Operator effective hyperfine

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. 

Figure 1. Diagrammatic representation of the effective hyperfine operator up to second order. Single/double arrows represent core/valence orbitals in the expansion and the dashed lines symbolize electrostatic interactions.

Nevertheless, calculation of such properties as spin-dependent electronic densities near nuclei, hyperfine constants, P,T-parity nonconservation effects, chemical shifts etc. with the help of the two-component pseudospinors smoothed in cores is impossible. We should notice, however, that the above core properties (and the majority of other properties of practical interest which are described by the operators heavily concentrated within inner cores or on nuclei) are mainly determined by electronic densities of the valence and outer core shells near to, or on, nuclei. The valence shells can be open or easily perturbed by external fields, chemical bonding etc., whereas outer core shells are noticeably polarized (relaxed) in contrast to the inner core shells. Therefore, accurate calculation of electronic structure in the valence and outer core region is of primary interest for such properties. 

Two-step calculation of molecular properties. To evaluate one-electron core properties (hyperfine structure, P,T-odd effects etc.) employing the above restoraton schemes it is sufficient to obtain the one-particle density matrix, Dpq, after the molecular RECP calculation in the basis of pseudospinors p. At the same time, the matrix elements Wpq of a property operator W(x) should be calculated in the basis of equivalent four-component spinors p. The mean value for this operator can be then evaluated as ... 

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-... 

Practical studies of the hyperfine structure of the levels of atoms and ions reveal the importance of relativistic effects for this phenomenon. Therefore, we need the corresponding relativistic formulas as well. A relativistic hyperfine structure operator has the form... 

In this chapter we introduce and derive the effective Hamiltonian for a diatomic molecule. The effective Hamiltonian operates only within the levels (rotational, spin and hyperfine) of a single vibrational level of the particular electronic state of interest. It is derived from the Ml Hamiltonian described in the previous chapters by absorbing the effects of off-diagonal matrix elements, which link the vibronic level of interest to other vibrational and electronic states, by a perturbation procedure. It has the same eigenvalues as the Ml Hamiltonian, at least to within some prescribed accuracy. 

It is well known from the Bom-Oppenheimer separation [1] that the pattern of energy levels for a typical diatomic molecule consists first of widely separated electronic states (A eiec 20000 cm-1). Each of these states then supports a set of more closely spaced vibrational levels (AEvib 1000 cm-1). Each of these vibrational levels in turn is spanned by closely spaced rotational levels ( A Emt 1 cm-1) and, in the case of open shell molecules, by fine and hyperfine states (A Efs 100 cm-1 and AEhts 0.01 cm-1). The objective is to construct an effective Hamiltonian which is capable of describing the detailed energy levels of the molecule in a single vibrational level of a particular electronic state. It is usual to derive this Hamiltonian in two stages because of the different nature of the electronic and nuclear coordinates. In the first step, which we describe in the present section, we derive a Hamiltonian which acts on all the vibrational states of a single electronic state. The operators thus remain explicitly dependent on the vibrational coordinate R (the intemuclear separation). In the second step, described in section 7.55, we remove the effects of terms in this intermediate Hamiltonian which couple different vibrational levels. The result is an effective Hamiltonian for each vibronic state. 

The first-order contribution of these hyperfine interactions to the effective electronic Hamiltonian involves the diagonal matrix elements of the individual operator terms over the electronic wave function, see equation (7.43). As before, we factorise out those terms which involve the electronic spin and spatial coordinates. For example, for the Fermi contact term we need to evaluate matrix elements of the type ... 

The nuclear hyperfine operators therefore have essentially the same form in the effective Hamiltonian as they do in the full Hamiltonian, certainly as far as the nuclear spin terms are concerned. Throughout our derivation, we have assumed that the electronic state r/, A) which is to be described by our effective Hamiltonian has a well-defined spin angular momentum S. It is therefore desirable to write the effective Hamiltonian in terms of the associated operator S rather than the individual spin angular momenta s,. We introduce the projection operators (P] for each electron i,... 

Table 11.7. Matrix representation of the fine and hyperfine effective Hamiltonian (11.79) operating within the IG, /, G, N, F, M) basis set...

Effective Hamiltonians and effective operators are used to provide a theoretical justification and, when necessary, corrections to the semi-empirical Hamiltonians and operators of many fields. In such applications, Hq may, but does not necessarily, correspond to a well defined model. For example. Freed and co-workers utilize ab initio DPT and QDPT calculations to study some semi-empirical theories of chemical bonding [27-29] and the Slater-Condon parameters of atomic physics [30]. Lindgren and his school employ a special case of DPT to analyze atomic hyperfine interaction model operators [31]. Ellis and Osnes [32] review the extensive body of work on the derivation of the nuclear shell model. Applications to other problems of nuclear physics, to solid state, and to statistical physics are given in reviews by Brandow [33, 34], while... 

A fundamental aspect of semi-empirical chemical bonding theories is their requirement that the model operators be state independent [56]. This property is, of course, not required of effective operators if only the numerical values are desired for the matrix elements of operators. Indeed, some semi-empirical theories, used in other areas of physics, do not impose the requirement of state independence. For instance, LS-dependent parameters are employed in describing the hyperfine coupling of two-electron atoms [31]. However, whenever effective operators themselves are the quantities of interest, as when studying semi-empirical theories of chemical bonding, state independence of effective operators becomes a necessity. This paper thus examines conditions leading to the generation of state-independent effective operators. 

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