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been chosen to be the eigenvectors of with eigenvalues – see relation (26). We just replace in (56), Ĥ₀ by , and obtain:

      (59)

Regrouping all these results and using relation (36), we can write the variational grand potential as the sum of three terms:

      (60)

      with:

(61)

2-d. Optimization

      We now vary the eigenenergies and eigenstates |θk〉 of to find the value of the density operator that minimizes the average value of the potential. We start with the variations of the eigenstates, which induce no variation of . The computation is actually very similar to that of Complement E_XV, with the same steps: variation of the eigenvectors, followed by the demonstration that the stationarity condition is equivalent to a series of eigenvalue equations for a Hartree-Fock operator (a one-particle operator). Nevertheless, we will carry out this computation in detail, as there are some differences. In particular, and contrary to what happened in Complement E_XV, the number of states |θi〉 to be varied is no longer fixed by the particle number N; these states form a complete basis of the individual state space, and their number can go to infinity. This means that we can no longer give to one (or several) state(s) a variation orthogonal to all the other |θj〉; this variation will necessarily be a linear combination of these states. In a second step, we shall vary the energies .

       α. Variations of the eigenstates

      As the eigenstates |θi〉 vary, they must still obey the orthogonality relations:

      (62)

      The simplest idea would be to vary only one of them, |θl〉 for example, and make the change:

      (63)

      The orthogonality conditions would then require:

      (64)

preventing |dθl〉 from having a component on any ket |θi〉 other than |θl〉: in other words, |dθl〉 and |θl〉 would be colinear. As |θl〉 must remain normalized, the only possible variation would thus be a phase change, which does not affect either the density operator (1) or any average values computed with . This variation does not change anything and is therefore irrelevant.

      It is actually more interesting to vary simultaneously two eigenvectors, which will be called |θl〉 and |θm〉, as it is now possible to give |θl〉 a component on |θm〉, and the reverse. This does not change the two-dimensional subspace spanned by these two states; hence their orthogonality with all the other basis vectors is automatically preserved. Let us give the two vectors the following infinitesimal variations (without changing their energies and ):

(65)

      where da is an infinitesimal real number and χ an arbitrary but fixed real number. For any value of χ, we can check that the variation of 〈θl |θl〉 is indeed zero (it contains the scalar products 〈θl |θm〉 or 〈θm |θl〉 which are zero), as is the symmetrical variation of 〈θm |θm〉, and that we have:

      (66)

The variations (65) are therefore acceptable, for any real value of χ.

      We now compute how they change the operator defined in (40). In the sum over k, only the k = l and k = m terms will change. The k = l term yields a variation:

      (67)

      whereas the k = m term yields a similar variation but where is replaced by . This leads to:

(68)

We now include these variations in the three terms of (61); as the distributions f are unchanged, only the terms and will vary. The infinitesimal variation of is written as:

      (69)

      As for , it contains two contributions, one from and one from . These two contributions

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