Orthonormality Constraints and Total Energy Expressions

In order to calculate expectation values for a wave function of the structure given in Eq. (8.100), it is convenient to introduce orthonormality constraints on the one-electron spinors 

we can benefit largely from factors that become zero upon resolution of an expectation value over two Slater determinants because of the orthogonality of two different spinors. Also, the Slater determinants turn out to be orthonormal then. 

Additionally, orthonormal Slater determinants result in orthonormal CSFs, 

The orthonormality restriction for the spinors, Eq. (8.108), may be split into two parts in the case of atoms. The product ansatz for the spinor automatically yields orthonormal angular parts (coupled spherical harmonics cf. chapter 9). But these do not contain information about the principal quantum numbers in the composite indices i and j. For this reason, the restriction to orthonormal spinors results in the orthonormality restriction for radial functions 

This holds true only if Kj = Kj since the spin and angular parts guarantee orthogonality in all other cases. If one wants to calculate excited electronic states using the same set of radial functions optimized for both the ground state and the excited state within the same symmetry, the orthonormality requirement for these states leads to orthogonal vectors of Cl coefficients. 

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