Constrained nuclear-electronic orbital second-order Moller-Plesset perturbation theory

TL;DR

The paper addresses the challenge of incorporating nuclear quantum effects into electronic structure calculations without a costly post hoc treatment. It develops cNEO-MP2 by combining the constrained nuclear-electronic orbital framework with multicomponent Hylleraas MP2, including electronic, electronic-nuclear, and nuclear correlation terms via $ ext{J}_2[ extbf{t}] = ext{J}_e[ extbf{t}^e] + ext{J}_{en}[ extbf{t}^{en}] + ext{J}_n[ extbf{t}^n]$ and energy corrections $E^{(2)}_{ee}$, $E^{(2)}_{en}$, and $E^{(2)}_{nn}$. It demonstrates that cNEO-MP2 captures vibrational averaging, isotopic effects, and zero-point energy within a single calculation, and validates the approach across diatomics, small polyatomic ions, and the Zundel cation where it improves vibrational frequencies relative to MP2 and often to VPT2-MP2. The method is implemented in a PySCF-based framework with noncanonical MP2 treatment and constrained correlated density, enabling accurate, efficient inclusion of nuclear quantum effects and paving the way for future multicomponent CC extensions. Overall, cNEO-MP2 offers a robust starting point for quantitatively including nuclear quantum effects in ab initio calculations, potentially reducing the computational burden of traditional vibrational analyses while improving property predictions.

Abstract

A multicomponent second-order Møller-Plesset perturbation theory (MP2) method is derived and implemented within the constrained nuclear-electronic orbital (cNEO) framework from a multicomponent generalization of the Hylleraas functional. The cNEO-MP2 method includes electronic-nuclear and nuclear correlation in the calculation of vibrationally averaged molecular properties, and is the first post Hartree-Fock wavefunction-based cNEO method. Nuclear quantum effects like vibrational averaging, isotopic effects, and zero-point energy can be captured in a single calculation or geometry optimization with cNEO-MP2, eliminating the need to perform costly subsequent calculations to determine higher order force constants as required with many existing methods used to determine vibrational effects upon molecular properties. The cNEO-MP2 method is benchmarked on a test set of diatomic and small polyatomic molecules and ions. Herein, we present internuclear distances, bond angles, potential energy surfaces, and vibrational frequencies calculated with the cNEO-MP2 method to demonstrate that it correctly captures the effects of nuclear vibrational motion upon molecular properties.

Figures (9)

• Figure 1: Single-component MP2 potential energy surface of the bifluoride anion (FHF$^{-}$) as a function of the position of the hydrogen atom in the xz-plane. The fluorine atoms are fixed at 1.146 and -1.146 Å.
• Figure 2: Single-component MP2 potential energy surface of the deuterium bifluoride anion (FDF$^{-}$) as a function of the position of the deuterium atom in the xz-plane. The fluorine atoms are fixed at 1.146 and -1.146 Å.
• Figure 3: Multicomponent cNEO-MP2 potential energy surface of the bifluoride anion (FHF$^{-}$) as a function of the position of the hydrogen atom in the xz-plane. Here the fluorine atoms are fixed at 1.146 and -1.146 Å.
• Figure 4: Multicomponent cNEO-MP2 potential energy surface of the deuterium bifluoride anion (FDF$^{-}$) as a function of the position of the deuterium atom in the xz-plane. Here the fluorine atoms are fixed at 1.146 and -1.146 Å.
• Figure 5: Geometry of the Zundel (H$_{5}$O$_{2}^{+}$) cation, calculated with cNEO-MP2, using the aug-cc-pVTZ electronic basis set, and the PB4-D nuclear basis set for the shared proton. The optimized structure has C$_{2}$ symmetry. This visualization was generated with Avogadro 2.0 Avogadro. The blue arrow corresponds to the positive z-axis, the red arrow to the positive x-axis, and the green arrow to the positive y-axis.