Basis-set extrapolation techniques for the accurate calculation of molecular equilibrium geometries using coupled-cluster theory

To reduce remaining basis-set errors in the determination of molecular equilibrium geometries, a basis-set extrapolation (BSE) scheme is suggested for the forces used in geometry optimizations. The proposed BSE scheme is based on separating the Hartree-Fock and electron-correlation contributions and uses expressions obtained by straightforward differentiation of well established extrapolation formulas for energies when using basis sets from Dunning's hierarchy of correlation-consistent basis sets. Comparison with reference data obtained at the R12 coupled-cluster level [CCSD(T)-R12] demonstrates that BSE significantly accelerates the convergence to the basis-set limit, thus leading to improvements comparable to or even better than those obtained by increasing the cardinal number in the used basis set by one. However, BSE alone is insufficient to improve agreement with experiment, even after additional consideration of inner-shell correlation and quadruple-excitation effects (mean error and standard deviation with extrapolation are -0.014 and 0.047 pm in comparison with mean error and standard deviation of -0.002 and 0.036 pm without extrapolation). Improvement is obtained only when other contributions of similar magnitude as the BSE contributions (e.g., pentuple-excitation effects and relativistic effects) are also considered. A rather large discrepancy (of the order of a few tenths of a picometer) is observed for the F(2) molecule indicating an enhanced basis-set requirement for the various contributions in this case.