发表机构
Hetao Institute of Mathematics and Interdisciplinary Science; University of Bristol（河套数学与交叉科学研究院; 布里斯托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在两个解析透明的基准模型中，比较玻恩-奥本海默、玻恩-黄与精确因式分解的势能面构造，揭示量子几何特性与能量排序规律，分离近似误差等并给出可验证形式。
AI 中文摘要
“势能面”这一术语指代几种不同的物理对象。玻恩-奥本海默构造给出核固定时的电子本征值，单表面玻恩-黄构造添加对角修正项，精确因式分解则给出一个与态相关的精确标量势。这些构造回答不同的问题，不应被视为对某一普适表面的竞争性定义。我们在两个解析透明的基准模型中对它们进行比较。对于费尔南德斯（Fernández）双线性耦合振子模型，我们以闭式形式得到精确分子光谱、玻恩-奥本海默光谱、玻恩-黄光谱以及基态精确因式分解表面。我们证明了对于所有可容许的质量比和耦合参数，这些能量的基态排序，并说明为何该排序无法一致推广到激发态。对角玻恩-黄修正与质量加权量子几何张量等价，费尔南德斯的六阶结果被重新表述为一个包含几何张量和同导数耦合的间隙加权谱矩的主导几何误差预算。在线性振动耦合模型中，几何张量在避免交叉处局域化，而总富比尼-施图迪（Fubini-Study）长度保持为π/2，将电子态变化的局域化与其总幅度分离开来。精确因式分解对于无节点的基态是光滑的，但当激发态核边际分布变得很小时，其条件数会变差但仍保持有限。这些模型将近似误差、几何修正和条件数以可直接验证的形式分离开来。
英文摘要
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.
CommentsRevised 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
