:wq

Like what we did in the previous post, we would like to do the same thing to the crystal, but in a quantum way, and also including the anharmonic potential, i.e. the third-order term in the Taylor expansion of the interatomic potential energy.

Consider a crystal in which unit cells are labeled by \(\boldsymbol{l}\), and atoms in each cell are labeled by \(\boldsymbol{b}\). The total energy of the system, noted by \(\mathcal{V}\), can be expanded using Taylor's expansion (greek letters mean direction): \[ \begin{aligned} \mathcal{V} = & \mathcal{V}_0 + \underbrace{\sum_{\boldsymbol{l},\boldsymbol{b}}\sum_{\alpha} \Phi_{\alpha}(\boldsymbol{l}\boldsymbol{b}) u_{\alpha}(\boldsymbol{l}\boldsymbol{b})}_{\mathcal{V}_1} + \underbrace{\frac{1}{2}\sum_{\boldsymbol{l},\boldsymbol{b},\boldsymbol{l}',\boldsymbol{b}'}\sum_{\alpha,\beta} \Phi_{\alpha,\beta}(\boldsymbol{l}\boldsymbol{b};\boldsymbol{l}'\boldsymbol{b}')u_{\alpha}(\boldsymbol{l}\boldsymbol{b})u_{\beta}(\boldsymbol{l}'\boldsymbol{b}')}_{\mathcal{V}_2} + \cdots\\ & \quad + \frac{1}{n!}\sum_{\boldsymbol{l}_1,\boldsymbol{b}_1,\boldsymbol{l}_2,\boldsymbol{b}_2,\cdots,\boldsymbol{l}_n,\boldsymbol{b}_n}\sum_{\alpha_1,\alpha_2,\cdots,\alpha_n}\Phi_{\alpha_1,\alpha_2,\cdots,\alpha_n}(\boldsymbol{l}_1,\boldsymbol{b}_1;\boldsymbol{l}_2,\boldsymbol{b}_2;\cdots;\boldsymbol{l}_n,\boldsymbol{b}_n)u_{\alpha_1}(\boldsymbol{l}_1,\boldsymbol{b}_1)\cdots u_{\alpha_n}(\boldsymbol{l}_n \boldsymbol{b}_n) \end{aligned} \] Where we note the nth order interatomic force constants (IFCs) using \(\Phi\), and we note \(u_{\alpha}(\boldsymbol{l}\boldsymbol{b})\) represent the spatial deviation of atom \(\boldsymbol{b}\) in cell \(\boldsymbol{l}\) along \(\alpha\) direction.

Welcome to Hexo! This is your very first post. Check documentation for more info. If you get any problems when using Hexo, you can find the answer in troubleshooting or you can ask me on GitHub.