Four-component Dirac operator

In a rigorous treatment, one replaces the one-electron operator h by the four-component Dirac-operator hjj and perhaps supplement the two-electron operator by the Breit interaction term [15]. Great progress has been made in such four-component ab initio and DPT methods over the past decade. However, they are not yet used (or are not yet usable) in a routine way for larger molecules. 

In principle, all four-component molecular electronic structure codes work like their nonrelativistic relatives. This is, of course, due to the formal similarity of the theories where one-electron Schrbdinger operators are replaced by four-component Dirac operators enforcing a four-component spinor basis. Obviously, the spin symmetry must be treated in a different way, i.e. it is replaced by the time-reversal symmetry being the basis of Kramers theorem. Point group symmetry is replaced by the theory of double groups, since spatial and spin coordinates cannot be treated separately. 

The eigenvalue equation for the four-component Dirac operator represented in a basis set reads... 

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). 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

To construct the Dirac-Fock equations, it is assumed that the wave function for an atom having N electrons may be expressed as an antisymmetrized product of four-component Dirac spinors of the form shown in Eq. (9). For cases where a single antisymmetrized product is an eigenfunction of the total angular momentum operator J2, the JV-electron atomic wave function may be written... 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

These atomic orbitals are four-component Dirac spinors. The symmetriza-tion coefficients are obtained by the use of group theoretical projection operators [21]... 

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. 

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-... 

The expression for the lowest order contribution to the parity violating potential within the Dirac Hartree-Fock framework is identical to that within the relativistically parameterised extended Hiickel approach in eq. (146). The difference is, however, that in DHF typically atomic basis sets with fixed radial functions are employed (see [161]) and that the molecular orbital coefficients are obtained in a self-consistent Dirac Hartree-Fock procedure. Computations of parity violating potentials along these lines have occasionally been called fully relativistic, although this term is rather unfortunate. In the four-component Dirac Hartree-Fock calculations by Quiney, Skaane and Grant [155] as well as in those by Schwerdtfeger, Laerdahl and coworkers [65,156,162,163] the Dirac-Coulomb operator has been employed, which is for systems with n electrons given by... 

The combination of the Dirac-Kohn-Sham scheme with non-relativis-tic exchange-correlation functionals is sometimes termed the Dirac-Slater approach, since the first implementations for atoms [13] and molecules [14] used the Xa exchange functional. Because of the four-component (Dirac) structure, such methods are sometimes called fully relativistic although the electron interaction is treated without any relativistic corrections, and almost no results of relativistic density functional theory in its narrower sense [7] are included. For valence properties at least, the four-component structure of the effective one-particle equations is much more important than relativistic corrections to the functional itself. This is not really a surprise given the success of the Dirac-Coulomb operator in wave function based relativistic ab initio theory. Therefore a major part of the applications of relativistic density functional theory is done performed non-rela-tivistic functionals. 

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,... 

Figure 16.4 Total electronic densities of M(C2H2) with M=Ni,Pt from Hartree-Fock calculations with two-component ZORA, scalar-relativistic DKH10, and nonrelativistic Schrodinger one-electron operators subtracted from the four-component Dirac-Hartree-Fock reference densities (data taken from Ref. [880]). The molecular structure of the complexes is indicated by element symbols and lines positioned just below the atomic nuclei (top panel). Asymmetries in the plot are due to the discretization of the density on a cubic grid of points. The DKH densities have not been corrected for the picture-change effect and, hence, deviate from the four-component reference density in the closest proximity to the nuclei. But these effects can hardly be resolved on the numerical grid employed to represent the densities.

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... 

NR - nonrelativistic, PT-MVD - pCTturbative treatment of mass-velocity and Darwin operators (only SCF), DKH - Douglas-Kroll-Hess, RECP — relativistic effective core potential, DC - four-component Dirac-Coulomb, Exp - experiment. 

Liu W, Peng D. Exact two-component Hamiltonians revisited. J Chem Phys. 2009 131 031104. Nakajima T, Hirao K. The Douglas-Kroll-Hess Approach. Chem Rev. 2011 112 385-402. Belpassi L, Storchi L, Quiney HM, Tarantelli F. Recent advances and perspectives in four-component Dirac-Kohn-Sham calculations. Phys Chem Chem Phys. 2011 13 12368-12394. Peng D, Reiher M. Exact decoupling of the relativistic Fock operator. Theor Chem Acc. 2012 131 1081. 

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