Configuration Interaction and
Coupled Cluster Methods 		previous next

An Exponential Ansatz

Before embarking on a discussion of CC theory it is beneficial to take a step back and reformulate CI theory in a manner which should ease the introduction to CC theory and highlight the major difference between the CI and CC theories. In CI theory, the wave function may be written as a linear combination of all excited states relative to a chosen reference configuration, equation (2.11). Using the notation of second quantization, one can introduce an excitation operator $\hat{C_n}$ which, when acted on the reference, generates a linear combination of all possible n-tuply excited configurations:

\begin{displaymath}C_n = \left( \frac{1}{n!}^2 \right) \sum_{ij \cdots ab \cdots... ...cdots}^{ab \cdots} a_a^{\dagger} a_b^{\dagger} \cdots a_j a_i. \end{displaymath} (51)

The $c_{ij \cdots}^{ab \cdots}$ coefficients are the CI coefficents for the configurations produced by the action of the string of creation and annihilation operators on the reference. Making use of these excitation operators, the CI wave function given by equation (2.11) may be rewritten as
$\displaystyle \Psi_{CI}$ = $\displaystyle (1+ \hat{C}) \vert \Phi_0 \rangle$ (52)
$\displaystyle {\rm where} \; \hat{C}$ = $\displaystyle \hat{C_1} + \hat{C_2} + \cdots = \sum_n^N \hat{C_n}.$ (53)

In this notation, it is possible to constuct any CI wave function which is truncated solely on the basis of excitation level by including only the desired $\hat{C_n}$ excitation operators in $\hat{C}$. However, this notation becomes impractical when attempting to describe complex MRCI and RAS CI wave functions.

The CC method employs an excitation operator $\hat{T}$ which is identical in form to the $\hat{C}$ operator of CI theory,

$\displaystyle \hat{T}$ = $\displaystyle \hat{T_1} + \hat{T_2} + \cdots = \sum_n^N \hat{T_n}.$ (54)
$\displaystyle {\rm with} \; T_n$ = $\displaystyle \left( \frac{1}{n!}^2 \right) \sum_{ij \cdots ab \cdots}^{n} t_{ij \cdots}^{ab \cdots} a_a^{\dagger} a_b^{\dagger} \cdots a_j a_i,$ (55)

but instead of acting on the reference in a linear fashion, the $\hat{T}$operator of CC theory acts exponentially:

\begin{displaymath}\Psi_{CC} = e^{\hat{T}} \vert \Phi_0 \rangle. \end{displaymath} (56)

For historical reasons, the $t_{ij \cdots}^{ab \cdots}$ coefficients in the $\hat{T_n}$ operators are known as cluster amplitudes. Analagous to CI theory, an excitation truncated CC method may be constructed by including only the desired excitation operators within $\hat{T}$. For example, the popular CCSD method is realized when only the $\hat{T_1}$ and $\hat{T_2}$operators are included within $\hat{T}$. Finally, the theoretical basis for employing the exponential formalism instead of remaining with the linear ansatz of CI theory will not be discussed in this dissertation except to note that the exponential approach produces a method which is both size consistent and size extensive, provided the reference function possesses these qualities, even when $\hat{T}$ is truncated at a chosen excitation level.
This page maintained by Brian C. Hoffman
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