The zeroth-order perturbation wave function (10.40) uses the full nuclear charge $(Z=3)$ for both the $1 s$ and $2 s$ orbitals of lithium. We expect that the $2 s$ electron, which is partially shielded from the nucleus by the two $1 s$ electrons, will see an effective nuclear charge that is much less than 3 . Even the $1 s$ electrons partially shield each other (recall the treatment of the helium ground state). This reasoning suggests the introduction of two variational parameters $b{1}$ and $b{2}$ into (10.40).
Instead of using the $Z=31$ s function in Table 6.2, we take
\(
\begin{equation}
f \equiv \frac{1}{\pi^{1 / 2}}\left(\frac{b{1}}{a{0}}\right)^{3 / 2} e^{-b{1} r / a{0}} \tag{10.52}
\end{equation}
\)
where $b_{1}$ is a variational parameter representing an effective nuclear charge for the $1 s$ electrons. Instead of the $Z=32 s$ function in Table 6.2, we use
\(
\begin{equation}
g=\frac{1}{4(2 \pi)^{1 / 2}}\left(\frac{b{2}}{a{0}}\right)^{3 / 2}\left(2-\frac{b{2} r}{a{0}}\right) e^{-b{2} r / 2 a{0}} \tag{10.53}
\end{equation}
\)
Our trial variation function is then
\(
\phi=\frac{1}{\sqrt{6}}\left|\begin{array}{lll}
f(1) \alpha(1) & f(1) \beta(1) & g(1) \alpha(1) \tag{10.54}\
f(2) \alpha(2) & f(2) \beta(2) & g(2) \alpha(2) \
f(3) \alpha(3) & f(3) \beta(3) & g(3) \alpha(3)
\end{array}\right|
\)
The use of different charges $b{1}$ and $b{2}$ for the $1 s$ and $2 s$ orbitals destroys their orthogonality, so (10.54) is not normalized. The best values of the variational parameters are found by setting $\partial W / \partial b{1}=0$ and $\partial W / \partial b{2}=0$, where the variational integral $W$ is given by the left side of Eq. (8.9). The results are [E. B. Wilson, Jr., J. Chem. Phys., 1, 210 (1933)] $b{1}=2.686, b{2}=1.776$, and $W=-201.2 \mathrm{eV} . W$ is much closer to the true value -203.5 eV than the result -192.0 eV found in the last section. The value of $b_{2}$ shows substantial, but not complete, screening of the $2 s$ electron by the $1 s$ electrons.
We might try other forms for the orbitals besides (10.52) and (10.53) to improve the trial function. However, no matter what orbital functions we try, if we restrict ourselves to a trial function of the form of (10.54), we can never reach the true ground-state energy. To do this, we can introduce $r{12}, r{23}$, and $r_{13}$ into the trial function or use a linear combination of several Slater determinants corresponding to various configurations (configuration interaction).