Four-component Dirac spinor

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... 

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 Dirac-Fock one-centre method was the first approximation used for relativistic molecular structure calculations and is now only of historical importance. In this method the electron-electron interaction is handled exactly and the one-electron wave functions are four component Dirac spinors. On the other hand both the nuclear potentials and all the one-electron orbitals are expanded about a single common centre taken to be the position of the nucleus of the heaviest atom of the system under consideration. Because of this expansion, the method is restricted to hydrides XH and even for them the expansion is only slowly convergent. Nevertheless, experience gained with non-relativistic calculations has shown surprisingly good results for equilibrium distances of X-H bonds and for force constants. 

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-... 

We now discuss how at the DKee level of theory the transformed Hartree potential can be conveniently evaluated also for a density constructed from four-component Dirac spinors see Eqs. (6) and (26). The fitting-function (FF) technique, e.g. with an auxiliary Gaussian basis / [20,32],... 

Four-component Dirac spinor quantities are given in the standard representation in which the Dirac matrices are (in terms of 2 x 2 blocks)... 

The equivalence of the lOTC method to the four-component Dirac approach has been documented by calculations of spin orbital energies in several papers [18,20, 63]. The unitary transformation does not affect the energy eigenspectrum, though it reduces the four-component bi-spinors to two-component spinor solutions. Due to this fact the two-component methods are frequently addressed as being quasi-relativistic and it is assumed that some information is lost. It can be demonstrated [22] that the two-component lOTC wave function which is the upper component of the unitarly transformed four-component Dirac spinor I ... 

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... 

There is increasing interest in the relativistic treatment of atoms/ molecules/ and solids. A relativistic Hartree-Fock scheme [Hartree-Fock-Dirac (HFD) method] based on the variation in the total energy obtained with a single Slater determinant (in which the one-electron orbitals are four-component Dirac spinors), using a Dirac-type Hamiltonian for each electron and including Coulomb interaction, was developed some time ago.< For the remaining interaction terms the first-order perturbation of the Breit interaction operato reduced to large components (Pauli approximation) is usually taken into account (see, however, the work of Mann and Johnson ). 

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