Hyperfine operators/terms

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,... 

There are numerous interactions which are ignored by invoking the Born-Oppenheimer approximation, and these interactions can lead to terms that couple different adiabatic electronic states. The full Hamiltonian, H, for the molecule is the sum of the electronic Hamiltonian, the nuclear kinetic energy operator, Tf, the spin-orbit interaction, H, and all the remaining relativistic and hyperfine correction terms. The adiabatic Born-Oppenheimer approximation assumes that the wavefunctions of the system can be written in terms of a product of an electronic wavefunction, (r, R), a vibrational wavefunction, Xni( )> rotational wavefunction, and a spin wavefunction, Xspin- However, such a product wave-function is not an exact eigenfunction of the full Hamiltonian for the... 

The second-order effects comprise three contributions. The second-order Zee-man and hyperfine interactions involve obvious extensions of second-order Rayleigh-Schrodinger perturbation theory using the appropriate operators. If we introduce an arbitrary gauge origin Ra, and the variable tq — r — Rq, the second-order term involving both Zeeman and hyperfine operators is... 

The wave functions correct to first-order perturbation theory are just the product functions of the respective electron and nuclear spin combinations. The hyperfine interaction term involves only the z-components of the electron and nuclear spin angular momentum operators when treated by first-order perturbation theory. 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

If we expand the hyperfine term of the spin Hamiltonian and write the operators in terms of raising and lowering operators ... 

The 113Cd Ti values estimated for the various peaks varied from 10 to 50 ms and obeyed the qualitative dependence upon 1/R6 (R = Mn-Cd distance) of the dipolar relaxation mechanism expected to be operative. The broad line widths were also shown to have significant contributions from the T2 relaxation induced by Mn++, with both dipolar and contact terms contributing. The 113Cd shifts of the peaks assigned to different shells were measured as a function of temperature, and observed to follow a linear 1/T dependence characteristic of the Curie-Weiss law, with slopes proportional to the transferred hyperfine interaction constant A. 

This shows that the FC operator arises as an artifact if one wants to describe the hyperfine interaction by first-order perturbation theory in terms of two-component spinors. In other words, when singular functions are involved, the boundary conditions cannot be ignored. These give rise to the FC operator. 

Hyperfine tensors are given in parts B and C of Table II. Although only the total hyperfine interaction is determined directly from the procedure outlined above, we have found it useful to decompose the total into parts in the following approximate fashion a Fermi term is defined as the contribution from -orbitals (which is equivalent to the usual Fermi operator as c -> < ) a spin-dipolar contribution is estimated as in non-relativistic theory from the computed expectation value of 3(S r)(I r)/r and the remainder is ascribed to the "spin-orbit" contribution, i.e. to that arising from unquenched orbital angular momentum. 

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. 

In a non-relativistic approximation the usual fine structure (splitting) of the energy terms is considered as a perturbation whereas the hyperfine splitting - as an even smaller perturbation, and they both are calculated as matrix elements of the corresponding operators with respect to the zero-order wave functions. 

As usual, the operators describing hyperfine interactions are to be expressed in terms of irreducible tensors. Then we are in a position to find formulas for their matrix elements. The corresponding operator, caused... 

Even in fairly small applied magnetic fields, say B = 20 mT, the terms of He are much larger than the hyperfine terms. This implies that the expectation value of the electronic spin, (S, alSIS, a) = (S), is determined by He. Under these circumstances, we can replace the spin operator S in the magnetic hyperfine term by its expectation value, (S), obtaining from Hm the nuclear Hamiltonian Hn... 

The first term is the Zeeman interaction depending upon the g(RS OW, q ) tensor, external magnetic field B0 and electron spin momentum operator S the second term is the hyperfine interaction of the th nucleus and the unpaired electron, defined in terms hyperfine tensor A (Rsklw, qj) and nuclear spin momentum operator n. The following terms do not affect directly the magnetic properties and account for probe-solvent [tfprobe—solvent (Rsiow, qJ)l ld solvent-solvent //solvent ( qj)] interactions. An explicit... 

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. 

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