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Practical CC Theory and the Hausdorff Expansion

In the previous 3 sections, formal CC theory was introduced, but as stated at the end of Section 2.6.2 the equations which result are impossible to solve or program due to the coupling of the amplitude equations and their dependence upon the energy. Therefore, in practical CC derivations, theorists exploit a bit of mathematical experience and multiply equation (2.57) through on the left by $e^{-\hat{T}}$. Subsequent left-projection by the reference and excited determinants produces the following new set of energy and amplitude equations:

\begin{displaymath}E = \langle \Phi_0 \vert e^{-\hat{T}} \hat{H} e^{\hat{T}} \vert \Phi_0 \rangle \end{displaymath} (63)

and
$\displaystyle \langle \Phi_{ij \cdots}^{ab \cdots} \vert e^{-\hat{T}} \hat{H} ^{\hat{T}} \vert \Phi_0 \rangle$ = $\displaystyle E \langle \Phi_{ij \cdots}^{ab \cdots} \vert e^{-\hat{T}} e^{\hat{T}} \vert \Phi_0 \rangle$  
  = $\displaystyle E \langle \Phi_{ij \cdots}^{ab \cdots} \vert \Phi_0 \rangle = 0,$ (64)

respectively. Notice that the introduction of the $e^{-\hat{T}}$ operator cancels out its $e^{\hat{T}}$ counterpart in the amplitude equations and guarantees that the right hand side vanishes, taking any dependence of the amplitudes on the energy with it. The similarity transformed Hamiltonian, $e^{-\hat{T}} \hat{H} e^{\hat{T}}$, employed in the above energy and amplitude equations is not a Hermitian operator; therefore, the energy equation does not satisfy any variational conditions where the energy is derived from the Average Value Theorem. Despite this disadvantage, which is considered to be small by a number of theorists, the use of this similarity transformed Hamiltonian has as a second benefit which makes this formulation of the CC equations both practical and desirable: the $e^{-\hat{T}} \hat{H} e^{\hat{T}}$ operator may be expanded as a linear combination of nested commutators
$\displaystyle e^{-\hat{T}} \hat{H} e^{\hat{T}}$ = $\displaystyle \hat{H} + [\hat{H}, \hat{T}] + \frac{1}{2!} [[\hat{H},\hat{T}], \hat{T}] + \frac{1}{3!} [[[\hat{H}, \hat{T}], \hat{T}], \hat{T} ] +$  
    $\displaystyle \frac{1}{4!} [[[[\hat{H}, \hat{T}], \hat{T}], \hat{T}], \hat{T}] + \cdots$ (65)

according to the Campbell-Baker-Hausdorff formula.

While the expansion of the similarity transformed Hamiltonian given above in equation (2.65) may not, at first glance, appear to be a simplification, the sequence of nested commutators truncates due to the structure of the electronic Hamiltionian. The second quantized form of the Hamiltonian, equation (2.35), includes strings containing at most a total of four general-index creation and annihilation operators, and when one evaluates the commutator between the Hamiltonian and the $\hat{T}$ operator, one replaces one of these operators by a Kronecker delta function and, thereby, reduces the number of available general-index operators in the Hamiltonian by one. Therefore, for commutators nested more than four deep, all of the available general-index creation and annihilation operators in the Hamiltonian will be consumed, converted to delta functions, and all that remains are commutations between the $\hat{T}$excitation operators, which commute. Thus, the sequence of nested commutators in equation (2.65) must truncate after the five terms explicitly written. Using this truncated Hausdorff expansion, it is possible to obtain analytic expressions for the commutators which may be inserted into both the energy and amplitude equations. Finally, these equations may then be reduced into expressions that depend only on the amplitudes and the known one- and two-electron integrals. This reduction will not be covered in this dissertation, and so our discussion of basic CC theory comes to a close.

This page maintained by Brian C. Hoffman
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