Spin-orbit interaction matrix elements

In the lowest optieally excited state of the molecule, we have one eleetron (ti ) and one hole (/i ), each with spin 1/2 which couple through the Coulomb interaetion and can either form a singlet 5 state (5 = 0), or a triplet T state (S = 1). Since the electric dipole matrix element for optical transitions — ep A)/(me) does not depend on spin, there is a strong spin seleetion rule (AS = 0) for optical electric dipole transitions. This strong spin seleetion rule arises from the very weak spin-orbit interaction for carbon. Thus, to turn on electric dipole transitions, appropriate odd-parity vibrational modes must be admixed with the initial and (or) final electronic states, so that the w eak absorption below 2.5 eV involves optical transitions between appropriate vibronic levels. These vibronic levels are energetically favored by virtue... 

Both the Slater and the rrkm treatments are inappropriate for calculations of °°, since the dissociation is not characterized by a critical extension of one bond, but rather by the transition from one potential surface to another. In such a case the observed activation energy at high pressures will be lower than the energy threshold for reaction110. From their high-pressure data Olschewski et a/.109 calculate that E0 = 63 kcaLmole-1 and that the transition matrix-element is 100 caLmole-1, which is in good agreement with the spin-orbit interaction term for O atoms. 

Using these coordinate values we may now evaluate the matrix elements of Eq. (14.12) by substituting for the dipole moments the dipole matrix elements between the initial and final states. This procedure yields explicitly time dependent matrix elements VAB(r). It is particularly interesting to consider the (0,0) resonances, for two reasons. First, the (0,0) resonances have no further splitting due to the spin orbit interaction and are therefore good candidates for detailed experimental study. Second, since these resonances only involve the matrix... 

For pN shells the effective Hamiltonian Heff contains two parameters F2 and 4>i, as well as the constant of spin-orbit interaction. The term with k = 0 causes a general shift of all levels, which is usually taken from experimental data in semi-empirical calculations, and can therefore be neglected. The coefficient at 01 is proportional to L(L + 1). Therefore, to find the matrix elements of the effective Hamiltonian it is enough to add the term aL(L + 1) to the matrix elements of the energy of electrostatic and spin-orbit interactions. Here a stands for the extra semi-empirical parameter. 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

Equation [172] or related expressions (Table 10) are applied extensively when evaluating the spin part of spin-orbit matrix elements, for configuration interaction (Cl) wave functions. The latter are usually provided for a single Ms component only. 

Apart from the selection rules for the electronic coupling matrix element, spin-forbidden and spin-allowed nonradiative transitions are treated completely analogously. Nonradiative transitions caused by spin-orbit interaction are mostly calculated in the basis of pure spin Born-Oppenheimer states. With respect to spin-orbit coupling, this implies a diabatic behavior, meaning that curve crossings may occur in this approach. The nuclear Schrodinger equation is first solved separately for each electronic state, and the rovibronic states are spin-orbit coupled then in a second step. 

Rates for nonradiative spin-forbidden transitions depend on the electronic spin-orbit interaction matrix element as well as on the overlap between the vibrational wave functions of the molecule. Close to intersections between potential energy surfaces of different space or spin symmetries, the overlap requirement is mostly fulfilled, and the intersystem crossing is effective. Interaction with vibrationally unbound states may lead to predissociation. 

We now turn our attention to the second-order contributions. In order to see how these are derived, let us consider in particular the contributions of the spin-orbit interaction, 3u o. Before we can use second-order perturbation theory to evaluate these contributions, we need to write down the general matrix elements of this operator. We can do this easily if we write the expression in equation (7.72) in the simplified form... 

Energies of spin-orbital levels by tensor operators The matrix elements of spin-orbit interaction in an lN configuration are shown to be... 

With the basis functions of Table 8.46 and the one-electron matrix elements of equation (8.103), it is a simple matter to calculate the crystal field energies. We present in Table 8.48 the complete crystal-field and spin-orbit energy matrices of f1 in D3h the splitting of the 2F of f1 configuration under spin-orbit interaction and further in D3h crystalline field is shown in Fig. 8.44. 

However, numerical estimates of the effect of frequency dependence of the coefficients of depolarization based on the calculation of molecular cross sections Gj(na>, Aw) are rather difficult. But it can be shown that in the cases when different Gj magnitudes are nonzero by symmetry selection rules (neglecting the spin-orbital interaction), they are of the same order of magnitude, since they are determined by the same energy denominators and reduced matrix elements of the operator of the dipole moment. 

Since the operators f and f2 occur only at the level of the calculation of the spatial spin-orbit integrals over atomic orbitals, Breit-Pauli spin-orbit coupling operators and DKH spin-orbit coupling operators can be discussed on the same footing as far as their matrix elements between multi-electron wave functions are concerned. These terms constitute, by definition, the spin-orbit interaction part of the operator H+ (Hess etal. 1995). The spin-independent terms characteristic of relativistic kinematics define the scalar relativistic part of the operator, and terms with more than one cr matrix (not considered here) contribute to spin-spin coupling phenomena. 

---