Analytical nuclear gradients for second-order Møller–Plesset perturbation theory using pair-natural orbitals based on localized virtual molecular orbitals

Accelerating analytical nuclear energy gradient calculations within wavefunction-based quantum chemical frameworks and extending their applicability to large systems remain significantly challenging. Recently, Pinski and Neese developed fully analytical nuclear energy gradients for second-order Møller-Plesset perturbation theory (MP2) using pair natural orbitals (PNOs) based on projected atomic orbitals (PAOs) [P. Pinski and F. Neese, J. Chem. Phys. 148, 031101 (2018); P. Pinski and F. Neese, J. Chem. Phys. 150, 164102 (2019)]. Here, we present an analytical nuclear gradient method based on the PNO-MP2 theory using orthonormal and non-redundant localized virtual molecular orbitals (LVMOs) instead of conventional PAOs. Compared to PAO-based approaches, our method exhibits two main differences. First, constraints from the construction of LVMOs are incorporated into the Lagrangian, requiring the solution of an additional Z-vector equation-the coupled-perturbed virtual localization (CP-VL) equation. Second, owing to the orthonormality of the LVMOs, the derivatives of the PNO coefficients with respect to both the MO coefficients and the nuclear coordinates vanish. The latter property significantly simplifies the formulation of analytical energy gradients. In addition, to mitigate the technical complexity of deriving gradient formulas with the PNO treatment, we used an extended scheme implemented in our automatic derivation program. Benchmark calculations on phenylalkane chains, docetaxel, and a host-guest complex demonstrated high computational scalability of our nuclear gradient implementations, additionally confirming that the computational effort required to solve the CP-VL equation is relatively minor. We illustrated that the energy gradients as functions of nuclear coordinates are smooth and that the errors in the predicted structures are small.