Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji

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and (18), the θ being replaced by the φ:

(42)

      This expression does not yield the stationary value of the total energy, but rather a sum where the particle interaction energy is counted twice. From a physical point of view, it is clear that if each particle energy is computed taking into account its interaction with all the others, and if we then add all these energies, we get an expression that includes twice the interaction energy associated with each pair of particles.

The sum of the does contain, however, useful information that enables us to avoid computing the interaction energy contribution to the variational energy. Eliminating between (40) and (42), we get:

      (43)

where the interaction energy is no longer present. One can then compute 〈Ĥ₀〉_HF and using the solutions of the Hartree-Fock equations (38), without worrying about the interaction energy. Using (15) and (17) in this relation, we can write the total energy as:

(44)

      The total energy is thus half the sum of the , of the average kinetic energy, and finally of the one-body average potential energy.

1-f. Hartree-Fock equations

      Equation (38) may be written as:

(45)

where the direct V^dlr(r) and exchange V^ex(r, r′) potentials are defined as:

(46)

      Note that the terms p = n coming from the two potentials cancel each other; hence they can be eliminated from the two summations, without changing the final result. The contribution of the direct potential is sometimes called the “Hartree term”, and the contribution of the exchange potential, the “Fock term”. The first is easy to understand: with the exception of the term p = n, it corresponds to the interaction of a particle at point r with all the others at points r′, averaged for each of them by its density distribution |φp(r′)|². As for the exchange potential, and in spite of its name, this term is not, strictly speaking, a potential; it is not diagonal in the position representation, even though it basically comes from a particle interaction which is diagonal in that representation. This peculiar non-diagonal form actually comes from the combination of the fermion antisymmetrization and the variational approximation. This exchange potential is homogeneous to a potential divided by the cube of a length. It is obviously a Hermitian operator as it is derived from a potential W₂ (r, r′) which is real and symmetric with respect to r and r′.

A more intuitive and simplified version of these equations was suggested by Hartree, in which the exchange potentials are ignored in (45). Without the integral term, these equations become very similar to a series of Schrödinger equations for independent particles, each of them moving in the mean potential created by all the others (still with the exception of the term p = n in the summation). Including the Fock term should, however, lead to more precise calculations.

Using for the potentials their expressions (46), the Hartree-Fock equations (45) become a set of N coupled equations. They are nonlinear, since the direct and exchange potentials depend on the functions φp(r). Even though they look like linear eigenvalue equations with eigenfunctions φn(r) as solutions, a linear resolution would actually require knowing in advance the solutions, since these functions also appear in the potentials (46). The term “self-consistent” is used to characterize this type of situation and the solutions φn(r) it leads to.

There are no general analytical methods to solve nonlinear self-consistent equations of this type, even in their simplified Hartree version, and numerical methods using successive approximations are commonly used. We start from a series of plausible functions , and compute with (46) the associated potentials. Considering them to be fixed, we obtain linear eigenvalue equations which can be solved quite readily with computers (the single very complicated equation in a 3N-dimensional space has been replaced by N independent 3-dimensional equations); we have to diagonalize a Hermitian operator to get a new series of orthonormal functions, resulting from the first iteration, and called and . The second iteration starts from these , to compute the new potential values, and get new linear differential equations. Solving these equations yields the next order , , etc. After a few iterations, one expects the and to vary only slightly with the iteration order (n), in which case the Hartree-Fock equations have been solved to a good approximation. Using (44) we can then compute the energy we were looking for. It is also possible that physical arguments can help us choose directly adequate trial functions φn(r) without any iteration. Inserting them in (44)

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