Obtaining Spin-Orbit Matrix Elements

In this equation, y is a so-called Massey parameter that is defined as 

METHODOLOGIES EOR OBTAINING SPIN-ORBIT MATRIX ELEMENTS 

In this section, relativistic quantum mechanics and methods for calculating the spin-orbit matrix elements are presented. Excellent reviews of this material have been published by Almlof and Gropen and Hess et al. The 

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Methodologies for Obtaining Spin-Orbit Matrix Elements 111... 

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

This last approach involves calculating spin-orbit matrix elements for the entire set of A -particle states, and although this is probably less time-consuming than the one-step methods, would nevertheless be quite time-consuming. Which one of these schemes to use would depend on the actual system under consideration. For light elements, a sufficiently accurate description might be obtained using the first approach. 

To obtain the basic matrix element expressions, assuming orthonormal orbitals, we again start from (7.2.5), and systematically evaluate the matrix elements (7.2.7) of the spin permutations. For the moment, we restrict the discussion to the singlet structures of one orbital configuration. We write (7.2.5) as... 

As a first step in applying these rules, one must examine > and > and determine by how many (if any) spin-orbitals > and > differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to aehieve maximal eoineidenee with those in the other determinant it is essential to keep traek of the number of permutations ( Np) that one makes in aehieving maximal eoineidenee. The results of the Slater-Condon rules given below are then multiplied by (-l) p to obtain the matrix elements between the original > and >. The final result does not depend on whether one ehooses to permute ... 

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

With the further condition that the spin-orbitals are orthogonal the special cases, Slater s rules, for matrix elements between determinants are obtained from this formula by inspection. The general formula can be written... 

Let us now look at one-particle operators in the second-quantization representation, defined by (13.22). Substituting into (13.22) the one-electron matrix element and applying the Wigner-Eckart theorem (5.15) in orbital and spin spaces, we obtain by summation over the projections... 

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. 

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