2 On Quantum Exchange Energy
Abstract
As is known,1 the Hartree equations with the quantum exchange can be derived from the variation principle δW = 0, where W is the energy of an atom, which has the form W = ∫ n+1∑ ϕi (x)H (x)ϕi (x) dx + + e2 ∫ ∫ (xx) (x′x′)− | (xx′) |2 | r− r′ | dxdx ′. (1) In this formula, n + 1 is the number of electrons in the atom and x is the set of the variables corresponding to a separate electron, i.e., x = (x, y, z, σ) and ∫ f (x) dx = ∑ σ ∫ f (x) dτ, H (x) is the energy operator of a separate electron (it does not include the interaction energy with remaining electrons), ϕi (x) (i = 1, 2, . . . , n + 1) are the one-electron wave functions of atomic electrons that are supposed to be mutually orthogonal and normalized, so that∫ ϕi (x)ϕj (x) dx = δij , (2) and, finally, (x, x′) denotes the “mixed” density, i.e., (x, x′) = n+1∑ i=1 ϕi (x)ϕi (x′) . (3) When varying expression (1), one has to take into account the additional conditions (2).
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