Atomic Many-Electron Wave Function and -Coupling

Atomic Many-Electron Wave Function and //-Coupling 

Since the structure of the many-electron Dirac Hamiltonian operator for atoms allows us to separate radial and angular degrees of freedom (as in the case of the one-electron atoms solved analytically in chapter 6), the coordinates in the 4-spinor s components may be separated into radial and spherical spinor parts. 

Because the one-electron operators are identical in form to the one-electron operator in hydrogen-like systems, we use for the independent particle model of Eq. (8.96) for the basis of the many-electron wave function a product consisting of N such hydrogen-like spinors. This ansatz allows us to treat the nonradial part analytically. The radial functions remain unknown. In principle, they may be expanded into a set of known basis functions, but we focus in this chapter on numerical methods, which can be conveniently employed for the one-dimensional radial problem that arises after integration of all angular and spin degrees of freedom. 

Recall section 4.4.3 and Eq. (4.143) for the definition of the Clebsch-Gordan coefficients (vector coupling coefficients) and Wigner-3)-symbols. Of course, the Clebsch-Gordan coefficients finally define the expansion coefficients of the pre-contracted CSF in Eq. (8.99). 

For shell occupations 1,2j (one hole) and 2 +1 (closed-shell) only one Slater determinant represents the total symmetry ( is the total angular momentum for an electron in a particular shell j = I s). Furthermore, for shell occupations 2 and 2j — 1 (two holes) Slater determinants result which are independent of the coupling scheme (for k 1). Two such Slater determinants, Fi(n) and 02(m), with n and m electrons can be coupled 

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