Density Functional Approaches to Relativistic Quantum Mechanics

The wave function is an elusive and somewhat mysterious object. Nobody has ever observed the wave function directly: rather, its existence is inferred from the various experiments whose outcome is most rationally explained using a wave function interpretation of quantum mechanics. Further, the N-particle wave function is a rather complicated construction, depending on 3N spatial coordinates as well as N spin coordinates, correlated in a manner that almost defies description. By contrast, the electron density of an N-electron system is a much simpler quantity, described by three spatial coordinates and even accessible to experiment. In terms of the wave function, the electron density is expressed as . . . ρ(r) = N ∫ Ψ* (r1,r2,...,rN)Ψ (r1,r2,...,rN)dr2dr3 ...drN (14.1) . . . where the sum over spin coordinates is implicit. It might be much more convenient to have a theory based on the electron density rather than the wave function. The description would be much simpler, and with a greatly reduced (and constant) number of variables, the calculation of the electron density would hopefully be faster and less demanding. We also note that given the correct ground state density, we should be able to calculate any observable quantity of a stationary system. The answer to these hopes is density functional theory, or DFT. Over the past decade, DFT has become one of the most widely used tools of the computational chemist, and in particular for systems of some size. This success has come despite complaints about arbitrary parametrization of potentials, and laments about the absence of a universal principle (other than comparison with experiment) that can guide improvements in the way the variational principle has led the development of wave-function-based methods. We do not intend to pursue that particular discussion, but we note as a historical fact that many important early contributions to relativistic quantum chemistry were made using DFT-like methods. Furthermore, there is every reason to try to extend the success of nonrelativistic DFT methods to the relativistic domain. We suspect that their potential for conquering a sizable part of this field is at least as large as it has been in the nonrelativistic domain.

Basis-Set Expansions of Relativistic Electronic Wave Functions

There have been several successful applications of the Dirac–Hartree–Fock (DHF) equations to the calculation of numerical electronic wave functions for diatomic molecules (Laaksonen and Grant 1984a, 1984b, Sundholm 1988, 1994, Kullie et al. 1999). However, the use of numerical techniques in relativistic molecular calculations encounters the same difficulties as in the nonrelativistic case, and to proceed to general applications beyond simple diatomic and linear molecules it is necessary to resort to an analytic approximation using a basis set expansion of the wave function. The techniques for such calculations may to a large extent be based on the methods developed for nonrelativistic calculations, but it turns out that the transfer of these methods to the relativistic case requires special considerations. These considerations, as well as the development of the finite basis versions of both the Dirac and DHF equations, form the subject of the present chapter. In particular, in the early days of relativistic quantum chemistry, attempts to solve the DHF equations in a basis set expansion sometimes led to unexpected results. One of the problems was that some calculations did not tend to the correct nonrelativistic limit. Subsequent investigations revealed that this was caused by inconsistencies in the choice of basis set for the small-component space, and some basic principles of basisset selection for relativistic calculations were established. The variational stability of the DHF equations in a finite basis has also been a subject of debate. As we show in this chapter, it is possible to establish lower variational bounds, thus ensuring that the iterative solution of the DHF equations does not collapse. There are two basically different strategies that may be followed when developing a finite basis formulation for relativistic molecular calculations. One possibility is to expand the large and small components of the 4-spinor in a basis of 2-spinors. The alternative is to expand each of the scalar components of the 4-spinor in a scalar basis. Both approaches have their advantages and disadvantages, though the latter approach is obviously the easier one for adapting nonrelativistic methods, which work in real scalar arithmetic.

Operators, Matrix Elements, and Wave Functions under Time-Reversal Symmetry

We now take on the task of developing the theory and methods for a relativistic quantum chemistry. The aim is to arrive at a qualitative as well as a quantitative understanding of the relativistic effects in molecules. We must be able to predict the effects of relativity on the wave functions and electron densities of molecules, and on the molecular properties arising from these. And we must develop methods and algorithms that enable us to calculate the properties and interactions of molecules with an accuracy comparable to that achieved for lighter systems in a nonrelativistic framework. Parts of this development follow fairly straightforwardly from our considerations of the atomic case in part II, but molecular systems represent challenges of their own. This is particularly true for the computational techniques. From the nonrelativistic experience we know that present-day quantum chemistry owes much of its success to the enormous effort that has gone into developing efficient methods and algorithms. This effort has yielded powerful tools, such as the use of basis-set expansions of wave functions, the exploitation of molecular symmetry, the description of correlation effects by calculations beyond the mean-field approximation, and so on. In developing a relativistic quantum chemistry, we must be able to reformulate these techniques in the new framework, or replace them by more suitable and efficient methods. In nonrelativistic theory, spin symmetry provides one of the biggest reductions in computational effort, such as in the powerful and elegant Graphical Unitary Group Approach (GUGA) for configuration interaction (CI) calculations (Shavitt 1988). For relativistic applications, time-reversal symmetry takes the place of spin symmetry, and this chapter is devoted to developing a formalism for efficient incorporation of this symmetry in our theory and methods. Time-reversal symmetry includes the spin symmetry of nonrelativistic systems, but there are significant differences from spin symmetry for systems with a Hamiltonian that is spin-dependent. The development of techniques that incorporate time-reversal symmetry presented here are primarily aimed at four-component calculations, but they are equally applicable to two-component calculations in which the spin-dependent operators are included at the self-consistent field (SCF) stage of a calculation.

Properties of Relativistic Mean-Field Theory

We have previously seen how the Dirac equation for one particle requires some rather special consideration and interpretation in order to arrive at a form that is able to treat electrons and positrons on an equal footing. These problems persist also when we go to systems with more than one electron. One might think that the extension to several electrons should not introduce dramatic changes. After all, we noted that even the one-electron problem must be viewed as a many-electron (and -positron) system in order to arrive at a consistent description. The problem with introducing more electrons is that electron–electron interactions that were previously small—for the one-electron case typically arising from vacuum polarization and self-interaction—now occur to the same order as the kinetic energy and the interaction with the potential. So while a perturbative approach such as QED can use the solutions of the one-electron Dirac equations as a very good starting approximation to a more accurate description of the full system, the same would not work for a system with more electrons because it would mean neglecting interactions of the same magnitude as the zeroth-order energy. For applications to quantum chemistry, the treatment of the entire electron–electron interaction as a perturbation would be hopelessly impractical, as it is even in manyelectron relativistic atomic structure calculations. The technique for dealing with this problem is well known from nonrelativistic calculations on many-electron systems. One-particle basis sets are developed by considering the behavior of the single electron in the mean field of all the other electrons, and while this neglects a smaller part of the interaction energy, the electron correlation, it provides a suitable starting point for further variational or perturbational treatments to recover more of the electron–electron interaction. It is only natural to pursue the same approach for the relativistic case. Thus one may proceed to construct a mean-field method that can be used as a basis for the perturbation theory of QED.

Regular Approximations

Perturbation theory is a useful tool for evaluating small corrections to a system, and as we noted in the preceding chapter, relativity is a small correction for much of the periodic table. If we can use perturbation theory based on an expansion in 1/c we assign much of the work associated with a more complete relativistic treatment to the end of an otherwise nonrelativistic calculation. The problems with the Pauli Hamiltonian—the singular operators and the questionable validity of an expansion in powers of p/mc—are essentially circumvented by the use of direct perturbation theory. For systems containing heavier atoms it is necessary to go to higher order in 1/c perturbation theory, and possibly even abandon perturbation theory altogether. If we could perform an expansion that yielded a zeroth-order Hamiltonian incorporating relativistic effects to some degree and that was manifestly convergent, it might be possible to use perturbation theory to low order for heavy elements. If we wish to incorporate some level of relativistic effects into the zeroth-order Hamiltonian, we cannot start from Pauli perturbation theory or direct perturbation theory. But can we find an alternative expansion that contains relativistic corrections and is valid for all r: that is, can we derive a regular expansion that is convergent for all reasonable values of the parameters? The expansion we consider in this chapter has roots in the work by Chang, Pélissier, and Durand (1986) and Heully et al. (1986), which was developed further by van Lenthe et al. (1993, 1994). These last authors coined the term “regular approximation” because of the properties of the expansion. The Pauli expansion results from taking 2mc2 out of the denominator of the equation for the elimination of the small component (ESC). The problem with this is that both E and V can potentially be larger in magnitude than 2mc2 and so the expansion is not valid in some region of space. In particular, there is always a region close to the nucleus where |V − E|/2mc2 > 1. An alternative operator to extract from the denominator is the operator 2mc2 − V , which is always positive definite for the nuclear potential and is always greater than 2mc2.

Perturbation Methods

Perturbation theory has been one of the most frequently used and most powerful tools of quantum mechanics. The very foundations of relativistic quantum theory—quantum electrodynamics—are perturbative in nature. Many-body perturbation theory has been used for electron correlation treatments since the early days of quantum chemistry, and in more recent times multireference perturbation theories have been developed to provide quantitative or semiquantitative information in very complex systems. In the beginnings of relativistic quantum mechanics, perturbation methods based on an expansion in powers of the fine structure constant, α = 1/c, were used extensively to obtain operators that would provide a connection with nonrelativistic quantum mechanics and permit some evaluation of relativistic corrections, in days well before the advent of the computer. This seems a reasonable approach, considering the small size of the fine structure constant—and for light elements it has been found to work remarkably well. Relativity is a small perturbation for a good portion of the periodic table. Perturbation expansions have their limitations, however, and as well as successes, there have been failures due to the highly singular or unbounded nature of the operators in the perturbation expansions. Therefore, in recent times other perturbation approaches have been developed to provide alternatives to the standard Breit–Pauli approach. This chapter is devoted to the development of perturbation expansions in powers of 1/c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy–Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit–Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties.

Unitary Transformations of the Dirac Hamiltonian

The separation of the spin-dependent terms in the Dirac Hamiltonian enables us to make an approximation in which the spin-free terms are included in the orbital optimization and the spin-dependent terms may be treated later as a perturbation. In this process, the parameter space required to treat the large and small components has not changed. Even with the extraction of (σ ·p) from the small component, we still have to calculate integrals involving pfL, which essentially regenerates the original small component space and so the integral work has not really changed. What has been achieved is the ability to use the machinery of spin algebra from nonrelativistic theory, but we are left with a large and a small component. The obvious next step is to separate the large and small components, or the positiveand negative-energy states. The small component can be eliminated from the Dirac equation by algebraic manipulation, but this leaves the energy in the denominator. It would be preferable to obtain an energy-independent Hamiltonian that acted only on positive-energy states and that could therefore be represented as two-component spinors. If, following this separation, it were possible to separate out the spin-free and spin-dependent terms, we would have a spin-free Hamiltonian that would operate on a one-component wave function, and we would then be able to use all the machinery of nonrelativistic quantum chemistry but with modified one- and two-electron integrals. The matrix form of the Dirac Hamiltonian suggests that we should seek a unitary transformation that will make it diagonal with respect to the large- and small-component spinor spaces. Such a transformation is called a Foldy–Wouthuysen transformation (Foldy and Wouthuysen 1950). Although in their original paper only the free-particle transformation was derived, together with an iterative decoupling procedure that will be described later in this chapter, the term Foldy–Wouthuysen transformation has come to mean any unitary transformation that decouples the large and small components, either exactly or approximately, and we will use it in this sense.

Molecular Properties

Strictly speaking, in quantum mechanics a measurable property is defined as an observable connected to a self-adjoint operator. However, in common usage the term molecular property is loosely taken to mean any physical attribute of a molecule, preferably amenable to experimental measurement. Common examples of properties of interest to chemists are molecular structure, thermodynamic quantities, spectroscopic transition energies and intensities, and various electric and magnetic moments. The amenability to experiment may exist only in principle—one of the strong points of modern computational chemistry is the possibility of studying phenomena occurring under conditions that lie beyond the present experimental capabilities. Sometimes, differential effects between different theoretical models are also regarded as properties: thus the correlation energy is generally considered to be the difference between the Hartree–Fock energy and the energy obtained from a complete many-electron treatment (e.g. full CI or MBPT to all orders). At best only the latter of these is accessible to experiment. Similarly, certain relativistic effects (e.g. bond contraction) only appear as the difference between results from a relativistic and a nonrelativistic calculation. The calculation of molecular properties in a relativistic framework follows the same principles as for the nonrelativistic case once a wave function or electron density of adequate quality is available. Our aim here is therefore not to provide explicit expressions and formulas for the calculation of a more or less complete catalog of properties. However, in relativistic calculations of molecular properties there are some aspects of the theory that warrant special care and consideration. In particular, we need to know how to handle features such as Lorentz invariance, gauge invariance, and negativeenergy states. Moreover, the electric and magnetic fields appear as natural parts of the relativistic Hamiltonian, and we therefore expect that properties involving these may require a different treatment from the nonrelativistic case where terms involving external fields are grafted onto the nonrelativistic Hamiltonian, often based on some reduction or approximation from the relativistic case.

Matrices and Wave Functions under Double-Group Symmetry

Symmetry is one of the most versatile theoretical tools of physics and chemistry. It provides qualitative insight into the wave functions and properties of systems, and it has also been used successfully to obtain great savings in computational efforts. In the preceding chapter we examined time-reversal symmetry, and now we turn to the more familiar point-group symmetry. We show how relativity requires special consideration and extensions of the concepts developed for the nonrelativistic case, and how time-reversal symmetry and double-group symmetry are connected. Although the techniques that incorporate double-group symmetry presented here are primarily aimed at four-component calculations, they are equally applicable to two-component calculations in which the spin-dependent operators are included at the SCF stage of a calculation. In the preceding chapter, we have shown how the use of time-reversal symmetry can lead to considerable reduction in the number of unique matrix elements that appear in the operator expressions. However, we are also interested in the overall structure of the matrices of the operators. In particular, we are interested in possible block structures, where classes of matrix elements may be set to zero a priori. If the matrices can be cast in block diagonal form, we may save on storage as well as computational effort in solving eigenvalue problems, for example. Matrix blocking will already be effected by the point-group symmetry of the molecule.

The Dirac Equation

The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.