Brian Hoffman's Quantum Chemistry Notes

Configuration Interaction and
Coupled Cluster Methods

Power Series Expansion and Truncation

The CC method depends upon the action of the exponential excitation operator $e^{\hat{T}}$ upon the reference, and so far a practical means of excecuting this operation has not been adressed. The trick to employing this excitation operator is to realize that it may be expanded as the power series

$\begin{displaymath}e^{\hat{T}} = 1 + \hat{T} + \frac{\hat{T}^2}{2!} + \frac{\hat{T}^3}{3!} + \cdots . \end{displaymath}$

(60)

As a matter of fact, the equivalence of $e^{\hat{T}}$ and this power series is commonly used in the various arguments employed to justify the exponential ansatz. If one inserts the exponential opertator into the expression and then separates out the various terms, one obtains the expression

$\begin{displaymath}E= \langle \Phi_0 \vert \hat{H} \vert \Phi_0 \rangle + \langl... ...ert \hat{H} \frac{\hat{T}^3}{3!} \vert \Phi_0 \rangle + \cdots \end{displaymath}$

(61)

from which one can find another benefit of the exponential formalism. Recall that the Hamiltonian operator only includes one- and two- particle operators, and thus, according to Slater's rules, matrix elements of the Hamiltonian between determinants which differ by more than two spin orbitals must vanish. Therefore, the third and subsequent terms in the above expansion, in which the $\hat{T}$ operator is raised to the third or higher power and can thus produce only triply or higher excited determinants when operated upon the reference, must also vanish and the energy expression is neatly and rigidly truncated to

$\begin{displaymath}E= \langle \Phi_0 \vert \hat{H} \vert \Phi_0 \rangle + \langl... ...hi_0 \vert \hat{H} \frac{\hat{T}^2}{2!} \vert \Phi_0 \rangle . \end{displaymath}$

(62)

This is a natural truncation of the CC equations due to the nature of the Hamiltonian and also applies to the amplitude equations, although the exact range of allowed powers of $\hat{T}$ will vary from that seen for the energy expression.

This page maintained by Brian C. Hoffman