New Insights into the Born Oppenheimer Approximation and "Potential Effects" with No Ordinary Potentials

关于封闭动力系统中Born-Oppenheimer方法的注记

Notes On The Born-Oppenheimer Approach In A Closed Dynamical System

最近的各种研究中的应用玻恩-奥本海默方法在一个封闭的重力物质系统进行检查。指出在没有先验时间的情况下，Born-Oppenheimer方法可能会产生潜在的新结果。

Born-Oppenheimer approximation for open quantum systems within the quantum trajectory approach

Based on the quantum trajectory approach, we extend the Born-Oppenheimer (BO) approximation from closed quantum system to open quantum system, where the open quantum system is described by a master equation in Lindblad form. The BO approximation is defined and the validity condition is derived. We find that the dissipation in fast variables benefits the BO approximation that is different from the dissipation in slow variables. A detailed comparison between this extension and our previous approximation (that is based on the effective Hamiltonian approach, see X. L. Huang and X. X. Yi, Phys. Rev…

Born-Oppenheimer近似中的Berry相和拓扑性质

Berry's phases and topological properties in the Born-Oppenheimer approximation

能级交叉问题是整齐地制定了第二个量化的配方，它表现出隐藏的本地规范对称性。几何相位的分析被简化为一个简单的对角化的哈密顿量。如果对角化水平交叉的无穷小邻域中的几何项，则对于任意大但有限的时间间隔$T$，几何相位变得平凡（因此没有奇异性）。例如，隆格-希金斯相变规则的拓扑证明在实际的玻恩-奥本海默近似中失败了，在玻恩-奥本海默近似中，$T$被等同于较慢系统的周期。Aharonov-Bohm相位和几何相位之间的关键区别进行了解释。我们还注意到绝热和非绝热几何相所涉及的规范对称性是完全不同的。

On the status of the Born-Oppenheimer expansion in molecular systems theory

It is shown that the adiabatic Born-Oppenheimer expansion does not satisfy the necessary condition for the applicability of perturbation theory. A simple example of an exact solution of a problem that can not be obtained from the Born-Oppenheimer expansion is given. A new version of perturbation theory for molecular systems is proposed.

玻恩-奥本海默波函数及其他模型中的电子-核纠缠

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

该研究分析分子波函数中电子与核运动的纠缠，通过玻恩-奥本海默分离、玻恩-黄展开等近似，发现基态近乎非纠缠、振动激发增强纠缠，避免交叉和玻恩-黄展开对纠缠有显著影响。

Is the molecular Berry phase an artifact of the Born-Oppenheimer approximation?

We demonstrate that the molecular Berry phase and the corresponding non-analyticity in the electronic Born-Oppenheimer wavefunction is, in general, not a true topological feature of the exact solution of the full electron-nuclear Schrödinger equation. For a numerically exactly solvable model we show that a non-analyticity, and the associated geometric phase, only appear in the limit of infinite nuclear mass, while a perfectly smooth behavior is found for any finite nuclear mass.

双结超导电路低能谱的Born-Oppenheimer近似

Low-energy spectrum of double-junction superconducting circuits in the Born-Oppenheimer approximation

研究双结超导电路中低能谱的Born-Oppenheimer近似模型，推导包含内部模式修正项的有效单模模型，并探讨其对周期边界条件和噪声灵敏度的影响。

Many-body Green's function theory of electrons and nuclei beyond the Born-Oppenheimer approximation

The method of many-body Green's functions is developed for arbitrary systems of electrons and nuclei starting from the full (beyond Born-Oppenheimer) Hamiltonian of Coulomb interactions and kinetic energies. The theory presented here resolves the problems arising from the translational and rotational invariance of this Hamiltonian that afflict the existing many-body Green's function theories. We derive a coupled set of exact equations for the electronic and nuclear Green's functions and provide a systematic way to approximately compute the properties of arbitrary many-body systems of electrons…

Born-Oppenheimer description of two atoms in a combined oscillator and lattice trap

We analyze the quantum states of two atoms in a combined harmonic oscillator and periodic lattice trap in one spatial dimension. In the case of tight-binding and only nearest neighbor tunneling, the equations of motion are conveniently represented in the momentum representation. We show that in the case of strong attraction between the particles, the different time scales of relative and center-of-mass motion validate a separation of the problem similar to the Born-Oppenheimer approximation applied in the description of electronic and nuclear motion in molecules.

Adiabatic electronic flux density: a Born-Oppenheimer Broken Symmetry ansatz

The Born-Oppenheimer approximation leads to the counterintuitive result of a vanishing electronic flux density upon vibrational dynamics in the electronic ground state. To circumvent this long known issue, we propose using pairwise anti-symmetrically translated vibronic densities to generate a symmetric electronic density that can be forced to satisfy the continuity equation approximately. The so-called Born-Oppenheimer broken symmetry ansatz yields all components of the flux density simultaneously while requiring only knowledge about the nuclear quantum dynamics on the electronic adiabatic gr…

关于含时Born-Oppenheimer近似

On the time-dependent Born-Oppenheimer Approximation

本文针对含大量核和电子的量子分子，提出含时Born-Oppenheimer近似的任意阶可迭代方案，并推导出等价于原Schrödinger方程且不含电子变量的核约化动力学有效方程，给出超越原始近似的可处理近似。

在量子计算机上高效模拟预玻恩-奥普芬贝格动力学

Efficient Simulation of Pre-Born-Oppenheimer Dynamics on a Quantum Computer

本文提出了一种在量子计算机上高效模拟预玻恩-奥普芬贝格动力学的量子算法，通过优化方法显著降低了光化学反应模拟的资源需求。

Pre-Born-Oppenheimer energies, leading-order relativistic and QED corrections for electronically excited states of molecular hydrogen

For rovibronic states corresponding to the $B$ and $B'\ ^1Σ_\text{u}^+$ electronic states of the hydrogen molecule, the pre-Born--Oppenheimer (four-particle) non-relativistic energy is converged to a 1-3 parts-per-billion relative precision. The four-particle non-relativistic energy is appended with leading-order relativistic, leading- and estimated higher-order quantum-electrodynamics corrections. The resulting term values referenced to the rovibronic ground state are obtained in an excellent agreement with the experimental results. Further results are reported and discussed for other rovibro…

Particle correlation from uncorrelated non Born-Oppenheimer SCF wavefunctions

We analyse a nonadiabatic self-consistent field method by means of an exactly-solvable model. The method is based on nuclear and electronic orbitals that are functions of the cartesian coordinates in the laboratory-fixed frame. The kinetic energy of the center of mass is subtracted from the molecular Hamiltonian operator in the variational process. The results for the simple model are remarkably accurate and show that the integration over the redundant cartesian coordinates leads to couplings among the internal ones.

On Corrections to the Born-Oppenheimer Approximation

This report presents a new approach for treating the coupling of electrons and nuclei in quantum mechanical calculations for molecules and condensed matter. It includes the standard "Born-Oppenheimer approximation" as a special case but treats both adiabatic and non-adiabatic corrections using perturbation theory. The adiabatic corrections include all terms that do not explicitly involve the nuclear wavefunctions, so that the nuclei move on a single electronic potential surface. The non-adiabatic corrections, which allow the nuclei to move on more than one potential surface, include coupling b…

能级交叉附近的Born-Oppenheimer近似

Born-Oppenheimer Approximation near Level Crossing

我们考虑二维圆锥相交附近的Born-Oppenheimer问题。对于能量接近的交叉能量，我们描述的波函数附近的各向同性交叉，并表明它是广义超几何函数0 F3。这个函数对于圆锥交点的作用就像艾里函数对于经典转折点的作用一样。作为应用，我们计算了交叉点附近振动能级的反常塞曼位移。

Shortcuts to Adiabaticity Assisted by Counterdiabatic Born-Oppenheimer Dynamics

Shortcuts to adiabaticity (STA) provide control protocols to guide the dynamics of a quantum system through an adiabatic reference trajectory in an arbitrary prescheduled time. Designing STA proves challenging in complex quantum systems when the dynamics of the degrees of freedom span different time scales. We introduce Counterdiabatic Born-Oppenheimer Dynamics (CBOD) as a framework to design STA in systems with a large separation of energy scales. CBOD exploits the Born-Oppenheimer approximation to separate the Hamiltonian into effective fast and slow degrees of freedom and calculate the corr…

Variational calculations for the hydrogen-antihydrogen system with a mass-scaled Born-Oppenheimer potential

The problem of proton-antiproton motion in the ${\rm H}$--${\rm \bar{H}}$ system is investigated by means of the variational method. We introduce a modified nuclear interaction through mass-scaling of the Born-Oppenheimer potential. This improved treatment of the interaction includes the nondivergent part of the otherwise divergent adiabatic correction and shows the correct threshold behavior. Using this potential we calculate the vibrational energy levels with angular momentum 0 and 1 and the corresponding nuclear wave functions, as well as the S-wave scattering length. We obtain a full set o…

The use of the Born-Oppenheimer factorization in the phase-space representation of the time-independent Schrödinger equation for bilinearly coupled harmonic oscillators

A system of two bilinearly coupled harmonic oscillators has been solved analytically by using the Born-Oppenheimer (BO) product wavefunction ansatz and the phase-space bound trajectory approach [J. S. Molano et al., Chem. Phys. Lett. \textbf{76}(12), 138171 (2021)]. The bilinearly coupled oscillator system allows to obtain the analytical expression of the quantum system, facilitating comparison with the results of using the BO ansatz product. The analytical and BO wavefunctions are obtained as a product of parabolic cylinder functions. The arguments of the parabolic cylinder functions of the e…

水平交叉点附近的玻恩-奥本海默波函数

The Born Oppenheimer wave function near level crossing

标准的玻恩-奥本海默理论没有给出水平交叉点附近的波函数的精确描述。我们给这样一个描述附近的各向同性圆锥交叉，能量接近交叉能量。这导致了两个耦合的二阶常微分方程的研究，其解决方案是描述在广义超几何函数的种类0 F3（;a，b，c;z）。我们发现，在低角动量时，由于交叉而产生的混合非常大，其比例类似于μ^（1/6），其中μ是电子与核的质量比。

$H_2^+$分子离子薛定谔方程在Born-Oppenheimer近似下的解析解

The analytical solution of the Schrödinger equation in Born-Oppenheimer approximation for $H_2^+$ molecular ion

对$H_2^+$的薛定谔方程（它是一个二阶微分方程）的解析解的分析表明，该方程的第二线性独立解是一个平方可积函数，因此基态总波函数是两个具有不同空间对称性的线性独立波函数的线性组合：圆柱形和球形。圆柱对称的波函数是众所周知的。它在原子核的位置有极大值。位于键中心的$R\neq0$处的球对称波函数和相应的球对称电子分布代表了一个无核联合原子电子密度的准原子。在此基础上，重新研究了$H_2^+$的基态波函数和电子密度的定性行为。解析地表明，具有两个最大值的总分子波函数到具有一个最大值的总分子波函数的变换通过平坦的波函数。所给出的三维电子密度图直观地反映了总波函数的球形分量及其随核间距的变化。

Pre-Born-Oppenheimer Molecular Structure Theory

In pre-Born-Oppenheimer (pre-BO) theory a molecule is considered as a quantum system as a whole, including the electrons and the atomic nuclei on the same footing. This approach is fundamentally different from the traditional quantum chemistry treatment, which relies on the separation of the motion of the electrons and the atomic nuclei. A fully quantum mechanical treatment of molecules has a great promise for future developments and applications. Its most accurate versions may contribute to the definition of new schemes for metrology and testing fundamental physical theories; its approximate …

腔玻恩-奥本海默哈特里-福克拟设：强耦合分子系综的光-物质性质

Cavity-Born-Oppenheimer Hartree-Fock Ansatz: Light-matter Properties of Strongly Coupled Molecular Ensembles

本文提出腔玻恩-奥本海默哈特里-福克拟设，用于非微扰自洽描述强耦合分子系综，并通过氟化氢系综验证了腔介导分子间偶极-偶极相互作用对单分子能量的影响。

三粒子量子系统中Born-Oppenheimer近似下的零范围模型用于Efimov效应

A Zero-Range Model for the Efimov Effect in the Born-Oppenheimer Approximation

本文基于Born-Oppenheimer近似，研究了三粒子系统中零范围相互作用下的Efimov效应，并证明了其无限多个负本征值的普遍几何规律。

Dynamical features of interference phenomena in the presence of entanglement

A "strongly" interacting, and entangling, heavy, non recoiling, external particle effects a significant change of the environment. Described locally, the corresponding entanglement event is a generalized electric Aharonov Bohm effect, that differs from the original one in a crucial way. We propose a gedanken interference experiment. The predicted shift of the interference pattern is due to a self induced or "private" potential difference experienced while the particle is in vacuum. We show that all non trivial Born Oppenheimer potentials are "private" potentials. We apply the Born Oppenheimer …

当我们讨论 Born-Oppenheimer 近似时，我们在讨论什么？

What are we talking about when we discuss the Born-Oppenheimer approximation?

本文批判性分析了 HLT 关于 Born-Oppenheimer Approximation 的文章，坚持认为该近似包含与 Heisenberg Uncertainty Principle 不相容的经典假设，并反驳了 HLT 认为无此假设且支持化学还原为物理学的观点。

Entanglement in the Born-Oppenheimer Approximation

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 …

两体系统的Pre-Born-Oppenheimer Dirac-Coulomb-Breit计算

Pre-Born-Oppenheimer Dirac-Coulomb-Breit computations for two-body systems

使用高斯型基函数求解两体系统的16分量无对Dirac-Coulomb-Breit方程，其变分能量对精细结构常数的依赖与微扰非相对论QED框架高度一致，为计算相对论QED方法的发展提供了坚实参考。

关于本生-奥本海默近似数学处理的探讨

On the mathematical treatment of the Born-Oppenheimer approximation

本文综述了数学家在分子束缚态、散射理论、共振理论及短时间演化中对本生-奥本海默近似准确性的严谨结果，指出数学方法与物理化学中常用方法的差异，并探讨未来发展方向。

超越玻恩-奥本海默近似的、由误差缓解技术赋能的多组分量子模拟

Error-Mitigation Enabled Multicomponent Quantum Simulations Beyond the Born-Oppenheimer Approximation

针对超越玻恩-奥本海默近似的分子量子模拟需求，提出多组分幺正耦合簇框架，结合误差缓解技术在超导量子硬件上实现了符合化学精度的多组分关联模拟。

Electric Transition Dipole Moment in pre-Born-Oppenheimer Molecular Structure Theory

This paper presents the calculation of the electric transition dipole moment in a pre-Born-Oppenheimer framework. Electrons and nuclei are treated equally in terms of the parametrization of the non-relativistic total wave function, which is written as a linear combination of basis functions constructed with explicitly correlated Gaussian functions and the global vector representation. The integrals of the electric transition dipole moment are derived corresponding to these basis functions in both the length and the velocity representation. The complete derivation and the calculations are perfo…

Reexamination of the ground state Born-Oppenheimer Yb$_2$ potential

The precision of the photoassociation spectroscopy of Yb dimer in degenerate gases is enough to improve the constraints on the new short-range gravity-like forces if the theoretical knowledge of the Born-Oppenheimer interatomic potential and non-Born-Oppenheimer interactions is refined [M. Borkowski et al. Sci. Rep. A {\bf 9}, 14807 (2019)]. The ground-state interaction potential of ytterbium dimer is investigated at the eXact 2-component core-correlated CCSD(T) level of {\it ab initio} theory in the complete basis set limit with extensive augmentation by diffuse functions. For the small basis…

Regularized Born-Oppenheimer molecular dynamics

While the treatment of conical intersections in molecular dynamics generally requires nonadiabatic approaches, the Born-Oppenheimer adiabatic approximation is still adopted as a valid alternative in certain circumstances. In the context of Mead-Truhlar minimal coupling, this paper presents a new closure of the nuclear Born-Oppenheimer equation, thereby leading to a molecular dynamics scheme capturing geometric phase effects. Specifically, a semiclassical closure of the nuclear Ehrenfest dynamics is obtained through a convenient prescription for the nuclear Bohmian trajectories. The conical int…

On the four-body problem in the Born-Oppenheimer approximation

The quantum problem of four particles in $\mathbb{R}^d$ ($d\geq 3$), with arbitrary masses $m_1,m_2,m_3$ and $m_4$, interacting through an harmonic oscillator potential is considered. This model allows exact solvability and a critical analysis of the Born-Oppenheimer approximation. The study is restricted to the ground state level. We pay special attention to the case of two equally heavy masses $m_1=m_2=M$ and two light particles $m_3=m_4=m$. It is shown that the sum of the first two terms of the Puiseux series, in powers of the dimensionless parameter $σ=\frac{m}{M}$, of the exact phase $Φ$ …

Beyond the Born-Oppenheimer approximation with quantum Monte Carlo

In this work we develop tools that enable the study of non-adiabatic effects with variational and diffusion Monte Carlo methods. We introduce a highly accurate wave function ansatz for electron-ion systems that can involve a combination of both fixed and quantum ions. We explicitly calculate the ground state energies of H$_{2}$, LiH, H$_{2}$O and FHF$^{-}$ using fixed-node quantum Monte Carlo with wave function nodes that explicitly depend on the ion positions. The obtained energies implicitly include the effects arising from quantum nuclei and electron-nucleus coupling. We compare our results…

超越Born-Oppenheimer格林函数理论：绝对与关系

Beyond Born-Oppenheimer Green's function theories: absolute and relational

本文提出一种关系性的约化格林函数理论，以解决超越Born-Oppenheimer近似下的库仑多体问题，强调可观测量的相对性定义，并阐明声子等实体可在关系理论中自然解释。

A correlated-polaron electronic propagator: open electronic dynamics beyond the Born-Oppenheimer approximation

In this work we develop a theory of correlated many-electron dynamics dressed by the presence of a finite-temperature harmonic bath. The theory is based on the ab-initio Hamiltonian, and thus well-defined apart from any phenomenological choice of collective basis states or electronic coupling model. The equation-of-motion includes some bath effects non-perturbatively, and can be used to simulate line- shapes beyond the Markovian approximation and open electronic dynamics which are subjects of renewed recent interest. Energy conversion and transport depend critically on the ratio of electron-el…

On the inclusion of the diagonal Born-Oppenheimer correction in surface hopping methods

The diagonal Born-Oppenheimer correction (DBOC) stems from the diagonal second derivative coupling term in the adiabatic representation, and it can have an arbitrary large magnitude when a gap between neighbouring Born-Oppenheimer (BO) potential energy surfaces (PESs) is closing. Nevertheless, DBOC is typically neglected in mixed quantum-classical methods of simulating nonadiabatic dynamics (e.g., fewest-switch surface hopping (FSSH) method). A straightforward addition of DBOC to BO PESs in the FSSH method, FSSH+D, has been shown to lead to numerically much inferior results for models containi…

Cavity Born-Oppenheimer Approximation for Correlated Electron-Nuclear-Photon Systems

In this work, we illustrate the recently introduced concept of the cavity Born-Oppenheimer approximation for correlated electron-nuclear-photon problems in detail. We demonstrate how an expansion in terms of conditional electronic and photon-nuclear wave functions accurately describes eigenstates of strongly correlated light-matter systems. For a GaAs quantum ring model in resonance with a photon mode we highlight how the ground-state electronic potential-energy surface changes the usual harmonic potential of the free photon mode to a dressed mode with a double-well structure. This change is a…

玻恩-奥本海默分子动力学中的马德隆变换与变分渐近法

Madelung transform and variational asymptotics in Born-Oppenheimer molecular dynamics

该研究基于哈密顿变分原理与欧拉-庞加莱约化构建BOMD的几何表述，借助马德隆变换的拉格朗日流体路径，通过渐近方法从连续模型变分原理推导BOMD，验证bohmion方案的即时实现，并拓展至混合量子-经典动力学模型。

运动布里渊近似

The Moving Born-Oppenheimer Approximation

运动布里渊近似通过混合量子-经典框架，揭示慢自由度与快自由度的复杂相互作用，应用于量子化学和材料科学等领域。

Efimov effect in a $D$-dimensional Born-Oppenheimer approach

We study a three-body system, formed by two identical heavy bosons and a light particle, in the Born-Oppenheimer approximation for an arbitrary dimension $D$. We restrict $D$ to the interval $2\,<\,D\,<\,4$, and derive the heavy-heavy $D$-dimensional effective potential proportional to $1/R^2$ ($R$ is the relative distance between the heavy particles), which is responsible for the Efimov effect. We found that the Efimov states disappear once the critical strength of the heavy-heavy effective potential $1/R^2$ approaches the limit $-(D-2)^2/4$. We obtained the scaling function for the $^{133}$C…

玻恩-奥本海默、玻恩-黄与精确因式分解：解析透明基准模型中的量子几何与误差

What solvable models reveal about Born-Oppenheimer, Born-Huang, and exact factorization

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。

理解空腔 Born-Oppenheimer 近似

Understanding the Cavity Born-Oppenheimer Approximation

该研究揭示了空腔 Born-Oppenheimer (CBO) 近似背后的物理效应，提出仅利用标准电子结构计算的非空腔物理量即可高精度恢复 CBO 能量和光谱，为完整 CBO 计算提供了实用高效的替代方案。

无玻恩-奥本海默框架下振动能量的相空间视角

A phase-space view of vibrational energies without the Born-Oppenheimer framework

本研究提出相空间电子哈密顿量$\hat H_{\rm PS}(\bm R,\bm P)$，通过部分维格纳变换严格建立其非微扰基础，在成本接近BO方法的情况下，能产生有意义的电子动量并显著提升振动能量计算精度，为非玻恩-奥本海默框架下的势能面构建提供更优方案。

超越Born-Oppenheimer动力学的光晶格量子模拟器

Optical lattice quantum simulator of dynamics beyond Born-Oppenheimer

提出基于光晶格中超冷费米子分子的量子模拟器，通过转动态模拟电子与核自由度，验证了非绝热效应，并展示了可调质量比下的电子交换与电离现象，且可推广至其他原子平台。

分子和材料的腔体-Born Oppenheimer近似方法：通过电场响应

Cavity-Born Oppenheimer Approximation for Molecules and Materials via Electric Field Response

本文提出了一种基于腔体Born-Oppenheimer近似的从头计算方法，用于计算分子和固体与光学腔耦合的声极子和声极子谱，利用电场响应性质提高计算效率和可解释性。

Elimination of the Translational Kinetic Energy Contamination in pre-Born-Oppenheimer Calculations

In this paper we present a simple strategy for the elimination of the translational kinetic energy contamination of the total energy in pre-Born--Oppenheimer calculations carried out in laboratory-fixed Cartesian coordinates (LFCCs). The simple expressions for the coordinates and the operators are thus preserved throughout the calculations, while the mathematical form and the parametrisation of the basis functions are chosen so that the translational and rotational invariances are respected. The basis functions are constructed using explicitly correlated Gaussian functions (ECGs) and the globa…

Ab initio linear-response approach to vibro-polaritons in the cavity Born-Oppenheimer approximation

Recent years have seen significant developments in the study of strong light-matter coupling including the control of chemical reactions by altering the vibrational normal modes of molecules. In the vibrational strong coupling regime the normal modes of the system become hybrid modes which mix nuclear, electronic, and photonic degrees of freedom. First principles methods capable of treating light and matter degrees of freedom on the same level of theory are an important tool in understanding such systems. In this work, we develop and apply a generalized force constant matrix approach to the st…

核电子轨道（NEO）理论的量子计算实现：迈向分子量子系统的精确前玻恩-奥本海默表述

A Quantum Computing Implementation of Nuclear-Electronic Orbital (NEO) Theory: Towards an Exact pre-Born-Oppenheimer Formulation of Molecular Quantum Systems

本文提出基于核电子轨道理论的量子计算方法，通过对称性缩减和参数初始化技术，在近期量子计算机上高效模拟电子-核量子系统，显著降低资源需求并保持高精度。