Introduction

The conventional, single-reference, coupled-cluster method is very effective for electronic states dominated by a single determinant, such as most molecular ground states near their equilibrium geometry. Such states are predominantly closed-shell singlet states, and CC calculations on them produce pure singlet wave functions. But even these states become dominated by more than one determinant when one or more bonds are stretched close to breaking, so that single-reference CC based on RHF orbitals is then not usually appropriate for the calculation of entire potential-energy surfaces. While such problems can be partially treated by using UHF reference functions, which usually separate correctly, the UHF approach makes use of symmetry breaking and is poor in the spin-recoupling region.

Most excited, ionized and electron-attached states are open-shell states, and CC calculations on them using UHF or ROHF orbitals do not usually result in pure-spin wave functions. Furthermore, such states often involve large contributions from more than one determinant and thus do not respond well to conventional single-reference treatments.

One solution to these problems is to resort to multireference methods, such as those described in Chapters 8 and 14, but such treatments are still quite difficult to apply at a high enough level. An effective alternative in many cases is provided by the equation-of-motion coupled-cluster (EOM-CC) method (Emrich 1981, Sekino and Bartlett 1984, Comeau and Bartlett 1993, Stanton and Bartlett 1993a). A closely related approach is the coupled-cluster linear response (CCLR) method (Monkhorst 1977, Dalgaard and Monkhorst 1983, Koch and Jørgensen 1990).

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).

This chapter addresses several more subtle but nevertheless important aspects of coupled-cluster MBPT theory.

Spin summations and computational considerations

The formalism described in the previous sections was presented in terms of spinorbitals, without regard to integration over spin coordinates. Even in the case of unrestricted Hartree–Fock (UHF) reference functions, in which the spatial orbitals for α and β spin are different, integration over spin is absolutely necessary to eliminate many integrals and to allow the introduction of constraints over the summation indices, achieving a computational effort of no more than three times that of comparable RHF calculations. Furthermore, all amplitudes in which the number of α and β spinorbitals is different for the hole and particle indices vanish, preserving the MS, but not the S, quantum number. In the restricted closed-shell Hartree–Fock (RHF) case, spin integration is used to combine contributions from α and β spinorbitals, deriving expressions in terms of spatial orbitals only and thus reducing the range of all indices by about a factor 2 (see Section 7.3). Restricted open-shell Hartree–Fock (ROHF) calculations are usually performed as UHF, despite double occupancy, because the most effective algorithms are still of the spin-integrated, spatial-orbital, form. The double occupancy cannot be exploited further without special effort.

The incorporation of spin integration can be done algebraically or, in some cases, diagrammatically. As an example of the diagrammatic treatment of spin summation in coupled-cluster calculations we shall consider the case of the CCD equation with an RHF reference function. The diagrammatic representation of this equation in a spinorbital basis was given in Fig. 9.2 in terms of antisymmetrized Goldstone diagrams.

“What are the electrons really doing in molecules?” This famous question was posed by R. A. Mulliken over a half-century ago. Accurate quantitative answers to this question would allow us, in principle, to know all there is to know about the properties and interactions of molecules. Achieving this goal, however, requires a very accurate solution of the quantum-mechanical equations, primarily the Schrödinger equation, a task that was not possible for most of the past half-century. This situation has now changed, primarily due to the development of numerically accurate many-body methods and the emergence of powerful supercomputers.

Today it is well known that the many-body instantaneous interactions of the electrons in molecules tend to keep electrons apart; this is manifested as a correlation of their motions. Hence a correct description of electron correlation has been the focal point of atomic, molecular and solid state theory for over 50 years. In the last two decades the most prominent methods for providing accurate quantum chemical wave functions and using them to describe molecular structure and spectra are many-body perturbation theory (MBPT) and its coupled-cluster (CC) generalizations. These approaches have become the methods of choice in quantum chemistry, owing to their accuracy and their correct scaling with the number of electrons, a property known as extensivity (or size-extensivity). This property distinguishes many-body methods from the configuration-interaction (CI) tools that have commonly been used for many years. However, maintaining extensivity – a critical rationale for all such methods – requires many-body methods that employ quite different mathematical tools for their development than those that have been customary in quantum chemistry.