Relativistic Hartree-Fock calculations

Relativistic calculations allow more detailed predictions of the chemical properties of transactinides compared with those of their lighter homologues. Electi onic configurations and oxidation states predicted for the transactinide elements 104 to 120 on the basis of relativistic Hartree-Fock calculations are listed in Table 14.6. An important result of these calculations is the splitting of the p levels into a pi/2 sub-level for 2 electrons and a P3/2 sublevel for 4 electrons. 

Fig. 4. Relativistic shifts for s d"-s d" excitation energies of transition-metal atoms. Differences between excitation energies from Hartree-Fock and relativistic Hartree-Fock calculations are plotted. (Reproducedfrom Ref 174 by permission of the authors and the American Institute of Physics.)...

Visscher, L., Aerts, P. J. C. and Visser, O. (1991a) General contraction in four-component relativistic Hartree-Fock calculations. In Wilson et al. (1991), pp. 197-205. 

A central feature in the chemistry of No is the dominance of the divalent oxidation state (26). In this respect, No is unique within the lanthanide and actinide series, since none of the other twenty-seven members possess a highly-stable divalent ion. The electronic configuration of the neutral atom obtained from relativistic Hartree-Fock calculations is 5f 7s 2 (j>). Clearly, the special stability of No + must arise from the difficulty in ionizing an f valence electron from the completed 5f shell. Thus, pairing of the last electron, to close the shell, results in the f electron levels taking a rather abrupt drop in energy below the Fermi surface. 

In relativistic Hartree-Fock calculations a wavefunction correct to 0(c ) yields a total energy correct to 0 c ), but orbital energies (which anyway have no rigorous physical meaning) only correct to 0 c ) [17, 18]. 

This strategy had been tried successfially in relativistic Hartree-Fock calculations. They found, and Boettger verified independently [77], that the scheme also works well with a DFT Hamiltonian, even though the theoretical underpinnings are very different fi om Hartree-Fock. 

Such a regularity was also exposed in the numerical relativistic Hartree-Fock calculations of Pyper and Grant (1978), though our arguments at this stage are purely non-relativistic. 

The relativistic form of the one-electron Schrodinger equation is the Dirac equation. One can do relativistic Hartree-Fock calculations using the Dirac equation to modify the Fock operator, giving a type of calculation called Dirac-Fock (or Dirac-Hartree-Fock). Likewise, one can use a relativistic form of the Kohn-Sham equations (15.123) to do relativistic density-functional calculations. (Relativistic Xa calculations are called Dirac-Slater or Dirac-Xa calculations.) Because of the complicated structure of the relativistic KS equations, relatively few all-electron fully relativistic KS molecular calculations that go beyond the Dirac-Slater approach have been done. [For relativistic DFT, see E. Engel and R. M. Dreizler, Topics in Current Chemistry, 181,1 (1996).]... 

Another approach is to do a nonrelativistic calculation, using, for example, the Hartree-Fock method, and then use perturbation theory to correct for relativistic effects. For perturbation-theory formulations of relativistic Hartree-Fock calculations and relativistic KS DFT calculations, see W. Kutzelnigg, E. Ottschofski, and R. Franke, J. Chem. Phys., 102,1740 (1995) and C. van Wiillen, J. Chem. Phys., 103,3589 (1995) 105,5485 (1996). 

While accurate, direct calculation of f-state energy structure is not generally feasible for lanthanide and actinide ions, parametric models for f-state energy-level structure determination have been developed that take advantage of relationships established by relativistie Hartree-Fock calculations (Crosswhite and Crosswhite 1984). Works by Cowan (1981) and Szasz (1992) should be consulted for additional information on theoretical atomic spectroscopy and relativistic Hartree-Fock calculations. 

Fig. 1. Charge densities P r) [J P(r) dr = 1] of the 4f5,2, 5si relativistic Hartree-Fock calculations for the 4f 5d 6s configuration of atomic samarium.

Mann, J.B., Waber, J.T. SCF relativistic Hartree-Fock calculations on the superheavy elements 118-131. J. Chem. Phys. 53, 2397-2406 (1970)... 

Fig. 15.9 Relativistic Hartree-Fock calculations of some actinide radial integrals (a) f (//) (i>) (c) G (/

Figure 3.4 Orbital radial expectation values (in ao) of (left) the 4f elements cerium through lutetium and (right) the 5f elements thorium through lawrencium from 4-component relativistic Hartree-Fock calculations averaging over the (n - 2)f (/= 1,14) valence configuration of the +3 cations...

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