Energy and Analytic Gradients for the Orbital-Optimized Coupled-Cluster Doubles Method with the Density-Fitting Approximation: An Efficient Implementation

Efficient implementations of the orbital-optimized coupled-cluster doubles [or simply ``optimized CCD'', OCCD, for short] method and its analytic energy gradients with the density-fitting (DF) approach, denoted by DF-OCCD, are presented. In addition to the DF approach, the Cholesky-decomposed variant (CD-OCCD) is also implemented for energy computations. The computational cost of the DF-OCCD method {(available in a plugin version of the {\sc DFOCC} module of {\sc Psi4})} is compared with that of the conventional OCCD {(from the {\sc Q-Chem} package)}. The OCCD computations were performed with the {\sc Q-chem} package, in which it is denoted by OD. In the conventional OCCD, one needs to perform four-index integrals transformations at each CCD iterations, which limits its applications to large chemical systems. Our results demonstrate that DF-OCCD provides dramatically lower computational costs compared to OCCD, there are almost 8-fold reductions in the computational time for the \ce{C6H14} molecule with the cc-pVTZ basis set. For open-shell geometries, interaction energies, and hydrogen transfer reactions, DF-OCCD provides significant improvements upon DF-CCD. {Further, the performance of the DF-OCCD method is substantially better for harmonic vibrational frequencies in the case of symmetry breaking problems. Moreover,} several factors make DF-OCCD more attractive compared to CCSD: (1) for DF-OCCD there is no need for orbital relaxation contributions in analytic gradient computations (2) active spaces can readily be incorporated into DF-OCCD (3) DF-OCCD provides accurate vibrational frequencies when symmetry-breaking problems are observed (4) in its response function, DF-OCCD avoids artificial poles; hence, excited-state molecular properties can be computed via linear response theory (5) Symmetric and asymmetric triples corrections based on DF-OCCD [DF-OCCD(T)] has a significantly better performance in near degeneracy regions.