We reconsider the time-dependent Born-Oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands. Band crossings are allowed and our results are local in the sense that they hold up to the first time when a band crossing is encountered. The adiabatic decoupling leads to an effective Schroedinger equation for the nuclei, including contributions from the Berry connection.
A Time-Dependent Born-Oppenheimer Approximation with Exponentially Small Error Estimates
本文构造了分子量子力学的指数精度含时Born-Oppenheimer近似。我们研究的分子系统的电子质量是固定的，其核质量是成正比的$ε^{-4}$，其中$ε$是一个小的扩展参数。通过对渐近展开式的最佳截断，我们构造了含时薛定谔方程的近似解，对于某些C和$γ>0$，该近似解与精确的归一化解一致，直到其范数由$\ds C \exp(-γ/ε^2)$限制的误差为止.
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…
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 $Φ$ …
Graph-based linear scaling electronic structure theory for quantum-mechanical molecular dynamics simulations is adapted to the most recent shadow potential formulations of extended Lagrangian Born-Oppenheimer molecular dynamics, including fractional molecular-orbital occupation numbers, which enables stable simulations of sensitive complex chemical systems with unsteady charge solutions. The proposed formulation includes a preconditioned Krylov subspace approximation for the integration of the extended electronic degrees of freedom, which requires quantum response calculations for electronic s…
Born-Oppenheimer-type Approximations for Degenerate Potentials: Recent Results and a Survey on the area
本文研究了当势函数V在无穷远处不趋于无穷大时，薛定谔算子本征值的渐近性。这种势称为简并势。关键是相空间中的一组相关联的哈密顿量小于一个固定的能量E可能有一个无限大的体积，使外尔公式给出的行为的计数功能必须重新审视。我们回顾在这方面的各种结果，在经典的背景下，以及在半经典的和评论不同的方法。在第3，4节中，我们提出了我们与A Morame（Université de Nantes）的联合工作，涉及退化势V（x）=f（y）g（z），其中g被假定为m个变量的齐次正函数，f是n个变量的光滑严格正函数，最小值在0。当f在无穷远处趋于无穷大时，我们给出了小于固定能量的特征值个数的半经典渐近性质。然后，我们给出了一个尖锐的估计的低本征值，使用玻恩奥本海默近似。通过一种改进的方法，我们也定位了更高的能量。最后，我们将前面的方法应用到一类在正则超曲面上为零的势函数上。
Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces
Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）；University of Oslo（奥斯陆大学）；Kungl. Tekniska Högskolan（皇家理工学院）；University of Delaware（特拉华大学）
