Variation principle minimum-energy requirement

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

Here af and cf for the cases n = l + 1 are found from the variational principle requiring the minimum of the non-relativistic energy, whereas cf (n > l + 1) - form the orthogonality conditions for wave functions. More complex, but more accurate, are the analytical approximations of numerical Hartree-Fock wave functions, presented as the sums of Slater type radial orbitals (28.31), namely... 

For the inequality between ground state energies, required for the second step, a minimum principle for the ground state energy (2.40) is used. However, while the Ritz variational principle is well established in the non-relativistic... 

In other words, the functional equivalence of Eq. (8) can be met just by requiring that D2 be TV-representable and this, in turn, means that one must determine the necessary and sufficient conditions for characterizing V% as a set containing TV-representable 2-matrices. This problem, however, is still unsolved. If not enough conditions are introduced in order to properly characterized V%, the minimum of Eq. (8) is not attained at E0 but at some other energy E 0 < E0. Thus, the upper-bound constraint of the quantum mechanical variational principle no longer applies and one can get variational" energies which are below the exact one[53]. 

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