Publisher Summary This chapter provides a summary of the relativistic calculations on multielectron or multicenter problems. The Dirac–Fock Hamiltonian and the main quantum electrodynamical (QED) corrections are discussed and the atomic and bandstructure calculations are reviewed. Then the construction of relativistic molecular orbitals and the solvable one-electron molecular and solid-state models are described. The simplest possible system for studying relativistic effects in chemical bonding is H2+. Several variational linear combinations of atomic orbitals (LCAO)-type solutions of the Dirac equation for H2+ show that the relativistic effects decrease the electronic energy by about –7 ╳ 10–6 a.u. The Dirac-Fock and Dirac-Slater molecular calculations, the relativistic semiempirical methods, and the perturbation treatments of relativistic effects are also described. In relativistic treatments of several spectroscopic properties, the entire formulation must be changed if relativistic wavefunctions are used. Some of its examples are considered. The chapter also presents a preliminary account of the relativistic effects on the chemical properties of the periodic system of elements.

Relativistic Quantum Chemistry

Relativistic effects can strongly influence the chemical and physical properties of heavy elements and their compounds. This influence has been noted in inorganic chemistry textbooks for a couple of decades. This review provides both traditional and new examples of these effects, including the special properties of gold, lead-acid and mercury batteries, the shapes of gold and thallium clusters, heavy-atom shifts in NMR, topological insulators, and certain specific heats.

Relativistic Effects in Chemistry: More Common Than You Thought

Multiconfiguration wave function expansions combined with configuration interaction methods are a method of choice for complex atoms where atomic state functions are expanded in a basis of configuration state functions. Combined with a variational method such as the multiconfiguration Hartree-Fock (MCHF) or multiconfiguration Dirac-Hartree-Fock (MCDHF), the associated set of radial functions can be optimized for the levels of interest. The present review updates the variational MCHF theory to include MCDHF, describes the multireference single and double process for generating expansions and the systematic procedure of a computational scheme for monitoring convergence. It focuses on the calculations of energies and wave functions from which other atomic properties can be predicted such as transition rates, hyperfine structures and isotope shifts, for example.

Advanced multiconfiguration methods for complex atoms : I. Energies and wave functions

We present a novel scheme to calculate electronic g-tensor values of doublet-state systems within a density functional method and discuss the implementation and results of first applications. The method employs two-component eigenfunctions of the Kohn−Sham equation where spin−orbit effects are taken into account self-consistently. Therefore, the g-tensor can be treated as first-order property with respect to the perturbation by external magnetic field alone. The Zeeman energy splitting, an inherently relativistic effect, is naturally and transparently determined by the two-component ground-state wave function (Kramers doublet) without invoking virtual states. Abandoning the widely accepted perturbative treatment of the spin−orbit term makes the present method also applicable to molecular systems with considerable spin−orbit interaction. Conceivable improvements of the method performance are proposed and discussed.

Calculation of Electronic g-Tensors Using a Relativistic Density Functional Douglas−Kroll Method

First-order relativistic corrections to the energy of closed-shell molecular systems are calculated, using all terms in the two-component Breit–Pauli Hamiltonian. In particular, we present the first implementation of the two-electron Breit orbit–orbit integrals, thus completing the first-order relativistic corrections within the two-component Pauli approximation. Calculations of these corrections are presented for a series of small and light molecules, at the Hartree–Fock and coupled-cluster levels of theory. Comparisons with four-component Dirac–Coulomb–Breit calculations demonstrate that the full Breit–Pauli energy corrections represent an accurate approximation to a fully relativistic treatment of such systems. The Breit interaction is dominated by the spin–spin interaction, the orbit–orbit interaction contributing only about 10% to the total two-electron relativistic correction in molecules consisting of light atoms. However, the relative importance of the orbit–orbit interaction increases with increa...

A closed-shell coupled-cluster treatment of the Breit-Pauli first-order relativistic energy correction.

1 Non-relativistic Quantum Mechanics.- 1.1 Formal quantum mechanics.- 1.2 The Schrodinger equation.- 1.3 Heisenberg's uncertainty principle and related topics.- 1.4 Angular momentum.- 1.5 Electron spin.- 1.6 The need for a relativistic theory.- 2 Vector and Matrix Algebra.- 2.1 Vectors and vector multiplication.- 2.2 The repeated subscript convention for summation.- 2.3 The Kronecker delta ?ij.- 2.4 The ?ijk notation.- 2.5 The ?ijk sum rules.- 2.6 Examples I.- 2.7 The vector operator ?.- 2.8 The gradient.- 2.9 The divergence.- 2.10 The curl.- 2.11 Examples II.- 2.12 Second derivatives in vector calculus.- 2.13 The Dirac delta function.- 2.14 Matrices and determinants: a summary.- 2.15 Vectors in four dimensions.- 3 Classical Mechanics.- 3.1 Inertial frames and Galileo's relativity principle.- 3.2 The principle of least action.- 3.3 Lagrange's equations of motion.- 3.4 The Lagrangian for a system of particles.- 3.5 Constants of motion.- 3.6 The Hamiltonian.- 4 Special Relativity.- 4.1 Einstein's principle of relativity.- 4.2 The interval.- 4.3 The Lorentz transformation.- 4.4 Contraction, dilation and paradoxes.- 4.5 The transformation of velocities.- 4.6 The relativistic mechanics of a free particle.- 4.7 Four-vectors.- 5 The Interaction of Charged Particles with Electromagnetic Fields.- 5.1 Units.- 5.2 The electromagnetic potentials.- 5.3 The field vectors.- 5.4 The Lorentz transformation of electric and magnetic fields.- 5.5 Gauge transformations.- 5.6 Maxwell's equations.- 5.7 The potentials and fields due to a stationary charge.- 5.8 The potentials due to a moving charge.- 5.9 The interaction of two charged particles.- 5.10 The Thomas precession.- 6 The Classical Theory of Electromagnetic Fields.- 6.1 Continuous mechanical systems.- 6.2 The Lagrangian density for an electromagnetic field.- 6.3 The current four-vector.- 6.4 The second pair of Maxwell's equations.- 6.5 Electromagnetic waves.- 6.6 Solution of the wave equation for free space.- 6.7 The characteristic vibrations of an electromagnetic field.- 7 Relativistic Wave Equations.- 7.1 Quantization of classical equations.- 7.2 Gauge invariance of quantum mechanical equations.- 7.3 The Klein-Gordon equation.- 8 The Dirac Equation.- 8.1 The Dirac equation for a free electron.- 8.2 The Dirac operators ? and ?.- 8.3 The introduction of an electromagnetic field.- 8.4 Electron spin.- 8.5 Lorentz invariance of the Dirac equation.- 8.6 The negative energy solutions - positrons.- 8.7 The non-relativistic approximation of the Dirac equation.- 8.8 The method of small components.- 8.9 The Foldy-Wouthuysen transformation.- 8.10 The free electron.- 9 The Wave Equation for Many Electrons.- 9.1 The electromagnetic potentials due to a moving electron.- 9.2 The Hamiltonian for two electrons.- 9.3 The Breit equation.- 9.4 Reduction of the Breit equation to non-relativistic form.- 9.5 Radiative corrections.- 9.6 The many-electron Hamiltonian.- 10 The Molecular Hamiltonian.- 10.1 The introduction of nuclei.- 10.2 Finite nuclear size effects.- 10.3 Spectroscopically useful Hamiltonians.- 10.4 Effective Hamiltonians.- 11 The Hydrogen Atom.- 11.1 Non-relativistic theory for a one-electron atom.- 11.2 The non-relativistic approximation of the Dirac equation.- 11.3 The simultaneous eigenfunctions of j2, jz, l2, s2 and K.- 11.4 Commutation relations for the Dirac Hamiltonian.- 11.5 The Dirac equation in polar coordinates.- 11.6 Solution of the radial equations.- 11.7 The energy levels.- 11.8 Comparison of Dirac and non-relativistic atomic orbitals.- 11.9 The Lamb shift.- 11.10 More complicated systems.- 12 Quantum Field Theory.- 12.1 Quantization of the electromagnetic field.- 12.2 Solution of the one-dimensional harmonic oscillator equation.- 12.3 Creation and annihilation operators.- 12.4 Photons.- 12.5 Zero-point energy and vacuum fluctuations.- 12.6 Fermions and second quantization.- 13 The Interaction of Radiation and Matter.- 13.1 The interaction Hamiltonian.- 13.2 Time-dependent perturbation theory.- 13.3 Matrix elements of the interaction Hamiltonian.- 13.4 Absorption and emission.- 13.5 Comparison of the semiclassical and quantized theories.- 13.6 Multi-photon processes.- 13.7 The scattering of photons by molecules.- 13.8 Line widths and resonance fluorescence.- Appendix A Units.- A.1 SI units.- A.2 Conversion from the mixed (Gaussian) CGS system to the SI system.- A.3 Recommended values of physical constants.- Appendix B Vector Relations in Three Dimensions.- Appendix C General Bibliography.- Author Index.

Advanced Molecular Quantum Mechanics: An Introduction to Relativistic Quantum Mechanics and the Quantum Theory of Radiation

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Advanced Molecular Quantum Mechanics: An Introduction to Relativistic Quantum Mechanics and the Quantum Theory of Radiation