Diagonal Spin-Orbit Parameters

A semiempirical calculation of molecular spin-orbit constants can be made, using the method of Ishiguro and Kobori (1967). 

In the simplest case where the bonding and antibonding 7r molecular orbitals can be expanded in terms of one atomic p orbital on each atom, we have 

This semiempirical approximation neglects two-center integrals such as 

Ca + CaCbSab the atom A population C + CaCbSab the atom B population 

For heteronuclear molecules, ab initio calculations of the Self-Consistent-Field (SCF) molecular orbitals are usually necessary to obtain the values of the Ck coefficients. The values of the spin-orbit constants depend on the effective charge on each atom, a positive charge causing a contraction of the atomic orbitals and an increase in C(atomic) a negative charge causes an opposite effect. In the simple case of an expansion of the molecular orbitals in terms of a large number of atomic orbitals associated with a single 1-value on each atom (Lefebvre-Brion and Moser, 1966), 

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A similar semiempirical method can be used to estimate off-diagonal spin-orbit parameters. Consider the a+ parameter, which occurs for the example in Eq. (3.4.18) ... 

This expression shows clearly that 1g depends on internuclear distance and that one-electron two-center integrals cannot be neglected in calculating matrix elements of 1q (Colbourn and Wayne, 1979). This situation is quite different from that for the off-diagonal spin-orbit parameters. The b parameters are given (Section 5.3.2) by ... 

It holds that F0 F2 > F4 and that the dominance of the spherical term shifts the levels with regard to their populations. The spin-orbit parameter Cso is comparable in size to F4 and enters with the off-diagonal elements M15 and M51 as well as in the diagonal ones with M55 and M99. 

Figure 5.16 Diagonal [A(R)] and off-diagonal spin-orbit ((AL+)) and orbital ((L+)) perturbation parameters for the OH X2n state (from Coxon and Foster, 1982) and the OD X2n A2E+ interaction. (From Coxon and Hammersley, 1975.)...

The zeroth-order Hamiltonian and the spin-orbit part of the perturbation are diagonal with respect to the quantum numbers K, E, P, Uj, It, Uc, and Ic-The terms of H involving the parameters aj, ac, and bo aie diagonal with respect to both the Ij and Ic quantum numbers, while the f>2 term connects with one another the basis functions with I j = Ij 2, 4- 2. The c terms... 

We now use the results we have obtained to calculate the energy levels in a magnetic field, determine the field values for the allowed electric dipole transitions, and compare the results with the experimental spectrum [56]. It is already clear that in the case (b) basis set we shall have to take note of the extensive mixing of different rotational levels by the AN = 2 off-diagonal matrix elements of the spin-spin interaction. In SO the spin spin parameter X is comparable with the rotational constant B0, and, as we shall see, in heavier molecules like SeO, X is so much larger than B0, because of spin orbit coupling, that a case (a) basis is more appropriate. 

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