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Why are CoupledCluster and MBPT Energies not Variational?
Electron correlation methods other than CI may not
be variational. For example, consider the coupledcluster
energy expression

(tex2html_deferredtex2html_deferred3.tex2html_deferred14 E = _0  e^T H e^T  _0
_0  _0
) 
If the operator
is not trunctated, then we know that
.
Generally, however, the
operator is truncated. Let us define
for our truncated .
Now define
.
Note that in general
,
which would have occured had
we used
on the left. Then the
energy expression is

(tex2html_deferredtex2html_deferred3.tex2html_deferred15 E = _B  H  _A _0  _0
) 
which, after expansion over the complete set of eigenvectors, becomes

(tex2html_deferredtex2html_deferred3.tex2html_deferred16 E = _ij c_i^* d_j _i  H  _j
_0  _0
) 
This simplifies to

(tex2html_deferredtex2html_deferred3.tex2html_deferred17E = _i c_i^* d_i E_i_0  _0
) 
At this point we can go no farther, because the terms
c_{i}^{*} d_{i} may
be negative, in contrast to the situation in equation (3.12).
For completeness, we also show that MBPT energies are not variational.
The nth order MBPT wavefunction may be written [12] as

(tex2html_deferredtex2html_deferred3.tex2html_deferred18 _MBPT^(n) =  _0 +
_k=1^n [ V (1   _0 _0 )
E_0  H_0 ] ^k  _0 _L
) 
where the sum is over ``linked diagrams'' only. The nth order energy
is then given by

(tex2html_deferredtex2html_deferred3.tex2html_deferred19E_MBPT^(n) = _0  H  _MBPT^(n1)
) 
Since this integral is not symmetric, the energy is not variational.
Only the firstorder perturbation theory energy (which is also the
HartreeFock energy) is variational, since it uses
.
Next: Application of the Variational
Up: The Variational Theorem
Previous: Variational Theorem for the
C. David Sherrill
20000418