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.

Formal quasidegenerate perturbation theory (QDPT)

As is well known, ordinary Rayleigh–Schrödinger perturbation theory breaks down when applied to a state that is degenerate in zero order, unless spin or symmetry restrictions eliminate all but one of the degenerate determinants from the expansion. The breakdown is due to singularities arising from the vanishing of denominators involving differences in energy between the reference determinant and determinants that are degenerate with it. Even when exact zero-order degeneracies are not present but two or more closelying zero-order states contribute strongly to the wave function, as is the case for many excited states or in situations involving bond breaking, the RSPT expansions tend either to diverge or to converge very slowly.

These problems commonly arise in the case of open-shell states because different distributions of the open-shell electrons among the open-shell orbitals, all with the same or very similar total zero-order energies, are possible. Many open-shell high-spin states can be treated effectively with singlereference- determinant methods using either unrestricted or restricted openshell HF reference determinants because the spin restrictions exclude alternative assignment of the electrons to the open-shell orbitals; however, low-spin states, such as open-shell singlets, require alternative approaches.

Several common series-extrapolation techniques can be used to speed up the convergence of a perturbation expansion or to obtain an approximate limit of a divergent series. The results of such an extrapolation usually improve as more of the early terms of the series become available. Approaches based on Padé approximants (closely related to continued fractions) have been applied in some studies (e.g. Reid 1967, Goscinski 1967, Brändas and Goscinski 1970, Bartlett and Brändas 1972, Bartlett and Shavitt 1977b, Swain 1977).

Introduction

As in the case of quasidegenerate perturbation theory (Chapter 8), multireference coupled-cluster (MRCC) theory is designed to deal with electronic states for which a zero-order description in terms of a single Slater determinant does not provide an adequate starting point for calculating the electron correlation effects. As already discussed in Chapters 8 and 13, these situations arise primarily for certain open-shell systems that are not adequately described by a high-spin single determinant (such as transitionmetal atoms), for excited states in general and for studies of bond breaking on potential-energy surfaces; they arise usually because of the degeneracy or quasidegeneracy of the reference determinants. While single-reference coupled-cluster (SRCC) methods are very effective in treating dynamic electron correlation, the conditions discussed here involve nondynamic correlation effects that are not described well by truncated SRCC at practical levels of treatment.

As shown in Section 13.4, many open-shell and multireference states can be treated by EOM-CC methods, including a single excitation from a closed shell state to an open-shell singlet state, which normally requires two equally weighted determinants in its zero-order description. Furthermore, doubleionization and double-electron-attachment EOM-CC, as well as spin-flip CC (Krylov 2001), allow the treatment of many inherently multireference target states. These methods have the advantage of being operationally of single-reference form, since then the only choices that need to be made are of the basis set and the level of correlation treatment. Although, they require an SRCC solution for an initial state (not necessarily the ground state) to initiate the procedure, once initiated multireference target states are available by the diagonalization of an effective Hamiltonian matrix in a determinantal representation.

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).

Introduction

Although having some distinct limitations (e.g., relatively weak gradients and poor directionality), B1-based measurements have some particular advantages over B0 gradient-based methods. However, B1-based techniques have so far received only limited usage and consequently in this chapter we provide only a cursory coverage of these techniques and the interested reader is referred to the pertinent reviews on the subject.

B1 gradients

B1 gradients are more complex than B0 gradients. Apart from purely technical considerations, there are three main differences between B0 and B1 gradients: (i) A B0 field couples only into the spin system along the z-axis, thus the effective gradient tensor is always truncated into an effective vector (see Section 2.2.2). Radio frequency fields, however, couple into the spin system from any orientation within the transverse plane. As a result the B1 gradient generally retains its tensor form when it couples into the spin system. (ii) When the same rf coil is used for both excitation and detection, any phase variation is cancelled during the measurement. But when an experiment involves two rf fields at the same frequency this cancellation no longer occurs and phase variations need to be considered. This spatial dependence of the phase difference between the two rf fields presents an additional complication (or opportunity). (iii) The third difference is that B1 fields are non-secular and so do not commute with internal Hamiltonians. Thus, unlike a B0 gradient, a B1 gradient cannot be treated additively with respect to internal Hamiltonians.

Introduction

This chapter primarily deals with specialised NMR pulse sequences for measuring diffusion and flow. Sequences for MRI applications are given in Chapter 9. Steady gradient methods and especially those involving the stray field of superconducting magnets are outside the scope of the present work and so only a brief coverage is given in Section 8.2. Multiple-quantum and heteronuclear measurements are covered in Section 8.3. There has been considerable development of fast diffusion pulse sequences and these are covered in Section 8.4. Methods for handling samples that contain overlapping resonances with differences in relaxation time are considered in Section 8.5. Multi-dimensional methods for mixture separation and diffusion editing are presented in Section 8.6. Double PGSE and multi-dimensional motional correlation experiments are discussed in Section 8.7. Flow and Electrophoretic NMR are covered in Sections 8.8 and 8.9, respectively. Finally, the use of long-range dipolar interactions and miscellaneous sequences are presented in Section 8.10.

Steady gradient and stray field measurements

The earliest gradient-based diffusion measurements were based on the (technically simple) steady gradient experiments as discussed in Chapter 2. However, due to the limitations mentioned in Section 2.2.4, PGSE has generally overshadowed SGSE.