Correlated spinors

The twin facts that heavy-atom compounds like BaF, T1F, and YbF contain many electrons and that the behavior of these electrons must be treated relati-vistically introduce severe impediments to theoretical treatments, that is, to the inclusion of sufficient electron correlation in this kind of molecule. Due to this computational complexity, calculations of P,T-odd interaction constants have been carried out with relativistic matching of nonrelativistic wavefunctions (approximate relativistic spinors) [42], relativistic effective core potentials (RECP) [43, 34], or at the all-electron Dirac-Fock (DF) level [35, 44]. For example, the first calculation of P,T-odd interactions in T1F was carried out in 1980 by Hinds and Sandars [42] using approximate relativistic wavefunctions generated from nonrelativistic single particle orbitals. 

Spin-dependent operators are now introduced. The external potential can be an operator Vext acting on the two-component spinors. The exchange-correlation potential is defined as in Eq. [27], although Exc is now a functional Exc = Exc[pap] of the spin-density matrix. The exchange-correlation potential is then... 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

The basis consisted of 21sl7plld7/ Gaussian spinors [52], and the 4spd/5spd6s electrons were correlated. Table 1 shows the nonrelativistic, Dirac-Coulomb,... 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

This appendix contains tables of characters for vector and spinor representations of the point groups G that are encountered most commonly in practical applications of group theory in chemical physics. Correlation tables are given separately in Appendix A4. 

Our results suggest that the spin correlation functions decay exponentially with a correlation length 1 for an arbitrary parameter a. We also assume that the decay of the correlation function is of the exponential type for the 14 parameter model as well, i.e., for any choice of site spinor I>A/u/p. This assumption is supported in special cases 1) the partition of the system into one-dimensional chains with exactly known exponentially decaying correlation functions 2) the two-dimensional AKLT model, for which the exponential character of the decay of the correlation function has been rigorously proved [32], Further evidence of the stated assumption lies in the numerical results obtained for various values of the parameter in the one-parameter model. 

The correct nonrelativistic limit as far as the basis set is concerned is obtained for uncontracted basis sets, which obey the strict kinetic balance condition and where the same exponents are used for spinors to the same nonrelativistic angular momentum quantum number for examples, see Parpia and Mohanty (1995) and also Parpia et al. (1992a) and Laaksonen et al. (1988). The situation becomes more complicated for correlated methods, since usually many relativistic configuration state functions (CSFs) have to be used to represent the nonrelativistic CSF analogue. This has been discussed for LS and j j coupled atomic CSFs (Kim et al. 1998). 

The first order correction simply corrects the double counting of electron interactions in the zero order energy expression. The second order correction gives the correlation energy that we are interested in. For convenience we defined a shorthand notation for spinor energy differences... 

Crucial is the selection of the active spinor space that determines the model space. It should be large enough to take all important interactions between nearby states into account, but also be small enough to keep the formalism computationally tractable and numerically stable. Direct diagonalization of this model space would be equivalent to a small Cl expansion and does of course not take dynamic electron correlation into account. Correlation is included by defining the so-called wave operator U... 

If spin-orbit effects are considered in ECP calculations, additional complications for the choice of the valence basis sets arise, especially when the radial shape of the / -f-1/2- and / — 1/2-spinors differs significantly. A noticeable influence of spin-orbit interaction on the radial shape may even be present in medium-heavy elements as 53I, as it is seen from Fig. 21. In many computational schemes the orbitals used in correlated calculations are generated in scalar-relativistic calculations, spin-orbit terms being included at the Cl step [244] or even after the Cl step [245,246]. It therefore appears reasonable to determine also the basis set contraction coefficients in scalar-relativistic calculations. Table 9 probes the performance of such basis sets for the fine structure splitting of the 531 P ground state in Kramers-restricted Hartree-Fock [247] and subsequent MRCI calculations [248-250], which allow the largest flexibility of... 

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