Orbital hyperfine constant

As we have discussed many times elsewhere, the magnetic and electric nuclear hyperfine constants provide information about the electronic structure of the molecule. The latter was outlined and summarised briefly in (8.391), where the unpaired electron is placed in a 7r-type molecular orbital, which may be regarded as a linear combination of the N and O atomic 2p orbitals. Dousmanis [ 140] was among to first to show the relationships between the dipolar hyperfine constants and the electronic wave function. The orbital hyperfine constant, a, which in NO is found to have the value 84.20378 MHz, is given by the expression... 

Comparison with the value of the orbital hyperfine constant a therefore suggests that to and I2 shouldhave the values —16.8 and 67.4 MHz, which are in quite good agreement with the experimental values —19.6273 and 75.064 79 MHz, considering the simplicity of the model. 

The orbital hyperfine constant a is the same in both (8.509) and (8.510), but the second term in (8.510) describes only the Fermi contact interaction, the constant hp being equal to the first part of b in (8.509). It is clear from a comparison of the two equations that... 

Besides the hyperfine constants for the muonium impurity itself, one can also investigate the so-called superhyperfine interaction for the neighboring 29Si atoms. These values have also been accurately measured (Kiefl et al., 1988) with level-crossing resonance.For the anisotropic parameters, it is customary to compare b with Aj,ree, which is an average of r 3 determined for the valence p-orbital. The results are given in Table II. Both... 

In this paper, the main features of the two-step method are presented and PNC calculations are discussed, both those without accounting for correlation effects (PbF and HgF) and those in which electron correlations are taken into account by a combined method of the second-order perturbation theory (PT2) and configuration interaction (Cl), or PT2/CI [100] (for BaF and YbF), by the relativistic coupled cluster (RCC) method [101, 102] (for TIF, PbO, and HI+), and by the spin-orbit direct-CI method [103, 104, 105] (for PbO). In the ab initio calculations discussed here, the best accuracy of any current method has been attained for the hyperfine constants and P,T-odd parameters regarding the molecules containing heavy atoms. 

We note here that Eq. (2.8) holds for a single electron in an orbital which is well separated by any other excited level. In the case of multiple unpaired electrons in different molecular orbitals, Eq. (2.8) still may hold in the absence of strong spin-orbit coupling effects but the interpretation of the hyperfine constant becomes complicated the hyperfine coupling is the sum of that for each molecular orbital. Indeed, each metal orbital which contains an unpaired electron is involved in a molecular orbital and provides a contribution to the total p for the various nuclei. The experimental data, however, provides through Eq. (2.8) the sum of the A values and therefore the sum of p. In order to make the spin density or contact constants comparable for different systems independent of the value of 5, i.e. independent of the number of electrons, the value of p is normalized to one electron, i.e. it is divided by the number of electrons which is just 2S (in such a way that p, /2S =1). Eq. (2.2) becomes... 

The simplest description of the LiO molecule would be in terms of an ionic complex, Li+O, so that the orbital and dipolar hyperfine constants would be those of the O atomic ion. Given a suitable wave fimction for an atomic 2p orbital located on the oxygen atom, it is then a straightforward matter to calculate the dipolar constants... 

The axial component of the magnetic hyperfine interaction for the 2 n 3/2 component is designated //3/2 in terms of the original Frosch and Foley constants [25] h n is equal to a + (1 /2)(b + c), and in terms of our preferred hyperfine constants it is a + (l/2)(fa + 21), the latter constants describing the orbital, Fermi contact and dipolar hyperfine interactions separately. Specifically, our constants are given by,... 

A further refinement in the analysis of the case (a) spectra was described by Carrington and Howard [28] in their study of the CF radical. We have already pointed out that in CIO, for example, it is only possible to determine the total axial magnetic hyperfine constant A 3/2 from studies of the J = 3/2 level alone. In CF, however, the rotational constant Bo is relatively large and the spin-orbit constant A is relatively small. This means that the ratio B0/A is considerably larger for CF (0.0183) than for CIO (0.0022), so that the rotational mixing of the 2n3/2 and 2ni/2 states (equation (9.38)) is more important. This fact, taken with additional measurements of the J = 5/2 level in the 2 n3/2 state, enables both /i3/2 and b to be determined uniquely, though with less accuracy than one would wish. 

We will not pursue the analysis in detail any further. It does illustrate the power of spherical tensor methods, and one can only shudder at the possibility of developing the theory in a cartesian coordinate system, with direction cosines. We list the final values of the molecular parameters for 14N35C1 in table 10.12. The values of the hyperfine constants may be interpreted semi-empirically in the following way. The outmost pair of electrons occupy a 3 /rT molecular orbital and the Fermi contact constants, given in table 10.12, may be compared with the atomic values [144] of 1811 and 5723 MHz for the nitrogen and chlorine atoms respectively one concludes that the s electron character... 

Comparison of the observed value of by with the known contact interaction constant of the Mn+ ion gives a value for cf of0.573. In other words the 3da and 4s hybridisation is a nearly perfect one-to-one mixture. The dipolar hyperfine constant c depends upon a sum of contributions from the 3d unpaired electrons in the 9atomic orbitals, a value which agrees well with the measured value of —48.199 MHz. As we have commented elsewhere, the quadrupole coupling constant involves all of the electrons, and is not readily amenable to a simple semi-empirical treatment. 

As we have shown in Appendix 8.5, and elsewhere, to is the axial component of the dipolar interaction obtained from the fourth term in equation (11.2). The large value of the Fermi contact constant is consistent with a model in which the unpaired electron occupies a a-type molecular orbital which has 45% N atom, v character. Radford produced convincing arguments to show that the model is also consistent with the small dipolar hyperfine constant, and also the electric quadrupole coupling constant. 

It should be remembered that tbese constants are for the v" = 1 level. A Hartree-Fock wave ftmction for Nj has been calculated by Cade, Sales and Wahl [101], from which the spin rotation and magnetic hyperfine constants were calculated by Rosner, Gaily and Holt [102] they were in excellent agreement with experiment. The dominant molecular orbital configuration, given at the beginning of this section, is... 

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