Orbital variation. Stationary-value conditions

We start from the energy functional (5.4.16), corresponding to use of a Cl-type variation function 

When the integrals are defined over occupied orbitals the A and B coefficients are elements of the spinless density matrices P and /7 but if instead we define the integrals with respect to. ypm-orbitals then the appropriate coefficients will be elements of the density matrices p and n (includmg spin). Thus both variation problems may be discussed using the same formalism, merely by changing the interpretation of the integrals and their coefficients. In both cases, the A and B coefficients are defined in terms of similar (transition) quantities for all pairs of CFs i) in the expansion (8.2.1) for example. 

We know that for E to be stationary under variation of Cl coefficients it is necessary to satisfy secular equations of the usual form (2.3.4) the problem that concerns us now is how to make E stationary under orbitalt variations. We assume that the orbitals used in the expansion (8.2.1) are orthonormal. 

First we note that when 0, -F 64 r (all orbitals) the first-order variation is 

Such operators are reminiscent of the Fock operator F used in Chapter 6 but there is one for every pair (r, s) of occupied orbitals. 

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Attaching the stationary condition for the minimal variation of orbital (pi (pi -> (pi + S(pi), SE/S(pi = 0, to the expectation value of the Hamiltonian operator containing one- and two-electron integrals, we obtain... 

However, the periodicity condition (4.518) for paths is to be maintained and properly implemented in approximating the effective-classical partition function (4.525) being, nevertheless, closely and powerfully related with the quantum beloved concept of stationary orbits defined/described by periodic quantum waves/paths. This way, the effective-classical path integral approach appears as the true quantum justification of the quantum atom and of the quantum stabilization of matter in general, providing reliable results without involving observables or operators relaying on special quantum postulates other than the variational principles - with universal (classical or quantum) value. 

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