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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
can always be expressed as a
linear combination of the exact nonrelativistic
eigenvectors
,
which span the entire
Nelectron space.

(tex2html_deferredtex2html_deferred3.tex2html_deferred9  = _i c_i  _i
) 
and the energy is given by

(tex2html_deferredtex2html_deferred3.tex2html_deferred10 E =  H  
) 
Now if
is normalized, and we substitue the expansion over
exact eigenfunctions (equation 3.9) into the equation above,
we obtain

(tex2html_deferredtex2html_deferred3.tex2html_deferred11 E = _i c_i^* c_i E_i
) 
where
is the ith energy eigenvalue, i.e.
.
Now subtract
,
the exact nonrelativistic ground state energy, from both sides to obtain

(tex2html_deferredtex2html_deferred3.tex2html_deferred12 E  E_0 = _i c_i^* c_i E_i  E_0
) 
or

(tex2html_deferredtex2html_deferred3.tex2html_deferred13E  E_0 = _i c_i^* c_i ( E_i  E_0 )
) 
since
is normalized and
.
Since
are greater than or equal to
for all
values of i and the coefficients
c_{i}^{*} c_{i} are necessarily nonnegative,
the right hand side of equation (3.12) is nonnegative, so that
.
We should also point out that the variational theorem
holds for the HartreeFock method as well as for CI, since equation
(3.10) is valid for the HartreeFock energyfor 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
in equation (3.9), we could also
have used an expansion over the exact eigenfunctions within the
oneelectron space spanned by
(i.e. we could expand the
approximate CI wavefunction
in terms of the full CI wavefunctions).
This means that not only is the approximate energy an upper bound to the
exact nonrelativistic groundstate energy, but it is also an upper bound
to the full CI energy in the given oneelectron basis.
Next: Why are CoupledCluster and
Up: The Variational Theorem
Previous: The Method of Linear
C. David Sherrill
20000418