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Application of the Variational Theorem to Other States

In this section, we parallel the arguments of Pauling and Wilson [13], p. 186. So far, we have shown only that the energy calculated as the expectation value of some trial function must be an upper bound to the true ground state energy ${\cal E}_0$. In certain cases, we may derive a similar result for other states. If we take a trial function $\vert \Phi_0 \rangle$ such that the first k coefficients in equation (3.9) are zero, then we may subtract ${\cal E}_k$ from equation (3.11) to obtain

 \begin{displaymath}
E - {\cal E}_k = \sum_i c_i^{*} c_i ({\cal E}_i - {\cal E}_k)
\end{displaymath} (tex2html_deferredtex2html_deferred3.tex2html_deferred20 E - E_k = _i c_i^* c_i (E_i - E_k) )

Now since we've assumed $c_j = 0, j = 0, 1, \ldots, k$, this simplifies to

 \begin{displaymath}
E - {\cal E}_k = \sum_{i=k} c_i^{*} c_i ({\cal E}_i - {\cal E}_k)
\end{displaymath} (tex2html_deferredtex2html_deferred3.tex2html_deferred21 E - E_k = _i=k c_i^* c_i (E_i - E_k) )

Once again, we can see that every term on the right side is nonnegative, so $E - {\cal E}_k \geq 0$.

There are any number of cases in which we have a trial function of the form just described. Consider, for example, a calculation on a triplet state for a molecule which has a singlet ground state. If our trial function is constrained to be a triplet, then all singlet eigenfunctions $\vert \Psi_i \rangle$ will have zero coefficients in the expansion of the trial function. In this case, the energy we minimize from the triplet trial function will be an upper bound to the lowest triplet energy, even though there is a lower-lying singlet state. Similar arguments can be made for spatial symmetry.


next up previous contents index
Next: Convergence of the Wavefunction Up: The Variational Theorem Previous: Why are Coupled-Cluster and
C. David Sherrill
2000-04-18