Four-Component Dirac Hamiltonian

RELATIVISTIC MULTIREFERENCE PERTURBATION THEORY COMPLETE ACTIVE-SPACE SECOND-ORDER PERTURBATION THEORY (CASPT2) WITH THE FOUR-COMPONENT DIRAC HAMILTONIAN... 

K. Hirao. Relativistic Multireference Perturbation Theory Complete Active-Space Second-Order Perturbation Theory (CASPT2) With The Four-Component Dirac Hamiltonian. In Radiation Induced Molecular Phenomena in Nucleic Adds, Volume 4 of Challenges and Advances in Computational Chemistry and Physics, p. 157-177. Springer, 2008. 

The extension to the case of the four-component Dirac Hamiltonian above follows readily by noting that the spin operator and the orbital angular momentum operator for this case are... 

Abe M, Nakajima T, Hirao K. The relativistic complete active-space second-order perturbation theory with the four-component Dirac Hamiltonian. J ChemPhys. 2006 125 234110. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

Spin-orbit interaction Hamiltonians are most elegantly derived by reducing the relativistic four-component Dirac-Coulomb-Breit operator to two components and separating spin-independent and spin-dependent terms. This reduction can be achieved in many different ways for more details refer to the recent literature (e.g., Refs. 17-21). 

The use of non-relativistic basis functions in (a) requires that the SO interaction can be considered as a relatively weak perturbation of the non-relativistic Hamiltonian, which typically is the case for second- and third-row atoms and transition metals. For systems with heavier atoms, two-component relativistic electronic basis functions should be employed or the analysis should be based on the four-component Dirac-Coulomb Hamiltonian. 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

Relativistic PPs to be used in four-component Dirac-Hartree-Fock and subsequent correlated calculations can also be successfully generated and used (Dolg 1996a) however, the advantage of obtaining accurate results at a low computational cost is certainly lost within this scheme. Nevertheless, such potentials might be quite useful for modelling a chemically inactive environment in otherwise fully relativistic allelectron calculations based on the Dirac-Coulomb-(Breit) Hamiltonian. 

The second possibility to reduce the four-component Dirac spinor to two-component Pauli form is to decouple the Dirac equation, i.e., to transform the Dirac Hamiltonian to block-diagonal form by a suitably chosen unitary transformation U,... 

The transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wavefunction. As discussed in detail in section 2, the four-component Dirac spinor will have only two nonvanishing components, as soon as the complete decoupling of the electronic and positronic degrees of freedom is achieved, and can thus be used as a two-component spinor. This feature can be exploited to calculate expectation values of operators in an efficient manner. However, this procedure requires that some precautions need to be taken care of with respect to the representation of the operators, i.e., their transition from the original (4 x 4)-matrix representation (often referred to as the Dirac picture) to a suitable two-component Pauli repre-... 

Dirac-Hartree-Fock and Dirac-Kohn-Sham methods By an application of an independent-particle approximation with the DC or DCB Hamiltonian, the similar derivation of the non-relativistic Hartree-Fock (HF) method and Kohn-Sham (KS) DFT yields the four-component Dirac-Hartree-Fock (DHF) and Dirac-Kohn-Sham (DKS) methods with large- and small-component spinors. 

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. 

For the DKeel and DKee2 models, this equivalence holds because the terms of both Hamiltonians related to the Hartree self-interaction are limited to the fpFW and first-order DKH transforms of the Hartree potential, Eqs. (23) and (24). Thus, the terms jointly notated by the symbols [mn k rei can consistently be used to determine the fitting coefficients of the density in the four-component Dirac picture, to build the Hartree part of the relativistic Hamiltonian at DKeel and DKee2 levels, and to evaluate the total energy. 

Abstract In this chapter I demonstrate a series of examples showing the importance of relativistic quantum chemistry to the proper description of variety of molecular and atomic properties including valence and core ionization potentials, electron affinities, chemical reactions, dissociation energies, spectroscopic parameters and other properties. An overview of basic principles of the relativistic quantum chemistry and the reduction of relativistic quantum chemistry to two-component form is also presented. I discuss the transition of the four-component Dirac theory to the infinite-order two-component (lOTC) formalism through the unitary transformation which decouples exactly the Hamiltonian. 

By the mid-1960 s it was recognized that this simple picture was not adequate. Sandars and Beck (1965) showed how relativistic effects of the type first described by Casimir (1963) could be accommodated by generalizing the non-relativistic Hamiltonian to the form given by (108). A rather profound mental adjustment was required instead of setting the relativistic Hamiltonian between products of four-component Dirac eigenfunctions, they asked for the effective operator that accomplishes the same result when set between non-relativistic states. The coefficients ujf now involve sums over integrals of the type dr, where Fj and Gj,... 

The purpose of this section is to show how the problem of passing from the four-component Dirac equation to two-component Pauli-like equations can be systematically investigated within the framework of the theory of effective Hamiltonians.Beyond the above-mentioned difficulties, we will be able to derive energy-independent two-component effective Hamiltonians that can be used for variational atomic and molecular calculations. To introduce the subject and the notation, let us first consider the simple case of a free electron. 

All exact-decoupling approaches can be related to the modified Dirac equation and we closely follow here the work presented in Refs. [16,647]. Two-component electrons-only Hamiltonians can be obtained from block-diagonalizing the four-component (one-electron) modified Dirac equation in matrix representation. As we have discussed in chapters 8 and 10 for four-component Dirac-Hartree-Fock-Roothaan calculations, basis functions for the small component must fulfill certain constraints as otherwise variational instability and a wrong nonrelativistic limit [547] would result. The correct nonrelativistic limit will be obtained if the kinetic-balance condition,

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The unitary transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wave function. As discussed in detail in chapters 11 and 12, the four-component Dirac spinor ip will... 

An important aspect of the DKH approach to molecular properties is to understand the necessity to start at the four-component Dirac framework with a Hamiltonian containing the property X under investigation. The evaluation of X within this four-component picture may then be accomplished either varia-tionally or by means of perturbation theory up to some well-defined order as discussed in section 15.1. The reduction to two-component formulations can be realized by suitably chosen DKH transformations for both the variational and the perturbative treatment of X. However, the unitary transformations to be applied are different for both schemes [764], which is to be shown in the following. Of course, this distinction holds irrespective of the specific features of X. The differences will only vanish for infinite-order perturbation theory. 

If the property is incorporated variationally into the four-component Dirac picture (as in section 15.1), the A-dependent energy is the reference value which has to be reproduced by the DKH calculation. In order to evaluate this energy within a two-component framework, the perturbed Hamiltonian... 

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