Matrix elements many-electron spin-orbit

Since the many-electron wave function can be expanded in a linear combination of Slater determinants, its matrix element with a spin-orbit coupling operator of the form of Equation (3.6) can be expressed as a sum of matrix elements of the operator between Slater determinants. For a matrix element between Slater determinants which differ in exactly one spin orbital (i.e. which are singly excited from i — a with respect to each other), the matrix element is... 

Since spin-orbit coupling is very important in heavy element compounds and the structure of the full microscopic Hamiltonians is rather complicated, several attempts have been made to develop approximate one-electron spin-orbit Hamiltonians. The application of an (effective) one-electron spin-orbit Hamiltonian has several computational advantages in spin-orbit Cl or perturbation calculations (1) all integrals may be kept in central memory, (2) there is no need for a summation over common indices in singly excited Slater determinants, and (3) matrix elements coupling doubly excited configurations do not occur. In many approximate schemes, even the tedious four-index transformation of two-electron integrals ceases to apply. The central question that comes up in this context deals with the accuracy of such an approximation, of course. 

As was mentioned in Chapter 2, there exists another method of constructing the theory of many-electron systems in jj coupling, alternative to the one discussed above. It is based on the exploitation of non-relativistic or relativistic wave functions, expressed in terms of generalized spherical functions [28] (see Eqs. (2.15) and (2.18)). Spin-angular parts of all operators may also be expressed in terms of these functions (2.19). The dependence of the spin-angular part of the wave function (2.18) on orbital quantum number is contained only in the form of a phase multiplier, therefore this method allows us to obtain directly optimal expressions for the matrix elements of any operator. The coefficients of their radial integrals will not depend, except phase multipliers, on these quantum numbers. This is the case for both relativistic and non-relativistic approaches in jj coupling. 

The shared features of quantum cell models are specified orbitals, matrix elements and spin conservation. As emphasized by Hubbard[5] for d-electron metals and by Soos and Klein [11] for organic crystals of 7r-donors or 7r-acceptors, the operators o+, and apa in (1), (3) and (4) can rigorously be identified with exact many-electron states of atoms or molecules. The provisos are to restrict the solid-state basis to four states per site (empty, doubly occupied, spin a and spin / ) and to stop associating the matrix elements with specific integrals. The relaxation of core electrons is formally taken into account. Such generalizations increase the plausibility of the models and account for their successes, without affecting their solution or interpretation. 

In the present treatment, we retain essentially all the diagonal matrix elements of X these are the first-order contributions to the effective electronic Hamiltonian. There are many possible off-diagonal matrix elements but we shall consider only those due to the terms in Xrot and X o here since these are the largest and provide readily observable effects. The appropriate part of the rotational Hamiltonian is —2hcB(R)(NxLx + NyLy). The matrix elements of this operator are comparatively sparse because they are subject to the selection rules AA = 1, A,Y=0 and AF=0. The spin-orbit coupling term, on the other hand, has a much more extensive set of matrix elements allowed... 

These rules are a consequence of the fact that the spin-orbit operator for the many-electron states is a sum of one-particle operators according to (5) and the Slater-Condon rules for matrix elements between states of such operators [121]. 

Suppose states D > and A > are one-determinant many-electron functions, which are written in terms of (real) molecular orbitals and where a is the spin index, a a, p. These are the optimized canonical orbitals obtained from Hartree-Fock calculations of states D and A. Using the standard rules of matrix element evaluations[18], one can obtain an appropriate expression for Eq. (1) in terms of MO s of the system. 

The presence of two-electron operators in the Breit-Pauli and similar expressions for makes their use computationally quite demanding, because such operators have nonvanishing matrix elements even between Slater determinants that differ in two spin-orbital occupancies and because there are many two-electron integrals. This is especially true in studies of photochemical reaction paths, where information about spin-orbit coupling is needed at many geometries. Several simplifications have been quite popular. 

Numerical ab initio calculations for selected examples with polarized basis sets and Cl of reasonably size confirmed that the size of the matrix elements within the active space matrix is negligible. In contrast, the elements of that involve both the active and inner shells are large, since is primeuily due to the shielding of nuclei by inner-shell electrons [11]. It is therefore common practice in many semiquantitative applications, to account for the effect of the fixed-core electrons by replacing the factor gPgZ r in by the empirical value of the atomic spin-orbit coupling constant valence p orbitals on... 

Values of the spin-orbit coupling constants for different atomic states have been determined fi om atomic spectra, or may be calculated from relativistic quantum mechanics. A more detailed examination of the matrix elements reveals that die spin-orbit coupling operator mixes states of different L and 5 values spin-orbit matrices for many of the most common electron configurations have been given by Condon and Shortley [2]. 

---