The Hartree-Fock equation for 1s2p 3 P/.

Our first example is the one treated in detail in chap:CSF. In this case, the only unknown functions are $\P{1s}{}$and $\P{2p}{}$. According to sec:variational, the variational principle requires that the energy functional be stationary with respect to small variations in these functions. Combining the results of eq:oneelect and eq:twoelect we get

$${\cal E}(1s2p\;\Term 3 P/) = I(1s,1s) + I(2p,2p) + F^{0}(1s,2p)- (1/3) G^{1}(1s,2p) .$$ (8)

However, this expression was derived on the assumption that all radial functions were normalized. For such constrained variation, Lagrange multipliers need to be introduced and the variational principle applied to

$${\cal F}(P(1s;r), P(2p;r)) = {\cal E}(1s2p\;\Term 3 P/)+\lamb......vert 1s \rangle+ \lambda_{2p,2p}\langle 2p \vert 2p \rangle$$ (9)

where $\langle nl \vert n^{\prime}l\rangle= \int_0^{\infty}P(nl;r)P(n^{\prime}l;r)dr$. Let us consider the variation of ${\cal F}$ with respect to a variation in $\P{1s}{}$. Clearly, the variation of terms independent of$\P{1s}{}$ is zero and the variation of the rest can be expressed in the form

$$\int_0^\infty \delta P(1s;r)Q(1s;r)\,dr =0.$$ (10)

The contributions to Q(1s;r) are as follows:

$$\begin{array}{l l c}\mbox{From} & I(1s,1s): & -{\cal L}P(1s;......rt 1s \rangle: &2\lambda_{1s,1s}P(1s;r)\end{array} \nonumber$$

eq:1sstat will be satisfied for all allowed variations $\delta \P{1s}{}$ only if Q(1s;r) = 0. Collecting terms and changing sign we get

$$\left({\cal L} -\frac{2}{r} Y^{0}({2p}{2p};r) -\varepsilon_{1s,1s} \right)P(1s;r)+ \frac{2}{3r}Y^{1}({1s}{2p};r)P(2p;r)=0$$ (11)

where $\varepsilon_{1s,1s} = 2\lambda_{1s,1s}$. Applying the stationary condition to variations in P(2p;r) leads to the condition

$$\left({\cal L} -\frac{2}{r} Y^{0}({1s}{1s};r) -\varepsilon_{2p,2p} \right)P(2p;r)+\frac{2}{3r}Y^{1}({1s}{2p};r)P(1s;r)=0 .$$ (12)

Each of these equations may be written in the form

$$\left(\frac{d^2}{dr^2} + \frac{2}{r}\left[Z - Y(nl;r)\right] ......r^2} - \varepsilon_{nl,nl}\right)P(nl;r) =\frac{2}{r}X(nl;r)$$ (13)

where Y(nl;r) accounts for the screening of the nucleus by the presence of the other electrons and X(nl;r) arises from antisymmetry and is called the exchange term. The parameter $\varepsilon_{nl,nl}$ is called the diagonal energy parameter. The boundary conditions are P(nl;0) = 0 and $P(nl;r)\rightarrow0$ as $r \rightarrow\infty$. Written in this form, the equation is a second-order differential equation of boundary value type, but analyzing X(nl;r) more carefully, it becomes clear that the equation is an integro-differential equation. For example, suppose we define the Y^k function of eq:yk in terms of an operator

$$Y^{k}({a}{\bullet};r) = r\int_0^\infty\frac{r_<^{k}}{r_>^{k+1}} P(a;s)\bullet\, ds$$ (14)

so that $Y^{k}({a}{b};r) = Y^{k}({a}{\bullet};r)P(b;r)$. In other words, the integral operator operating on a function includes the function in the integrand. Then eq:hf1s for the radial function P(1s;r) becomes

$$\left({\cal L} -\frac{2}{r} Y^{0}({2p}{2p};r) -\frac{1}{3r}......{1}({2p}{\bullet};r) -\varepsilon_{1s,1s} \right)P(1s;r)=0 .$$ (15)

In terms of this operator, which combines the properties of a differential equation with that of an integral equation, the equation is an eigenvalue problem. Such equations are called ``integro-differential''. In the present case, the equation for $\P{1s}{}$ is a linear, homogeneous equation; that is, if $\P{nl}{}$ is a solution, $c\P{nl}{}$ is also a solution. We will see later that, in some cases, the integro-differential eigenvalue problem is non-linear. The nuclear potential together with the Y^k functions that contribute to the direct potential is sometimes referred to as the local potential whereas Y^k functions that contribute to the exchange function and are of the integral equation type define what is referred to as the non-local potential.

2001-01-09