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we shall see, all the average values useful in our calculation can be simply expressed as a function of this operator.

2-a. Average energy

      We now evaluate the different terms included in the average energy, starting with the terms containing one-particle operators.

       α. Kinetic and external potential energy

      Using relation (B-12) of Chapter XV, we obtain for the average kinetic energy 〈Ĥ₀〉:

(51)

The same argument as that for the evaluation of the matrix elements (49) shows that the average value in the state (47) is only different from zero if r = s; in that case, it is equal to one when r ≤ N, and to zero otherwise. This leads to:

      (52)

The subscript 1 was added to the trace to underline the fact that this trace is taken in the one-particle state space and not in the Fock space. The two operators included in the trace only act on that same particle, numbered arbitrarily 1; the subscript 1 could obviously be replaced by the subscript of any other particle, since they all play the same role. The average potential energy coming from the external potential is computed in a similar way and can be written as:

(53)

       β. Average interaction energy, Hartree-Fock potential operator

      The average interaction energy can be computed using the general expression (C-16) of Chapter XV for any two-particle operator, which yields:

      (54)

      For the average value in the Fock state to be different from zero, the operator must leave unchanged the populations of the individual states |θn〉 and |θq〉. As in § C-5-b of Chapter XV, two possibilities may occur: either r = n and s = q (the direct term), or r = q and s = n (the exchange term). Commuting some of the operators, we can write:

      (55)

where nr and ns are the respective populations of the states |θr〉 and |θs〉. Now these populations are different from zero only if the subscripts r and s are between 1 and N, in which case they are equal to 1 (note also that we must have r ≠ s to avoid a zero result). We finally get⁵:

      (56)

      (the constraint i ≠ j may be ignored since the right-hand side is equal to zero in this case). Here again, the subscripts 1 and 2 label two arbitrary, but different particles, that could have been labeled arbitrarily. We can therefore write:

(57)

where P_ex(1,2) is the exchange operator between particle 1 and 2 (the transposition which permutes them). This result can be written in a way similar to (53) by introducing a “Hartree-Fock potential” WHF, similar to an external potential acting in the space of particle 1; this potential is defined as the operator having the matrix elements:

(58)

      This operator is Hermitian, since, as the two operators P_ex and W₂ are Hermitian and commute, we can write:

      (59)

Furthermore, we recognize in (58) the matrix element of a partial trace on particle 2 (Complement E_III, § 5-b):

(60)

where the projector PN has been introduced inside the trace to limit the sum over j to its first N terms, as in (57). The one-particle operator WHF(1) is thus the partial trace over a second particle (with the arbitrary label 2) of a product of operators acting on both particles. As the summation over j is now taken into account, we are left in (57) with a summation over i, which introduces a trace over the remaining particle 1, and we get:

(61)

This average value depends on the subspace chosen with the variational ket in two ways: explicitly as above, via the projector PN(1) that shows up in the average value (61), but also implicitly via the definition of the Hartree-Fock potential in (60).

       ϒ. Role of the one-particle reduced density operator

All the average values can be expressed in terms of the projector PN onto the subspace of the space the individual states spanned by the N individual states |θ₁〉, |θ₂〉, ….|θN〉, which means, according to (50), in terms of the one-particle reduced density operator Скачать книгу