Many-electron Relativistic Hamiltonians

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [1]. The simplest form is the Dirac-Coulomb (DC) Hamiltonian, where the nonrelativistic one-electron terms in the Schrodinger equation are replaced by the one-electron Dirac operator hj). 

The third term in Eq. (8) is the sum over all electron-electron repulsion operators abbreviated by g(i,j), which is in the case of the relativistic many-electron Hamiltonian equal to... 

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

While there are serious difficulties to define a relativistic many-electron Hamiltonian, its non-relativistic limit is, fortunately, well-defined, because in this limit the magnetic interaction of the moving electrons and the retardation of the Coulomb interaction vanish, and there is no quantization of the electromagnetic field (and, of course, no absorption or emission of light). So if relativistic effects are small, one is close to a well-defined situation, namely the non-relativistic limit, and one need not worry much about the appropriate choice of a relativistic many-electron 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)... 

Extended Douglas—Kroll transformations applied to the relativistic many-electron Hamiltonian... 

By an apphcation of the DK transformation to the relativistic many-electron Hamiltonian, recently, we have shown that the many-electron DK Hamiltonian also gives satisfactory results for a wide variety of atoms and molecules compared with... 

The reduction of the relativistic many-electron hamiltonian by expansion in powers of the external field is the second-order Douglas-Kroll transformation [29], and has been used with success by Hess and co-workers [30]. The operators which result from this transformation are non-singular, but the integrals over the resulting operators are complicated and have to be approximated, even for finite basis set expansions. The reduction of the Dirac-Coulomb-Breit equation to two-component form using direct perturbation theory has been described by Kutzelnigg and coworkers [26, 27, 31], Rutkowski [32], and van Lenthe et al. [33]. 

Although in the Dirac-Coulomb Hamiltonian the one-particle operator is correct to all orders in a, the two-particle interaction is only correct to a°. The Dirac-Coulomb Hamiltonian is not invariant under Lorentz transformations, however it can be considered as the leading term of a yet unknown relativistic many-electron Hamiltonian which fulfills this requirement. An operator which also takes into account the leading relativistic corrections for the two-electron terms is the Coulomb-Breit term (Breit 1929,1930,1932, 1938),... 

J. Sucher. Relativistic Many-Electron Hamiltonians. Phys. Scr., 36 (1987) 271-281. 

In most cases, however, the relativistic effects are rather weak and may be separated into spin-orbit coupling effects and scalar effects. The latter lead to compression and/or expansion of electron shells and can rather accurately be treated by modifying the one-electron part of the non-relativistic many-electron Hamiltonian. With this scalar-relativistic Hamiltonian the (modified) energies and wave functions are computed and subsequently an effective spin-orbit part is added to the Hamiltonian. The effects of the spin-orbit term on the energies and wave functions are commonly estimated using second-order perturbation theory. More information for the interested reader can be found in excellent textbooks on relativistic quantum chemistry [2, 3]. 

With these provisions the optimization process may proceed in analogy with any of the various schemes developed for nonrelativistic MCSCF. Jensen et al. (1996) have shown in detail how this may be done for one particular algorithm—the norm-extended optimization. The only added complication for the relativistic case arises from the need to use complex arithmetic. The implementation of time-reversal and doublegroup symmetry follows from the discussions of the symmetry of Fock matrices and of the relativistic many-electron Hamiltonian in earlier chapters. 

The isolated cluster Hamiltonian can be either the non-relativistic many-electron Hamiltonian or a suitable relativistic choice, both in their all-electron versions or in any effective core potential version. We will discuss later. The embedding potential acting on the cluster electrons reads ... 

---