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## 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):

 (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,
 (70)

for
 (71) (72)

By minimizing the HF energy (70) with respect to the one-body density matrix, one obtains
 (73)

which is usually solved by finding the HF s.p. orbitals that diagonalize the HF Hamiltonian (72),
 (74)

and then constructing the one-body density matrix from these orbitals:
 (75)

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

From Eq. (75) it is clear that not the real interaction , but the effective interaction , 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.

Next: Conserved and Broken Symmetries Up: MANY-NUCLEON SYSTEMS Previous: Effective Interactions (II)
Jacek Dobaczewski 2003-01-27