Coupled-Cluster Theory for Molecular Structure and Spectra: The Challenges Posed By Molecules and the Coupled-Cluster Solutions

Coupled-cluster (CC) theory derives from the ansatz that the n-particle wavefunction is $\vert \Psi >$=exp(T)$\vert $0$>$, where T is an excitation operator with $\vert $0$>$ some choice of mean-field wavefunction. That is sufficient to obtain energies. But to obtain anything else, we use the CC functional, E=$<$0$\vert $(!+$\Lambda )$exp(-T)Hexp(T)$\vert $0$>$, whose left and right hand eigenvectors provide energies and associated density matrices for the treatment of properties in CC theory. The introduction of $\Lambda $ makes it possible to obtain the $\sim $3N forces associated with N atoms in the same time as the energy itself. This is essential information for indentifying the critical points on a potential energy surface and their associated Hessians, for either the prediction of vibrational spectra or to characterize a saddle point (transition state) for a reaction. A generalization of the functional to $\omega $(k) =$<$0$\vert $L(k) exp(-T)Hexp(T)R(k)$\vert $0$>$, provides excitation energies, $\omega $(k) along with excited state left- and right-hand wavefunctions, Finally, with the response functions obtained from these left- and right-hand eigenfunctions, used in closed form, higher-order properties like NMR coupling constants are obtained. In this way, coupled-cluster theory provides a method that addresses all the properties of interest for molecules and their interactions. This development will be the topic of our contribution. For details please see, R. J. Bartlett and M Musial, ``Coupled-cluster theory in quantum chemistry'', Revs. of Modern Phys. \textbf{79, }291-352 (2007).