Relativistic Quantum Chemistry: An Advanced Approach to the Construction of the Green Function of the Dirac Equation with Complex Energy and Mean-Field Nuclear Potential

The development of quantum chemistry, that is, the solution of the Schrödinger equation for molecules, is almost exclusively founded on the expansion of the molecular electronic wave function as a linear combination of atom-centered functions, or atomic orbitals—the LCAO approximation. These orbitals are usually built up out of some set of basis functions. The properties of the atomic functions at large and small distances from the nucleus determines to a large extent what characteristics the basis functions must have, and for this purpose it is sufficient to examine the properties of the hydrogenic solutions to the Schrödinger equation. If we are to do the same for relativistic quantum chemistry, we should first examine the properties of the atomic solutions to determine what kind of basis functions would be appropriate. However, the atomic solutions of the Dirac equation provide more than merely a guide to the choice of basis functions. The atoms in a molecule retain their atomic identities to a very large extent, and the modifications caused by the molecular field are quite small for most properties. In order to arrive at a satisfactory description of the relativistic effects in molecules, we must first of all be able to treat these effects at the atomic level. The insight gained into the effects of relativity on atomic structure is therefore a necessary and useful starting point for relativistic quantum chemistry. As in the nonrelativistic case, most of the salient features of the atomic systems are exposed in the treatment of the simplest of these, the hydrogen-like one-electron atoms. In Hartree atomic units the time-independent Dirac equation yields the coupled equations where we have shifted the energy by −mc2 (with m = 1), as discussed in section 4.6. We will use this shifted energy scale for the rest of the book unless otherwise explicitly indicated. V is here a scalar, central potential.

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Introduction to Relativistic Quantum Chemistry

10.1093/oso/9780195140866.001.0001 ◽

2007 ◽

Cited By ~ 30

Author(s):

Kenneth G. Dyall ◽

Knut Faegri

Keyword(s):

Dirac Equation ◽

Quantum Chemistry ◽

Relativistic Theory ◽

Density Functional ◽

Relativistic Quantum ◽

Relativistic Effects ◽

Basis Sets ◽

Spin Orbit ◽

Approximate Methods ◽

Nonrelativistic Theory

This book provides an introduction to the essentials of relativistic effects in quantum chemistry, and a reference work that collects all the major developments in this field. It is designed for the graduate student and the computational chemist with a good background in nonrelativistic theory. In addition to explaining the necessary theory in detail, at a level that the non-expert and the student should readily be able to follow, the book discusses the implementation of the theory and practicalities of its use in calculations. After a brief introduction to classical relativity and electromagnetism, the Dirac equation is presented, and its symmetry, atomic solutions, and interpretation are explored. Four-component molecular methods are then developed: self-consistent field theory and the use of basis sets, double-group and time-reversal symmetry, correlation methods, molecular properties, and an overview of relativistic density functional theory. The emphases in this section are on the basics of relativistic theory and how relativistic theory differs from nonrelativistic theory. Approximate methods are treated next, starting with spin separation in the Dirac equation, and proceeding to the Foldy-Wouthuysen, Douglas-Kroll, and related transformations, Breit-Pauli and direct perturbation theory, regular approximations, matrix approximations, and pseudopotential and model potential methods. For each of these approximations, one-electron operators and many-electron methods are developed, spin-free and spin-orbit operators are presented, and the calculation of electric and magnetic properties is discussed. The treatment of spin-orbit effects with correlation rounds off the presentation of approximate methods. The book concludes with a discussion of the qualitative changes in the picture of structure and bonding that arise from the inclusion of relativity.

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Essentials of relativistic quantum chemistry

The Journal of Chemical Physics ◽

10.1063/5.0008432 ◽

2020 ◽

Vol 152 (18) ◽

pp. 180901 ◽

Cited By ~ 6

Author(s):

Wenjian Liu

Keyword(s):

Quantum Chemistry ◽

Relativistic Quantum ◽

Relativistic Quantum Chemistry

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Relativistic Quantum Chemistry and Spectroscopy of Kaonic Atomic Systems with Accounting for Radiative and Strong Interaction Effects

Advances in Quantum Systems in Chemistry, Physics, and Biology - Progress in Theoretical Chemistry and Physics ◽

10.1007/978-3-030-34941-7_2 ◽

2020 ◽

pp. 33-50

Author(s):

Olga Yu. Khetselius ◽

Andrey A. Svinarenko ◽

Valentin B. Ternovsky ◽

Yuliya V. Dubrovskaya ◽

Inga N. Serga

Keyword(s):

Quantum Chemistry ◽

Strong Interaction ◽

Relativistic Quantum ◽

Interaction Effects ◽

Atomic Systems ◽

Relativistic Quantum Chemistry

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Infinite-order two-component theory for relativistic quantum chemistry

The Journal of Chemical Physics ◽

10.1063/1.1436462 ◽

2002 ◽

Vol 116 (7) ◽

pp. 2696-2704 ◽

Cited By ~ 221

Author(s):

Maria Barysz ◽

Andrzej J. Sadlej

Keyword(s):

Quantum Chemistry ◽

Infinite Order ◽

Relativistic Quantum ◽

Component Theory ◽

Two Component ◽

Relativistic Quantum Chemistry

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

Introduction to Relativistic Quantum Chemistry ◽

10.1093/oso/9780195140866.003.0015 ◽

2007 ◽

Author(s):

Kenneth G. Dyall ◽

Knut Faegri

Keyword(s):

Quantum Chemistry ◽

Time Reversal ◽

Relativistic Quantum ◽

Spin Symmetry ◽

Computational Effort ◽

Wave Functions ◽

Time Reversal Symmetry ◽

Molecular Systems ◽

Self Consistent Field ◽

Relativistic Quantum Chemistry

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.

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