The Spin-Orbit Operator

The form of the spin-orbit Hamiltonian has been given by Van Vleck (1951) and is an extension to diatomic molecules of the solution for the relativistic equation originally derived for a two-electron atom  

The first part of the Eq. (3.4.2) form of Hso represents the spin-orbit coupling of each electron in the field of the two bare nuclei with charges ZA and Zb- It is a single-electron operator. 

The second part, the spin-other-orbit interaction, is due to interelectronic interactions and has the effect of partially counterbalancing the field of the bare nuclei. Its sign is opposite to that of the first part. This operator is a two-electron operator because of the r term. 

It can be shown (Veseth, 1970) that all electron-nuclear distances, r ) can be referred to a common origin, and, neglecting only the contribution of spin-other-orbit interactions between unpaired electrons, the two-electron part of the spin-orbit Hamiltonian can be incorporated into the first one-electron part as a screening effect. The spin-orbit Hamiltonian of Eq. (3.4.2) can then be written as 

In the form of Eq. (3.4.3), Hso is a single-electron operator. It is prudent to avoid discussion of further simplified effective forms of this operator, such as AL-S, because these forms of Hso have led to many errors in the literature. The selection rules for the microscopic Hso operator are based on its symmetry with respect to the operations of the point group of the diatomic molecule, Coou(heteronuclear) or T oo/i (homonuclear). The a factor of Hso is invariant under all symmetry operations. The lj-Sj term may be expanded, 

---

We shall consider only the leading part of the spin-orbit operator assumed in the phenomenological fonn... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

The labelling of terms as S,L,J,Mj) is preferable when one takes into account the effect of spin-orbit coupling, since / and Mj remain good quantum numbers even after this perturbation is accounted for. In detail, the effect of spin-orbit coupling over a many-electron atomic term is evaluated by writing the spin-orbit operator in terms of the total angular and spin momentum, L and 5 ... 

At this stage we will take the spin-orbit operator to have the general form... 

Here the components of excited state J are expressed in a representation that diagonalizes the spin-orbit operator. In general, this will be a complex representation. The principle of spectroscopic stability can again be used to express the components of Jin a representation that we denote jM. This representation is made up of space and spin parts where the spin part diagonalizes the spin operator. 

Spin-orbit operator and the part of the spin-orbit operator that acts on the spatial part of the wave function. 

The spin-orbit operator LS given in Eq. (67) is expressed in terms of the individual electron-orbital and spin-momentum operators rather than the total momentum operators L and S. It can be shown (/, 5) that when evaluating integrals involving only LS functions of the same configuration, ls can be replaced by... 

An example of an operator which does not commute with SjijF is the spin-orbit operator ns0. For instance, for a two-electron atom... 

The spin-orbit does not interact for the A-term or E-term manifolds as these kets do not involve the angular momentum (in the cubic groups). Consequently all the g-limes degenerate energy levels within the model space possess zero matrix elements of the spin-orbit operator. 

The scalar product (L S) occurring in the spin-orbit operator can be manipulated with the help of the escalator operators... 

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... 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

For a matrix element of the spin-orbit operator to be different from zero, it is in principle sufficient that the direct products of the irreps of space and spin functions on both sides of the matrix element be equal. Thus,... 

It is the change of sign of individual integrals due to spin integration that lets the matrix element survive. In other cases, where s+i or s i are involved, spin flips of individual orbitals occur in addition. Because of these individual spin flips, the spin-orbit operator can alter not only the Mg quantum number by AMs = 0, 1, but also may change the spin quantum number S by at most one unit (i.e., AS = 0, 1). 

The spatial parts of the a, b XA states can couple to X 3Bi via the y component of the spin-orbit operator. The operator Sy couples the singlet spin function So (Ai) to the Bi triplet function. 

To determine the first-order spin-orbit splitting pattern of an orbitally degenerate electronic state, we shall make use of the energy expression obtained from the phenomenological operator, which in this case reduces to Aso A S because only the z component of the spin-orbit operator is involved. 

For compounds containing heavy atoms, spin-orbit and electron correlation energies are approximately of the same size, and one cannot expect these effects to be independent of each other. A variational approach that treats both interactions at the same level is then preferable to a perturbation expansion. Special care is advisable in the choice of the spin-orbit operator in this case. The variational determination of spin-orbit coupling requires a spin-orbit... 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

It appears, therefore, that it is possible to obtain accurate expectation values of the spin-orbit operators for diatomic molecules. Matcha et a/.112-115 have provided general expressions for the integrals involved and from their work Hall, Walker, and Richards116 derived the diagonal one-centre matrix elements of the spin-other-orbit operator for linear molecules. Provided good Hartree-Fock wavefunctions are available, these should be sufficient for most calculations involving diatomic molecules. 

---