Effects in heavy-atom molecules

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. 

Study of P- and T-parity nonconservation effects in heavy-atom molecules Historical background... 

Whilst this demonstrates that calculations using the methods of this paper may prove very useful in studies of molecules containing only low-Z atoms, a major objective has been to study systems containing heavier atoms. So far, only a limited number of molecular calculations have been carried out with BERTHA at the DHF level, mainly in connection with studies of hyperfine and PT-odd effects in heavy polar molecules such as YbF [33] and TIF [13]. The reader is referred to the literature for an assessment of these calculations and for technical details on the construction of basis sets which must not only describe molecular bonding properly but also give a good representation of spinors close to the heavy nuclei to handle the short-range electron-nuclear electroweak interactions. 

P,T-PARITY VIOLATION EFFECTS IN POLAR HEAVY-ATOM MOLECULES... 

It is also critical to have a high value of the effective electric held IF, acting on the electron. The only way to know that parameter is to perform relativistic calculations. It is notable that the first semiempirical estimates of this kind were performed by Sandars in [16, 15] for Cs and TIF, correspondingly. In these papers the importance of accounting for relativistic effects and using heavy atoms and heavy-atom molecules in EDM experiments was first understood. 

The shape-consistent (or norm-conserving ) RECP approaches are most widely employed in calculations of heavy-atom molecules though ener-gy-adjusted/consistent pseudopotentials [58] by Stuttgart team are also actively used as well as the Huzinaga-type ab initio model potentials [66]. In plane wave calculations of many-atom systems and in molecular dynamics, the separable pseudopotentials [61, 62, 63] are more popular now because they provide linear scaling of computational effort with the basis set size in contrast to the radially-local RECPs. The nonrelativistic shape-consistent effective core potential was first proposed by Durand Barthelat [71] and then a modified scheme of the pseudoorbital construction was suggested by Christiansen et al. [72] and by Hamann et al. [73]. 

Generalized RECP When core electrons of a heavy-atom molecule do not play an active role, the effective Hamiltonian with RECP can be presented in the form... 

The P,T-parity nonconservation parameters and hyperfine constants have been calculated for the particular heavy-atom molecules which are of primary interest for modern experiments to search for PNC effects. It is found that a high level of accounting for electron correlations is necessary for reliable calculation of these properties with the required accuracy. The applied two-step (GRECP/NOCR) scheme of calculation of the properties described by the operators heavily concentrated in atomic cores and on nuclei has proved to be a very efficient way to take account of these correlations with moderate efforts. The results of the two-step calculations for hyperfine constants differ by less than 10% from the corresponding exper-... 

In this simple case there is no advantage to the pseudopotential calculation (the 3-21G( ) geometry is actually better ), but more challenging calculations on very-heavy-atom molecules, particularly transition metal molecules, rely heavily on ab initio or DFT (Chapter 7) calculations with pseudopotentials. Nevertheless, ordinary nonrelativistic all-electron basis sets sometimes give good results with quite heavy atoms [64]. A concise description of pseudopotential theory and specific relativistic effects on molecules, with several references, is given by Levine [65]. Reviews oriented toward transition metal molecules [66a,b,c] and the lanthanides [66d] have appeared, as well as detailed reviews of the more technical aspects of the theory [67]. See too Section 8.3. 

Two methods are mainly responsible for the breakthrough in the application of quantum chemical methods to heavy atom molecules. One method consists of pseudopotentials, which are also called effective core potentials (ECPs). Although ECPs have been known for a long time, their application was not widespread in the theoretical community which focused more on all-electron methods. Two reviews which appeared in 1996 showed that well-defined ECPs with standard valence basis sets give results whose accuracy is hardly hampered by the replacement of the core electrons with parameterized mathematical functions" . ECPs not only significantly reduce the computer time of the calculations compared with all-electron methods, they also make it possible to treat relativistic effects in an approximate way which turned out to be sufficiently accurate for most chemical studies. Thus, ECPs are a very powerful and effective method to handle both theoretical problems which are posed by heavy atoms, i.e. the large number of electrons and relativistic effects. 

Because this chapter is a follow-up of previous work in the field it is not necessary to repeat the basics of ab initio methods. This has been done in detail by Basch and Hoz, who also discuss the most important atomic properties of Ge, Sn and Pb. We also recommend the theoretical section in the chapter by Apeloig about organosilicon compounds in this series who gave an excellent overview about the most important aspects of ab initio, semiempirical and force-field methods. The reader will find there an explanation of the most common standard methods which will be mentioned in this review without further explanation. We will focus in the following on those theoretical and computational aspects of methods which are particularly important for heavy-atom molecules that have been advanced in the last decade, i.e. ECPs and DFT. We also briefly discuss relativistic effects. We point out that semiempirical methods" and force field parameters are available for the elements Ge, Sn and Pb. However, the application of the two methods has not gained much popularity and not many papers have been published in the field. Most reports are restricted to special problems. ... 

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