Separation of electron-electron and electron-proton correlation in multicomponent orbital-optimized perturbation theory

The multicomponent orbital-optimized second-order MOller-Plesset perturbation theory (OOMP2) method is the first multicomponent MP2 method that is able to calculate qualitatively accurate protonic densities, protonic affinities, and geometrical changes due to nuclear quantum effects in multicomponent systems. In this study, two approximations of the multicomponent OOMP2 method are introduced in an effort to demonstrate that, in orbital-optimized multicomponent methods, performing the orbital-optimization process with only electron-proton correlation is sufficient to obtain accurate protonic properties. Additionally, these approximations should reduce the computational expense of the multicomponent OOMP2 method. In the first approximation, the first-order wave function is written as a linear combination of one-electron one-proton excitations rather than as a linear combination of one-electron one-proton and two-electron excitations as in the original multicomponent OOMP2 method. Electron-electron correlation is included perturbatively after the orbital-optimization procedure has converged. In the second approach, the first approximation is further modified to neglect all terms in the orbital-rotation gradients that depend on the two-electron molecular-orbital integrals, which, assuming a fixed-sized protonic basis set, reduces the computational scaling for the orbital-optimization iterations to Ne3, where N-e is a measure of the electronic system size, compared to the Ne5 scaling of the original multicomponent OOMP2 method. The second approximation requires one Ne5 step after orbital convergence to compute the electron-electron correlation energy. The accuracy of the calculated protonic densities, protonic affinities, and optimized geometries of these approximations is similar or improved relative to the original multicomponent OOMP2 method.