Spin-orbit perturbation matrix elements

Table 5.11 Some Spin-Orbit and Orbital Perturbation Matrix elements...

Theory can now provide much valuable guidance and interpretive assistance to the mechanistic photochemist, and the evaluation of spin-orbit coupling matrix elements has become relatively routine. For the fairly large molecules of common interest, the level of calculation cannot be very high. In molecides composed of light atoms, the use of effective charges is, however, probably best avoided, and a case is pointed out in which its results are incorrect. It seems that the mean-field approximation is a superior way to simplify the computational effort. The use of at least a double zeta basis set with a method of wave function computation that includes electron correlation, such as CASSCF, appears to be imperative even for calculations that are meant to provide only semiquantitative results. The once-prevalent degenerate perturbation theory is now obsolete for quantitative work but will presumably remain in use for qualitative interpretations. 

In perturbed biradicals, the 5[y]S character is shared among two or all three singlet states as shown in equation (44) and the spin-orbit coupling matrix elements for each of these will then be simply proportional to the coefficient of the 5[y]S part of the singlet wave function. For a singlet wave function Sj defined in equation (44), the matrix elements of spin-orbit coupling are... 

Since J+L = J+ 17) the L-uncoupling operator is a one-electron operator, and consequently, in the single-configuration approximation, the configurations describing the two interacting states can differ by no more than one spin-orbital. The electronic part of the perturbation matrix element is then proportional to the same (7r+ l+ spin-electronic perturbation. However, owing to the presence of the J+ operator, the total matrix element of the B(.R)J+L operator between (fi — 1 and ft) is proportional to [J(J + 1) — 0(0 — 1)]1 2-... 

Heavy molecules are, in principle, more favorable for detecting spin-orbit perturbations and, in this way, locating metastable states. In the NS molecule, which is isovalent with NO, a perturbation matrix element between b4E and B2n of 8 cm-1 has allowed the 4E state to be located (Jenouvrier and Pascat, 1980). In the NSe molecule, the 4n state has been detected by its interaction... 

The zeroth-order Hamiltonian and the spin-orbit part of the perturbation are diagonal with respect to the quantum numbers K,H,P, )t,It, uc and lc. The matrix elements (— lc oc h I t °c +) °f remaining part of the... 

Restricting our discussion to the subspace spanned by the terms 6Aig and 4 Tig, the matrix element of the spin-orbit operator have been evaluated by Weissbluth [59] using the formalism pioneered by Griffith [56] and ending at the eigenvalue problem of the 18 x 18 dimension (which is partly factored— Table 34). Then the second-order perturbation theory yields the energies of the lowest multiplets as... 

Several methods exist for calculating g values. The use of crystal field wave functions and the standard second order perturbation expressions (22) gives g = 3.665, g = 2.220 and g = 2.116 in contrast to the experimentaf values (at C-band resolution) of g = 2.226 and g 2.053. One possible reason for the d screpancy if the use of jperfXirbation theory where the lowest excited state is only 5000 cm aboye the ground state and the spin-orbit coupling constant is -828 cm. A complete calculation which simultaneously diagonalizes spin orbit and crystal field matrix elements corrects for this source of error, but still gives g 3.473, g = 2.195 and g = 2.125. Clearly, covalent delocalization must also be taken into account. 

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