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.

Introduction

This chapter is concerned with the practical issues and key considerations involved in setting up PGSE experiments and the subsequent data analysis. Selection of PGSE parameters is discussed in Section 6.2 and sample preparation is discussed in Section 6.3. The various methods of gradient calibration are considered in Section 6.4. Finally, PGSE data analysis and display are considered in Section 6.5. Under favourable conditions it is possible to measure diffusion coefficients with greater than 99% accuracy. Indeed simple PGSE experiments have been shown to be reasonably robust with respect to experimental parameters (e.g., rf pulse flip angle). It cannot be overemphasised that the overall accuracy of a diffusion measurement is intimately connected to the accuracy of the gradient calibration. It is too easy to confuse the apparent precision of a diffusion measurement obtained from analysing the PGSE data with the true overall accuracy. For example, the PGSE data obtained from an experiment may be highly single exponential, but the gradient calibration or temperature control may have been inaccurate such that the analysis of the PGSE data leads to a highly precise but unfortunately a highly inaccurate diffusion coefficient.

Irrespective of the aim of the PGSE experiment, the analysis is always simplified by starting with a distortion-free data set with good signal-to-noise and, especially when the system has multiple components, good resolution.