Hartree-Fock method

The Hartree-Fock (HF) approach relies on assuming that the ground state of a many-fermion system can be uniquely characterized by the one-body density matrix (59). There are many ways of deriving the HF equations; the simplest one is to use the variational principle together with the following approximation of the two-body density matrix (60):

$$\rho_{x'y'xy} = \rho_{x'x}\rho_{y'y} - \rho_{x'y}\rho_{y'x} .$$ (69)

This equation expresses the two-body density matrix by the one-body density matrix, and hence the total energy (58) becomes a functional of the one-body density matrix only,

$$E_{\mbox{\scriptsize {HF}}} = T_{xy}\rho_{yx} + {\textstyle{\frac{1}{4}}}G_{xyx'y'} \left(\rho_{x'x}\rho_{y'y} - \rho_{x'y}\rho_{y'x}\right)$$

$$= T_{xy}\rho_{yx} + {\textstyle{\frac{1}{2}}}\Gamma_{xx'}\rho_{x'x} = {\textstyle{\frac{1}{2}}}\left(T_{xy} + h_{xy}\right)\rho_{yx} ,$$ (70)

for

$$\Gamma_{xx'}= G_{xyx'y'}\rho_{y'y} \quad \Longleftarrow\quad \mbox{HF potential} ,$$ (71)

$$h_{xy} = T_{xy} + \Gamma_{xy} \quad \Longleftarrow\quad \mbox{HF Hamiltonian} .$$ (72)

By minimizing the HF energy (70) with respect to the one-body density matrix, one obtains

$$h_{xy}\rho_{yz} - \rho_{xy}h_{yz} = 0 \quad \Longleftarrow\quad \mbox{HF equation} ,~~~~~$$ (73)

which is usually solved by finding the HF s.p. orbitals that diagonalize the HF Hamiltonian (72),

$$\int\hspace{-1.2em}\sum {\rm d}{y}\, h_{xy}\phi_i(y) = \epsilon_i \phi_i(x) ,$$ (74)

and then constructing the one-body density matrix from these orbitals:

$$\rho_{xy} = \sum_{i\in{\mbox{\scriptsize {occ}}}} \phi_i(x) \phi^*_i(y).$$ (75)

Equations (74) and (75) guarantee that the HF condition (73) is fulfilled (because $h_{xy}$ and $\rho_{xy}$ are then diagonal in the common basis), so the HF solution is found whenever, for a given set of occupied orbitals, $i\in{\mbox{occ}}$, the density matrix self-consistently reproduces the HF potential (71).

From Eq. (75) it is clear that not the real interaction $V_{xyx'y'}$, but the effective interaction $G_{xyx'y'}$, must be used in the HF method. Indeed, when the density-matrix (75) is inserted in the expression for the HF energy (70), one recovers the action of the effective interaction on the two-body product wave functions (61). It is now obvious that the determination of the effective interaction must be coupled to the solution of the HF equations, and performed self-consistently. Namely, for a given effective interaction one solves the HF equations, and the obtained HF orbitals (74) are in turn used in the Bethe-Goldstone equation to find effective interaction. Such a doubly self-consistent procedure is called the Brueckner-Hartree-Fock method.

Modern understanding of the HF approximation is not directly based on the variational method applied to Slater determinants. Certainly, the basic approximation for the two-body density matrix (69) is an exact result for a Slater determinant, but the key element of the approach is expression (70), which states that the ground-state energy can be approximated by a functional of the one-body density matrix.