Spin-rotational Hamiltonian

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. 

The first two-step calculations of the P,T-odd spin-rotational Hamiltonian parameters were performed for the PbF radical about 20 years ago [26, 27], with a semiempirical accounting for the spin-orbit interaction. Before, only nonrelativistic SCF calculation of the TIF molecule using the relativistic scaling was carried out [86, 87] here the P,T-odd values were underestimated by almost a factor of three as compared to the later relativistic Dirac-Fock calculations. The latter were first performed only in 1997 by Laerdahl et al. [88] and by Parpia [89]. The next two-step calculation, for PbF and HgF molecules [90], was carried out with the spin-orbit RECP part taken into account using the method suggested in [91]. 

E are the projections of the electron orbital angular momentum on the molecular axis and the subscript is the projection of the total electron angular momentum on the molecular axis directed from the heavy atom to fluorine. It is convenient to describe the spin-rotational spectrum of the ground electronic state in terms of the effective spin-rotational Hamiltonian following [90, 117] ... 

In [90] the conclusion was made, that the accuracy in calculations of the parameters of the effective spin-rotational Hamiltonian is close to 20%. However, further ab initio calculations showed the situation is more complicated. 

In a case (a) basis the component of the electron spin along the intemuclear axis, E, is a specified quantum number. The matrix elements of the spin spin and spin-rotation terms in the effective Hamiltonian were given in chapter 9 in our discussion of the spectrum of SeO. The rotational and spin-rotation Hamiltonians in the case (a) basis are... 

The effective form for the spin-rotation Hamiltonian is given by Brown and Watson (1977) as... 

The effective spin-rotational Hamiltonian ifgr reads... 

In 1992 Dmitriev, Khait, Kozlov, Labzowsky, Mitrushenkov, Shtoff and Titov [151] used shape consistent relativistic effective core potentials (RECP) to compute the spin-dependent parity violating contribution to the effective spin-rotation Hamiltonian of the diatomic molecules PbF and HgF. Their procedure involved five steps (see also [32]) i) an atomic Dirac-Hartree-Fock calculation for the metal cation in order to obtain the valence orbitals of Pb and Hg, ii) a construction of the shape consistent RECP, which is divided in a electron spin-independent part (ARECP) and an effective spin-orbit potential (ESOP), iii) a molecular SCF calculation with the ARECP in the minimal basis set consisting of the valence pseudoorbitals of the metal atom as well as the core and valence orbitals of the fluorine atom in order to obtain the lowest and the lowest H molecular state, iv) a diagonalisation of the total molecular Hamiltonian, which... 

Y. Dmitriev, Y. Khait, M. Kozlov, L. Labzowsky, A. Mitrushenkov, A. Shtoff, A. Titov, Calculation of the spin-rotational Hamiltonian including P- and P, T-odd weak interaction terms for HgF and PbF molecules, Phys. Lett. A 167 (1992) 280-286. 

A. Titov, N. Mosyagin, V. Ezhov, P, T-odd spin-rotational Hamiltonian for YbF molecule, Phys. Rev. Lett. 77 (1996) 5346-5349. 

The effective spin-rotation Hamiltonian for asymmetric top molecules has been discussed by several authors [51 Van, 61 Cur, 79Bro]. In general it takes the form... 

In quoting the results of fits of experimental data to a Hamiltonian, it is important to specify the representation employed (usualla F for a molecule nearer the prolate symmetric top limit and IIF or III for a molecule near the oblate limit) and the reduction(s) used in the centrifugal distortion and spin-rotation Hamiltonians (A or S in the present compilation). 

The variety of coupling energies in open-shell molecules is such that the analysis of LMR spectra is accomplished by numerical fitting of the observed transitions to the eigenvalues of a model hamiltonian, with many more parameters than for equivalent closed shell singlet states. Nevertheless, the availability of a large number of laser lines both in the mid infrared and in the far infrared has permitted the full parameterisation of the spin/rotational hamiltonian in many cases. For example, for NH2 there are 3 rotational constants, 5 centrifugal... 

NH2 Radical. The NH2 radical Is an asymmetric top with the asymmetry parameter k = (2 B-A-C)/(A-C)= -0.38 (axes b C2, c molecular plane). An increase of the rotational quantum number N leads to a change from prolate- to oblate-top behavior. The rotational constants A, B, and C, the centrifugal distortion constants Ak, A k, A, 5k, and 5, and the spin-rotational coupling constants Ag, Bg, and Cg, for the vibrational ground and excited states are listed in Table 10, p. 182. The rotational Hamiltonian used for fitting the spectroscopic data is a combination of the A-reduced asymmetric rotor Hrot [1] and the spin-rotation Hamiltonian figR [2] ... 

Spin-rotation coupling constants " ND2 and " NHD were calculated [28] using a reduced spin-rotation Hamiltonian [2]. Rotational, centrifugal distortion, and spin-rotation coupling constants of " ND2 and " NHD in the (0,0,0) state which were derived from an ab initio-calculated general valence force field of 4th order [22] agree well with experimentally derived constants. 

Equation (5.3) is not valid when Ag is large [23, 24] and it has been shown in the literature how Eq.(5.3) can be understood in terms of the adiabatic rotation of effective spin orientations (ARES) [23, 24]. The authors show that the term (Ag Ag) in Eq. (5.3) should be replaced by the y tensor (which relates L -i- S to with representing the effective spin angular momentum, L and S are the vector operators of the orbital and spin angular momentum respectively). The y tensor therefore is a measure of how effectively S follows the molecular axis. Only in the weak spin-orbit coupling case (i.e. where the effective spin follows the rotation of the molecular axis weakly) does the traditional spin-rotation Hamiltonian follow the ARES Hamiltonian. (i.e. Eq.(5.3) is obtained.)... 

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