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

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.