The Hartree-Fock approximation

When Hartree-Fock theory was first under development, Hartree-Fock approximations to many-electron wave functions often were assumed to be single Slater determinants. As explained in Chapter 1, such wave functions, in general, are not eigenfunctions of the total angular momentum operators. In current atomic physics it is customary to define the Hartree-Fock approximation as a stationary solution of the variational problem eq:varprob for a single configuration state function and the Hartree-Fock energy, the stationary energy associated with this solution. A special case is the fixed- or frozen-core Hartree-Fock (FCHF) approximation where orbitals defining the core are not allowed to vary.

As shown in the previous chapter, the energy functional associated with a configuration state function can be written as a linear combination of one- and two-electron radial integrals. Thus, we need to consider variations of these integrals with respect to a variation in the radial functions from $\P{nl}{}$ to $\P{nl}{}+\delta\P{nl}{}$.

Starting with the one-electron integral

$$I(nl,nl) = -\frac{1}{2}\int_0^\infty\P{nl}{} {\cal L} \P{nl}{}\,dr,$$ (1)

we have

$$\delta I(nl,nl) -\frac{1}{2}\int_0^\infty\delta\P{nl}{} {\cal L} \P{nl}{}\,dr-\frac{1}{2}\int_0^\infty\P{nl}{} {\cal L} \delta\P{nl}{}\,dr$$ =

$$- \int_0^\infty\delta\P{nl}{} {\cal L} \P{nl}{}\,dr$$ = (2)

where the latter follows from integration by parts and the zero boundary conditions at the origin and at infinity.

Next we consider the radial integrals arising from the two-electron Coulomb operator, the F^k and G^k Slater integrals. In order to derive the first-order variation of these integrals (and R^k in general) it is convenient to first replace variables (r₁,r₂) by (r,s) and introduce the functions

$$r\int_0^\infty\frac{r_<^{k}}{r_>^{k+1}} P(a;s)P(b;s)\, ds$$ Y^k(ab;r) =

$$\int_0^r \left(\frac{s}{r}\right)^k P(a;s)P(b;s)\,ds$$ =

$$\quad + \int_r^\infty \left(\frac{r}{s}\right)^{k+1}P(a;s)P(b;s)\, ds .$$ (3)

Then

$$\int_0^\infty P^2(a;r) \left(\frac{1}{r}\right)Y^{k}({b}{b};r)\, dr$$ F^k(a,b) =

$$\int_0^\infty P^2(b;s) \left(\frac{1}{s}\right)Y^{k}({a}{a};s)\, ds$$ = (4)

since the order of integration in the definition of F^k may be reversed. Similarly

$$G^{k}(a,b) = \int_0^\infty P(a;r)P(b;r) \left(\frac{1}{r}\right)Y^{k}({a}{b};r)\,dr .$$ (5)

It is then straightforward to show that

$$\delta F^{k}(a,b) = 2(1+\delta_{ab}) \int_0^\infty\delta P(a;r) P(a;r) \left( \frac{1}{r}\right) Y^{k}({b}{b};r)\,dr$$ (6)

and

$$\delta G^{k}(a,b) = 2 \int_0^\infty\delta P(a;r) P(b;r)\left( \frac{1}{r}\right) Y^{k}({a}{b};r)\,dr .$$ (7)

Instead of proceeding directly to the derivation of the general Hartree-Fock equations, let us start with some simple cases, each illustrating certain points, and concluding with the general case.

2001-01-09