Generalized relativistic effective core

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... 

The electron density i/ (0)p at the nucleus primarily originates from the ability of s-electrons to penetrate the nucleus. The core-shell Is and 2s electrons make by far the major contributions. Valence orbitals of p-, d-, or/-character, in contrast, have nodes at r = 0 and cannot contribute to iA(0)p except for minor relativistic contributions of p-electrons. Nevertheless, the isomer shift is found to depend on various chemical parameters, of which the oxidation state as given by the number of valence electrons in p-, or d-, or /-orbitals of the Mossbauer atom is most important. In general, the effect is explained by the contraction of inner 5-orbitals due to shielding of the nuclear potential by the electron charge in the valence shell. In addition to this indirect effect, a direct contribution to the isomer shift arises from valence 5-orbitals due to their participation in the formation of molecular orbitals (MOs). It will be shown in Chap. 5 that the latter issue plays a decisive role. In the following section, an overview of experimental observations will be presented. 

This chapter introduces the core concepts of what is now called classical physics (mechanics, electricity, magnetism, and properties of waves). Today we think of classical physics as a special case in a more general framework which would include relativistic effects (for particles with velocities which approach the speed of light) and quantum effects, which are needed for a complete description of atomic behavior. Nonetheless, we will find that this classical perspective (with a few minor corrections) serves as an excellent starting point for understanding many atomic and molecular properties. 

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