6.11.2 (2) Triples and Quadruples Corrections

While the (T) corrections discussed above have been extraordinarily successful, there is nonetheless still room for further improvements in accuracy, for at least some important classes of problems. They contain judiciously chosen terms from 4th- and 5th-order Møller-Plesset perturbation theory, as well as higher order terms that result from the fact that the converged cluster amplitudes are employed to evaluate the 4th- and 5th-order order terms. The (T) correction therefore depends upon the bare reference orbitals and orbital energies, and in this way its effectiveness still depends on the quality of the reference determinant. Since we are correcting a coupled-cluster solution rather than a single determinant, this is an aspect of the (T) corrections that can be improved. Deficiencies of the (T) corrections show up computationally in cases where there are near-degeneracies between orbitals, such as stretched bonds, some transition states, open shell radicals, and diradicals.

Prof. Steve Gwaltney, while working at Berkeley with Martin Head-Gordon, has suggested a new class of non iterative correction that offers the prospect of improved accuracy in problem cases of the types identified above.⁴⁰⁷ Q-Chem contains Gwaltney’s implementation of this new method, for energies only. The new correction is a true second-order correction to a coupled-cluster starting point, and is therefore denoted as (2). It is available for two of the cluster methods discussed above, as OD(2) and CCSD(2).^{407, 408} Only energies are available at present.

The basis of the (2) method is to partition not the regular Hamiltonian into perturbed and unperturbed parts, but rather to partition a similarity-transformed Hamiltonian, defined as $\tilde{H}=e^{-\widehat{T}}\widehat{H}{e}^{\widehat{T}}$. In the truncated space (call it the $p$-space) within which the cluster problem is solved (e.g., singles and doubles for CCSD), the coupled-cluster wave function is a true eigenvalue of $\tilde{H}$. Therefore we take the zero order Hamiltonian, ${\tilde{H}}^{(0)}$, to be the full $\tilde{H}$ in the p-space, while in the space of excluded substitutions (the q-space) we take only the one-body part of $\tilde{H}$ (which can be made diagonal). The fluctuation potential describing electron correlations in the $q$-space is $\tilde{H}-{\tilde{H}}^{(0)}$, and the (2) correction then follows from second-order perturbation theory.

The new partitioning of terms between the perturbed and unperturbed Hamiltonians inherent in the (2) correction leads to a correction that shows both similarities and differences relative to the existing (T) corrections. There are two types of higher correlations that enter at second-order: not only triple substitutions, but also quadruple substitutions. The quadruples are treated with a factorization ansatz, that is exact in 5th order Møller-Plesset theory,⁶⁰⁵ to reduce their computational cost from $N^9$ to $N^6$. For large basis sets this can still be larger than the cost of the triples terms, which scale as the 7th power of molecule size, with a factor twice as large as the usual (T) corrections.

These corrections are feasible for molecules containing between four and ten first row atoms, depending on computer resources, and the size of the basis set chosen. There is early evidence that the (2) corrections are superior to the (T) corrections for highly correlated systems.⁴⁰⁷ This shows up in improved potential curves, particularly at long range and may also extend to improved energetic and structural properties at equilibrium in problematical cases. It will be some time before sufficient testing on the new (2) corrections has been done to permit a general assessment of the performance of these methods. However, they are clearly very promising, and for this reason they are available in Q-Chem.