Dirac-Hartree-Fock calculation

Figure 4.13 Excitation energies for the s-d and s-p gaps of the Group 11 elements. Experimental (Cu, Ag and Au) and coupled cluster data (Rg) are from Refs. [4, 91]. For the s-p gap of Rg we used Dirac-Hartree-Fock calculations including Breit and QED corrections.

Numbers in parentheses give the relativistic effects. See also Table 6. Dirac-Hartree-Fock calculations, Reference 61. 

The use of systematic sequences of even-tempered basis sets in mar trix Dirac-Hartree-Fock calculations for the argon atom ground state. 

Dyall, K. G., Faegri, K. and Taylor, P. R. (1991a) Polyatomic Molecular Dirac-Hartree-Fock Calculations with Gaussian Basis Sets. In Wilson et al. (1991), pp. 167-184. 

As long as one is interested only in the total energy of the atomic electron system, the change from the simple but unrealistic PNC to a roughly realistic FNC is much more important than finer details due to variation of the finite nucleus model. This can be seen also from a recently published comparative study on numerical Dirac-Hartree-Fock calculations for... 

Spectroscopic constants for TlAt, Tl(117) (113)At and (113)(117) from Dirac-Hartree-Fock calculations using both relativistic and non-relativistic basis sets. Re - bond distance, k - force constant, v - vibrational frequency. 

Figure 5. Relativistic effects on bond lengths and binding energies of group 4 tctrahydrides XH. The bond length contraction (in A) and bond destabilization (in eV) were obtained as the difference between relativistic Dirac-Hartree-Fock calculations based on the Dirac-Coulomb-Gaunt Hamiltonian and corresponding nonrelativistic Hartree-Fock calculations [28,29].

Figure 6. Influence of relativistic corrections to the electron-electron interaction on the bond length contraction and bond destabilization of the group 4 tetrahydrides XH. The percentage of results obtained with the Dirac-Coulomb-Gaunt (DCG) Hamiltonian wrt. those obtained with the Dirac-Coulomb (DC) Hamiltonian has been derived from Dirac-Hartree-Fock calculations [28,29].

Due to the energy-dependence of the Hamiltonian the Wood-Boring approach leads to nonorthogonal orbitals and has been mainly used in atomic finite difference calculations as an alternative to the more involved Dirac-Hartree-Fock calculations. The relation... 

In the most recent version of the energy-consistent pseudopotential approach the reference data is derived from finite-dilference all-electron multi-configuration Dirac-Hartree-Fock calculations based on the Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian. As an example the first parametrization of such a potential,... 

Figure 13. Valence spinors of the Db atom in the 6d 7s ground state configuration from average-level all-electron (AE, dashed lines) multiconfiguration Dirac-Hartree-Fock calculations and corresponding valence-only calculations using a relativistic energy-consistent 13-valence-electron pseudopotential (PP, solid lines). A logarithmic scale for the distance r from the (point) nucleus is us in order to resolve the nodal structure of the all-electron spinors. The innermost parts have been truncated.

In 1992 Dmitriev, Khait, Kozlov, Labzowsky, Mitrushenkov, Shtoff and Titov [151] used shape consistent relativistic effective core potentials (RECP) to compute the spin-dependent parity violating contribution to the effective spin-rotation Hamiltonian of the diatomic molecules PbF and HgF. Their procedure involved five steps (see also [32]) i) an atomic Dirac-Hartree-Fock calculation for the metal cation in order to obtain the valence orbitals of Pb and Hg, ii) a construction of the shape consistent RECP, which is divided in a electron spin-independent part (ARECP) and an effective spin-orbit potential (ESOP), iii) a molecular SCF calculation with the ARECP in the minimal basis set consisting of the valence pseudoorbitals of the metal atom as well as the core and valence orbitals of the fluorine atom in order to obtain the lowest and the lowest H molecular state, iv) a diagonalisation of the total molecular Hamiltonian, which... 

The expression for the lowest order contribution to the parity violating potential within the Dirac Hartree-Fock framework is identical to that within the relativistically parameterised extended Hiickel approach in eq. (146). The difference is, however, that in DHF typically atomic basis sets with fixed radial functions are employed (see [161]) and that the molecular orbital coefficients are obtained in a self-consistent Dirac Hartree-Fock procedure. Computations of parity violating potentials along these lines have occasionally been called fully relativistic, although this term is rather unfortunate. In the four-component Dirac Hartree-Fock calculations by Quiney, Skaane and Grant [155] as well as in those by Schwerdtfeger, Laerdahl and coworkers [65,156,162,163] the Dirac-Coulomb operator has been employed, which is for systems with n electrons given by... 

Multi-configuration Dirac-Hartree-Fock calculations... 

The previous section outlined the development of correlation consistent basis sets involving mostly light, p-block elements. In the extension of these ideas to heavier elements, the effects of relafivify on fhe basis set should be introduced. In addition to relativistic effects, the influence of low-lying electronic states must also be considered in the cases of fhe fransifion metals. For most cases only scalar relativistic effects will be considered, i.e., even if spin-orbit coupling is included in the calculations, each i component will be described by the same contracted basis set. The exceptions are the correlation consistent basis sets of Dyall [29-33], which were developed in fully-relafivisfic, 4-component Dirac-Hartree-Fock calculations. These basis sets, which are of DZ-QZ quality, are currently available for the heavier p-block elements, as well as the 4d and 5d transition metals. 

Table 5 Snapshot of the percentage of integrals calculated in a Dirac-Hartree-Fock calculation of Ge(cp)2. The labels LL, LS and SS refer to the blocks of the Fock matrix which receive contributions from the specified classes of integral.

The chemistry of superheavy elements has been theoretically investigated already during the years 1970-1971 by B. Fricke and W. Greiner [15]. Utilizing Dirac-Hartree-Fock calculations the Periodic System obtained is shown in Figure 8.45. 

FIGURE 8.45 The chemical periodic system according tho the Dirac-Hartree-Fock calculations by Fricke and Greiner (1970). The different orbitals and their occupation are depicted by shading. Thus, element 112 is eka-mercury and element 114 is eka-lead. 

Orbital energies e (a.u.) and radial expectation values (r) (a,u.) for the valence shells of Ce and Lu from multi-conflguration Dirac-Hartree-Fock calculations for the average of the 4f 5d 6s and 4f 5d 6s configurations, respectively. The ratio of relativistic and corresponding nonrelativistic values is given in parentheses, Data taken... 

Numerical discretization methods pose an interesting consequence for fully numerical Dirac-Hartree-Fock calculations. These grid-based methods are designed to directly calculate only those radial functions on a given set of mesh points that occupy the Slater determinant. It is, however, not possible to directly obtain any excess radial functions that are needed to generate new CSFs as excitations from the Dirac-Hartree-Fock Slater determinant. Hence, one cannot directly start to improve the Dirac-Hartree-Fock results by methods which capture electron correlation effects based on excitations that start from a single Slater determinant as reference function. This is very different from basis-set expansion techniques to be discussed for molecules in the next chapter. The introduction of a one-particle basis set provides so-called virtual spinors automatically in a Dirac-Hartree-Fock-Roothaan calculation, which are not produced by the direct and fully numerical grid-based approaches. 

Each atomic spinor tp r) = tp r,RA) has its center at the position of the nucleus Ra of some atom A. In a first step, we include only those atomic spinors (r) which would be considered in an atomic Dirac-Hartree-Fock calculation on every atom of the molecule. Of course, if a given atom occurs more than once in the molecule, a set of atomic spinors of this atom is to be placed at every position where a nucleus of this t) e of atom occurs in the molecule. The number of basis spinors m is then smallest for such a minimal basis set. In this case, it can be calculated as the number of shells s per atom times the degeneracy d of these shells times the number of atoms M in the molecule, m = s A) x d s) x M. 

Figure 16.1 Relativistic Dirac-Hartree-Fock calculation for Kr-like Rn of Figure 9.3 compared to a nonrelativistic calculation where the speed of light has been set to c = 10. Depicted are the large-component radial functions Pmcif) only. While the small-component radial function Is negligible in the nonrelativistic calculation, it is of non-negligible size in the relativistic calculation, which is important for considerations based on the electronic density where the small component contributes. Note the relativistic contraction of the Is and 4s shells and the negligible effect on the 3d valence shell.

J. B. Mann, J. T. Waber. Self-consistent Relativistic Dirac-Hartree-Fock Calculations of Lanthanide Atoms. Atomic Data, 5(2) (1973) 201-229. 

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