The relativistic Hamiltonian

The no-virtual-pair approximation (NVPA) is invoked so that negative-energy solutions of the SCF equations are discarded. 

The relativistic many-electron Hamiltonian cannot be written in closed form it may be derived perturbatively from quantum electrodynamics [l]. 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 ho, 

Usually, the frequency-independent Breit operator is taken, 

The DC or DCB Hamiltonians may lead to the admixtm-e of negative-energy eigenstates of the Dirac Hamiltonian in an erroneous way [3,4]. The no-virtual-pair approximation [5,6] is often invoked to correct this problem the negative-energy states are eliminated by the projection operator A , leading to the projected Hamiltonians 

---

In this section, we review the relativistic Hamiltonian with special emphasis on how interactions are introduced. Within the Bom-Oppenheimer approximation the electronic Hamiltonian, relativistic or not, has the generic form... 

We now consider how to eliminate either all relativistic effects or exclusively the spin-orbit interaction from the relativistic Hamiltonian. We start from the Dirac equation in the molecular field... 

All potentials satisfy Coulomb gauge. The corresponding potentials are introduced into the relativistic Hamiltonian, and the perturbation operators are obtained as... 

If the relativistic effects are sufficiently large and therefore cannot be accounted for as corrections, then as a rule one has to utilize relativistic wave functions and the relativistic Hamiltonian, usually in the form of the so-called relativistic Breit operator. In the case of an N-electron atom the latter may be written as follows (in atomic units, in which the absolute value of electron charge e, its mass m and Planck constant h are equal to one, whereas the unit of length is equal to the radius of the first Bohr orbit of the hydrogen atom) ... 

In order to improve the theoretical description of a many-body system one has to take into consideration the so-called correlation effects, i.e. to deal with the problem of accounting for the departures from the simple independent particle model, in which the electrons are assumed to move independently of each other in an average field due to the atomic nucleus and the other electrons. Making an additional assumption that this average potential is spherically symmetric we arrive at the central field concept (Hartree-Fock model), which forms the basis of the atomic shell structure and the chemical regularity of the elements. Of course, relativistic effects must also be accounted for as corrections, if they are small, or already at the very beginning starting with the relativistic Hamiltonian and relativistic wave functions. 

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

Hence the relativistic Hamiltonian Xdoes not commute with either Lz or combining equations (3.48) and (3.51) we find that Xdoes commute with... 

The formalism parallels to some extent the nonrelativistic arguments of Kahn et al. (23). We begin with the relativistic Hamiltonian... 

This analysis has been tested for Xe2+ and Au2 within the effective core potential approximation (65). Four sets of calculations have been carried out nonrelativistic, first-order relativistic, fully relativistic, and first-order nonrelativistic (relativistic wave function with nonrelativistic Hamiltonian). The computed Rc values were 3.01, 2.67, 2.58, and 3.14 A, respectively, for Au2 and 3.24, 3.18, 3.19, and 3.24 A for Xe2+. For these cases then, the analysis of Schwarz et al. is clearly inappropriate. It may be the case that the nonrelativistic electronic contraction stabilized in first-order calculations is independent of the usual relativistic AO contraction. However since the contraction is only stable in the presence of the relativistic Hamiltonian, it is still a relativistic orbital contraction, but now at a molecular level. 

The formulation of the relativistic CASPT2 method is almost the same as the nonrelativistic CASPT2 in the second quantized form. In this section, firstly we express the relativistic Hamiltonian in the second quantized form, and then, we give a summary of the CASPT2 method [11, 12],... 

To elucidate the nature of chemical bonding in metal carbides with the NaCl structure, the valence electronic states for TiC and UC have been calculated using the discrete-variational (DV) Xa method. Since relativistic effects on chemical bonding of compounds containing uranium atom become significant, the relativistic Hamiltonian, i.e., the DV-Dirac-Slater method, was used for UC. The results... 

For heavy elements, all of the above non-relativistic methods become increasingly in error with increasing nuclear charge. Dirac 47) developed a relativistic Hamiltonian that is exact for a one-electron atom. It includes relativistic mass-velocity effects, an effect named after Darwin, and the very important interaction that arises between the magnetic moments of spin and orbital motion of the electron (called spin-orbit interaction). A completely correct form of the relativistic Hamiltonian for a many-electron atom has not yet been found. However, excellent results can be obtained by simply adding an electrostatic interaction potential of the form used in the non-relativistic method. This relativistic Hamiltonian has the form... 

Hartree—Fock calculations are done using the coordinate—spin representation for the orbitals. For the relativistic Hamiltonian this is written in two-component form (4.182) as... 

For the relativistic Hamiltonian the procedure is called multiconfiguration Dirac—Fock. A computer program for structure calculations in this approximation has been described by Grant et al. (1980). The non-relativistic procedure has been described by Froese-Fischer (1977) and implemented by the same author (Froese-Fischer, 1978). 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

Studies of rare earth or transition metal complexes often necessitate use of multireference wave functions. Among the Coupled Cluster type methods one can distinguish two main lines of approach to incorporate multireference character in the reference wave function. In the Hilbert space method one computes a single wave function for a particular state, while in the Fock space method one tries to obtain a manifold of states simultaneously. Since the latter method [40] has recently been implemented and applied in conjunction with the relativistic Hamiltonian [48-50] we will focus on this approach. 

In the nonrelativistic Hamiltonian in the presence of a magnetic field (115) we ought to choose a system of units for which 6 = 1. It is recommended to do this also for the relativistic Hamiltonian (116). 

Relativistic effects in a certain property may be defined as the difference between the results obtained from a relativistic and a nonrelativistic calculation. Clearly, using this definition the quantitative study of relativistic effects will depend on the choice of the (relativistic) Hamiltonian as well as on the qual-... 

The relativistic Hamiltonian may be defined by adding Hso to Hel. The eigenfunctions of this new Hamiltonian are the relativistic wavefunctions, i,n, which define the relativistic potential curves... 

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. 

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,... 

---