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 that define the coupled cluster wavefunction, e l o However, the asymmetric energy formula shown in Eq. [50] does not conform to any variational conditions in which 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, 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  

Unfortunately, this equation is considerably more complex than the projective energy expression given in Eq. [50], since there is no natural truncation of its power series expansion, 

For example, in the term (OolT HTIa)o) which is included in Eq. [57], as T creates an excited determinant from IOq) on the right, T creates an excited determinant from (OqI 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 regarding 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 

On the other hand, the inverse of the exponentiated excitation operator, e T is also an excitation operator, as can be seen from its power series expansion. 

---