Relativistic effective core potential generalized

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. 

Titov, A.V., Mosyagin, N.S. Generalized relativistic effective core potentials Theoretical grounds. Int. J. Quant. Chem. 71, 359-401 (1999)... 

A. V. Titov and N. S. Mosyagin, The generalized relativistic effective core potential method Theory and calculations, Russ. J. Phys. Chem., 74, S376-S387 (2(X)0). 

Fig. 6. Average relativistic effective core potential and relativistic effective core potential energy curves for two states of Bi2. HF, Hartree-Fock GVB(pp), eight-configuration perfect-pairing generalized valence bond FVCI, full-valence Cl based on the GVB(pp) wave functions FV7R, full-valence Cl plus single and double promotions to virtual MOs relative to seven-dominant configurations. (The FVCI and FV7R calculations include the REP-based spin-orbit operator.)...

Finally, transition metals are heavy atoms and one should consider whether relativistic effects need to be considered [45]. In general, apart from using a relativistic effective core potential, relativity is not included for first and second-row metals but must be treated for third-row species. The relativistic effect can be broken down into three components—the Darwin correction (mass-velocity), the Zitterbewegung and spin-orbit coupling. The former two are more or less straightforward to implement and capture most of the relativistic energy. 

The relativistic calculations on the electronic structure of actinide compounds were reviewed by Pyykko (1987). He also reviewed relativistic quantum chemistry in 1988, whereas the relativistic calculations were limited to small molecules containing one heavy atom only (Pyykko 1988). Calculations on the uranyl and neptunyl ions were introduced in the review article. The general information on the computational chemistry of heavy elements and relativistic calculation techniques appear in the book written by Balasubramanian (1997). There are several first-principle approaches to the electronic structure of actinide compounds. The relativistic effective core potential (ECP) and relativistic density functional methods are widely used for complex systems containing actinide elements. Pepper and Bursten (1991) reviewed relativistic quantum chemistry, while Schreckenbach et al. (1999) reviewed density functional calculations on actinide compounds in which theoretical background and application to actinide compounds were described in detail. The Encyclopedia of computational chemistry also contains examples including lanthanide and actinide elements (Schleyer et al. 1998). The various methods for the computational approach to the chemistry of transuranium elements are briefly described and summarized below. 

In this chapter we tried to point out certain general observations and our personal preferences for choosing computational tools that represent a good compromise between accuracy and computational cost. When dealing with the heavy elements, relativistic corrections are imperative and, to a good approximation one can include them via small-core relativistic effective core potentials. All electron calculations with relativistic Hamiltonians are becoming more doable for large molecular systems, but with the exception of certain problems they do not necessarily afford an increase in accuracy worth the extra cost. Density functional theory has been proven to be a useful tool to probe the electronic structures and... 

A remark should be made here with respect to the generation and adjustment of the widely used effective core potentials (ECP, or pseudopotentials) [85] in standard non-relativistic quantum chemical calculations for atoms and molecules. The ECP, which is an effective one-electron operator, allows one to avoid the explicit treatment of the atomic cores (valence-only calculations) and, more important in the present context, to include easily the major scalar relativistic effects in a formally non-relativistic approach. In general, the parameters entering the expression for the ECP are adjusted to data obtained from numerical atomic reference calculations. For heavy and superheavy elements, these reference calculations should be performed not with the PNC, but with a finite nucleus model instead [86]. The reader is referred to e.g. [87-89], where the two-parameter Fermi-type model was used in the adjustment of energy-conserving pseudopotentials. 

In recent years, there has been an increasing interest in the inclusion of relativistic effects for molecules containing heavy atoms. One of the most practical yet reliable methods is to use relativistically derived effective core potentials. Major relativistic effects such as the Darwin and mass-velocity effects are easily taken into account in the form of a spin-free (SF) one-electron operator. The spin-orbit (SO) interaction is in general too strong to be considered as a small perturbation, and therefore should be treated explicitly as a part of the total Hamiltonian. 

In molecular quantum chemistry Gaussian-fxmction-based computations, effective core potentials were originally derived from a reference calculation of a single atom within the nonrelativistic Hartree-Fock or relativistic Dirac-Fock (see Sect. 8.3) approximations, or from some method including electron correlations (Cl, for instance). A review of these methods, as well as a general theory of ECPs is provided in [480,481]. 

To formalize these concepts, we first develop the frozen-core approximation, and then examine pseudoorbitals or pseudospinors and pseudopotentials. From this foundation we develop the theory for effective core potentials and ab initio model potentials. The fundamental theory is the same whether the Hamiltonian is relativistic or nonrela-tivistic the form of the one- and two-electron operators is not critical. Likewise with the orbitals the development here will be given in terms of spinors, which may be either nonrelativistic spin-orbitals or relativistic 2-spinors derived from a Foldy-Wouthuysen transformation. We will use the terminology pseudospinors because we wish to be as general as possible, but wherever this term is used, the term pseudoorbitals can be substituted. As in previous chapters, the indices p, q, r, and s will be used for any spinor, t, u, v, and w will be used for valence spinors, and a, b, c, and d will be used for virtual spinors. For core spinors we will use k, I, m, and n, and reserve i and j for electron indices and other summation indices. 

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