive the basic perturbation-theory equations and their many-body representations in several complementary ways. Classical derivation of Rayleigh–Schrödinger perturbation theory The perturbation Ansatz We begin with a classical textbook derivation of formal Rayleigh–Schrödinger perturbation theory (RSPT).
This chapter examines applications drawn from perturbation theory. The main topic in perturbation theory is the energy and spontaneous decay rate of the 21-cm hyperfine line in atomic hydrogen. Before there were electronic computers, people had quite an accurate theoretical understanding of the energy levels in helium and more complicated systems. The trick was (and is) to find approximation schemes that treat unimportant parts of a physical system in quite crude approximations while reducing the interesting parts to a problem simple enough that it is feasible to compute but yet detailed enough to yield accurate results. The approximation methods in the chapter deal with the effects of small changes in the Hamiltonian, resulting for example from the application of a static or time variable electric or magnetic field. This may cause small changes in energy levels, and it may induce transitions among eigenstates of the original Hamiltonian.
Modern computers have made possible the evaluation of higher order terms of perturbation series. Apart from the empirical work involving numerical calculations, much formal mathematical work has been done on the convergence properties of perturbation series and of Pade approximants which represent them. Some of the modern work in perturbation theory is reviewed in the context of traditional results from the theory of real and complex variables. The two major versions of time-independent perturbation theory, the Rayleigh-Schrodinger (RS) and Brillouin-Wigner (BW) theories, are compared and contrasted, and the alternative techniques for evaluating the terms in the energy series are discussed. The sum-over-states method, the differential equation method and the variational principle method are treated, with emphasis on their inter-relations.
Scope The book focuses primarily on many-body (or better, many-electron) methods for electron correlation. These include Rayleigh–Schrödinger perturbation theory (RSPT) , particularly in its diagrammatic representation (referred to as many-body perturbation theory , or MBPT ), and coupled-cluster (CC) theory; their relationship to configuration interaction (CI) is included. Further extensions address properties other than the energy, and also excited states and multireference CC and MBPT methods. The many-body algebraic and diagrammatic methods used in electronic structure theory have their origin in quantum field theory and in the study of nuclear matter and nuclear structure. The second-quantization formalism was first introduced in a treatment of quantized fields by Dirac (1927) and was extended to fermion systems by Jordan and Klein (1927) and by Jordan and Wigner (1928). This formalism is particularly useful in field theory, in scattering problems and in the study of infinite systems because it easily handles problems involving infinite, indefinite or variable numbers of particles. The diagrammatic approach was introduced into field theory by Feynman (1949a,b) and applied to many-body systems by Hugenholtz (1957) and by Goldstone (1957). Many-body perturbation theory and its linked-diagram formalism were first introduced by Brueckner and Levinson (1955) and by Brueckner (1955), and were formalized by Goldstone (1957). Other important contributions to the methodology, first in field theory and then in the theory of nuclear structure, are due to Dyson (1949a,b), Wick (1950), Hubbard (1957, 1958a,b) and Frantz and Mills (1960). Applications to the electronic structure of atoms and molecules began with the work of Kelly (1963, 1964a,b, 1968), and molecular applications using finite analytical basis sets appeared in the work of Bartlett and Silver (1974a, b).
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.