PHYS 5120 - Computational Energy Materials and Electronic Structure Simulations-W4

PHYS 5120 - 计算能源材料和电子结构模拟 Lecture-4

Lecturer: Prof.PAN DING

1 Monte Carlo (MC) Method:

• 内容:

This whiteboard provides a concise but detailed overview of two important and related simulation techniques in computational physics and chemistry: the Metropolis Monte Carlo (MC) method and Hamiltonian (or Hybrid) Monte Carlo (HMC). Here is a detailed breakdown of the concepts presented.

1. Metropolis Monte Carlo (MC) Method

The heading “Metropolis MC method” introduces a foundational algorithm in statistical mechanics. Metropolis Monte Carlo is a method used to generate a sequence of states for a system, allowing for the calculation of average properties. 左上角的这一部分介绍了基础的 Metropolis Monte Carlo 算法。它是一种生成状态序列的方法，使得处于任何状态的概率都符合期望的概率分布（在物理学中通常是玻尔兹曼分布）。

• Conceptual Diagram: The small box with numbered sites (0-5) and an arrow showing a move from state 0 to 2, and then to 3, illustrates a “random walk.” In Metropolis MC, the system transitions from one state to another by making small, random changes. 小方框中标有编号的位点（0-5），箭头表示从状态 0 到状态 2，再到状态 3 的移动，代表“随机游走”。在 Metropolis MC 中，系统通过进行微小的随机变化从一个状态过渡到另一个状态。
• Random Number Generation: The notation rand t \in (0,1) indicates the use of a random number \(t\) drawn from a uniform distribution between 0 and 1. This is a core component of the algorithm, used to decide whether to accept or reject a proposed new state. 符号 rand t \in (0,1) 表示使用从 0 到 1 之间的均匀分布中抽取的随机数 \(t\)。这是算法的核心部分，用于决定是否接受或拒绝提议的新状态。
• Detailed Balance Condition: The equation \(P_o T(o \to n) = P_n T(n \to o)\) is the principle of detailed balance. It states that in a system at equilibrium, the probability of being in an old state (\(o\)) and transitioning to a new state (\(n\)) is equal to the probability of being in the new state and transitioning back to the old one. This condition is crucial because it ensures that the simulation will eventually sample states according to their correct thermodynamic probabilities (the Boltzmann distribution). 方程 \(P_o T(o \to n) = P_n T(n \to o)\) 是详细平衡的原理。它指出，在平衡系统中，处于旧状态 (\(o\)) 并转变为新状态 (\(n\)) 的概率等于处于新状态并转变回旧状态的概率。此条件至关​​重要，因为它确保模拟最终将根据正确的热力学概率（玻尔兹曼分布）对状态进行采样。
• Acceptance Rate: The note \sim 30\%? likely refers to the target acceptance rate for an efficient Metropolis MC simulation. If new states are accepted too often or too rarely, the exploration of the system’s possible configurations is inefficient. While the famous optimal acceptance rate for certain high-dimensional problems is around 23.4%, a range of 20-50% is often considered effective. 注释“30%？”指的是高效 Metropolis 蒙特卡罗模拟的目标接受率。如果新状态接受过于频繁或过于稀少，系统对可能配置的探索就会变得低效。虽然某些高维问题的最佳接受率约为 23.4%，但通常认为 20-50% 的范围是有效的。

2. Hamiltonian / Hybrid Monte Carlo (HMC)

The second topic, “Hamiltonian/Hybrid MC (HMC),” is a more advanced Monte Carlo method that uses principles from classical mechanics to propose new states more intelligently than the simple random-walk approach of the standard Metropolis method. This often leads to a much higher acceptance rate and more efficient exploration of the state space. 第二个主题“哈密顿/混合蒙特卡罗 (HMC)”是一种更先进的蒙特卡罗方法，它利用经典力学原理，比标准 Metropolis 方法中简单的随机游走方法更智能地提出新状态。这通常会带来更高的接受率和更高效的状态空间探索。

The whiteboard outlines a four-step HMC algorithm:

Step 1: Randomize Velocities The first step is to randomize the velocities: \(\vec{v}_i \sim \mathcal{N}(0, k_B T)\). 第一步是随机化速度：\(\vec{v}_i \sim \mathcal{N}(0, k_B T)\)。 * This step introduces momentum into the system. For each particle \(i\), a velocity vector \(\vec{v}_i\) is randomly drawn from a normal (Gaussian) distribution with a mean of 0 and a variance related to the temperature \(T\) and the Boltzmann constant \(k_B\). 此步骤将动量引入系统。对于每个粒子 \(i\)，速度矢量 \(\vec{v}_i\) 会随机地从正态（高斯）分布中抽取，该分布的均值为 0，方差与温度 \(T\) 和玻尔兹曼常数 \(k_B\) 相关。 * The full formula for this probability distribution, \(f(\vec{v})\), is the Maxwell-Boltzmann distribution, which is written out further down the board. 该概率分布的完整公式 \(f(\vec{v})\) 是麦克斯韦-玻尔兹曼分布。

Step 2: Molecular Dynamics (MD) Integration The board notes this as t=0 \to h \text{ or } mh MD and mentions the Verlet algorithm.

• This is the “Hamiltonian dynamics” part of the algorithm. Starting from the current positions and the newly randomized velocities, the system’s trajectory is calculated for a short period of time (\(h\) or \(mh\)) using Molecular Dynamics (MD). 这是算法的“哈密顿动力学”部分。从当前位置和新随机化的速度开始，使用分子动力学 (MD) 计算系统在短时间内（\(h\) 或 \(mh\)）的轨迹。
• The name Verlet refers to the Verlet integration algorithm, a numerical method used to solve Newton’s equations of motion. It is popular in MD simulations because it is time-reversible and conserves energy well over long simulations. 指的是 Verlet 积分算法，这是一种用于求解牛顿运动方程的数值方法。它在 MD 模拟中很受欢迎，因为它具有时间可逆性，并且在长时间模拟中能量守恒效果良好。

Step 3: Calculate Total Energy The third step is to calculate total energy: \(E_n = K_n + V_n\). 第三步是“计算总能量”：\(E_n = K_n + V_n\)。 * After the MD trajectory, the system is in a new state \(n\). The total energy of this new state, \(E_n\), is calculated as the sum of its kinetic energy (\(K_n\), from the velocities) and its potential energy (\(V_n\), from the positions). MD 轨迹之后，系统处于新状态 \(n\)。新状态的总能量 \(E_n\) 等于其动能 (\(K_n\)，由速度计算得出）和势能 (\(V_n\)，由位置计算得出)之和。

Step 4: Acceptance Test The final step is the acceptance criterion: \(\text{acc}(o \to n) = \min(1, e^{-\beta(E_n - E_o)})\). 最后一步是验收标准：\(\text{acc}(o \to n) = \min(1, e^{-\beta(E_n - E_o)})\)。 * This is the Metropolis acceptance criterion. The algorithm decides whether to accept the new state \(n\) or reject it and stay in the old state \(o\). 这是 Metropolis 验收标准。算法决定是接受新状态 \(n\) 还是拒绝它并保持旧状态 \(o\)。 * The probability of acceptance depends on the change in total energy (\(E_n - E_o\)). If the new energy is lower, the move is always accepted. If the new energy is higher, it might still be accepted with a probability \(e^{-\beta(E_n - E_o)}\), where \(\beta = 1/(k_B T)\). This allows the system to escape from local energy minima. 验收概率取决于总能量的变化 (\(E_n - E_o\))。如果新能量较低，则始终接受该移动。如果新的能量更高，它仍然可能以概率 \(e^{-\beta(E_n - E_o)}\) 被接受，其中 \(\beta = 1/(k_B T)\)。这使得系统能够摆脱局部能量最小值。

Key Formulas and Notations

• Maxwell-Boltzmann Distribution麦克斯韦-玻尔兹曼分布: The formula for the velocity distribution is given as: \(f(\vec{v}) = \left(\frac{m}{2\pi k_B T}\right)^{3/2} \exp\left(-\frac{m v^2}{2 k_B T}\right)\) This gives the probability density for a particle of mass \(m\) to have a velocity \(\vec{v}\) at a given temperature \(T\).质量为 \(m\) 的粒子速度为 的概率密度

• Energy Conservation and Acceptance Rate: The notes \(E_n \approx E_o\) and \(75\%\) highlight a key advantage of HMC. Because the Verlet integrator approximately conserves energy, the final energy \(E_n\) after the MD trajectory is usually very close to the initial energy \(E_o\). This means the term \((E_n - E_o)\) is small, and the acceptance probability is high. The \(75\%\) indicates a typical or target acceptance rate for HMC, which is significantly higher than for standard Metropolis MC. 注释 \(E_n \approx E_o\) 和 \(75\%\) 凸显了 HMC 的一个关键优势。由于 Verlet 积分器近似地守恒能量，MD 轨迹后的最终能量 \(E_n\) 通常非常接近初始能量 \(E_o\)。这意味着 \((E_n - E_o)\) 项很小，接受概率很高。\(75\%\) 表示 HMC 的典型或目标接受率，明显高于标准 Metropolis MC。

• Hamiltonian Operator: The symbol \(\hat{H}\) written on the adjacent board represents the Hamiltonian operator, which gives the total energy of the system. The note Δ Adiabatic suggests that the MD evolution is ideally an adiabatic process (no heat exchange), during which the total energy (the Hamiltonian) is conserved. 相邻板上的符号 \(\hat{H}\) 代表哈密顿算符，它给出了系统的总能量。注释“Δ Adiabatic”表明 MD 演化在理想情况下是一个绝热过程（无热交换），在此过程中总能量（哈密顿量）守恒。

This whiteboard displays the fundamental equation of quantum chemistry: the time-dependent Schrödinger equation, along with the detailed breakdown of the molecular Hamiltonian operator. This equation is the starting point for almost all ab initio (first-principles) quantum mechanical calculations of molecular systems. 这块白板展示了量子化学的基本方程：含时薛定谔方程，以及分子哈密顿算符的详细分解。该方程是几乎所有分子系统从头算（第一性原理）量子力学计算的起点。

3. The Time-Dependent Schrödinger Equation

At the top of the board, the fundamental equation governing the evolution of a quantum mechanical system is presented: 白板顶部显示了控制量子力学系统演化的基本方程： \(i\hbar \frac{\partial \Psi}{\partial t} = \hat{\mathcal{H}} \Psi\)

• \(\Psi\) (Psi) is the wave function of the system. It contains all the information that can be known about the system (e.g., the positions and momenta of all particles). 是系统的波函数。它包含了关于系统的所有已知信息（例如，所有粒子的位置和动量）。

• \(\hat{\mathcal{H}}\) is the Hamiltonian operator, which represents the total energy of the system. 是哈密顿算符，表示系统的总能量。

• \(i\) 是虚数单位。

• \(i\) is the imaginary unit.

• \(\hbar\) is the reduced Planck constant.是约化普朗克常数。

• \(\frac{\partial \Psi}{\partial t}\) represents how the wave function changes over time.表示波函数随时间的变化。

This equation states that the time evolution of the quantum state is dictated by the system’s total energy operator, the Hamiltonian. The note “Δ Adiabatic process” likely connects to the context of the Born-Oppenheimer approximation, where the electronic Schrödinger equation is solved for fixed nuclear positions, assuming the electrons adjust adiabatically (instantaneously) to the motion of the nuclei. 该方程表明，量子态的时间演化由系统的总能量算符——哈密顿算符决定。注释“Δ绝热过程”与玻恩-奥本海默近似相关，在该近似中，电子薛定谔方程是针对固定原子核位置求解的，假设电子以绝热方式（瞬时）调整以适应原子核的运动。

4. The Full Molecular Hamiltonian (\(\hat{\mathcal{H}}\))

The main part of the whiteboard is the detailed expression for the non-relativistic, time-independent molecular Hamiltonian. It is the sum of the kinetic and potential energies of all the nuclei and electrons in the system. The equation can be broken down into five distinct terms: 白板的主要部分是非相对论性、时间无关的分子哈密顿量的详细表达式。它是系统中所有原子核和电子的动能和势能之和。

该方程可以分解为五个不同的项：

\(\hat{\mathcal{H}} = -\sum_{I=1}^{P} \frac{\hbar^2}{2M_I}\nabla_I^2 - \sum_{i=1}^{N} \frac{\hbar^2}{2m}\nabla_i^2 + \frac{e^2}{2}\sum_{I=1}^{P}\sum_{J \neq I}^{P} \frac{Z_I Z_J}{|\vec{R}_I - \vec{R}_J|} + \frac{e^2}{2}\sum_{i=1}^{N}\sum_{j \neq i}^{N} \frac{1}{|\vec{r}_i - \vec{r}_j|} - e^2\sum_{I=1}^{P}\sum_{i=1}^{N} \frac{Z_I}{|\vec{R}_I - \vec{r}_i|}\)

Let’s analyze each component:

A. Kinetic Energy Terms 动能项

1. Kinetic Energy of the Nuclei 原子核的动能: \(-\sum_{I=1}^{P} \frac{\hbar^2}{2M_I}\nabla_I^2\) This term is the sum of the kinetic energy operators for all the nuclei in the system.此项是系统中所有原子核的动能算符之和。
   • The sum is over all nuclei, indexed by \(I\) from 1 to \(P\).该和涵盖所有原子核，索引为 \(I\)，从 1 到 \(P\)。
   • \(M_I\) is the mass of nucleus \(I\).是原子核 \(I\) 的质量。
   • \(\nabla_I^2\) is the Laplacian operator, which involves the second spatial derivatives with respect to the coordinates of nucleus \(I\).是拉普拉斯算符，它涉及原子核 \(I\) 坐标的二阶空间导数。
2. Kinetic Energy of the Electrons 电子的动能: \(-\sum_{i=1}^{N} \frac{\hbar^2}{2m}\nabla_i^2\) This is the corresponding sum of the kinetic energy operators for all the electrons.这是所有电子的动能算符的对应和。
   • The sum is over all electrons, indexed by \(i\) from 1 to \(N\).该和是针对所有电子的，索引为 \(i\)，从 1 到 \(N\)。
   • \(m\) is the mass of an electron.是电子的质量。
   • \(\nabla_i^2\) is the Laplacian operator with respect to the coordinates of electron \(i\).是关于电子 \(i\) 坐标的拉普拉斯算符。

B. Potential Energy Terms (Electrostatic Interactions) 势能项（静电相互作用）

3. Nuclear-Nuclear Repulsion 核间排斥力: \(+\frac{e^2}{2}\sum_{I=1}^{P}\sum_{J \neq I}^{P} \frac{Z_I Z_J}{|\vec{R}_I - \vec{R}_J|}\) This term represents the potential energy from the electrostatic (Coulomb) repulsion between all pairs of positively charged nuclei.该项表示所有带正电原子核对之间静电（库仑）排斥力产生的势能。
   • The double summation runs over all unique pairs of nuclei (\(I, J\)).对所有唯一的原子核对 (\(I, J\)) 进行双重求和。
   • \(Z_I\) is the atomic number (i.e., the charge) of nucleus \(I\).是原子核 \(I\) 的原子序数（即电荷）。
   • \(\vec{R}_I\) is the position vector of nucleus \(I\).是原子核 \(I\) 的位置矢量。
   • \(e\) is the elementary charge.是基本电荷。
4. Electron-Electron Repulsion 电子间排斥力: \(+\frac{e^2}{2}\sum_{i=1}^{N}\sum_{j \neq i}^{N} \frac{1}{|\vec{r}_i - \vec{r}_j|}\) This term represents the potential energy from the electrostatic repulsion between all pairs of negatively charged electrons.该项表示所有带负电的电子对之间静电排斥的势能。
   • The double summation runs over all unique pairs of electrons (\(i, j\)).对所有不同的电子对 (\(i, j\)) 进行双重求和。
   • \(\vec{r}_i\) is the position vector of electron \(i\).是电子 \(i\) 的位置矢量。
5. Nuclear-Electron Attraction 核-电子引力: \(-e^2\sum_{I=1}^{P}\sum_{i=1}^{N} \frac{Z_I}{|\vec{R}_I - \vec{r}_i|}\) This final term represents the potential energy from the electrostatic attraction between the nuclei and the electrons.这最后一项表示原子核和电子之间静电引力的势能。
   • The summation runs over all nuclei and all electrons.该求和适用于所有原子核和所有电子。

5. Notations and Conventions

• Atomic Units: The note \(\frac{1}{4\pi\epsilon_0} = k = 1\) is a key indicator of the convention being used. This sets the Coulomb constant to 1, which is a hallmark of Hartree atomic units. In this system, the elementary charge (\(e\)), electron mass (\(m\)), and reduced Planck constant (\(\hbar\)) are also set to 1. This simplifies the Hamiltonian significantly, removing the physical constants and making the equations easier to work with computationally. 是所用约定的关键指标。这将库仑常数设置为 1，这是Hartree 原子单位的标志。在这个系统中，基本电荷 (\(e\))、电子质量 (\(m\)) 和​​约化普朗克常数 (\(\hbar\)) 也设为 1。这显著简化了哈密顿量，消除了物理常数，使方程更易于计算。
• Interaction Terms: The notations \(\{i, j\}\), \(\{i, j, k\}\), etc., refer to the “many-body” problem. The Hamiltonian contains two-body terms (interactions between pairs of particles), and solving the Schrödinger equation exactly is extremely difficult because the motion of every particle is correlated with every other particle. Computational methods are designed to approximate these interactions. 符号 \(\{i, j\}\)、\(\{i, j, k\}\) 等指的是“多体”问题。哈密顿量包含二体项（粒子对之间的相互作用），而精确求解薛定谔方程极其困难，因为每个粒子的运动都与其他粒子相关。计算方法旨在近似这些相互作用。

This whiteboard presents the mathematical foundation for non-adiabatic molecular dynamics, a sophisticated method in theoretical chemistry and physics used to simulate processes where the Born-Oppenheimer approximation breaks down. This typically occurs in photochemistry, electron transfer reactions, and when molecules interact with intense laser fields. 这块白板展示了非绝热分子动力学的数学基础，这是理论化学和物理学中一种复杂的方法，用于模拟玻恩-奥本海默近似失效的过程。这通常发生在光化学、电子转移反应以及分子与强激光场相互作用时。

6. Topic: Non-Adiabatic Molecular Dynamics (MD) 非绝热分子动力学 (MD)

The title “Δ non-adiabatic MD” indicates that the topic moves beyond the standard Born-Oppenheimer approximation. In this approximation, it is assumed that the light electrons adjust instantaneously to the motion of the heavy nuclei, allowing the system to be described by a single potential energy surface. Non-adiabatic methods, by contrast, account for the quantum mechanical coupling between multiple electronic states.

标题“Δ 非绝热 MD”表明该主题超越了标准的玻恩-奥本海默近似。在该近似中，假设轻电子会根据重原子核的运动进行瞬时调整，从而使系统可以用单个势能面来描述。相比之下，非绝热方法则考虑了多个电子态之间的量子力学耦合。

7. The Born-Huang Ansatz 玻恩-黄拟设

The starting point for this method is the “ansatz” (an educated guess for the form of the solution). This is the Born-Huang expansion for the total molecular wave function, \(\Psi\). 该方法的起点是“拟设”（对解形式的合理猜测）。这是分子总波函数 \(\Psi\) 的玻恩-黄展开式。

\(\Psi(\vec{R}, \vec{r}, t) = \sum_{n} \Theta_n(\vec{R}, t) \Phi_n(\vec{R}, \vec{r})\)

• \(\Psi(\vec{R}, \vec{r}, t)\) is the total wave function for the entire molecule. It depends on the coordinates of all nuclei (\(\vec{R}\)), all electrons (\(\vec{r}\)), and time (\(t\)). 是整个分子的总波函数。它取决于所有原子核 (\(\vec{R}\))、所有电子 (\(\vec{r}\)) 和时间 (\(t\)) 的坐标。

• \(\Phi_n(\vec{R}, \vec{r})\) are the electronic wave functions. They are the solutions to the electronic Schrödinger equation for a fixed nuclear geometry \(\vec{R}\) and form a complete basis set. The index \(n\) labels the electronic state (e.g., ground state, first excited state, etc.). 它们是给定原子核几何构型 \(\vec{R}\) 的电子薛定谔方程的解，并构成一个完整的基组。下标 \(n\) 标记电子态（例如，基态、第一激发态等）。

• \(\Theta_n(\vec{R}, t)\) are the nuclear wave functions. Each \(\Theta_n\) describes the motion of the nuclei on the potential energy surface of the corresponding electronic state, \(\Phi_n\). Crucially, they depend on time. 是核波函数。每个 \(\Theta_n\) 描述原子核在相应电子态 \(\Phi_n\) 势能面上的运动。至关重要的是，它们依赖于时间。

This ansatz expresses the total molecular state as a superposition of electronic states, where the coefficients of the superposition are the nuclear wave functions. 该拟设将总分子态表示为电子态的叠加，其中叠加的系数是核波函数。

8. The Partitioned Molecular Hamiltonian 分割分子哈密顿量

The total molecular Hamiltonian, \(\hat{\mathcal{H}}\), is partitioned into terms that act on the nuclei and electrons separately. 总分子哈密顿量 \(\hat{\mathcal{H}}\) 被分割成分别作用于原子核和电子的项。

\(\hat{\mathcal{H}} = -\sum_{I} \frac{\hbar^2}{2M_I}\nabla_I^2 + \hat{\mathcal{H}}_e + \hat{V}_{nn}\)

• \(-\sum_{I} \frac{\hbar^2}{2M_I}\nabla_I^2\): This is the kinetic energy operator for the nuclei, often denoted as \(\hat{T}_n\).这是原子核的动能算符，通常表示为 \(\hat{T}_n\)。

• \(\hat{\mathcal{H}}_e\): This is the electronic Hamiltonian, which includes the kinetic energy of the electrons and the potential energy of electron-electron and electron-nuclear interactions. 这是电子哈密顿量，包含电子的动能以及电子-电子和电子-核相互作用的势能。

• \(\hat{V}_{nn}\): This is the potential energy operator for nuclear-nuclear repulsion.这是核-核排斥的势能算符。

9. The Electronic Schrödinger Equation 电子薛定谔方程

The electronic basis functions, \(\Phi_n\), are defined as the eigenfunctions of the electronic Hamiltonian (plus the nuclear repulsion term) for a fixed nuclear configuration \(\vec{R}\). 电子基函数 \(\Phi_n\) 定义为对于固定的核构型 \(\vec{R}\)，电子哈密顿量（加上核排斥项）的本征函数。

\((\hat{\mathcal{H}}_e + \hat{V}_{nn}) \Phi_n(\vec{R}, \vec{r}) = E_n(\vec{R}) \Phi_n(\vec{R}, \vec{r})\)

• \(E_n(\vec{R})\) are the eigenvalues, which are the potential energy surfaces (PES). Each electronic state \(n\) has its own PES, which dictates the forces acting on the nuclei when the molecule is in that electronic state. 是特征值，即势能面 (PES)。每个电子态 \(n\) 都有其自身的势能面，它决定了分子处于该电子态时作用于原子核的力。

10. Deriving the Equations of Motion for the Nuclei 推导原子核运动方程

The final part of the whiteboard begins the derivation of the time-dependent Schrödinger equation for the nuclear wave functions, \(\Theta_k\). The process starts with the full time-dependent Schrödinger equation, \(i\hbar \frac{\partial \Psi}{\partial t} = \hat{\mathcal{H}} \Psi\). To find the equation for a specific nuclear wave function \(\Theta_k\), this main equation is projected onto the corresponding electronic basis state \(\Phi_k\). 白板的最后一部分开始推导原子核波函数 \(\Theta_k\) 的含时薛定谔方程。该过程从完整的含时薛定谔方程 \(i\hbar \frac{\partial \Psi}{\partial t} = \hat{\mathcal{H}} \Psi\) 开始。为了找到特定原子核波函数 \(\Theta_k\) 的方程，需要将这个主方程投影到相应的电子基态 \(\Phi_k\) 上。

This is done by multiplying from the left by the complex conjugate of the electronic wave function, \(\Phi_k^*\), and integrating over all electronic coordinates, \(d\vec{r}\). 可以通过从左边乘以电子波函数 \(\Phi_k^*\) 的复共轭，然后在所有电子坐标 \(d\vec{r}\) 上积分来实现。

\(\int \Phi_k^* i\hbar \frac{\partial}{\partial t} \Psi \,d\vec{r} = \int \Phi_k^* \hat{\mathcal{H}} \Psi \,d\vec{r}\)

The board then shows the result of substituting the Born-Huang ansatz for \(\Psi\) and the partitioned Hamiltonian for \(\hat{\mathcal{H}}\) into this projected equation: 然后，黑板显示将 Born-Huang 拟设式代入 \(\Psi\)，将分块哈密顿量代入以下投影方程的结果：

\(i\hbar \frac{\partial}{\partial t} \Theta_k(\vec{R}, t) = \int \Phi_k^* \left( -\sum_{I} \frac{\hbar^2}{2M_I}\nabla_I^2 + \hat{\mathcal{H}}_e + \hat{V}_{nn} \right) \sum_n \Theta_n \Phi_n \,d\vec{r}\)

• Left Hand Side: The left side of the projection has been simplified. Because the electronic basis functions \(\Phi_n\) form an orthonormal set (\(\int \Phi_k^* \Phi_n d\vec{r} = \delta_{kn}\)), the sum collapses to a single term for \(n=k\). 投影左侧已简化。由于电子基函数 \(\Phi_n\) 构成一个正交集 (\(\int \Phi_k^* \Phi_n d\vec{r} = \delta_{kn}\)，因此当 \(n=k\) 时，和将折叠为一个项。

• Right Hand Side: This complex integral is the core of non-adiabatic dynamics. When the nuclear kinetic energy operator, \(\nabla_I^2\), acts on the product \(\Theta_n \Phi_n\), it acts on both functions (via the product rule). The terms that arise from \(\nabla_I\) acting on the electronic wave functions \(\Phi_n\) are known as non-adiabatic coupling terms. These terms are responsible for enabling transitions between different electronic potential energy surfaces, which is the essence of non-adiabatic dynamics. 这个复积分是非绝热动力学的核心。当核动能算符 \(\nabla_I^2\) 作用于乘积 \(\Theta_n \Phi_n\) 时，它会作用于这两个函数（通过乘积规则）。由 \(\nabla_I\) 作用于电子波函数 \(\Phi_n\) 而产生的项称为非绝热耦合项。这些术语负责实现不同电子势能面之间的转变，这是非绝热动力学的本质。

This whiteboard continues the mathematical derivation for non-adiabatic molecular dynamics started in the previous image. It focuses on expanding the nuclear kinetic energy term to reveal the crucial couplings between different electronic states.这块白板延续了上一张图片中非绝热分子动力学的数学推导。它着重于扩展核动能项，以揭示不同电子态之间的关键耦合。

11. Starting Point: The Projected Schrödinger Equation 起点：投影薛定谔方程

The derivation picks up from the equation for the time evolution of the nuclear wave function, \(\Theta_k\). The right-hand side of this equation is being evaluated. 推导过程取自核波函数 \(\Theta_k\) 的时间演化方程。该方程的右边正在求值。

\(= \int \Phi_k^* \left( -\sum_{I} \frac{\hbar^2}{2M_I}\nabla_I^2 \right) \sum_n \Theta_n \Phi_n \,d\vec{r} + E_k \Theta_k\)

This equation separates the total energy into two parts 该方程将总能量分为两部分 : * The first term is the contribution from the nuclear kinetic energy operator, \(-\sum_{I} \frac{\hbar^2}{2M_I}\nabla_I^2\). 第一项是核动能算符的贡献 * The second term, \(E_k \Theta_k\), is the contribution from the potential energy. This term arises from the action of the electronic Hamiltonian part \((\hat{\mathcal{H}}_e + \hat{V}_{nn})\) on the basis functions. Due to the orthonormality of the electronic wavefunctions (\(\int \Phi_k^* \Phi_n \,d\vec{r} = \delta_{kn}\)), the sum over \(n\) collapses to a single term for the potential energy. 第二项，\(E_k \Theta_k\)，是势能的贡献。这一项源于电子哈密顿量部分 \((\hat{\mathcal{H}}_e + \hat{V}_{nn})\) 对基函数的作用。由于电子波函数（\(\int \Phi_k^* \Phi_n \,d\vec{r} = \delta_{kn}\)）的正交性，\(n\)项的和会坍缩为势能的一项。

The challenge, and the core of the physics, lies in evaluating the first term, as the nuclear derivative \(\nabla_I\) acts on both the nuclear wave function \(\Theta_n\) and the electronic wave function \(\Phi_n\). 难点在于，也是物理的核心在于如何计算第一项，因为核导数 \(\nabla_I\) 同时作用于核波函数 \(\Theta_n\) 和电子波函数 \(\Phi_n\)。

12. Applying the Product Rule for the Laplacian 应用拉普拉斯算子的乘积规则

To expand the kinetic energy term, the product rule for the Laplacian operator acting on two functions (A and B) is used. The board writes this rule as: 为了展开动能项，我们利用了拉普拉斯算子作用于两个函数（A 和 B）的乘积规则。棋盘上将这条规则写成： \(\nabla^2(AB) = (\nabla^2 A)B + 2(\nabla A)\cdot(\nabla B) + A(\nabla^2 B)\)

In our case, \(A = \Theta_n(\vec{R}, t)\) and \(B = \Phi_n(\vec{R}, \vec{r})\). The derivative \(\nabla_I\) is with respect to the nuclear coordinates \(\vec{R}_I\). 在我们的例子中，\(A = \Theta_n(\vec{R}, t)\)，\(B = \Phi_n(\vec{R}, \vec{r})\)。导数 \(\nabla_I\) 是关于原子核坐标 \(\vec{R}_I\) 的。

13. Expanding the Kinetic Energy Term 展开动能项

Applying this rule, the integral containing the kinetic energy operator is expanded: 应用此规则，展开包含动能算符的积分： \(= -\sum_I \frac{\hbar^2}{2M_I} \int \Phi_k^* \sum_n \left( (\nabla_I^2 \Theta_n)\Phi_n + 2(\nabla_I \Theta_n)\cdot(\nabla_I \Phi_n) + \Theta_n(\nabla_I^2 \Phi_n) \right) d\vec{r} + E_k \Theta_k\)

This step explicitly shows how the nuclear kinetic energy operator gives rise to three distinct types of terms.此步骤明确展示了核动能算符如何产生三种不同类型的项。

14. Final Result and Identification of Coupling Terms 最终结果及耦合项的识别

The final step is to take the integral over the electronic coordinates (\(d\vec{r}\)) and rearrange the terms. The expression is simplified by again using the orthonormality of the electronic wave functions, \(\int \Phi_k^* \Phi_n \, d\vec{r} = \delta_{kn}\). 最后一步是对电子坐标 (\(d\vec{r}\)) 进行积分，并重新排列各项。再次利用电子波函数的正交性简化表达式，\(\int \Phi_k^* \Phi_n \, d\vec{r} = \delta_{kn}\)。

\(= -\sum_I \frac{\hbar^2}{2M_I} \left( \nabla_I^2 \Theta_k + \sum_n 2 \left( \int \Phi_k^* \nabla_I \Phi_n \, d\vec{r} \right) \cdot \nabla_I \Theta_n + \sum_n \left( \int \Phi_k^* \nabla_I^2 \Phi_n \, d\vec{r} \right) \Theta_n \right) + E_k \Theta_k\)

This final equation is profound. It represents the time-independent Schrödinger equation for the nuclear wave function \(\Theta_k\), but it is coupled to all other nuclear wave functions \(\Theta_n\). Let’s break down the key terms within the parentheses: 最后一个方程意义深远。它代表了核波函数 \(\Theta_k\) 的与时间无关的薛定谔方程，但它与所有其他核波函数 \(\Theta_n\) 耦合。让我们分解一下括号内的关键项：

• \(\nabla_I^2 \Theta_k\): This is the standard kinetic energy term for the nuclei moving on the potential energy surface of state \(k\). This is the only term that would remain in the simple Born-Oppenheimer (adiabatic) approximation. 这是原子核在势能面 \(k\) 上运动的标准动能项。这是在简单的 Born-Oppenheimer（绝热）近似中唯一保留的项。

• \(\left( \int \Phi_k^* \nabla_I \Phi_n \, d\vec{r} \right)\): This is the first-derivative non-adiabatic coupling term (NACT), often called the derivative coupling. This vector quantity determines the strength of the coupling between electronic states \(k\) and \(n\) due to the velocity of the nuclei. It is the primary term responsible for enabling transitions between different potential energy surfaces. 这是一阶导数非绝热耦合项 (NACT)，通常称为导数耦合。该矢量决定了由于原子核速度而导致的电子态 \(k\) 和 \(n\) 之间耦合的强度。它是实现不同势能面之间跃迁的主要项。

• \(\left( \int \Phi_k^* \nabla_I^2 \Phi_n \, d\vec{r} \right)\): This is the second-derivative non-adiabatic coupling term, a scalar quantity. While often smaller than the first-derivative term, it is also part of the complete description of non-adiabatic effects. 是二阶导数非绝热耦合项，一个标量。虽然它通常小于一阶导数项，但它也是非绝热效应完整描述的一部分。

In summary, this derivation shows mathematically how the motion of the nuclei (via the \(\nabla_I\) operator) can induce quantum mechanical transitions between different electronic states (\(\Phi_k \leftrightarrow \Phi_n\)). The strength of these transitions is governed by the non-adiabatic coupling terms, which depend on how the electronic wave functions change as the nuclear geometry changes. 总之，该推导从数学上展示了原子核的运动（通过 \(\nabla_I\) 算符）如何诱导不同电子态之间的量子力学跃迁（\(\Phi_k \leftrightarrow \Phi_n\)）。这些跃迁的强度由非绝热耦合项控制，而非绝热耦合项又取决于电子波函数如何随原子核几何结构的变化而变化。

This whiteboard concludes the derivation of the equations for non-adiabatic molecular dynamics by defining the coupling operator and then showing how different levels of approximation—specifically the Born-Huang and the more restrictive Born-Oppenheimer approximations—arise from neglecting certain coupling terms. 这块白板通过定义耦合算符，并展示不同程度的近似——特别是 Born-Huang 近似和更严格的 Born-Oppenheimer 近似——是如何通过忽略某些耦合项而产生的，从而推导出非绝热分子动力学方程的。

15. Definition of the Non-Adiabatic Coupling Operator 非绝热耦合算符的定义

The whiteboard begins by collecting all the non-adiabatic coupling terms derived previously into a single operator, \(C_{kn}\). 白板首先将之前推导的所有非绝热耦合项合并为一个算符 \(C_{kn}\)。

Let \(C_{kn} = -\sum_{I} \frac{\hbar^2}{2M_I} \left( 2 \left( \int \Phi_k^* \nabla_I \Phi_n \, d\vec{r} \right) \cdot \nabla_I + \left( \int \Phi_k^* \nabla_I^2 \Phi_n \, d\vec{r} \right) \right)\)

• This operator, \(C_{kn}\), represents the total effect of the coupling between electronic state \(k\) and electronic state \(n\), which is induced by the kinetic energy of the nuclei. 此算符 \(C_{kn}\) 表示由原子核动能引起的电子态 \(k\) 和电子态 \(n\) 之间耦合的总效应。
• The operator acts on the nuclear wave function that follows it in the full equation. The \(\nabla_I\) term acts as a derivative on that wave function. 该算符作用于完整方程中跟随它的核波函数。\(\nabla_I\) 项充当该波函数的导数。

16. The Coupled Equations of Motion 耦合运动方程

Using this compact definition, the full set of coupled time-dependent Schrödinger equations for the nuclear wave functions can be written as: 基于此简洁定义，核波函数的完整耦合含时薛定谔方程组可以写成：

\(i\hbar \frac{\partial}{\partial t} \Theta_k = \left( -\sum_{I} \frac{\hbar^2}{2M_I}\nabla_I^2 + E_k \right) \Theta_k + \sum_n C_{kn} \Theta_n\)

This is the central result. It shows that the time evolution of the nuclear wave function on a given potential energy surface \(k\) (described by \(\Theta_k\)) depends on two things: 这是核心结论。它表明，核波函数在给定势能面 \(k\)（用 \(\Theta_k\) 描述）上的时间演化取决于两个因素： 1. The motion on its own surface, governed by its kinetic energy and the potential \(E_k\). 其自身表面上的运动，由其动能和势能 \(E_k\) 控制。 2. The influence of the nuclear wave functions on all other electronic surfaces (\(\Theta_n\)), mediated by the coupling operators \(C_{kn}\). 核波函数对所有其他电子表面（\(\Theta_n\)）的影响，由耦合算符 \(C_{kn}\) 介导。

17. The Born-Huang Approximation 玻恩-黄近似

The first and most crucial approximation is introduced to simplify this complex set of coupled equations. 为了简化这组复杂的耦合方程，引入了第一个也是最重要的近似。

If \(C_{kn} = 0\) for \(k \neq n\) (Born-Huang approximation)

This approximation assumes that the off-diagonal coupling terms, which are responsible for transitions between different electronic states, are negligible. However, it retains the diagonal coupling term (\(C_{kk}\)). This leads to a simplified, uncoupled equation: 该近似假设导致不同电子态之间跃迁的非对角耦合项可以忽略不计。然而，它保留了对角耦合项（\(C_{kk}\)）。这可以得到一个简化的非耦合方程：

\(i\hbar \frac{\partial}{\partial t} \Theta_k = \left( -\sum_{I} \frac{\hbar^2}{2M_I}\nabla_I^2 + E_k + C_{kk} \right) \Theta_k\)

Substituting the definition of \(C_{kk}\): 代入 \(C_{kk}\) 的定义：

\(i\hbar \frac{\partial}{\partial t} \Theta_k = \left( -\sum_{I} \frac{\hbar^2}{2M_I}\nabla_I^2 + E_k - \sum_I \frac{\hbar^2}{2M_I} \left( 2 \left( \int \Phi_k^* \nabla_I \Phi_k \, d\vec{r} \right) \cdot \nabla_I + \int \Phi_k^* \nabla_I^2 \Phi_k \, d\vec{r} \right) \right) \Theta_k\)

The term \(C_{kk}\) is known as the diagonal Born-Oppenheimer correction (DBOC). It represents a small correction to the potential energy surface \(E_k\) that arises from the fact that the electrons do not adjust perfectly and instantaneously to the nuclear motion, even within the same electronic state. \(C_{kk}\) 项被称为对角玻恩-奥本海默修正 (DBOC)。它表示对势能面 \(E_k\) 的微小修正，其原因是即使在相同的电子态下，电子也无法完美且即时地适应核运动。

• Note on Real Wavefunctions 关于实波函数的注释: The board shows that for real wavefunctions, the first-derivative part of the diagonal correction vanishes: \(\int \Phi_k \nabla_I \Phi_k \, d\vec{r} = 0\). This is because the integral is related to the gradient of the normalization condition, \(\nabla_I \int \Phi_k^2 \, d\vec{r} = \nabla_I(1) = 0\), which expands to \(2\int \Phi_k \nabla_I \Phi_k \, d\vec{r} = 0\). 黑板显示，对于实波函数，对角修正的一阶导数部分为零：\(\int \Phi_k \nabla_I \Phi_k \, d\vec{r} = 0\)。这是因为积分与归一化条件的梯度有关，\(\nabla_I \int \Phi_k^2 \, d\vec{r} = \nabla_I(1) = 0\)，其展开为 \(2\int \Phi_k \nabla_I \Phi_k \, d\vec{r} = 0\)。

18. The Born-Oppenheimer Approximation 玻恩-奥本海默近似

The final and most widely used approximation is the Born-Oppenheimer approximation. It is more restrictive than the Born-Huang approximation. 最后一种也是最广泛使用的近似方法是玻恩-奥本海默近似。它比玻恩-黄近似更具限制性。

If \(C_{kk} = 0\) (Born-Oppenheimer approximation) 若\(C_{kk} = 0\)（玻恩-奥本海默近似）

This assumes that the diagonal correction term is also negligible. By setting all \(C_{kn}=0\) (both diagonal and off-diagonal), the equations become completely decoupled, and the nuclear motion evolves independently on each potential energy surface. 这假设对角修正项也可忽略不计。通过令所有\(C_{kn}=0\)（包括对角和非对角），方程组完全解耦，原子核运动在每个势能面上独立演化。

The result is the standard time-dependent Schrödinger equation for the nuclei: 由此可得标准的原子核的含时薛定谔方程：

\(i\hbar \frac{\partial}{\partial t} \Theta_k = \left( -\sum_{I} \frac{\hbar^2}{2M_I}\nabla_I^2 + E_k \right) \Theta_k\)

This equation is the foundation of most of quantum chemistry. It states that the nuclei move on a static potential energy surface \(E_k(\vec{R})\) provided by the electrons, without any possibility of transitioning to other electronic states or having the surface be corrected by their own motion.

该方程是大多数量子化学的基础。原子核在由电子提供的静态势能面 \(E_k(\vec{R})\) 上运动，不存在跃迁到其他电子态或因自身运动而修正势能面的可能性。