Dirac-Coulomb approximation

If one neglects the transverse contribution, one arrives at what is termed the Dirac-Coulomb approximation (a standard in quantum chemistry). Inclusion of the transverse term, which describes retardation and magnetic effects, in perturbation theory (weakly relativistic hmit) leads to the Dirac-Coulomb-Breit Hamiltonian. 

Starting from the Dirac-Coulomb approximation, a set of Dirac-Kohn-Sham equations may again be derived. In chapter 8, a spinor-rotation procedure was used to derive the relativistic Fock operator. A similar procedure applied to the present case shows that the gradient of the energy has elements the form... 

The superscripts give the order in perturbation theory of the spinor, the subscripts correspond to the superscripts on the operators and determine whether rif or ( ) is used for the density, and may take the values 0 and 2. The second and third of the exchange operators exchange the component types as well as the electron indices this is merely a convenient form of expression that reflects the fact that the spinors contributing to the density for any electron in the Dirac-Coulomb approximation must have the same component type. 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

The most straightforward method for electronic structure calculation of heavy-atom molecules is solution of the eigenvalue problem using the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonians [4f, 42, 43] when some approximation for the four-component wave function is chosen. 

EDE in the external Coulomb field in Fig. 1.6. The eigenfunctions of this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schrodinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions. 

In the Schrodinger-Coulomb approximation the expression in (6.33) reduces to the leading nuclear size correction in (6.3). New results arise if we take into account Dirac corrections to the Schrodinger-Coulomb wave functions of relative order (Za). For the nS states the product of the wave functions in (6.33) has the form (see, e.g, [17])... 

The Breit-Pauli Hamiltonian is an approximation up to 1/c2 to the Dirac-Coulomb-Breit Hamiltonian obtained from a free-particle Foldy-Wouthuysen transformation. Because of the convergence issues mentioned in the preceding section, the Breit-Pauli Hamiltonian may only be employed in perturbation theory and not in a variational procedure. The derivation of the Breit-Pauli Hamiltonian is tedious (21). 

Within the Born-Oppenheimer approximation, the total electronic Dirac-Coulomb Hamiltonian is written as... 

We may nevertheless ask whether it is possible to base RDFT on an approximate relativistic many-body approach, as, for example, the Dirac-Coulomb (DC) Hamiltonian,... 

Since this only affects the one-electron portion of the Hamiltonian, its implementation in DFT is straightforward for atomic calculations. However the eigenvalues of this relativistic Hamiltonian also correspond to a negative continuum [24]. A more sophisticated Hamiltonian is the non-virtual pair approximation or the projected Dirac-Coulomb-Breit Hamiltonian [24] ... 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

The simplest approximation is to combine the Dirac theory with the nrl of electrodynamics, which automatically leads to the Dirac Coulomb Hamiltonian... 

Since the Dirac equation is valid only for the one-electron system, the one-electron Dirac Hamiltonian has to be extended to the many-electron Hamiltonian in order to treat the chemically interesting many-electron systems. The straightforward way to construct the relativistic many-electron Hamiltonian is to augment the one-electron Dirac operator, Eq. (70) with the Coulomb or Breit (or its approximate Gaunt) operator as a two-electron term. This procedure yields the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonian derived from quantum electrodynamics (QED)... 

---