The analytic gradient for the coupled pair functional method: Formula and application for HCl, H2CO, and the dimer H2COṡṡṡHCl

A general analytic gradient expression is formulated for the coupled pair functional (CPF) electron correlation procedure. This method is implemented by modification of our existing configuration interaction (CI) gradient. It is demonstrated that since the CPF energy is not invariant to a unitary transformation of the occupied orbitals, additional self-consistent-field (SCF) redundant orbital rotation contributions must be included to evaluate the CPF energy gradient. The new procedure is used to rigorously determine the equilibrium structure, harmonic frequencies and dipole moment of the H₂COṡṡṡHCl hydrogen bonded complex and its constituent monomers. These results are compared with the predictions of SCF, second order perturbation theory (MP2), singles and doubles configuration interaction (CISD) and the experimental values. The CPF method in conjunction with a large basis set is shown to obtain highly accurate structures and molecular properties for the H₂CO and HCl molecules. The H-Cl frequency of the complex is the most intense band in the infrared spectrum. The harmonic frequency shift of the H-Cl stretch is predicted to be 151 cm^-1 with the double-zeta plus polarization basis for the gas phase dimer. Since it is well known that the structure of a dimer is dependent on the properties of the monomers and because the CPF triple zeta plus double polarization (TZ2P) level of theory obtains the most accurate results for the monomers, it is concluded that the CPF TZ2P method gives the most reliable prediction of the equilibrium structure of the complex. The hydrogen bond is predicted to have a length, r_OṡṡṡH of 1.952 Å and to be nonlinear, ∠OṡṡṡH-Cl=163.2°. The angular orientation of the two monomers in the complex, described by ∠C=OṡṡṡH, is determined to be 109.0° at equilibrium. Based upon the CPF TZ2P structure, the harmonic frequency shift of the H-Cl stretch is believed to be greater than 151 cm^-1.