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operator that is the pertinent variable to optimize rather than the set of individual states: certain variations of those states do not change PN, and are meaningless for our purpose.

Furthermore, the choice of the trial ket is equivalent to that of PN. In other words, the variational ket built in (1) does not depend on the basis chosen in the subspace : if we choose in this subspace any orthonormal basis {|uj〉} other than the {|θj〉} basis, and if we replace in (1) the by the , the ket will remain the same (to within a non-relevant phase factor) as we now show. As seen in § A-6 of Chapter XV, each operator is a linear combination of the , so that in the product of all the (j = 1, 2, ..N) we will find products of N operators . Relation (A-43) of Chapter XV however indicates that the squares of any creation operators are zero, which means that the only non-zero products are those including once and only once each of the N different operators . Each term is then proportional to the ket built from the . Consequently, the two variational kets built from the two bases are necessarily proportional. As definition (1) ensures they are also normalized, they can only differ by a phase factor, which means they are equivalent from a physical point of view. It is thus the operator that best embodies the trial ket .

2-b. Optimization of the one-particle density operator

      We now vary to look for the stationary conditions for the total energy:

      (62)

      We therefore consider the variation:

      (63)

      which leads to the following variations for the average values of the one-particle operators:

      (64)

      As for the interaction energy, we get two terms:

      (65)

      which are actually equal since:

      (66)

      and we recognize in the right-hand side of this expression the trace:

      (67)

      As we can change the label of the particle from 2 to 1 without changing the trace, the two terms of the interaction energy are equal. As a result, we end up with the energy variation:

(68)

      To vary the projector PN, we choose a value j₀ of j and make the change:

      (69)

where |δθ〉 is any ket from the space of individual states, and χ any real number; no other individual state vector varies except for |θ_j0〉. The variation of PN is then written as:

(70)

      We assume |δθ〉 has no components on any |θi〉, that is no components in , since this would change neither εN, nor the corresponding projector PN. We therefore impose:

(71)

which also implies that the norm of |θ_j0〉 remains constant⁶ to first order in |δθ〉. Inserting (70) into (68), we obtain:

      (72)

      For the energy to be stationary, this variation must remain zero whatever the choice of the arbitrary number χ. Now the linear combination of two exponentials e^iχ and e^–iχ will remain zero for any value of χ only if the two factors in front of the exponentials are zero themselves. As each term can be made equal to zero separately, we obtain:

      (73)

      This relation must be satisfied for any ket |δθ〉 orthogonal to the subspace . This means that if we define the one-particle Hartree-Fock operator as:

(74)

      the stationary condition for the total energy is simply that the ket HHF |θ_j0〉 must belong to :

(75)

As

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