The Born-Oppenheimer (BO) and single-surface Born-Huang (BH) approximations replace the full molecular problem by nuclear motion associated with one clamped electronic state, whereas the full BH expansion and exact factorization (EF) are exact representations. We use two analytically transparent models to identify what controls one-surface error. For Fernández's bilinearly coupled oscillators, the Brattsev ground-state bracket was already known; we derive closed-form differences proving strictness for every positive nuclear mass and nonzero stable coupling. Using Hunter's conditional-amplitude construction, we place the closed-form ground-state EF potential alongside BO and single-surface BH. Exact excited-state calculations show that the bracket does not extend through the spectrum. The diagonal BH correction is the quantum-metric coefficient of the retained electronic state divided by twice the nuclear mass. Fernández's sixth-order expansion shows that the next omitted correction contains the same derivative couplings, an additional inverse electronic-energy separation, and a factor depending on the nuclear state. We therefore add a two-state model with a position-dependent gap. Decreasing its minimum separation makes the metric higher and narrower while leaving the total Fubini-Study length equal to pi/2. The exact EF, BO, and BH potentials can differ visibly even when their nuclear ground-state amplitudes are nearly identical. The two models show that one-surface accuracy depends on nuclear mass, electronic-state variation, energy separation from omitted states, and the nuclear wavefunction.
Comments
Revised and expanded version; title changed from "Born-Oppenheimer, Born-Huang, and exact factorization: quantum geometry and error in analytically transparent benchmark models." Historical attribution and terminology have been clarified, and the exact and numerical comparisons and error-budget discussion have been strengthened. Main conclusions unchanged. 37 pages, 5 figures, 5 tables
The Born-Oppenheimer (BO) and single-surface Born-Huang (BH) approximations replace the full molecular problem by nuclear motion associated with one clamped electronic state, whereas the full BH expansion and exact factorization (EF) are exact representations. We use two analytically transparent models to identify what controls one-surface error. For Fernández's bilinearly coupled oscillators, the Brattsev ground-state bracket was already known; we derive closed-form differences proving strictness for every positive nuclear mass and nonzero stable coupling. Using Hunter's conditional-amplitude construction, we place the closed-form ground-state EF potential alongside BO and single-surface BH. Exact excited-state calculations show that the bracket does not extend through the spectrum. The diagonal BH correction is the quantum-metric coefficient of the retained electronic state divided by twice the nuclear mass. Fernández's sixth-order expansion shows that the next omitted correction contains the same derivative couplings, an additional inverse electronic-energy separation, and a factor depending on the nuclear state. We therefore add a two-state model with a position-dependent gap. Decreasing its minimum separation makes the metric higher and narrower while leaving the total Fubini-Study length equal to pi/2. The exact EF, BO, and BH potentials can differ visibly even when their nuclear ground-state amplitudes are nearly identical. The two models show that one-surface accuracy depends on nuclear mass, electronic-state variation, energy separation from omitted states, and the nuclear wavefunction.
Over the last few decades, Born-Oppenheimer (BO) and Born-Oppenheimer-Huang (BOH) treatments for electron-nuclear equation of motion have been considered as the cornerstone for qualitative as well as quantitative exploration and understanding of chemical and physical processes. The BO approximation due to the large mass difference between electrons and nuclei as well as the energy separation among the electronic states could able to predict some of the processes (chemical and physical) satisfactorily at lower energy regime. Such approximation (BO) quite often fails for those cases with the involvement of excited electronic states, in particular, due to the coupling among the states. On the other hand, the BO treatment provides two important quantities, namely, the adiabatic potential energy surfaces (PESs) and the nonadiabatic coupling terms (NACTs) determining the forces, which control the motion of the nuclei in a molecule. The NACTs play a major role in photoexcitation spectra, charge transfer chemical reactions, ferroelectric property and other processes, where significant anomalies are observed between single and multiple PES(s) results. Since the NACTs often exhibit singularity at the close vicinity of degenerate states forming conical intersections (CIs), such coupling terms could break the symmetry of the molecular processes changing the observables qualitatively.
We analyze the entanglement between electronic and nuclear motions in molecular wave functions widely used by theoretical chemists, namely, (i) Born-Oppenheimer factorization in the adiabatic picture, (ii) the transformation into a diabatic picture, (iii) the use of a Born-Huang expansion, and (iv) the eigenfunction of the full molecular Hamiltonian. Our showcase is based on two one-electron one-dimensional molecular Hamiltonians (H2+ and the Shin-Metiu model). We find that within the Born-Oppenheimer approximation, any molecular state (although factorizable) is always entangled, and its entanglement content may be assessed by the variation of the electronic wave function along the different nuclear geometries, with the nuclear wave function indeed playing the role of a tester. The presence of avoided crossings among the adiabatic potential energy curves brings about dramatic changes in the entanglement content of the wave function: sharp avoided crossings favor a diabatic picture (real crossings between potential energy curves), while in broad avoided crossings, the adiabatic picture prevails. The total eigenfunction of the molecular Shin-Metiu Hamiltonian indicates that nuclear densities accommodate well within the diabatic curves for strong adiabatic couplings but within adiabatic curves for weak ones. Consequently, we find that the electron-nuclei entanglement content is a valid witness to unveil strong or weak nonadiabatic couplings in molecules. In terms of entanglement, we also find that the Born-Huang expansion, based on Born-Oppenheimer adiabatic electronic states, does not provide a correct trend of entanglement compared with that of the total molecular eigenfunction, thus indicating a very slow convergence of this expansion.
The Born-Oppenheimer electronic wave function ${\mathrm{\ensuremath{\Phi}}}_{R}^{\text{BO}}(r)$ picks up a topological phase factor $\ifmmode\pm\else\textpm\fi{}1$, a special case of Berry phase, when it is transported around a conical intersection of two adiabatic potential energy surfaces in R space. We show that this topological quantity reverts to a geometric quantity ${e}^{i\ensuremath{\gamma}}$ if the geometric phase $\ensuremath{\gamma}=\ensuremath{\oint}\mathrm{Im}\ensuremath{\langle}{\mathrm{\ensuremath{\Phi}}}_{R}|{\mathbf{\ensuremath{\nabla}}}_{\ensuremath{\mu}}{\mathrm{\ensuremath{\Phi}}}_{R}\ensuremath{\rangle}\ifmmode\cdot\else\textperiodcentered\fi{}d{\mathbf{R}}_{\ensuremath{\mu}}$ is evaluated with the conditional electronic wave function ${\mathrm{\ensuremath{\Phi}}}_{R}(r)$ from the exact electron-nuclear factorization ${\mathrm{\ensuremath{\Phi}}}_{R}(r)\ensuremath{\chi}(R)$ instead of the adiabatic function ${\mathrm{\ensuremath{\Phi}}}_{R}^{\text{BO}}(r)$. A model of a pseudorotating triatomic molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides examples of nontrivial induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current that cannot be removed by a gauge transformation. The exact potential energy surface is calculated and found to contain a term depending on the Fubini-Study metric for the conditional electronic wave function.
Whenever a quantum chemist extracts the eigenstate of an electronic Hamiltonian, he makes, consciously or not, a decision concerning the phase of the wave function. This is done for each calculated state at each nuclear position. Thus he defines a Born–Oppenheimer (BO) frame of reference. There is no absolute phase just as there is no absolute position or time in mechanics. This leads naturally to the question: What are the quantities which do not depend on the arbitrary phases, i.e., what are the BO invariants? In this article we identify BO invariants with respect to an arbitrary path in nuclear configuration space. We identify invariant electronic states along these paths and their Aharonov–Anandan geometric phases. For closed loops not passing through electronic energy degeneracies these invariant states are the BO adiabatic wave functions and the phases are the Berry phases. The results establish rigorous relations between the full nonadiabatic couplings matrix and the geometric phases.
The role of electron-nuclear entanglement on the validity of the Born-Oppenheimer (BO) approximation is investigated. While nonadiabatic couplings generally lead to entanglement and to a failure of the BO approximation, surprisingly the degree of electron-nuclear entanglement is found to be uncorrelated with the degree of validity of the BO approximation. This is because while the degree of entanglement of BO states is determined by their deviation from the corresponding states in the crude BO approximation, the accuracy of the BO approximation is dictated, instead, by the deviation of the BO states from the exact electron-nuclear states. In fact, in the context of a minimal avoided crossing model, extreme cases are identified where an adequate BO state is seen to be maximally entangled, and where the BO approximation fails but the associated BO state remains approximately unentangled. Further, the BO states are found to not preserve the entanglement properties of the exact electron-nuclear eigenstates, and to be completely unentangled only in the limit in which the BO approximation becomes exact.
The role of electron-nuclear entanglement on the validity of the Born-Oppenheimer (BO) approximation is investigated. Although nonadiabatic couplings generally lead to entanglement and to a failure of the BO approximation, surprisingly the degree of electron-nuclear entanglement is found to be uncorrelated with the degree of validity of the BO approximation. This is because while the degree of entanglement of BO states is determined by their deviation from the corresponding states in the crude BO approximation, the accuracy of the BO approximation is dictated, instead, by the deviation of the BO states from the exact electron-nuclear states. In fact, in the context of a minimal avoided crossing model, extreme cases are identified where an adequate BO state is seen to be maximally entangled, and where the BO approximation fails but the associated BO state remains approximately unentangled. Further, the BO states are found to not preserve the entanglement properties of the exact electron-nuclear eigenstates, and to be completely unentangled only in the limit in which the BO approximation becomes exact.