Dirac-Hartree-Fock calculation relativistic

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

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].

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... 

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. 

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... 

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. 

A. D. McLean and Y. S. Lee, J. Ghent. Phys., 76, 735 (1982). Relativistic Effects on R, and D, in Silver(I) Hydride and Gold(I) Hydride from All-Electron Dirac-Hartree-Fock Calculations. 

The electronic structure of the ground state of neutral atoms of Lr has been predicted to be the level of the 5f 6d 7s configuration [64]. However, recent relativistic Dirac-Hartree-Fock calculations have been made for the low-lying electronic states of Lr, and the ground state of Lr could be either the 5f 6d 7s or the 5f 7p 7s configuration [65]. The separation in the calculated energy difference between these two levels is less than the uncertainty in their calculated energies. No experimental information is available. 

Figure 3 One-particle energies of orbitals and spinors of the (n — 2)f (n - I)d ns configuration of Ce (n = 6) and Th (n = 7) from average-level nonrelativistic (nrel) Hartree-Fock and relativistic (rel) Dirac-Hartree-Fock calculations...

Quantum chemistry for lanthanides and actinides is an active area of current research. The applicable methods range from relativistically parametrized semiempirical extended Hiickel-type approaches to fully relativistic allelectron Dirac-Hartree-Fock calculations with a subsequent correlation treatment. It is emphasized that electron correlation effects and relativistic effects including spin-orbit coupling have to be treated simultaneously in order to avoid errors arising from the nonadditivity of these effects. Considerable progress is expected, especially on the ab initio side of quantum chemical applications, for small lanthanide and actinide systems during the next few years. 

A relativistic Dirac-Hartree-Fock calculation is somewhat more complicated than the corresponding nonrelativistic calculation due to the fact that each wavefunction has a large and a small component. Thus for the n electron problem there are 2n coupled equations in the relativistic calculation rather than n as in the nonrelativistic calculation. There is an even more severe complication however, produced by the fact that each nonrelativistic (nl) orbital corresponds to two relativistic orbitals (n,l,j = 1+ J)and (n,l,j =1 -J) (except of course, if 1= 0). Consequently, what is a one configuration Hartree-Fock (HF) calculation non-relativistically usually corresponds to a multi-configuration Hartree-Fock (MCHF) relativistically. What this implies is that a single configuration Hartree-Fock calculation is usually less likely to give accurate results in the relativistic case than in the nonrelativistic case. 

There are several ways to include relativity in ah initio calculations more efficiently at the expense of a bit of accuracy. One popular technique is the Dirac-Hartree-Fock technique, which includes the one-electron relativistic terms. Another option is computing energy corrections to the nonrelativistic wave function without changing that wave function. 

DHF (Dirac -Hartree-Fock) relativistic ah initio method DHF (derivative Hartree-Fock) a means for calculating nonlinear optical properties... 

Apparently, a large number of successful relativistic configuration-interaction (RCI) and multi-reference Dirac-Hartree-Fock (MRDHF) calculations [27] reported over the last two decades are supposedly based on the DBC Hamiltonian. This apparent success seems to contradict the earlier claims of the CD. As shown by Sucher [18,28], in fact the RCI and MRDHF calculations are not based on the DBC Hamiltonian, but on an approximation to a more fundamental Hamiltonian based on QED which does not suffer from the CD. At this point, let us defer further discussion until we review the many-fermion Hamiltonians derived from QED. 

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