# Is there a rule of thumb to predict when coupled cluster might dip below the variational limit?

Coupled cluster is a non-variational method, meaning that it can give energies that are below the true FCI energy (the "variational limit" for variational methods).

Often coupled cluster still gives energies far above the true FCI energy, and sometimes it gives energies that dip below the variational limit.

Are there some guidelines on when this will happen and when we are safe to assume it will not? I'm aware that coupled cluster generally works best when there is one dominant determinant (single-reference cases), so I am asking about guidelines within the single-references case, when we might expect to get energies that violate the variational principle.

• This doesn't exactly address your question, which is a very good one. But, Sherrill and colleagues wrote a 2005 paper on diatomic carbon as a benchmark to compare the validity of CCSD(T) and MR-CI. Reviewing the abstract reveals that coupled clusters provided good agreement with MR-CI between 1.0 and 2.0 angstroms: doi.org/10.1063/1.1867379 This older presentation of Meissner has an excellent graph of deviation of CCSD and MR-CI from FCI, which could be considered as an exact solution to the equation: academia.edu/download/42658253/Meissner1.pdf – Bertram Dec 14 '19 at 4:05
• @Bertram, the second link leads (for me) to the message "Oops! It looks like you're in the wrong aisle". As for the Sherrill paper, I really don't see any connection between that paper and my question, at all. – user1271772 Dec 16 '19 at 3:10

Let us begin with the following principle: the CC ansatz always returns an appropriate electronic wavefunction (meaning a combination of Slater determinants). Still, at least in a non-relativistic approach, the Hamiltonian does commute with the Spin operator $$\begin{equation} S^{2}=S_{+}S_{-}-S_{z}+S_{z}^{2} \end{equation}$$ (I'm using atomic units here) and we want the wavefunction to be an eigenfunction of total spin. The difficulty in doing so is due to the fact that the excitation operator to ensure spin purity is not easy to parametrize unless you are lucky enough to have a $$S^{2}=0$$ state and your system wavefunction is represented by a $$S^{2}=0$$. So, when you're starting with a wavefunction which is not a spin eigenfunction (for example the ground state of the nitrogen atom with the unpaired electrons), CC might not be able to recover the spin purity and, by allowing superposition of functions with different spin, you might get an energy lower than the correct one because you're looking for the minimum in a space bigger than the one you are physically allowed to . Hope I was clear