Configuration Interaction and
Coupled Cluster Methods 		previous next

The CC Energy and Amplitude Equations

Having established that the CI and CC formalisms differ primarily in the method in which the excitation operators $\hat{C}$ and $\hat{T}$ operate on the reference, linearly for CI and exponentailly for CC, the next step is to examine how one derives working CC equations. Beginning from the universial starting point, the Schrödinger equation, one substitues in the form of the CC wave function given by equation (2.56) and finds

\begin{displaymath}\hat{H} e^{\hat{T}} \vert \Phi_0 \rangle = E e^{\hat{T}}. \end{displaymath} (57)

Projecting through on the left by the reference, $\vert \Phi_0 \rangle$, one can obtain an expression for the energy

\begin{displaymath}\langle \Phi_0 \vert \hat{H} e^{\hat{T}} \vert \Phi_0 \rangle = E \langle \Phi_0 \vert e^{\hat{T}} \vert \Phi_0 \rangle = E, \end{displaymath} (58)

provided one employs the technique of intermediate normalization and sets the overlap between the reference and the CC wave function, $\langle \Phi_0 \vert \Psi_{CC} \rangle$, to unity. Obtaining an energy exprssion is only the first step, however; one must also determine all of the cluster amplitudes which define the wave function with this energy. In order to accomplish this feat, one must left-project equation (2.57) by the excited determinants produced by the action of the $\hat{T}$ operator:

\begin{displaymath}\langle \Phi_{ij \cdots}^{ab \cdots} \vert \hat{H} e^{\hat{T}... ...ij \cdots}^{ab \cdots} \vert e^{\hat{T}} \vert \Phi_O \rangle. \end{displaymath} (59)

For example, one can produce an equation for the specific amplitude tij ab by left-projecting by the $\vert \Phi_{ij}^{ab} \rangle$ excited determinant. The mathematics for extracting this coefficient are slightly tedious, but primarily involve standard manipulations of second-quantized operators and, as such, will not be belabored here. Unfortunately, the resulting equation is a programmers nightmare because it is non-liner and depends upon the energy as well as other cluster amplitudes. However, these equations are exact, and if one were able to solve them with the full $\hat{T}$ operator, one would indeed obtain the full CI energy and wave function.
This page maintained by Brian C. Hoffman
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