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## Why are Coupled-Cluster and MBPT Energies not Variational?

Electron correlation methods other than CI may not be variational. For example, consider the coupled-cluster 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 ci* di 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^(n-1) )

Since this integral is not symmetric, the energy is not variational. Only the first-order perturbation theory energy (which is also the Hartree-Fock energy) is variational, since it uses .

Next: Application of the Variational Up: The Variational Theorem Previous: Variational Theorem for the
C. David Sherrill
2000-04-18