Appendix A — Mathematical Reference

B Mathematical Reference

B.1 Vector Calculus Identities

\[ \nabla \cdot (\nabla \times \mathbf{A}) = 0 \tag{B.1}\]

\[ \nabla \times (\nabla f) = \mathbf{0} \tag{B.2}\]

\[ \nabla^2 f = \nabla \cdot (\nabla f) \tag{B.3}\]

B.2 Fourier Transforms

The continuous Fourier transform convention used in these notes:

\[ \tilde{f}(\mathbf{k}) = \int f(\mathbf{r})\, e^{-i\mathbf{k}\cdot\mathbf{r}}\, d\mathbf{r} \tag{B.4}\]

\[ f(\mathbf{r}) = \frac{1}{(2\pi)^3}\int \tilde{f}(\mathbf{k})\, e^{i\mathbf{k}\cdot\mathbf{r}}\, d\mathbf{k} \tag{B.5}\]

B.3 Useful Constants

Table B.1: Physical constants used in atomistic simulations.

Constant	Symbol	Value
Boltzmann constant	\(k_B\)	\(8.617 \times 10^{-5}\) eV/K
Planck constant	\(h\)	\(4.136 \times 10^{-15}\) eV·s
Reduced Planck	\(\hbar\)	\(6.582 \times 10^{-16}\) eV·s
Electron mass	\(m_e\)	\(9.109 \times 10^{-31}\) kg
Bohr radius	\(a_0\)	\(0.529\) Å
Hartree	\(E_h\)	\(27.211\) eV

---
title: "Mathematical Reference"
---

# Mathematical Reference {#sec-mathref}

## Vector Calculus Identities

$$
\nabla \cdot (\nabla \times \mathbf{A}) = 0
$$ {#eq-div-curl}

$$
\nabla \times (\nabla f) = \mathbf{0}
$$ {#eq-curl-grad}

$$
\nabla^2 f = \nabla \cdot (\nabla f)
$$ {#eq-laplacian}

## Fourier Transforms

The continuous Fourier transform convention used in these notes:

$$
\tilde{f}(\mathbf{k}) = \int f(\mathbf{r})\, e^{-i\mathbf{k}\cdot\mathbf{r}}\, d\mathbf{r}
$$ {#eq-ft}

$$
f(\mathbf{r}) = \frac{1}{(2\pi)^3}\int \tilde{f}(\mathbf{k})\, e^{i\mathbf{k}\cdot\mathbf{r}}\, d\mathbf{k}
$$ {#eq-ift}

## Useful Constants

| Constant | Symbol | Value |
|----------|--------|-------|
| Boltzmann constant | $k_B$ | $8.617 \times 10^{-5}$ eV/K |
| Planck constant | $h$ | $4.136 \times 10^{-15}$ eV·s |
| Reduced Planck | $\hbar$ | $6.582 \times 10^{-16}$ eV·s |
| Electron mass | $m_e$ | $9.109 \times 10^{-31}$ kg |
| Bohr radius | $a_0$ | $0.529$ Å |
| Hartree | $E_h$ | $27.211$ eV |

: Physical constants used in atomistic simulations. {#tbl-constants}