Off-Diagonal Spin-Orbit Parameters

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 result may be extended to the case of a molecular orbital expanded in terms of many atomic orbitals similarly to that for diagonal spin-orbit constants 

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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 ... 

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.)...

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. 

Here, grs is a parameter that is quantified either from experimental data, or is calculated by an ab initio method as one-half of the singlet-triplet excitation energy gap of the r—s bond. In terms of the qualitative theory in Chapter 3, grs is therefore identical to the key quantity —2(3 5 - This empirical quantity incorporates the effect of the ionic components of the bond, albeit in an implicit way. (c) The Hamiltonian matrix element between two determinants differing by one spin permutation between orbitals r and s is equal to grs. Only close neighbor grs elements are taken into account all other off-diagonal matrix elements are set to zero. An example of a Hamiltonian matrix is illustrated in Scheme 8.1 for 1,3-butadiene. 

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. 

Effantin, et al., (1982), Amiot and Islami (1986), and Wada and Kanamori (1999), in their analysis of CO a3n a 3E+ perturbations, found it necessary to include in their Heff fit model, in addition to the spin-orbit ( o) and L-uncoupling (/ o) off-diagonal interaction parameters, a direct spin-spin interaction parameter (e) as well as a second-order Hso x HROT interaction term (p3). The p3 parameter is not a centrifugal correction to either o or (3o. It is a cross term between the implicit iZ-dependence of the spin-orbit coupling constant of the c3n state and the explicit iZ-dependence of the B(R)J-h perturbation operator. 

The electrostatic and spin-orbit interactions represent the major effects in the parametric free-ion Hamiltonian. Nevertheless the combined diagonalization of these matrices and fitting of the F and parameters sometimes results in calculated energy levels which are off by 100 cm or more from the experimental values. These deviations result from the neglect of configuration interaction. 

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