We present a novel theory and implementation for computing coupled electronic and quantal nuclear subsystems on a single potential energy surface, moving beyond the standard Born–Oppenheimer (BO) separation of nuclei and electrons. We formulate an exact self-consistent nucleus–electron embedding potential from the single product molecular wavefunction and demonstrate that the fundamental behavior of the correlated nucleus–electron can be computed for mean-field electrons that are responsive to a quantal anharmonic vibration of selected nuclei in a discrete variable representation. Geometric gauge choices are discussed and necessary for formulating energy invariant biorthogonal electronic equations. Our method is further applied to characterize vibrationally averaged molecular bonding properties of molecular energetics, bond lengths, and protonic and electron densities. Moreover, post-Hartree–Fock electron correlation can be conveniently computed on the basis of nucleus–electron coupled molecular orbitals, as demonstrated for correlated models of second-order Møllet–Plesset perturbation and full configuration interaction theories. Our approach not only accurately quantifies non-classical nucleus–electron couplings for revising molecular bonding properties but also provides an alternative time-independent approach for deploying non-BO molecular quantum chemistry.

中文翻译：

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从精确波函数分解修正分子键合指纹的核-电子相关性

我们提出了一种在单个势能面上计算耦合电子和量子核子系统的新理论和实现，超越了核和电子的标准波恩-奥本海默 (BO) 分离。我们从单乘积分子波函数中制定了一个精确的自洽核-电子嵌入电位，并证明了相关核-电子的基本行为可以计算为平均场电子，这些电子响应选定核的量子非谐振动离散变量表示。讨论了几何规范的选择，这对于制定能量不变的双正交电子方程是必要的。我们的方法进一步应用于表征分子能量、键长、以及质子和电子密度。此外，后 Hartree-Fock 电子相关性可以在核-电子耦合分子轨道的基础上方便地计算，如二阶 Møllet-Plesset 扰动和完整构型相互作用理论的相关模型所证明的那样。我们的方法不仅准确量化了非经典核-电子耦合以修正分子键合特性，而且还提供了一种用于部署非 BO 分子量子化学的与时间无关的替代方法。