Spin-orbit coupling integrals

Tj g state yields two different four-fold degenerate Vq states which mix with Eg, and accordingly, Equation 9 yields two distinct coefficients, denoted cj and C2 At the level of first-order perturbation theory, these two states lie at +/2 and 4 2+/3 ith respect to the ground state, where is the effective spin-orbit coupling integral for the Co " 3d orbital (17). 

We introduce the dimensionless bending coordinates qr = t/XrPr anti qc = tAcPc ith Xt = (kT -r) = PrOir, Xc = sJ kcPc) = Pc nc. where cor and fOc are the harmonic frequencies for pure trans- and cis-bending vibrations, respectively. After integrating over 0, we obtain the effective Hamiltonian H = Ho + H, which is employed in the perturbative handling of the R-T effect and the spin-orbit coupling. Its zeroth-order pait is of the foim... 

Table 3. Calculated Slater—Condon Integrals, Spin—Orbit Coupling Constants, and Ligand Field Parameters (in cm-1) for CsMgBr, Eu2+ Considering the Ground (GC) and Excited (EC) Configurations Local Structures of the Eu2+ Impurity...

A further term that can contribute to E(1)yAa is the ZFS (59,60). As implied by its name, ZFS splits the components of a state in the absence of a magnetic field. For states that are only spin degenerate, ZFS occurs when the spin S>l/2. Like the g-tensor, ZFS causes the axis of spin quantization to deviate from the direction of the magnetic field. The consequences with respect to spin integration and orientational averaging are similar to those caused by the use of a non-isotropic g-tensor. ZFS is made up of two terms, one second-order in spin-orbit coupling and the other from spin-spin coupling (59). The calculation of ZFS within DFT has been the subject of several recent publications (61-65). 

The term maia a(1) is the first-order correction to the integral of the electric dipole moment operator in the a direction over orbitals a and i. The perturbed integral will depend on the change of the orbitals in the presence of a magnetic field or spin-orbit coupling. 

If the perturbation is spin—orbit coupling then a field-independent basis set is used in the calculation of interest (yv( 1)=0) and the t/1) coefficients are all that is needed to calculate the perturbed integrals. If the perturbation is a magnetic field then a field-independent basis set can be used. However, it is often desirable to utilize a field-dependent basis set such as GIAOs (66-69) that reduces the origin dependence of the results obtained. With such a basis set the evaluation of perturbed integrals is somewhat more involved as the Xv(1) terms are no longer zero. 

Evaluate required integrals over the electric and magnetic dipole moment and spin-orbit coupling operators as well as GIAO contributions, if needed. 

The expression for the contribution to the spin-orbit induced MCD intensity from perturbation of the ground state is somewhat reminiscent of an expression for the Ag quantity of EPR spectroscopy. The similarity lies in the paramagnetic term, Agp. This term is composed of integrals of a spin-orbit operator over molecular orbitals similar to the expression for the perturbation of the ground state in the presence of spin-orbit coupling (Eqs. 52-56). The paramagnetic contribution to Ag dominates for blue copper proteins and it was suspected that the MCD parameters and Amay have some sort of relationship. It was found that many of the terms that make large contributions to AgP do play a role in the MCD intensity but no simple relationship was found (160). 

Only four-index two-electron integrals contribute to the spin-orbit coupling matrix element ... 

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