Explicit Dynamical Electron−Proton Correlation in the Nuclear−Electronic Orbital Framework

Abstract

A method that includes explicit electron-proton correlation directly into the nuclear-electronic orbital self-consistent-field framework is presented. This nuclear-electronic orbital explicitly correlated Hartree-Fock (NEO-XCHF) scheme is formulated using Gaussian basis functions for the electrons and the quantum nuclei in conjunction with Gaussian-type geminal functions. The NEO approach is designed for the quantum treatment of a relatively small number of nuclei, such as the hydrogen nuclei involved in key hydrogen bonding interactions or hydrogen transfer reactions. The conventional nuclear-electronic-orbital-based methods produce nuclear wave functions that are too localized, leading to severe overestimations of hydrogen vibrational frequencies, as well as inaccuracies in geometries, isotope effects, couplings, and tunneling splittings. The application of the NEO-XCHF approach to a model system illustrates that the description of the nuclear wave function is significantly improved by the inclusion of explicit electron-proton correlation. In contrast to the NEO-HF method, the NEO-XCHF method leads to hydrogen vibrational stretch frequencies that are in excellent agreement with those calculated from grid-based methods. This approach is computationally practical for many-electron systems because only a relatively small number of nuclei are treated quantum mechanically and only electron-proton correlation is treated explicitly. Electron-electron dynamical correlation can be included with density functional theory or perturbation theory methods.

Some implications of the Hartree product treatment of the quantum nuclei in the ab initio nuclear–electronic orbital methodology

“…In general various approximate methods have been introduced in order to solve the corresponding Schrödinger's equation [1,[10][11][12][13][14][15][16][17][18][19][20][21][22]. One of possibilities is using the following trial normalized wavefunction assuming the Hartree product type wavefunction for the quantum nuclei (the spin variables have been neglected for brevity) within the context of the variational principle [20]:…”

“…The lack of safe adiabatic background makes inclusion of the muon-electron correlation more subtle, therefore, the usual strategy to "tune" the explicitly correlated NEO-HF methods to reproduce the known adiabatic results on the nuclear distribution though applicable, is not in general trustable [10][11][12][13][14][15][16][17][18][19]. An alternative strategy is to adapt a more semi-empirical viewpoint and try to fix the exponents of the nuclear/muonic basis set on proper values instead of trying to deduce them variationally to avoid the well-known "overlocalization" of the nuclear/muonic distribution [11,14,18,19].…”

“…These methodologies are beyond the usual ab initio procedures; within the Born-Oppenheimer (BO) paradigm electrons are treated as quantum waves and nuclei as clamped point charges, including exclusively the kinetic energy operators of electrons into the Hamiltonian [9]. The Nuclear-electronic orbital (NEO) is particularly a promising non-BO ab initio methodology since various types of the NEO wavefunctions have been proposed with varying degree of complexity [1,[10][11][12][13][14][15][16][17][18][19][20][21][22]. Very recently, the NEO has also been successfully utilized for molecules containing exotic light quantum particles like the positively charged muons [23][24][25][26].…”

Including nuclear quantum effects into highly correlated electronic structure calculations of weakly bound systems

“…45 Starting with the HF approximation for both electrons and nuclei, 44,46,47 their current implementation allows to include the electron-nuclear correlation either with explicitly correlated Gaussians (the NEO-XCHF and NEO-XCHF2 versions 48,49 ) or using variational and perturbative post-HF methods such as nonorthogonal CI (NOCI), 50 multi-configurational self-consistent field (MCSCF), and MP2 approaches. 51 Aimed to describe large molecular aggregates, multi-component density functional theory (DFT), first proposed by Parr et al 52 in the earlier 1980s and later by Shigeta et al 53 and Kreibich and Gross, 54 has been recently included in NOMO, 55,56 NEO, 57 and MCMO 58 implementations.…”

When electrons and protons get excited

“…The proton is in a highly excited vibrational state immediately after optical excitation and subsequently relaxes to its equilibrium state. The investigation of the electronic and nuclear dynamics under these nonequilibrium conditions is particularly interesting because of the quantum mechanical behavior of hydrogen nuclei associated with their light mass and the complexity of electron-proton correlation arising from their attractive electrostatic interaction (10). The traditional BornOppenheimer separation between electronic and nuclear motions based on mass and timescale differences may break down in these cases (11,12).…”