Quantum Mechanics, Volume 3. Claude Cohen-Tannoudji
is the same as the subscript n (of the wave function we are looking for); the contributions n = p cancel each other exactly in the direct and exchange potentials.
Both the “Hartree term” giving the direct potential contribution, and the “Fock term “ giving the exchange potential, can be interpreted in the same way as above (§ 1-f). The Hartree term contains the contributions of all the other electrons to the mean potential felt by one electron. The exchange potential, on the other hand, only involves electrons in the same spin state, and this can be simply interpreted: the exchange effect only occurs for two totally indistinguishable particles. Now if these particles are in orthogonal spin states, and as the interactions do not act on the spins, one can in principle determine which is which and the particles become distinguishable: the quantum exchange effects cancel out. As we already pointed out, the exchange potential is not a potential stricto sensu. It is not diagonal in the position representation, even though it basically comes from a particle interaction that is diagonal in position. It is the antisymmetrization of the fermions, together with the chosen variational approximation, which led to this peculiar non-diagonal form. It is however a Hermitian operator, as can be shown using the fact that the initial potential W2(r, r′) is real and symmetric with respect to r and r′.
2-e. Discussion
The resolution of the nonlinear Hartree-Fock equations is generally done by the successive iteration approximate method discussed in § 1-f. There is no particular reason for the solution of the Hartree-Fock equations to be unique8; on the contrary, they can yield solutions that depend on the states chosen to begin the nonlinear iterations. They can actually lead to a whole spectrum of possible energies for the system. This is how the ground state and excited state energies of the atom are generally computed. The atomic orbitals discussed in Complement EVII, the central field approximation and the electronic “configurations” discussed in Complement BXIV can now be discussed in a more precise and quantitative way. We note that the exchange energy, introduced in this complement for a two-electron system, is a particular case of the exchange energy term of the Hartree-Fock potential. There exist however many other physical systems where the same ideas can be applied: nuclei (the Coulomb force is then replaced by the nuclear interaction force between the nucleons), atomic aggregates (with an interatomic potential having both repulsive and attractive components, see Complements CXI and GXI), and many others.
Once a Hartree-Fock solution for a complex problem has been found, we can go further. One can use the basis of the eigenfunctions just obtained as a starting point for more precise perturbation calculations, including for example correlations between particles (Chapter XI). In atomic spectra, we sometimes find cases where two configurations yield very close mean field energies. The effects of the interaction terms beyond the mean field approximation will then be more important. Perturbation calculations limited to the space of the configurations in question permits obtaining better approximations for the energy levels and their wave functions; one then speaks of “mixtures”, or of “interactions between configurations”.
Comment:
The variational method based on the Fock states is not the only one that leads to the Hartree-Fock equations. One could also start from an approximation of the two-particle density operator ρII by a function of the one-particle density operator ρI and write:
(97)
Expressing the energy of the N-particle system as a function of ρI, we minimize it by varying this operator, and find the same results as above. This method amounts to a closure of the hierarchy of the N-body equations (§ C-4 of Chapter XVI). We have in fact already seen with equation (21) and in § 2-a that the Hartree-Fock approximation amounts to expressing the two-particle correlation functions as a function of the one-particle correlation functions. In terms of correlation functions (Complement AXVI), this amounts to replacing the two-particle function (four-point function) by a product of one-particle functions (two-point function), including an exchange term. Finally, another method is to use the diagram perturbation theory; the Hartree-Fock approximation corresponds to retaining only a certain class of diagrams (class of connected diagrams).
Finally note that the Hartree-Fock method is not the only one yielding approximate solutions of Schrodinger’s equation for a system of interacting fermions; in particular, one can use the “electronic density functional” theory (a functional is a function of another function, as for instance the action S in classical lagrangian mechanics). The method is used to obtain the electronic structure of molecules or condensed phases in physics, chemistry, and materials science. Its study nevertheless lies outside the scope of this book, and the reader is referred to [6], which summarizes the method and gives a number of references.
1 1 We assume the nucleus mass to be infinitely larger than the electron mass. The electronic system can then be studied assuming the nucleus fixed and placed at the origin.
2 2 The Pauli exclusion principle is not sufficient to explain why an atom’s size increases with its atomic number Z. One can evaluate the approximate size of a hypothetical atom with non-interacting electrons (we consider the atom’s size to be given by the size of the outermost occupied orbit). The Bohr radius a0 varies as 1/Z, whereas the highest value of the principal quantum number n of the occupied states varies approximately as Z1/3. The size n2a0 we are looking for varies approximately as Z-1/3.
3 3 As any Hermitian operator can be diagonalized, we simply show that (36) leads to matrix elements obeying the Hermitian conjugation relation. Let us verify that the two integrals and are complex conjugates of each other. For the contributions to these matrix elements of the kinetic and potential (in V1) energy, we simply find the usual relations insuring the corresponding operators are Hermitian. As for the interaction term, the complex conjugation is obvious for the direct term; for the exchange term, a simple inversion of the integral variables d3r and d3r′, plus the fact that W2(r, r′) is equal to W2(r′,r) allows verifying the conjugation.
4 4 A determinant value does not change if one adds to one of its column a linear combination of the others. Hence we can add to the first column of the Slater determinant (2) the linear combination of the θ2(r), θ3(r), … that makes it proportional to φ1(r). One can then add to the second column the combination that makes it proportional to φ2(r), etc. Step by step, we end up with a new expression for the original wave function , which now involves the Slater determinant of the φi(r). It is thus proportional to this determinant. A demonstration of the strict equality (within a phase factor) will be given in § 2.
5 5 As in the previous complement, we have replaced W2(R1, R2) by W2(1, 2) to simplify the notation
6 6 Since (71) shows that |δθ〉 is orthogonal to any linear combinations of the |θi〉, we can write (〈θj0| + 〈δθ|)(|θj0〉 + |δθ〉) = 〈θj0 |θj0〉 + 〈δθ |δθ〉 = 1 + second order terms.
7 7