Born-Oppenheimer近似和失效:非绝热耦合

绝热表象

分子体系的Hamiltonian算符可写为:
\begin{equation} \hat{H}(\mathbf{r}, \mathbf{R}) = \hat{T}_\text{N} + \hat{H}_\text{e}(\mathbf{r}) + \hat{V}_{eN}(\mathbf{r}, \mathbf{R}) \end{equation}

其中电子的Hamiltonian为
\begin{equation} \hat{H}_\text{e}(\mathbf{r}) = \hat{T}_\text{e} + \hat{V}_\text{ee} \end{equation}

如果假定核坐标作为固定参数,我们可以写出电子的Schroedinger方程:
\begin{equation} \big[ \hat{H}_\text{e}(\mathbf{r}) + \hat{V}_{eN}(\mathbf{r}, \mathbf{R}) \big]\phi_n(\mathbf{r}; \mathbf{R}) = \varepsilon_n(\mathbf{R}) \phi_n(\mathbf{r}; \mathbf{R}) \end{equation}

\(\phi_n(\mathbf{r}; \mathbf{R})\) 和 \(\varepsilon_n(\mathbf{R})\) 分别称为固定核坐标下的电子绝热本征函数和绝热本征值。由于绝热本征函数 \(\phi_n(\mathbf{r}; \mathbf{R})\) 构成完备基,对于满足Schroedinger方程
\begin{equation} \hat{H}(\mathbf{r}, \mathbf{R})\Psi(\mathbf{r}, \mathbf{R}) = E\Psi(\mathbf{r}, \mathbf{R}) \end{equation}

的分子波函数 \(\Psi(\mathbf{r}, \mathbf{R})\) ,可用绝热基展开
\begin{equation} \Psi(\mathbf{r}, \mathbf{R}) = \sum\limits_n \chi_n(\mathbf{R})\phi_n(\mathbf{r}; \mathbf{R}) \end{equation}

\(\chi_n(\mathbf{R})\) 是绝热表象下的核波函数。将分子波函数代入到分子的Schroedinger方程中,两边乘以 \(\phi_m\) ,并对电子坐标积分
\begin{equation} \big[ \hat{T}_\text{N}(\mathbf{R}) + \varepsilon_m(\mathbf{R}) \big]\chi_m(\mathbf{R}) + \sum\limits_n \Lambda_{mn}(\mathbf{R})\chi_n(\mathbf{R}) = E\chi_m(\mathbf{R}) \end{equation}

\(\Lambda_{mn}(\mathbf{R})\) 是非绝热耦合矩阵算符
\begin{equation} \Lambda_{mn}(\mathbf{R}) = -\hbar^2\sum\limits_i \frac{1}{M_i}\bigg( A_{mn}^i\frac{\partial}{\partial \mathbf{R}_i} + \frac12 B_{mn}^i \bigg) \end{equation}

\begin{equation} A_{mn}^i = \Big\langle \phi_m \Big|\frac{\partial}{\partial \mathbf{R}_i} \Big|\phi_n \Big\rangle_\mathbf{r} \end{equation}

\begin{equation} B_{mn}^i = \Big\langle \phi_m \Big|\frac{\partial^2}{\partial \mathbf{R}_i^2} \Big|\phi_n \Big\rangle_\mathbf{r} \end{equation}

\begin{equation} \nabla_\mathbf{R}^2\chi_n(\mathbf{R})\phi_n(\mathbf{r}; \mathbf{R}) = \big[ \nabla_\mathbf{R}^2\chi_n(\mathbf{R}) \big] \phi_n(\mathbf{r}; \mathbf{R}) + \nabla_\mathbf{R}\chi_n(\mathbf{R}) \cdot \nabla_\mathbf{R}\phi_n(\mathbf{r}; \mathbf{R}) + \chi_n(\mathbf{R}) \big[ \nabla_\mathbf{R}^2\phi_n(\mathbf{r}; \mathbf{R}) \big] \end{equation}

分子的Schroedinger方程写成矩阵形式
\begin{equation} (\mathbf{T} + \mathbf{V})\mathbf{X}(\mathbf{R}) = E\mathbf{X}(\mathbf{R}) \end{equation}

对角的矩阵
\begin{equation} V_{mn}(\mathbf{R}) = \varepsilon_m(\mathbf{R})\delta_{mn} \end{equation}

称为绝热势能。非对角的动能矩阵为
\begin{equation} T_{mn}(\mathbf{R}) = T(\mathbf{R})\delta_{mn} + \Lambda_{mn}(\mathbf{R}) \end{equation}

绝热近似

不同绝热态之间的非绝热耦合由非绝热算符给出,它导致了非绝热跃迁。但是非绝热耦合矩阵的计算很困难,方程也很难解。

我们假定非对角元的耦合 \(\Lambda_{mn}\) ( \(m\neq n\) )可舍去。这一近似来自于核质量远大于电子,因此电子的动能比核动能大很多的事实。由于非绝热耦合矩阵 \(A_{mn}^i\) 和 \(B_{mn}^i\) 来自于核运动,所以也比较小。

因此,若忽略非对角的非绝热耦合, \(\Lambda_{mn}\) 可写成 \(\Lambda_{mn}\delta_{mn}\) ,因此有
\begin{equation} \big[ \hat{T}_\text{N}(\mathbf{R}) + \varepsilon_m(\mathbf{R}) \big]\chi_m(\mathbf{R}) + \Lambda_{mm}(\mathbf{R})\chi_m(\mathbf{R}) = E\chi_m(\mathbf{R}) \end{equation}

这个是Born-Huang近似,其中Hamiltonian定义为
\begin{equation} \hat{H}_n^\text{BH} = \hat{T}_\text{N} + \varepsilon_n(\mathbf{R}) + \Lambda_{nn}(\mathbf{R}) \end{equation}

电子的本征函数 \(\phi_n(\mathbf{r}; \mathbf{R})\) 可能含有一个关于 \(\mathbf{R}\) 的相位因子 \(\exp[if(\mathbf{R})]\) ,但实践中通常会令 \(\phi_n(\mathbf{r}; \mathbf{R})\) 为实函数。所以可以得到 \(A_{nn}^i(\mathbf{R})\) 等于零:
\begin{equation} \begin{aligned}
A_{mn}^i &= \bigg\langle \phi_m \bigg|\frac{\partial}{\partial \mathbf{R}_i}\bigg| \phi_n \bigg\rangle \\
&= \frac{\partial}{\partial \mathbf{R}_i} \langle \phi_m|\phi_n \rangle – \bigg\langle \frac{\partial}{\partial \mathbf{R}_i}\phi_m \bigg| \phi_n \bigg\rangle \\
&= -A_{nm}^i
\end{aligned} \end{equation}

在大多数情况下, \(B_{nn}(\mathbf{R})\) 与绝热势能 \(\varepsilon_n(\mathbf{R})\) 相比很小。因此导致 \(\Lambda_{nn}(\mathbf{R})\) 在绝热近似中可被忽略,这就是Born-Oppenheimer近似:
\begin{equation} \big[ \hat{T}_\text{N}(\mathbf{R}) + V_n(\mathbf{R}) \big]\chi_n(\mathbf{R}) = E\chi_n(\mathbf{R}) \end{equation}

我们将 \(\varepsilon_n\) 用 \(V_n\) 标记,此时Hamiltonian为
\begin{equation} \hat{H}_n^\text{BO} = \hat{T}_\text{N} + V_n(\mathbf{R}) \end{equation}

两种近似都被称为绝热近似。此时分子的波函数的绝热展开只有一项
\begin{equation} \Psi(\mathbf{r}, \mathbf{R}) = \chi_n(\mathbf{R})\phi_n(\mathbf{r}; \mathbf{R}) \end{equation}

非绝热耦合的性质

\begin{equation} \begin{aligned}
\frac{\partial}{\partial \mathbf{R}_i} A_{mn}^i &= \frac{\partial}{\partial \mathbf{R}_i} \bigg\langle \phi_m \bigg| \frac{\partial}{\partial \mathbf{R}_i} \bigg| \phi_n \bigg\rangle \\
&= \bigg\langle \frac{\partial}{\partial \mathbf{R}_i}\phi_m \bigg| \frac{\partial}{\partial \mathbf{R}_i} \phi_n \bigg\rangle + B_{mn}^i
\end{aligned} \end{equation}

\begin{equation} \begin{aligned}
\bigg\langle \frac{\partial}{\partial \mathbf{R}_i}\phi_m \bigg| \frac{\partial}{\partial \mathbf{R}_i} \phi_n \bigg\rangle &= \sum\limits_k \bigg\langle \frac{\partial}{\partial \mathbf{R}_i}\phi_m \bigg| \phi_k \bigg\rangle \bigg\langle \phi_k \bigg| \frac{\partial}{\partial \mathbf{R}_i} \phi_n \bigg\rangle \\
&= – \sum\limits_k A_{mk}^i A_{kn}^i
\end{aligned} \end{equation}

合并两式
\begin{equation} B_{mn}^i = \frac{\partial}{\partial \mathbf{R}_i} A_{mn}^i + \sum\limits_k A_{mk}^i A_{kn}^i \end{equation}

为了计算 \(A_{mn}^i\) ,我们考虑电子的Schroedinger方程
\begin{equation} \hat{H}_e\phi_n(\mathbf{r}; \mathbf{R}) = \varepsilon_n(\mathbf{R}) \phi_n(\mathbf{r}; \mathbf{R}) \end{equation}

这里的 \(\hat{H}_e\) 实际上是前面的 \(\hat{H}_\text{e}(\mathbf{r}) + \hat{V}_{eN}(\mathbf{r}, \mathbf{R})\) 。两边关于核坐标求导
\begin{equation} \nabla_\mathbf{R} \hat{H}_e\phi_n(\mathbf{r}; \mathbf{R}) = \nabla_\mathbf{R}\varepsilon_n(\mathbf{R}) \phi_n(\mathbf{r}; \mathbf{R}) \end{equation}

\begin{equation} (\nabla_\mathbf{R} \hat{H}_e)\phi_n(\mathbf{r}; \mathbf{R}) + \hat{H}_e \nabla_\mathbf{R}\phi_n(\mathbf{r}; \mathbf{R}) = (\nabla_\mathbf{R}\varepsilon_n(\mathbf{R})) \phi_n(\mathbf{r}; \mathbf{R}) + \varepsilon_n(\mathbf{R}) \nabla_\mathbf{R}\phi_n(\mathbf{r}; \mathbf{R}) \end{equation}

根据Hellmann-Feynman定理,能量的梯度为
\begin{equation} \nabla_\mathbf{R}\varepsilon_n(\mathbf{R}) = \langle \phi_n|\nabla_\mathbf{R}\hat{H}_e|\phi_n \rangle \end{equation}

左乘 \(\phi_m\) ( \(m \neq n\) ),并积分
\begin{equation} \langle \phi_m|\nabla_\mathbf{R}\hat{H}_e|\phi_n \rangle + \langle \phi_m|\hat{H}_e|\nabla_\mathbf{R}\phi_n \rangle = \langle \phi_n|\nabla_\mathbf{R}\hat{H}_e|\phi_n \rangle \langle \phi_m|\phi_n \rangle + \varepsilon_n\langle \phi_m|\nabla_\mathbf{R}\phi_n \rangle \end{equation}

\begin{equation} \langle \phi_m|\nabla_\mathbf{R}\hat{H}_e|\phi_n \rangle + \varepsilon_m\langle \nabla_\mathbf{R}\phi_n|\phi_m \rangle = \varepsilon_n\langle \phi_m|\nabla_\mathbf{R}\phi_n \rangle \end{equation}

得到非绝热耦合
\begin{equation} A_{mn} = \langle \phi_m|\nabla_\mathbf{R}\phi_n \rangle = \frac{\langle \phi_m|\nabla_\mathbf{R}\hat{H}_e|\phi_n \rangle}{\varepsilon_n – \varepsilon_m}\quad (m \neq n) \end{equation}

可以看出,当两个绝热电子势能面能量相差 \(\varepsilon_n – \varepsilon_m\) 很大时, \(A_{mn}\) 比较小;而当两个绝热势能面接近时,耦合就会变得很大,绝热近似不再可靠。

绝热近似的失效

我们将分子的Hamiltonian看作零阶Hamiltonian和作为微扰的非绝热耦合之和,绝热近似(Born-Huang近似)的解即为零阶波函数
\begin{equation} |\Psi_{n\nu}^{(0)}\rangle = \phi_n(\mathbf{r}; \mathbf{R})\chi_{n\nu}(\mathbf{R}) \end{equation}

\(n\) 指电子态、 \(\nu\) 指振动能级(也可以再加上转动能级的量子数)。我们观察以绝热波函数作为基时,分子Hamiltonian的矩阵元
\begin{equation} \begin{aligned}
\langle \Psi_{m\mu}^{(0)}|\hat{H}|\Psi_{n\nu}^{(0)} \rangle &= \langle \chi_{m\mu}| \langle \phi_m | \hat{T}_\text{N} + \hat{H}_\text{e} |\phi_n \rangle_\mathbf{r} |\chi_{n\nu} \rangle_\mathbf{R} \\
&= \langle \chi_{m\mu}| \langle \phi_m | \hat{T}_\text{N} |\phi_n \rangle_\mathbf{r} + \varepsilon_n\delta_{mn} |\chi_{n\nu} \rangle_\mathbf{R} \\
&= \langle \chi_{m\mu}| \hat{T}_\text{N} + \Lambda_{mn} + \varepsilon_n\delta_{mn} |\chi_{n\nu} \rangle_\mathbf{R} \\
&= \langle \chi_{m\mu}| \Lambda_{mn} |\chi_{n\nu} \rangle_\mathbf{R} + E_{n\nu}^{(0)}\delta_{mn}\delta_{\mu\nu}
\end{aligned} \end{equation}

我们将非对角的非绝热耦合看作微扰,因此波函数的一阶矫正为
\begin{equation} |\Psi_{n\nu}^{(1)}\rangle = |\phi_n\rangle |\chi_{n\nu}^{(1)} \rangle = \sum\limits_{m\neq n}\sum\limits_{\mu} \frac{\langle \chi_{m\mu} | \Lambda_{mn} | \chi_{n\nu} \rangle}{E_{n\nu}^{(0)} – E_{m\mu}^{(0)}} |\phi_m\rangle|\chi_{m\mu} \rangle \end{equation}

因此,对于某个电子振动态 \(|\phi_m\rangle|\chi_{m\mu} \rangle\) ,只有当
\begin{equation} \big| \langle \chi_{m\mu} | \Lambda_{mn} | \chi_{n\nu} \rangle \big| \ll \big| E_{n\nu}^{(0)} – E_{m\mu}^{(0)} \big| \end{equation}

时,绝热近似才有效。

绝热近似下的时间演化

含时的绝热展开
\begin{equation} \Psi(\mathbf{r}, \mathbf{R}, t) = \sum\limits_n \Theta_n(\mathbf{R}, t)\phi_n(\mathbf{r}; \mathbf{R}) \end{equation}

由于 \(\Psi(\mathbf{r}, \mathbf{R}, t)\) 和 \(\phi_n(\mathbf{r}; \mathbf{R})\) 都归一化,所以 \(P_n(t)=\int|\Theta_n(\mathbf{R}, t)|^2\,\mathrm{d}\mathbf{R}\) 可看作在 \(t\) 时刻处于电子态 \(\phi_n\) 的概率。代入到含时Schroedinger方程:
\begin{equation} i\hbar\frac{\mathrm{d}}{\mathrm{d}t}\Theta_m = (\hat{T}_\text{N} + \varepsilon_m + \Lambda_{mm})\Theta_m + \sum\limits_{n\neq m} \Lambda_{mn}\Theta_n \end{equation}

方程右边第一项是核波函数在势能面 \(\varepsilon_m + \Lambda_{mm}\) 上的演化,第二项则描述了布居转移到其他的电子态,即非绝热跃迁。

参考资料

  1. Theory and Application of Quantum Molecular Dynamics
  2. Photochemistry: A Modern Theoretical Perspective

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