Exx-Symmetry

记一次完不成的约定，一个永不再问的问题。

——那等我解决了symmetry=1的问题再来问你

——那希望我到时能有别的回答...

——嗯，拭目以待

本仓库是尝试为ABACUS中的EXX添加symmetry==1的支持时的试验代码。

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尝试用群论给出的波函数关系还原出DM(R)： $$S(gk)=\ket{\phi_{gk}}\bra{\phi_{gk}}=S^T(gk)$$ $$S(g,k)\equiv\ket{\phi_{gk}}\bra{g\phi_{k}}$$ 群论给出的波函数关系： $$c^T_{gk}\ket{\phi_{gk}}=\pm c^T_k\ket{g\phi_k}$$ 右乘行向量$\bra{\phi_{gk}}$得到$c^T_{gk}S^T(gk)=c^T_kS^T(g,k)$ 两边转置得 $$S(g,k)c_k=S(gk)c_{gk}, c_k=S^{-1}(g,k)S(gk)c_{gk}$$

然而，实现出$S(g,k)$，发现$k_1\neq k_2$时是零矩阵... $$S_{\mu\nu}(g,k)=\int{}d\mathbf{r}\sum_\mathbf{R_1}\phi_\mu(\mathbf{r}-\tau_\mu-\mathbf{R}1)e^{-i\alpha\mathbf{k}\cdot\mathbf{R_1}}\sum\mathbf{R_2}\phi_{\nu}(\mathbf{r}-\tau_{\nu}-\mathbf{R}_2)e^{i\mathbf{k}\cdot\mathbf{R}2}$$ ——$\ket{\phi{gk}}$和$\ket{g\phi_k}$是正交的？！

那么每个k点的信息都不可或缺，symmetry=1无解。

However, something may still be useful:

• calculate kstar
• Symmetry_Basic::atom_ordering_new: preserve index for atom-map
• doc: https://xmywuqhxb0.feishu.cn/wiki/A7ETwz0wSiOZILk8yEac6IWGnWg

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About ABACUS

ABACUS (Atomic-orbital Based Ab-initio Computation at UStc) is an open-source package based on density functional theory (DFT). The package utilizes both plane wave and numerical atomic basis sets with the usage of norm-conserving pseudopotentials to describe the interactions between nuclear ions and valence electrons. ABACUS supports LDA, GGA, meta-GGA, and hybrid functionals. Apart from single-point calculations, the package allows geometry optimizations and ab-initio molecular dynamics with various ensembles. The package also provides a variety of advanced functionalities for simulating materials, including the DFT+U, VdW corrections, and implicit solvation model, etc. In addition, ABACUS strives to provide a general infrastructure to facilitate the developments and applications of novel machine-learning-assisted DFT methods (DeePKS, DP-GEN, DeepH, etc.) in molecular and material simulations.

Online Documentation

For detailed documentation, please refer to our documentation website.