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Next: Why are Coupled-Cluster and Up: The Variational Theorem Previous: The Method of Linear

Variational Theorem for the Ground State

One particularly nice feature of the CI method is that the calculated lowest energy eigenvalue is always an upper bound to the exact ground state energy. Our approximate wavefunction $\vert \Theta \rangle$can always be expressed as a linear combination of the exact nonrelativistic eigenvectors $\vert \Psi_i \rangle$, which span the entire N-electron space.

 \begin{displaymath}
\vert \Theta \rangle = \sum_{i} c_i \vert \Psi_i \rangle
\end{displaymath} (tex2html_deferredtex2html_deferred3.tex2html_deferred9 | = _i c_i | _i )

and the energy is given by

 \begin{displaymath}
E = \frac{\langle \Theta \vert \hat{H} \vert \Theta \rangle}{\langle \Theta \vert \Theta \rangle}
\end{displaymath} (tex2html_deferredtex2html_deferred3.tex2html_deferred10 E = | H | | )

Now if $\vert \Theta \rangle$ is normalized, and we substitue the expansion over exact eigenfunctions (equation 3.9) into the equation above, we obtain

 \begin{displaymath}
E = \sum_i c_i^{*} c_i {\cal E}_i
\end{displaymath} (tex2html_deferredtex2html_deferred3.tex2html_deferred11 E = _i c_i^* c_i E_i )

where ${\cal E}_i$ is the ith energy eigenvalue, i.e. $\hat{H} \vert \Psi_i \rangle = {\cal E}_i \vert \Psi_i \rangle$. Now subtract ${\cal E}_0$, the exact nonrelativistic ground state energy, from both sides to obtain

 \begin{displaymath}
E - {\cal E}_0 = \sum_i c_i^{*} c_i {\cal E}_i - {\cal E}_0
\end{displaymath} (tex2html_deferredtex2html_deferred3.tex2html_deferred12 E - E_0 = _i c_i^* c_i E_i - E_0 )

or

\begin{displaymath}E - {\cal E}_0 = \sum_i c_i^{*} c_i ( {\cal E}_i - {\cal E}_0 )
\end{displaymath} (tex2html_deferredtex2html_deferred3.tex2html_deferred13E - E_0 = _i c_i^* c_i ( E_i - E_0 ) )

since $\vert \Theta \rangle$ is normalized and $\sum_i c_i^{*} c_i = 1$. Since ${\cal E}_i$ are greater than or equal to ${\cal E}_0$ for all values of i and the coefficients ci* ci are necessarily non-negative, the right hand side of equation (3.12) is non-negative, so that $E \geq {\cal E}_0$. We should also point out that the variational theorem holds for the Hartree-Fock method as well as for CI, since equation (3.10) is valid for the Hartree-Fock energy--for a given set of MO's, the HF energy can be formulated as a (trivial) 1 x 1 CI eigenvalue problem. In a similar manner, the MCSCF method (where MO's and CI coefficients are optimized) is also ``variational.''

It should be clear that instead of using the exact nonrelativistic eigenfunctions $\vert \Psi_i \rangle$ in equation (3.9), we could also have used an expansion over the exact eigenfunctions within the one-electron space spanned by $\vert \Theta \rangle$ (i.e. we could expand the approximate CI wavefunction $\vert \Theta \rangle$ in terms of the full CI  wavefunctions). This means that not only is the approximate energy an upper bound to the exact nonrelativistic ground-state energy, but it is also an upper bound to the full CI  energy in the given one-electron basis.


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