A Variational Coupled Cluster Theory?

The ``projective'' techniques described above for solving the coupled cluster equations represent a particularly convenient way of obtaining the amplitudes which define the coupled cluster wavefunction, $e^{\hat{T}} \Phi_0$. However, the asymmetric energy formula shown in Eq. [3.9] does not conform to any variational conditions where the energy is determined from an expectation value equation. As a result, the computed energy will not be an upper bound to the exact energy in the event that the cluster operator, $\hat{T}$, is truncated. But the exponential ansatz does not require that we solve the coupled cluster equations in this manner. We could, instead, construct a variational solution by requiring that the amplitudes minimize the expression[1,2]

$$E_{exact} \leq E = \frac{\langle \Phi_0 \vert(e^{\hat{T}})^{\... ...vert\hat{H}\vert \Psi \rangle}{\langle \Psi \vert\Psi\rangle}.$$ (3.15)

Unfortunately, this equation is considerably more complex than the projective energy expression given in Eq. [3.9] since there is no natural truncation of its power series expansion,

$$\langle \Phi_0 \vert(e^{\hat{T}})^{\dagger}\hat{H}e^{\hat{T}}... ...rac{1}{2}\left(\hat{T}\right)^2 + \ldots)\vert \Phi_0 \rangle.$$ (3.16)

For example, in the term $\langle \Phi_0 \vert \hat{T}^{\dagger} \hat{H} \hat{T} \vert \Phi_0 \rangle$, which is included in the above equation, as $\hat{T}$ creates an excited determinant from $\vert \Phi_0 \rangle$ on the right, $\hat{T}^{\dagger}$ creates an excited determinant from $\langle \Phi_0 \vert$ on the left. Thus, the Hamiltonian matrix elements will not vanish at some high excitation level, and the series will not terminate before the N-electron limit. Truncation of this expression for large numbers of terms appears to be arbitrary at best.

The ostensible impracticality of a variational coupled cluster theory raises an important question as to the physical reality of the coupled cluster energy as computed using projective, asymmetric techniques. Quantum mechanics dictates that physical observables (such as the energy) are expectation values of Hermitian operators. The coupled cluster energy expression contains the operator $e^{-\hat{T}} \hat{H} e^{\hat{T}}$, which is not Hermitian, regardless of the truncation of $\hat{T}$:^3.2

$$\left( e^{-\hat{T}} \hat{H} e^{\hat{T}} \right)^{\dagger} = \... ... e^{-\hat{T}^{\dagger}} \neq e^{-\hat{T}} \hat{H} e^{\hat{T}}.$$ (3.17)

However, if $\hat{T}$ is not truncated, the similarity transformed operator has an energy eigenvalue spectrum that is identical to the original Hermitian operator, $\hat{H}$, thus justifying its formal use in quantum mechanical models. Practically speaking, the coupled cluster energy tends to closely approximate the expectation value result even when $\hat{T}$ is truncated. Furthermore, one might speculate that some measure of the difference between the expectation value and asymmetric energies -- perhaps as measured by the asymmetry of the coupled cluster reduced density[65] -- might prove to be a useful diagnostic of the reliability of results obtained from the coupled cluster method for specific systems. This issue has been recently discussed by Kutzelnigg.[93]

Variational coupled cluster methods that make use of Eq. [3.16] have been studied by several researchers. The unitary coupled cluster (UCC) approach in which the cluster operator $\hat{T}$ is replaced by $\hat{T}-\hat{T}^{\dagger}$ (where $\hat{T}^{\dagger}$ indicates a de-excitation operator which is the Hermitian adjoint of $\hat{T}$) was pursued by Hoffmann and Simons[94,95]. The infinite series in this case is not truncated arbitrarily, but instead by identifying which terms are needed to complete the series through a particular order of perturbation theory. Bartlett and Noga have constructed an alternative theory, termed the expectation value coupled cluster (XCC) method[96], in which the usual definition of $\hat{T}$ is retained and Eq. [3.16] is used, but again the series truncation is based on perturbation theory arguments. Finally, we note the extended coupled cluster method (ECCM) of Arponen and Bishop[97,98], which uses a modified energy functional including an additional exponentiated deexcitation operator analogous to $e^{\hat{T}^{\dagger}}$. These as well as other variational and semi-variational approaches to the cluster expansion have been reviewed recently by Bartlett et al.[99] and by Szalay et al.[100]