Atoms Dirac calculations

Fig. 7.78 Linear relation of the quadmpole splitting A q = ( jl)eqQ (1 + j /3)l/2 and the isomer shift b for aurous (a) and auric (b) compounds. Also included is a correlation with the relative change in electron density at the gold nucleus, Ali/r(o)P, as derived from Dirac-Fock atomic structure calculations for several electron configurations of gold. An approximate scale of the EFG (in the principal axes system) is given on the right-hand ordinate (from [341])...

The molecular orbitals in the nonrelativistic and one-component calculations and the large component in the Dirac-Fock functions were spanned in the Cd s Ap9d)l[9slp6d basis of [63] and the H (5s 2p)/[35 l/>] set [61]. Contraction coefficients were taken from corresponding atomic SCF calculations. The basis for the small components in the Dirac-Fock calculations is derived by the MOLFDIR program from the large-component basis. The basis set superposition error is corrected by the counterpoise method [64]. The Breit interaction was found to have a very small effect and is therefore not included in the results. 

The Pauli operator of equations 2 to 5 has serious stability problems so that it should not, at least in principle, be used beyond first order perturbation theory (20). These problems are circumvented in the QR approach where the frozen core approximation (21) is used to exclude the highly relativistic core electrons from the variational treatment in molecular calculations. Thus, the core electronic density along with the respective potential are extracted from fully relativistic atomic Dirac-Slater calculations, and the core orbitals are kept frozen in subsequent molecular calculations. 

The case of hydrogen is peculiar in one respect. Experiment gives distinctly fewer terms than are specified in the term scheme of fig. 8 for = 2 only two terms are found, for n = 3 only three, and so on. The theoretical calculation shows that here (by a mathematical coincidence, so to speak) two terms sometimes coincide, the reason being that the relativity and spin corrections partly compensate each other. It is found that terms with the same inner quantum number j but different azimuthal quantum numbers I always strictly coincide, for instance, the ns and the np, term, the p. , and the d, term, and so on such pairs of terms are drawn close together in fig. 8. For the value of the terms a formula was given by Sommerfeld (1916), even before the introduction of wave mechanics the same formula is also obtained when the hydrogen atom is calculated by Dirac s relativistic (E908) 11... 

The best-known and widely-quoted tabulation of atomic Dirac-Hartree-Fock energies was published by Desclaux [11], covered elements in the range Z=1 to Z=120 using finite difference methods. A number of computer packages are available to perform MCDHF calculations [19]. Published DHF and Dirac-Fock-Slater (DFS) calculations for atoms are now too numerous to construct a comprehensive catalogue. It is, however, possible to sort the purposes for which these calculations have been performed into general classes. 

A couple of new quasi-relativistic Hamiltonians have been proposed and the methods have been implemented and tested on some one-electron atoms. The calculations show that the energies obtained with the present quasi-relativistic Hamiltonians are in fairly good agreement with the corresponding Dirac energies. The discrepancy between the quasi-relativistic and the Dirac energies scales with a Z, where Z is the the nuclear charge and a is the fine structure constant. 

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 results obtained by MBPT serve as a decisive basis for the selection of suitable orbital spaces in high-level ab initio calculations in order to account for the dominant contributions. The MBPT analysis applies equally well to four-component theory where a single configuration wave function built from Slater determinants of Dirac spinors is used as the reference wave function. In the following we will briefly outline the various theoretical approaches for relativistic atomic hfs calculations. 

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. 

The atomic potentials for the heavy atoms were calculated by Liberman and coworkers using the Dirac Hamiltonian and statistical exchange potential somewhat smaller than that suggested by Slater. (D. Liberman, J. T. Waber, and D. T. Cromer, Phys, Rev, 1965, 137, A27 R. D. Cowan, A. C. Larson, D. Liberman,. B. Mann, and J. Waber, ibid., 1966, 144, 5). 

From the atomic many-electron Hamiltonian we calculate the total electronic energy of an electronic state A according to Eq. (8.146). Then, along the lines of what has been presented in chapter 8, we obtain for the expectation value of an atomic Dirac-Coulomb Hamiltonian [351]... 

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. 

Y. Ishikawa. Atomic Dirac-Fock-Breit Self-Consistent Field Calculations. Int.. Quantum Chem., Quantum Chem. Symp., 24... 

G. L. MaUi, A. B. F. Da Silva, Y. Ishikawa. Universal Gaussian basis functions in relativistic quantum chemistry atomic Dirac-Fock-CoulQmb and Dirac-Fock-Breit calculations. Can. J. Chem., 70 (1992) 1822-1826. 

Dolg. Fully relativistic pseudopotentials for alkaline atoms Dirac-Hartree-Fock and configuration interaction calculations of alkaline monohydrides. Theor. Chim. Acta, 93 (1996) 141-156. 

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