Multiconfiguration Dirac—Fock calculations

Fig. 10.7. Nearest neighbour spacings in multiconfigurational Dirac-Fock calculations for the 4p spectrum of strontium (after J.-P. Connerade et al. [550]).

Fig. 11.4. The inner-shell absorption spectrum of Sm vapour. Also included are results of multiconfigurational Dirac-Fock calculations for the free atom (after B.K. Sarpal et al. [627]).

J. Bieron, I. P. Grant, C. FVoese Fischer, Multiconfiguration Dirac-Fock calculations of the hyperfine structure constants and determination of the nucleeir quadrupole moment of yttrium 90, Phys. Rev. A 58 (1998) 4401-4405. 

These elements possess well-developed multiplet structure in the soft X-ray spectra of the condensed phase, which can be compared with the spectrum of the fiee atom by performing ab initio Dirac—Fock calculations, the actual 4f occupancy in the solid can be deduced. An example is shown in figs. 4 and 5. The first point to note is that the multiplet structure of the atom survives in the solid, because of the strong localization of the 4f electrons, so that soft X-ray spectroscopy provides a useful probe of the 4f occupancy. To emphasize this point, we show, in fig. 4, a comparison between spectra of Sm vapour and Sm in the solid phase, which reveals the great similarity of structure between them. A multiconfigurational atomic structure calculation (fig. 5 Sarpal et al. 1991) demonstrates that, even in this complex situation, it is possible to deduce from ab initio atomic structure calculations what the 4f occupancy is. 

P. Indelicato. Projection operators in multiconfiguration Dirac-Fock calculations Application to the ground state of heliumlike ions. Phys. Rev. A, 51(2) (1995) 1132-1145. 

P. Indelicato, J. P. Desclaux. Multiconfiguration Dirac-Fock calculations of transition energies with QED corrections in three-electron ions. Phys. Rev. A, 42 (1990) 5139-5149. 

The experiments of Blancard et al. (1989, 1990a,b) and Connerade and Kamatak (1990) show clearly that Sm and Tm clusters imbedded in an Ar matrix (10 K) exhibit a +2 valence for very low dilutions (figs. 1-3). Two main conclusions arise from these results First, surface divalency in metallic Tm occurs only on rough surfaces where the atoms have reduced coordination numbers, as observed by Domke et al. (1986) for Tm sputtered films (100 inn thick) dq>osited onto copper substrates at 300 or 13 K. Second, the previous interpretation of Thole et al. (1985) for Sm is not consistent with the new results. Indeed, the Harttee-Fock calculation cannot explain transitions from the Sm 4f groimd state and the observed 3d-4f multiplet structure for Sm atoms and small clusters at low teperature. So, multiconfigurational Dirac-Fock calculations (unsealed ab initio calculations) have been performed to understand the 3d-4f spectra. It is also concluded that for Sm clusters the divalence is found to stabilize for a coordination of 8-10,... 

Johnson, E., Fricke, B., Keller Jr, O.L., Nestor Jr, C.W., Tucker, T.C. Ionization potentials and radii of atoms and ions of element 104 (unnilquadium) and of hafnium (2+) derived form multiconfiguration Dirac-Fock calculations. J. Chem. Phys. 93, 8041-8050 (1990)... 

Vilkas, M.J., Ishikawa, Y. and Koc, K. (1998) Quadratically convergent multiconfiguration Dirac-Fock and multireference relativistic configuration-interaction calculations for many-electron systems. Physical Review E, 58, 5096-5110. 

Of the five group-13 elements, only B and A1 have experimentally well characterized electron affinities. Lists of recommended EAs [50,51] show errors ranging from 50% to 100% for Ga, In, and T1. Very few calculations have appeared for the latter atoms. These include the multireference configuration interaction (MRCI) ofAmau etal. using pseudopotentials [52], our relativistic coupled cluster work on T1 [45], and the multiconfiguration Dirac-Fock (MCDF) computation of Wijesundera [53]. 

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). 

R(q) is momentum-dependent. Both the extended-average-level and the optimal-level multiconfiguration Dirac—Fock methods were tried. The results are illustrated in fig. 11.12. The sensitivity of electron momentum spectroscopy is shown by the fact that the former calculation is completely... 

Fig. 11.12. The experimental and theoretical branching ratios for the 1000 eV ionisation of lead to the 6ps/2 and 6pi/2 states of Pb+, plotted against recoil momentum p (Frost et al, 1986). The calculations with target-state correlations in the plane wave impulse approximation are indicated by MCDF, multiconfiguration Dirac—Fock EAL, extended average level OL, optimal level. Cl indicates ion-state configuration interaction.

Fig. 11.13. The 1000 eV noncoplanar-symmetric momentum profiles for lead (Frost et al., 1986). Curves show the plane-wave impulse approximation. The experiment is normalised at the peak of the 6p-manifold profile (a). The 14.6 eV and 18.4 eV states of the 6s manifold are indicated by (b) and (c). Spectroscopic factors are given in table 11.2. For (a), (b) and (c) respectively the Hartree—Fock calculation (broken curve) is normalised to multiconfiguration Dirac—Fock (solid curve) by factors 0.82, 0.70 and 0.64.

The emergence of Wigner statistics can be approached from the point of view of ab initio multiconfigurational Dirac-Fock theory. Although a full calculation for the 5p spectrum of Ba presents a formidable challenge and has not yet proved possible, fairly extensive calculations have proved fea-... 

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.

Atomic calculations. Most atomic calculations for the heaviest elements were performed by using Dirac-Fock (DF) and Dirac-Slater (DS) methods [20-24,58] and later by using multiconfiguration Dirac-Fock (MCDF) [64-72] and Dirac-Coulomb-Breit Coupled Cluster Single Double excitations (DCB CCSD) [73-85] methods, with the latter being presently the most accmate one. 

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