Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

      We have left out the condition i ≠ j no longer useful since the i = j terms are zero. The second line of this equation contains the sum of the direct and the exchange terms.

      The result can be written in a more concise way by introducing the projector PN over the subspace spanned by the N kets |θi〉:

      (19)

      Its matrix elements are:

(20)

This leads to:

(21)

       Comment:

      The matrix elements of PN are actually equal to the spatial non-diagonal correlation function G₁(r, r′), which will be defined in Chapter XVI (§ B-3-a). This correlation function can be expressed as the average value of the product of field operators Ψ(r):

      (22)

      For a system of N fermions in the states |θ₁〉, |θ₂〉, ..,|θN〉, we can write:

      (23)

Inserting this relation in (18) we get:

      (24)

      Comparison with relation (C-28) of Chapter XV, which gives the same average value, shows that the right-hand side bracket contains the two-particle correlation function G₂(r, r′). For a Fock state, this function can therefore be simply expressed as two products of one-particle correlation functions at two points:

      (25)

1-c. Optimization of the variational wave function

      We now vary to determine the conditions leading to a stationary value of the total energy :

(26)

where the three terms in this summation are given by (15), (16) and (18). Let us vary one of the kets |θk〉, k being arbitrarily chosen between 1 and N:

      (27)

      or, in terms of an individual wave function:

      (28)

      This will yield the following variations:

(29)

and:

(30)

      As for the variation of , we must take from (18) two contributions: the first one from the terms i = k, and the other from the terms j = k. These contributions are actually equal as they only differ by the choice of a dummy subscript. The factor 1/2 disappears and we get:

(31)

The variation of is simply the sum of (29), (30) and (31).

      We now consider variations δθk, which can be written as:

(32)

      (where δε is a first order infinitely small parameter). These variations are proportional to the wave function of one of the non-occupied states, which was added to the occupied states to form a complete orthonormal basis; the phase χ is an arbitrary parameter. Such a variation does not change, to first order, either the norm of |θk〉, or its scalar product with all the occupied states l ≤ N; it therefore leaves unchanged our assumption that the occupied states basis is orthonormal. The first order variation of the energy is obtained by inserting δθk and its complex conjugate into (29), (30) and (31); we then get terms in eiχ in the first case, and terms in e–iχ in the second. For to be stationary, its variation must be zero to first order for any value of χ; now the sum of a term in eiχ and another in eiχ will be zero for any value of χ only if both terms are zero. It follows that we can impose to be zero (stationary condition) considering the variations of δθk and to be independent. Keeping only the terms in