Electronic-nuclear entanglement in Born-Oppenheimer wave functions and beyond

TL;DR

This work analyzes electro-nuclear entanglement in molecular wave functions within BO, BH, and beyond using 1D models, applying Schmidt decomposition to quantify entanglement via von Neumann entropy. It demonstrates that the BO vibronic ground state is almost separable, while entanglement increases with vibrational excitation and becomes pronounced near avoided crossings, where adiabatic BO descriptions can fail. The Born–Hue expansion generally mirrors the entanglement of the dominant BO component but can exceed it when multiple BO states strongly mix, and diabatization often provides a more accurate entanglement picture for sharp crossings. The study advances understanding of how electronic and nuclear degrees of freedom become quantum mechanically correlated in simple yet representative molecular systems, with implications for non-adiabatic dynamics and entanglement-based control.

Abstract

We analyze the entanglement between electronic and nuclear motions in molecular wave functions, by using different widely used ansatzes in molecular Hamiltonian models (H$^+_2$ in 1D and the Shin-Metiu model); namely, i) Born-Oppenheimer separation in both adiabatic and ii) diabatic pictures and beyond it, iii) using a Born-Huang expansion, which involves non-adiabatic couplings in the solution. Any molecule can be considered a bipartite system in terms of electronic and nuclear halfspaces. Accordingly, the Schmidt decomposition theorem can be applied to each molecular ansatz to find a much shorter representation in terms of Schmidt basis and to estimate the entanglement content through the calculation of von Neumann entropies. Although here we justify that the ground BO vibronic state of any molecule may be regarded as an almost separable and non-entangled state, this property worsens with the vibrational excitation, which increases the entanglement monotonically. The interlace between electronic and nuclear wave functions for each vibronic state becomes even more critical for those electronic excited states whose electronic wave functions change drastically with the molecular geometry. We find that the entanglement may be quantified by the variation of the electronic wave function along the different nuclear geometries, and that the nuclear wave function indeed plays the role of a tester. In addition, the presence of avoided crossings among the potential energy curves brings about a strong enhancement of entanglement in the Born-Oppenheimer adiabatic picture, which instead is much reduced in a diabatic picture. Finally, the Born-Huang expansion uncovers some synergistic vibronic states whose entanglement content is larger than that of any Born-Oppenheimer component in the superposition.

Figures (13)

• Figure 1: (a) Schematic representation of the one-dimensional H$_2^+$ molecule, with motions constrained along the $x$-axis. The nuclei are located at positions $R_1$ and $R_2$ with respect to the nuclear center of mass, and $R$ denotes the internuclear distance. The variable $x$ represents the electronic degree of freedom relative to the nuclear center of mass. (b) Diagram of the Shin-Metiu model. Two ions with nuclear charges $Z_\alpha$ and $Z_\beta$ are fixed in space and separated by a distance $L$, while a third ion with charge $Z_\gamma$ moves freely. The electron also moves along the $x$-axis. For simplicity, all nuclear charges ($Z_\alpha$, $Z_\beta$, and $Z_\gamma$) in both models are set to unity.
• Figure 2: (a) Vibrational bound states shifted to their corresponding variational energy associated to the BO potential energy curves for the molecular orbitals $1\sigma_g$ and $2\sigma_g$. (b)-(c) Electron-nuclei and nuclei-nuclei Coulomb electrostatic potential $\hat{U}(x, R_i)$ and the corresponding electronic wave functions for the $1\sigma_g$ and $2\sigma_g$ molecular orbitals, evaluated at selected nuclear configurations $R_i$ near the equilibrium bond lengths $R_e^{(1)} = 2.0$ a.u. and $R_e^{(2)} = 8.4$ a.u., respectively. The chosen nuclear positions $R^{(m)}_i$ correspond to the coordinates where the vibrational wave functions with labels $m$=0, 1 and 2 show their first local maxima and minima labeled with $i$=0, 1 and 2 (see also Fig. \ref{['fig:vibrational_states']}).
• Figure 3: Vibrational wave functions for the one-dimensional H$_2^+$ molecular model. The numerical vibrational states $\chi_{1\sigma_g/2\sigma_g,m}$ are shown in black solid lines, while their corresponding simplified versions $\tilde{\chi}_{1\sigma_g/2\sigma_g,m}$, constructed using Eq. \ref{['eq:simplified_vibrationa_wave_function']}, are shown in blue for $m=$0, 1 and 2. The marked positions $R^{(m)}_i$ indicate the internuclear distance at which the vibrational wave function with label $m$ shows maxima or minima, with the ordering labels $i=0,1,2...$; (a) Lowest three vibrational wave functions of the electronic ground state $1\sigma_g$. (b) Lowest three vibrational wave functions of the electronic excited state $2\sigma_g$. Both $\chi_{1\sigma_g/2\sigma_g,m}$ and $\tilde{\chi}_{1\sigma_g/2\sigma_g,m}$ states are normalized to unity.
• Figure 4: (a) Von Neumann entropy as a function of the energy of the vibronic state for the $1\sigma_g$ PEC (blue dots) and for the $2\sigma_g$ PEC (red dots). Each plot includes the corresponding bound vibrational states, i.e. the states below the black dashed vertical lines for each case. The dashed blue and red lines correspond to the von Neumann entropies computed using the simplified model of the density matrix explained in Section \ref{['sec:simplified_density']} and the black doted lines denote the von Neumann entropies computed using the approach described by Eq. \ref{['eq:entropy_approach']}. (b) von Neumann entropy in 3D H$^+_2$ of rovibrational states as a function of the energy $W^J_{1s\sigma_g,m}$ (blue dots are the same as in Fig. \ref{['fig:entropy_H2+']}(a) for 1D). Solid grey lines connect the entropy pertaining to the same vibrational number $m$, for rotational states from $J=0$ up to $J=15$.
• Figure 5: Total wave function under the Born-Oppenheimer approximation for the one-dimensional H$_2^+$ model. The central panels display the full electron-nuclear wave function, while the side panels present the first two Schmidt modes for both the electronic and nuclear subsystems, along with their corresponding largest Schmidt coefficients. (a) Lowest and highest vibrational eigenstates associated with the $1\sigma_g$ PEC. (b) Lowest and highest vibrational eigenstates associated with the $2\sigma_g$ PEC. The reconstructed wave functions obtained through the Schmidt decomposition accurately reproduce the original BO wave functions.