Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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Consequently, for the energy to be stationary there is a simple condition: the invariance under the action of the integro-differential operator of the N-dimensional vector space , spanned by all the linear combinations of the functions θi(r) with i = 1, 2, ..N.

       Comment:

One could wonder why we limited ourselves to the variations δθk written in (32), proportional to non-occupied individual states. The reason will become clearer in § 2, where we use a more general method that shows directly which variations of each individual states are really useful to consider (see in particular the discussion at the end of § 2-a). For now, it can be noted that choosing a variation δθk proportional to the same wave function θk(r) would simply change its norm or phase, and therefore have no impact on the associated quantum state (in addition, a change of norm would not be compatible with our hypotheses, as in the computation of the average values we always assumed the individual states to remain normalized). If the state does not change, the energy must remain constant and writing a stationary condition is pointless. Similarly, to give θk(r) a variation proportional to another occupied wave function θl(r) (where l is included between 1 and N) is just as useless, as we now show. In this operation, the creation operator acquires a component on (Chapter XV, § A-6), but the state vector expression (1) remains unchanged. The state vector thus acquire a component including the square of a creation operator, which is zero for fermions. Consequently, the stationarity of the energy is automatically ensured in this case.

1-d. Equivalent formulation for the average energy stationarity

Operator can be diagonalized in the subspace , as can be shown³ from its definition (36) – a more direct demonstration will be given in § 2. We call φn(r) its eigenfunctions. These functions φn(r) are linear combinations of the θj(r) corresponding to the states appearing in the trial ket (1), and therefore lead to the same N-particle state, because of the antisymmetrization⁴. The basis change from the θi(r) to the φn(r) has no effect on the projector PN onto to the subspace , whose matrix elements appearing in (36) can be expressed in a way similar to those in (20):

      (37)

      Consequently, the eigenfunctions of the operator obey the equations:

(38)

      where are the associated eigenvalues. These relations are called the “Hartree-Fock equations”.

For the average total energy associated with a state such as (1) to be stationary, it is therefore necessary for this state to be built from N individual states whose orthogonal wave functions φ₁, φ₂, .. , φN are solutions of the Hartree-Fock equations (38) with n = 1, 2, .. , N. Conversely, this condition is sufficient since, replacing the θj(r) by solutions φn(r) of the Hartree-Fock equations in the energy variation (34) yields the result:

      (39)

      which is zero for all δφk(r) variations, since, according to (32), they must be orthogonal to the N solutions φn(r). Conditions (38) are thus equivalent to energy stationarity.

1-e. Variational energy

Assume we found a series of solutions for the Hartree-Fock equations, i.e. a set of N eigenfunctions φn(r) with the associated eigenvalues . We still have to compute the minimal variational energy of the N-particle system. This energy is given by the sum (26) of the three terms of kinetic, potential and interaction energies obtained by replacing in (15), (16) and (18) the θi(r) by the eigenfunctions φn(r):

(40)

(the subscripts HF indicate we are dealing with the average energies after the Hartree-Fock optimization, which minimizes the variational energy). Intuitively, one could expect this total energy to be simply the sum of the energies , but, as we are going to show, this is not the case. Multiplying the left-hand side of equation (38) by and after integration over d³r, we get:

      (41)

      We then take a summation over the subscript n, and use (15),

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