Convergence of the Wavefunction

A consequence of the variational theorem is that as the energy E of an approximate variational wavefunction approaches the exact energy ${\cal E}_0$, the approximate wavefunction $\vert \Phi \rangle$ approaches the exact one $\vert \Psi_0 \rangle$. This follows from equation (3.12), which shows that as the energy E is minimized (or, equivalently, as $E - {\cal E}_0$ is minimized), then $\sum_i c_i^{*} c_i ( {\cal E}_i - {\cal E}_0 )$ is minimized; that is, the sum of squares of the absolute values of the coefficients of excited states with weight factors $( {\cal E}_i - {\cal E}_0)$ is minimized. It is apparent that these weight factors might not be optimal if we want the $\vert \Phi \rangle$ which gives the best value for a property other than the energy, such as dipole moment. However, in the limit that E is minimized with a sufficiently large basis so that $E = {\cal E}_0$, then $\sum_{i=1} c_i^{*} c_i ( {\cal E}_i - {\cal E}_0 ) = 0$, or c₀ = 1, implying that $\vert \Phi \rangle = \vert \Psi_0 \rangle$. Once we have the exact wavefunction, then of course all properties can be computed exactly (again, within the Born-Oppernheimer approximation and neglecting relativistic effects). To summarize, variational improvements in the energy give improvements in the approximate wavefunction, which in turn improves the values of all other properties; however, these other properties will not necessarily converge as fast as the energy with respect to N-electron basis set improvement.