The Variational Method

By rearranging equation (1.7) it is possible to obtain an expression for the energy, E_app, associated with an approximate wavefunction, $\Psi_{app}$.
$$\begin{align}E_{app} = \frac{\int \Psi_{app}^{*} \hat{H} \Psi_{app} \, d\tau}{\i... ...t{H}\vert\Psi_{app}\rangle}{\langle\Psi_{app}\vert\Psi_{app}\rangle} \end{align}$$
This approximate wavefunction can also be written as a linear combination of the exact eigenstates of $\hat{H}$:
$$\begin{align}\Psi_{app} = \sum_{i} c_{i} \Psi_{i}. \end{align}$$
Now, consider the integral
$$\begin{align}\begin{split} \int \Psi_{app}^{*} (\hat{H} - E_{0}) \Psi_{app} \, d... ...au \\ = & \sum_{i} c_{i}^{*} c_{i} (E_{i} - E_{0}) \ge 0 \end{split}\end{align}$$
as $\vert c_{i}\vert^{2} \ge 0$ and $E_{i} \ge E_{0}$. Therefore,
$$\begin{align}\int \Psi_{app}^{*} (\hat{H} - E_{0}) \Psi_{app} \, d\tau \ge 0, \end{align}$$
from which follows the Variational Principle, that $E_{app} \ge E_{0}$. This also gives insight into how to find the best approximate wavefunction, as
$$\begin{align}E_{app} = E_{0} \Longleftrightarrow \Psi_{app} = \Psi_{0}. \end{align}$$
Hence, minimization of E_app with respect to all allowed $\Psi_{app}$ will give the exact ground state energy and wavefunction. Unfortunately, this is not practical and what is usually done is to expand the molecular orbitals as linear combinations of basis functions,
 $$\begin{align} \psi_{i} = \sum_{\mu} c_{\mu i} \phi_{\mu}, \end{align}$$
reducing the problem to finding the optimum set of coefficients, $c_{\mu i}$, which minimize E_app, something which can be achieved via matrix diagonalisation (see section 1.5). Obviously, the set of functions $\phi_{\mu}$ cannot be complete and so an approximation has been made. The number and type of functions chosen has a large effect on the overall accuracy (and speed) of a calculation. Some of the more common functions used will be given in the section on Basis Sets (section 1.7).