DHF calculations

The Dirac equation can be readily adapted to the description of one electron in the held of the other electrons (Hartree-Fock theory). This is called a Dirac-Fock or Dirac-Hartree-Fock (DHF) calculation. 

Systematic studies of reaction energies for the abstraction of H2 from MH4 (equation 5) for M = Si, Ge, Sn and Pb were reported by Dyall who also compared the results of the DHF calculations with those of other methods (ECP, PT)126, by Schwerdtfeger and coworkers48b who included also the eka-lead element 114 and by Thiel and coworkers105 who studied also the activation barriers for this reaction. More recent computations concentrated on the evaluation of the quality of the various theoretical approaches103106. The results of the calculations are collected in Table 4 and are shown graphically in Figure 3a. 

In the nonrelativistic limit, only the (LL LL) class of multi-centre integrals is needed this class is still the most important in relativistic calculations. In terms of the conventional expansion in powers of a, a DHF calculation at this level includes all the one-centre relativistic effects, screening by the Coulomb interaction, and some of the spin-other-orbit effects. The (LL 55) and (55 55)... 

Highly-ionized atoms DHF calculations on isoelectronic sequences of few-electron ions serve as the starting point of fundamental studies of physical phenomena, though many-body corrections are now applied routinely using relativistic many-body theory. Relativistic self-consistent field studies are used as the basis of investigations of systematic trends in ionization energies [137-144], radiative transition probabilities [145-148], and quantum electrodynamic corrections [149-151] in few-electron systems. Increased experimental precision in these areas has driven the development of many-body methods to model the electron correlation effects, and the inclusion of Breit interaction in the evaluation of both one-body and many-body corrections. 

Heavy neutral species In order to model the electron density of heavy elements for subsequent use in the construction of pseudopotential representations of the inner-shell regions near heavy nuclei, atomic DHF calculations are often employed as a starting point [158]. These pseudopotential approximations are used in order to reduce the computational cost of molecular or solid-state calculations of extended systems containing heavy elements. Similarly, atomic DHF calculations are used in the design of atom-centred basis sets, either by the direct... 

The number of reported molecular DHF calculations is sufficiently small that a fairly complete account is possible. The cases which have been studied in the DHF model all involve small molecules, or molecules which exhibit high spatial symmetry. Larger molecules have been studied using more approximate schemes, ranging from semi-empirical and pseudopotential methods to Dirac-Fock-Slater and density functional methods. These are discussed elsewhere in this book. 

The previous section considered the derivation of second quantized Hamiltonians that can be used in post-DHF calculations. From now on we will regard the matrix elements of h and g as (complex) numbers and direct the attention to the associated operators. By applying the no-pair approximation we retained only particle conserving operators in the Hamiltonian. Such operators can concisely be expressed using the replacement operators Eq = a p Q and... 

Explicit inclusion of relativistic effects in valence-only calculations has been by far less frequently attempted. Datta, Ewig and van Wazer [135] used a Phillips-Kleinman PP in a study of PbO, whereas Ishikawa and Malli [136] tested PPs of semilocal form in four-component atomic DHF finite difference calculations. This work was extended by Dolg [137] to four-component molecular DHF calculations with a subsequent correlation treatment. In addition a complicated form of Vcv based on the Foldy-Wouthuysen transformation [138] was derived by Pyper [139] and applied in atomic calculations [140]. For all these approaches the computational effort is significantly higher than for the implicit treatment of relativity, and the gain of computational accuracy is not obvious at all. 

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