Relativistic Formulation

The fact that an electron has an intrinsic spin comes out of a relativistic formulation of quantum mechanics. Even though the Schrodinger equation does not predict it, wave functions that are antisymmetric and have two electrons per orbital are used for nonreiativistic calculations. This is necessary in order to obtain results that are in any way reasonable. 

The relativistic formulation of Thomas-Fermi theory started at the same time as the original non-relativistic one, the first work being of Vallarta and Rosen [9] in 1932. The result they arrived at can be found by replacing the kinetic energy fimctional by the result of the integration of the relativistic kinetic energy in terms of the momentum p times the number of electrons with a given momentum p from /i = 0 to the Fermi momentum p = Pp. ... 

Quantum Systems in Chemistry and Physics is a broad area of science in which scientists of different extractions and aims jointly place special emphasis on quantum theory. Several topics were presented in the sessions of the symposia, namely 1 Density matrices and density functionals 2 Electron correlation effects (many-body methods and configuration interactions) 3 Relativistic formulations 4 Valence theory (chemical bonds and bond breaking) 5 Nuclear motion (vibronic effects and flexible molecules) 6 Response theory (properties and spectra atoms and molecules in strong electric and magnetic fields) 7 Condensed matter (crystals, clusters, surfaces and interfaces) 8 Reactive collisions and chemical reactions, and 9 Computational chemistry and physics. 

The first topic has an important role in the interpretation and calculation of atomic and molecular structures and properties. It is needless to stress the importance of electronic correlation effects, a central topic of research in quantum chemistry. The relativistic formulations are of great importance not only from a formal viewpoint, but also for the increasing number of studies on atoms with high Z values in molecules and materials. Valence theory deserves special attention since it improves the electronic description of molecular systems and reactions with the point of view used by most laboratory chemists. Nuclear motion constitutes a broad research field of great importance to account for the internal molecular dynamics and spectroscopic properties. 

Most modern band-calculations for non-magnetic solids are performed in the LDA approximation, which is extended also to the relativistic formulation (see Chap. F). Care is taken in the choice of the set of o )i s. A particular problem exists in connecting the atomic wave functions ipi s, calculated in a central potential, in the inter-core region (see Fig. 12) of the solid. It is beyond the scope of this Chapter to go deeper into these details, which will be discussed further in Chap. F. 

Relativistic quantum chemistry is currently an active area of research (see, for example, the review volume edited by Wilson [102]), although most of the work is beyond the scope of this course. Much of the effort is based on Dirac s relativistic formulation of the Schrodinger equation this results in wave functions that have four components rather than the single component we conventionally think of. As a consequence the mathematical and computational complications are substantial. Nevertheless, it is very useful to have programs for Dirac-Fock (the relativistic analogue of Hartree-Fock) calculations available, as they can provide calibration comparisons for more approximate treatments. We have developed such a program and used it for this purpose [103]. 

Relativistic Formulations. - Over the past twenty years or so, we have witnessed a continued and growing interest in relativistic quantum chemical methodology and the associated computational algorithms which facilitate... 

There are two key features which distinguish relativistic quantum mechanics from the non-relativistic formulation -... 

First, in the relativistic formulation the number of particles is not conserved. Electron-positron creation processes, which conserve the total charge of the system but not the number of particles, are permitted in the relativistic formalism. The use of second quantized methodology is therefore mandatory in a fully relativistic formulation of the molecular structure problem. 

Second, the Hamiltonian operator for a relativistic many-body system does not have the simple, well-known form of that for the non-relativistic formulation, i.e. a sum of a sum of one-electron operators, describing the electronic kinetic energy and the electron-nucleus interactions, and a sum of two-electron terms associated with the Coulomb repulsion between the electrons. The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics.46... 

Relativistic effects remarkably influence the electronic structure and the chemical bonding of heavy atoms [15]. In order to calculate the relativistic effects a four-component relativistic formulation by solving the Dirac equation is essential [16]. 

Similar experience has been made for other f-electron systems. Nevertheless, we should point out that by applying the LDA+U scheme we leave the framework of DFT. This does not apply to the SIC (self-interaction correction) formalism (Dreizler and Gross 1990), for which a proper relativistic formulation has been worked out recently (Forstreuter et al. 1997 Temmerman et al. 1997) and applied to magnetic solids (Temmerman et al. 1997). 

In proceeding to the relativistic description of molecular systems, one would like to be able to draw on the advances and developments of the non-relativistic case. However, as we shall show, the relativistic formulation as well as the effects that this formulation place demands on the basis sets that are not necessarily satisfied by a simple transfer of the non-relativistic framework. The subject of our presentation here is therefore to describe the special features and requirements for basis sets to be used in relativistic calculations. As this volume will show, there axe numerous approaches to describing relativity for molecular systems. Here we shall relate our discussion to the conceptually simplest of these, the... 

The restriction to scalar terms and hence to a one-component Hamiltonian is sometimes referred to as a spin averaged formulation. However, the names spin-free or scalar relativistic formulation seem to be more appropriate. The matrix representations of all terms occurring after these manipulations are already available within every quantum chemical program package except for the PVP expressions. Their evaluation can, however, be reduced to the representation of the external potential via the relation... 

We now have a well defined prescription for the calculation of the properties of atoms and molecules within a relativistic formulation. As in the non-relativistic case, the relativistic many-body perturbation theory becomes increasingly complicated in higher orders and in practice it is possible to take the expansion to about fourth order with the size of basis set that is required for calculations of useful accuracy. Sapirstein82 has recently re-iterated the view that... 

An overview of the salient features of the relativistic many-body perturbation theory is given here concentrating on those features which differ from the familiar non-relativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic many-body perturbation theory are considered in more detail below the representation of the Dirac spectrum in the algebraic approximation is discussed the non-additivity of relativistic and electron correlation effects is considered and the use of the Dirac-Hartree-Fock-Coulomb-Breit reference Hamiltonian demonstrated effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. 

Some work has been reported on relativistic coupled cluster methods most notably by Kaldor, Ishikawa and their collaborators.232 These calculations are carried out within the no virtual pair approximation and are therefore analogous to the non-relativistic formulation. Perturbative analysis of the relativistic electronic structure problem demonstrated the importance of the negative energy branch of the spectrum in the calculation of energies and other expectation values. 

For most chemical applications, one is not interested in negative energy solutions of a four-component Dirac-type Hamiltonian. In addition, the computational expense of treating four-component complex-valued wave functions often limited such calculations to benchmark studies of atoms and small molecules. Therefore, much effort was put into developing and implementing approximate quantum chemistry methods which explicitly treat only the electron degrees of freedom, namely two- and one-component relativistic formulations [2]. This analysis also holds for a relativistic DFT approach and the solutions of the corresponding DKS equation. 

At present, theoretical analyses of coupling tensors that are not calculated from a full relativistic formulation continue to be based on the four electron-nucleus interactions described by the Ramsey operators, Eqs. (1-4) ... 

The most serious reservation expressed about relativistic formulations of quantum chemistry in [2] is to know to what question they are supposed to be the answer, in the circumstances in which we And ourselves . This article is intended to demonstrate that one may formulate a valid theory of quantum chemistry without invoking the Schrodinger equation, and that the conventional quantum chemistry that we would And in, say, [3, 4] may be obtained as a limiting case of our formulation. We prefer to keep the relativistic structure of the Dirac theory intact at all times, which means that our formulation does not contain relativistic corrections . This is quite deliberate, and a feature which we will turn into a computational advantage. 

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