diff --git a/_posts/2024-12-12-wannier_elf.md b/_posts/2024-12-12-wannier_elf.md index 5e75ca7e..2d906c5b 100644 --- a/_posts/2024-12-12-wannier_elf.md +++ b/_posts/2024-12-12-wannier_elf.md @@ -18,6 +18,8 @@ toc: - In the event reading the following is not really needed, the implementation is available here: [Wannier-ELF](https://github.com/utksi/wannier_elf) +--- + ### **What, and Why?** The Electron Localization Function (ELF) is a valuable tool in computational chemistry and condensed matter physics for visualizing and understanding electron localization in atoms, molecules, and solids. @@ -36,7 +38,7 @@ $$ - $$t_W(\mathbf{r})$$: von Weizsäcker kinetic energy density. - $$t_h(\mathbf{r})$$: Kinetic energy density of a homogeneous electron gas. -Essentially, it is a three-dimensional scalar field that **tracks the variation in KE density** (Total - Von-Weizsäcker term) at some point $$\mathbf{\vec{r}}$$ in the cell, compared to if one had a homogeneous electron gas with the same electron density. (for which the gradient term: $$|\nabla n(\mathbf{r})|$$ should be exactly zero.) +Essentially, it is a three-dimensional scalar field that **tracks the variation in KE density** (Total - Von-Weizsäcker term) at some point $$\mathbf{\vec{r}}$$ in the cell, compared to if one had a homogeneous electron gas with the same electron density, for which the gradient term: $$\nabla n(\mathbf{r})$$ should be exactly zero. This ratio: $$\dfrac{ t_P(\mathbf{r}) }{ t_h(\mathbf{r}) }$$ is also sometimes called the **localization index** : $$\chi (r)$$. @@ -60,6 +62,8 @@ Given $$n(\mathbf{r})$$: the electron density. t_h(\mathbf{r}) = \dfrac{3}{5} (3\pi^2)^{2/3} [n(\mathbf{r})]^{5/3} $$ +--- + ### VASP/CASTEP expressions for ELF If you're reading this, then it's highly likely that you already know that ELF fields can be written very easily after a obtaining charge density in VASP/CASTEP. @@ -111,7 +115,7 @@ $$ -2A \sum_i \psi_i^*(\mathbf{r}) \nabla^2 \psi_i(\mathbf{r}) = 2A \sum_i |\nabla \psi_i(\mathbf{r})|^2 - 2A \sum_i \nabla \cdot \left( \psi_i^*(\mathbf{r}) \nabla \psi_i(\mathbf{r}) \right) $$ -Recognizing that $A = \dfrac{\hbar^2}{2m}$, the term $$2A \sum_i |\nabla \psi_i(\mathbf{r})|^2$$ corresponds to twice the total kinetic energy density: +Recognizing that $$A = \dfrac{\hbar^2}{2m}$, the term $$2A \sum_i |\nabla \psi_i(\mathbf{r})|^2$$ corresponds to twice the total kinetic energy density: $$ 2 t(\mathbf{r}) = 2A \sum_i |\nabla \psi_i(\mathbf{r})|^2 @@ -511,7 +515,7 @@ def validate_symmetry(self, field: np.ndarray, label: str) -> None: max_violation = max(max_violation, max_diff) ``` -**AND** +**AND** validating the fields were symmetrized correctly ```python def validate_field_properties( @@ -570,7 +574,7 @@ def validate_field_properties( --- -### Calculating $$ELF(r)$$ +### Calculating $$\mathrm{ELF(r)}$$ Finally, calculating ELF, which is straightforward: @@ -608,7 +612,7 @@ Using a `threshold`, masking values with `mask` seems to be important for stable --- -### Writing scalar fields $$F(r)$$ +### Writing scalar fields ```python self.write_field_xsf("density.xsf", density) @@ -641,7 +645,7 @@ def write_field_xsf(self, filename: str, field: np.ndarray) -> None: A good example is $$\mathrm{CeO_2}$$ where Cerium is supposed to have +4 and not +3 formal oxidation state. So the $$ELF(r)$$ field value near Cerium across all cross sections should be minimal.
- +
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