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A consequence of the variational theorem is
that as the energy *E* of an approximate variational wavefunction
approaches the exact energy
,
the approximate wavefunction
approaches the exact one
.
This follows from
equation (3.12), which shows that as the energy *E* is minimized
(or, equivalently, as
is minimized), then
is minimized; that is, the sum of squares
of the absolute values of the coefficients of excited states with weight
factors
is minimized. It is apparent that
these weight factors might not be optimal if we want the
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
,
then
,
or *c*_{0} = 1, implying that
.
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.

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** Up:** The Variational Theorem
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*C. David Sherrill*

*2000-04-18*