Formal Coupled Cluster Theory

The exponential ansatz described above is essential to coupled cluster theory, but we do not yet have a recipe for determining the so-called ``cluster amplitudes'' (t_i^a, t_ij^ab, etc.) which parameterize the power series expansion implicit in Eq. [2.31]. Naturally, the starting point for this analysis is the electronic Schrödinger equation,

$$\hat{H} \vert \Psi \rangle = E \vert \Psi \rangle,$$ (3.1)

where the coupled cluster wavefunction, $\Psi_{\rm CC} \equiv e^{\hat{T}}\Phi_0$, is used to approximate the exact solution, $\Psi$,

$$\hat{H} e^{\hat{T}} \vert \Phi_0 \rangle = E e^{\hat{T}} \vert \Phi_0 \rangle.$$ (3.2)

Using a ``projective'' technique, one may left-multiply this equation by the reference, $\Phi_0$, to obtain an expression for the energy,

$$\langle \Phi_0 \vert \hat{H} e^{\hat{T}} \vert \Phi_0 \rangle = E \langle \Phi_0 \vert e^{\hat{T}} \vert \Phi_0 \rangle = E,$$ (3.3)

where intermediate normalization, $\langle \Phi_0 \vert \Psi_{\rm CC} \rangle = 1$, is assumed. Additionally, one may obtain expressions for the cluster amplitudes by left-projecting the Schrödinger equation by the excited determinants produced by the action of the cluster operator, $\hat{T}$, on the reference,

$$\langle \Phi_{ij\ldots}^{ab\ldots} \vert \hat{H} e^{\hat{T}} ... ..._{ij\ldots}^{ab\ldots} \vert e^{\hat{T}} \vert \Phi_0 \rangle,$$ (3.4)

where $\vert \Phi_{ij\ldots}^{ab\ldots} \rangle$ represents an excited determinant in which orbitals $\phi_i$, $\phi_j$, etc. have been replaced with orbitals $\phi_a$, $\phi_b$, etc.^3.1 Projection by the determinant $\vert \Phi_{ij}^{ab} \rangle$, for example, will produce an equation for the specific amplitude t_ij^ab (coupled to other amplitudes). These equations are non-linear (due to the presence of $e^{\hat{T}}$) and energy dependent. Furthermore, they are formally exact; if the cluster operator, $\hat{T}$, is not truncated, the exact wavefunction within the space spanned by the set of orthogonal one-electron functions, $\phi_p$, may be obtained.