The Hausdorff Expansion

Although the energy and amplitudes expressions (Eqs. [3.3] and [3.4], respectively) are useful for gaining a formal understanding of the coupled cluster method, they are not amenable to practical computer implementation.[90] One must first rewrite these expressions in terms of the one- and two-electron integrals arising from the electronic Hamiltonian as well as the cluster amplitudes, which, apart from the energy itself, are the only unknown quantities. To that end, it is convenient to exercise mathematical foresight and multiply the Schrödinger equation (Eq. [3.2]) by the inverse of the exponential operator, $e^{-\hat{T}}$. Upon subsequent left-projection by the reference, $\Phi_0$, and the excited determinants, $\Phi_{ij\ldots}^{ab\ldots}$, one obtains modified energy and amplitude equations,

$$\langle \Phi_0 \vert e^{-\hat{T}}\hat{H}e^{\hat{T}}\vert \Phi_0 \rangle = E$$ (3.9)

and

$$\langle \Phi_{ij\ldots}^{ab\ldots} \vert e^{-\hat{T}}\hat{H}e^{\hat{T}}\vert \Phi_0 \rangle = 0,$$ (3.10)

respectively, which involve the similarity-transformed Hamiltonian, $e^{-\hat{T}} \hat{H} e^{\hat{T}}$. Equations [3.9] and [3.10] define the conventional coupled cluster method. It may be shown that these expressions are equivalent to Eqs. [3.3] and [3.4][5,80], but with two advantages. First, the amplitude equations (Eq. [3.10]) are now decoupled from the energy equation (Eq. [3.9]). Second, a simplification via the so-called Campbell-Baker-Hausdorff formula[91] of $e^{-\hat{T}} \hat{H} e^{\hat{T}}$ leads to a linear combination of nested commutators of $\hat{H}$ with the cluster operator, $\hat{T}$, viz.

 

$$e^{-\hat{T}}\hat{H}e^{\hat{T}} \hat{H} + \left[\hat{H},\hat{T}\right] + \frac{1}{2!}\left[\left[... ...{1}{3!}\left[\left[\left[\hat{H},\hat{T}\right],\hat{T}\right],\hat{T}\right] +$$ =

$$\frac{1}{4!}\left[\left[\left[\left[\hat{H},\hat{T}\right],\hat{T}\right], \hat{T}\right],\hat{T}\right] + \ldots.$$ (3.11)

This expression is usually referred to simply as the Hausdorff expansion, and although it may not immediately appear to be a simplification of the coupled cluster equations, the infinite series truncates naturally in a manner somewhat analogous to that described earlier for the operator, $\hat{H}e^{\hat{T}}$.

As shown explicitly in Refs. 80, 84, and 92, the creation and annihilation operators described earlier may be used to represent dynamical operators such as the electronic Hamiltonian:

$$\hat{H} = \sum_{pq} h_{pq} a_{p}^{\dagger} a_{q}^{\ } + \frac... ...\rangle a_{p}^{\dagger} a_{q}^{\dagger} a_{s}^{\ } a_{r}^{\ }.$$ (3.12)

In this expression, $h_{pq} \equiv \langle \phi_p \vert \hat{h} \vert \phi_q \rangle$represents a matrix element of the one-electron component of the Hamiltonian, $\hat{h}$, while $\langle pq \vert\vert rs \rangle \equiv \langle \phi_p \phi_q \vert \phi_r \phi_s \rangle - \langle \phi_p \phi_q \vert \phi_s \phi_r \rangle$ is its antisymmetrized two-electron counterpart. Equation [3.12] contains general annihilation and creation operators (e.g., $a_{p}^{\dagger}$ or $a_{q}^{\ }$) which may act on orbitals in either the occupied or virtual subspaces. The cluster operators, $\hat{T}_n$, on the other hand, contain operators which are restricted to act in only one of these spaces (e.g., $a_{b}^{\dagger}$ which may act only on the virtual orbitals). As pointed out earlier, the cluster operators therefore commute with one another, but not with the Hamiltonian, $\hat{H}$. For example, consider the commutator of the pair of general second-quantized operators from the one-electron component of the Hamiltonian in Eq. [3.12] with the single-excitation pair found in the cluster operator, $\hat{T}_1$:

$$\left[ a_{p}^{\dagger} a_{q}^{\ }, a_{a}^{\dagger} a_{i}^{\ }... ...^{\ } - a_{a}^{\dagger} a_{i}^{\ } a_{p}^{\dagger} a_{q}^{\ }.$$ (3.13)

The anticommutation relations of annihilation and creation operators given in Eqs. [2.19], [2.20], and [2.21] may be applied to the two terms on the right-hand side of this expression to give

$$\left[ a_{p}^{\dagger} a_{q}^{\ }, a_{a}^{\dagger} a_{i}^{\ } \right] a_{p}^{\dagger} \delta_{qa} a_{i}^{\ } - a_{a}^{\dagger} \delta_{ip} a_{q}^{\ },$$ = (3.14)

The Kronecker delta functions, $\delta_{qa}$ and $\delta_{ip}$, resulting from Eq. [2.21] cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of $\hat{H}$ and $\hat{T}$ serves to eliminate one of the electronic Hamiltonian's general-index annihilation or creation operators in favor of a simple delta function. Since $\hat{H}$ contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from $\hat{H}$ will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher-order terms will contain commutators of only the cluster operators, $\hat{T}$, and are therefore zero. Hence, Eq. [3.11] truncates itself naturally after the first five terms shown.[80] This convenient property results entirely from the two-electron property of the Hamiltonian and the fact that the cluster operators commute; it is not dependent on the number of electrons in the system, the level of substitution included in $\hat{T}$, or any consideration of the types of determinants upon which the operators act.

Using the truncated Hausdorff expansion, we may obtain analytic expressions for the commutators in Eq. [3.11] and insert these into the coupled cluster energy and amplitude equations (Eqs. [3.9] and [3.10], respectively). However, this is only the first step in obtaining expressions which may be efficiently implemented on the computer. We must next choose a truncation of $\hat{T}$ and then derive expressions containing only one- and two-electron integrals and cluster amplitudes. This is a formidable task to which we will return in later sections.