Coupled Cluster - Coupled-cluster Equations

Coupled-cluster Equations

Coupled-cluster equations are equations whose solution is the set of coefficients t. There are several ways of writing such equations but the standard formalism results in a terminating set of equations which may be solved iteratively. The naive variational approach does not take advantage of the connected nature of the cluster amplitudes and results in a non-terminating set of equations. The coupled cluster Schrödinger equation is formally:

Suppose there are q coefficients t to solve for. Therefore, we need q equations. It is easy to notice that each t-coefficient may be put in correspondence with a certain excited determinant: corresponds to the determinant obtained from by substituting the occupied orbitals i,j,k,... with the virtual orbitals a,b,c,... Projecting the Schrödinger equation above by q such different determinants from the left, we obtain the sought-for q equations:

where by we understand the whole set of the appropriate excited determinants. To manifest the connectivity of these equations, we can reformulate the above equation in a more convenient form. We apply to both sides of the coupled-cluster Schroedinger equations. After this we project the Schroedinger equation to and, and obtain:

,

the latter being the equations to be solved and the former the equation for the evaluation of the energy. Consider the standard CCSD method:

,

,

,

in which the similarity transformed Hamiltonian (defined as ) can be explicitly written down with the BCH formula:

.

The resulting similarity transformed Hamiltonian is not hermitian. Standard quantum chemistry packages (ACES II, NWChem, etc.) solve the coupled-equations iteratively using the Jacobi updates and the DIIS extrapolations of the t amplitudes.