This page establishes the notation, units, and conventions used throughout the course. It is intended as a standing reference: every symbol defined here retains its meaning in all subsequent chapters unless explicitly redefined in context.
Atomic Units
All equations in this course are written in atomic units (a.u.) from Chapter 3 onward, unless stated otherwise. The atomic unit system sets:
$$ \hbar = m_e = e = 4\pi\varepsilon_0 = 1, $$where $\hbar$ is the reduced Planck constant, $m_e$ the electron rest mass, $e$ the elementary charge, and $\varepsilon_0$ the permittivity of free space. The derived units are:
| Quantity | Atomic unit | SI value |
|---|---|---|
| Length | Bohr $a_0$ | $0.52918$ Å |
| Energy | Hartree $E_h$ | $27.211$ eV $= 2$ Ry |
| Mass | Electron mass $m_e$ | $9.109\times10^{-31}$ kg |
| Charge | Elementary charge $e$ | $1.602\times10^{-19}$ C |
| Time | $\hbar/E_h$ | $24.19$ as |
| Velocity | $a_0 E_h/\hbar$ | $2.187\times10^6$ m/s |
Practical conversions for common codes:
- VASP quotes energies in eV and plane-wave cutoffs in eV.
- Quantum ESPRESSO quotes energies in Ry and cutoffs in Ry.
- 1 Hartree = 2 Ry = 27.211 eV; 1 Bohr = 0.52918 Å.
In atomic units the kinetic energy operator is $-\frac{1}{2}\nabla^2$ (not $-\frac{\hbar^2}{2m_e}\nabla^2$), the Coulomb potential between charges $q_1$ and $q_2$ at separation $r$ is $q_1 q_2/r$ (not $q_1 q_2/4\pi\varepsilon_0 r$), and the Bohr magneton is $\mu_B = e\hbar/2m_e = 1/2$ a.u.
Chapter 1 retains $\hbar$ and $m_e$ explicitly in some equations so that connections to standard quantum mechanics textbooks remain transparent. From Chapter 3 onward the atomic unit form is used without further comment.
Mathematical Notation
Operators and Wavefunctions
| Symbol | Meaning |
|---|---|
| $\hat{H}$ | Many-body electronic Hamiltonian |
| $\hat{T}$ | Many-body kinetic energy operator |
| $\hat{T}_s$ | Non-interacting (KS) kinetic energy operator |
| $\hat{V}_{ee}$ | Bare electron–electron Coulomb repulsion operator |
| $\hat{V}_{\rm ext}$ | External potential operator (nuclei + applied fields) |
| $\Psi(\mathbf{r}_1,\ldots,\mathbf{r}_N)$ | Many-body electronic wavefunction |
| $\Psi^\lambda$ | Ground-state wavefunction at coupling strength $\lambda$ |
| $\phi_i(\mathbf{r})$ | $i$-th Kohn–Sham single-particle orbital |
| $\tilde\phi_i(\mathbf{r})$ | PAW pseudo (smooth) orbital |
| $\phi_n^a(\mathbf{r})$ | PAW all-electron partial wave on atom $a$ |
| $\epsilon_i$ | $i$-th Kohn–Sham eigenvalue (Lagrange multiplier) |
| $f_i$ | Occupation number of orbital $i$ ($0 \leq f_i \leq 1$) |
Densities and Potentials
| Symbol | Meaning |
|---|---|
| $\rho(\mathbf{r})$ | Electron number density; $\int\rho,d\mathbf{r} = N$ |
| $\rho_\sigma(\mathbf{r})$ | Spin-resolved density, $\sigma \in {\uparrow,\downarrow}$ |
| $m(\mathbf{r})$ | Magnetisation density $= \rho_\uparrow - \rho_\downarrow$ |
| $n_{\rm xc}(\mathbf{r},\mathbf{r}’)$ | Exchange-correlation hole |
| $V_{\rm ext}(\mathbf{r})$ | External scalar potential |
| $V_{\rm H}(\mathbf{r})$ | Hartree potential $= \int\rho(\mathbf{r}’)/|\mathbf{r}-\mathbf{r}’|,d\mathbf{r}'$ |
| $V_{\rm xc}(\mathbf{r})$ | Exchange-correlation potential $= \delta E_{\rm xc}/\delta\rho$ |
| $V_{\rm eff}(\mathbf{r})$ | KS effective potential $= V_{\rm ext} + V_{\rm H} + V_{\rm xc}$ |
| $\mathbf{B}_{\rm xc}(\mathbf{r})$ | XC magnetic field (non-collinear SDFT) |
Important distinction: $\hat{V}{ee}$ is the bare quantum-mechanical electron–electron repulsion in the many-body Hamiltonian. The Hartree energy $E{\rm H}[\rho]$ and its potential $V_{\rm H}(\mathbf{r})$ are the classical electrostatic self-energy of the continuous charge density $\rho(\mathbf{r})$; they approximate but are not equal to $\langle\hat{V}{ee}\rangle$. The difference is captured by $E{\rm xc}$.
Energy Functionals
| Symbol | Meaning |
|---|---|
| $E[\rho]$ | Total ground-state energy functional |
| $F[\rho]$ | Universal functional $= \langle\Psi[\rho]|\hat{T}+\hat{V}_{ee}|\Psi[\rho]\rangle$ |
| $T[\rho]$ | True (interacting) kinetic energy functional |
| $T_s[\rho]$ | Non-interacting KS kinetic energy $= -\frac{1}{2}\sum_i\langle\phi_i|\nabla^2|\phi_i\rangle$ |
| $E_{\rm H}[\rho]$ | Hartree energy $= \frac{1}{2}\iint\rho(\mathbf{r})\rho(\mathbf{r}’)/|\mathbf{r}-\mathbf{r}’|,d\mathbf{r},d\mathbf{r}'$ |
| $E_{\rm xc}[\rho]$ | Exchange-correlation energy (see Chapter 4) |
| $E_{\rm x}^{\rm exact}$ | Exact (Fock) exchange energy $= W_{\rm xc}^{\lambda=0}$ |
| $E_{\rm c}^{\rm GL2}$ | Second-order Görling–Levy correlation energy |
| $W_{\rm xc}^\lambda$ | Adiabatic connection integrand at coupling $\lambda$ |
XC Functional Ingredients
| Symbol | Meaning |
|---|---|
| $\varepsilon_{\rm xc}(\rho)$ | XC energy per electron of the uniform electron gas |
| $s(\mathbf{r})$ | Reduced density gradient $= |\nabla\rho|/(2k_F\rho)$ |
| $k_F(\mathbf{r})$ | Local Fermi wavevector $= (3\pi^2\rho)^{1/3}$ |
| $\tau(\mathbf{r})$ | KS kinetic energy density $= \frac{1}{2}\sum_i f_i|\nabla\phi_i|^2$ |
| $\tau^W(\mathbf{r})$ | von Weizsäcker KE density $= |\nabla\rho|^2/8\rho$ |
| $\tau^{\rm UEG}(\mathbf{r})$ | Thomas–Fermi KE density $= \frac{3}{10}(3\pi^2)^{2/3}\rho^{5/3}$ |
| $\alpha(\mathbf{r})$ | Iso-orbital indicator $= (\tau-\tau^W)/\tau^{\rm UEG}$ |
| $F_{\rm xc}(\rho,s)$ | GGA enhancement factor |
| $\lambda$ | Adiabatic connection coupling constant $\in [0,1]$ |
| $\omega$ | Range-separation parameter in hybrid functionals (Å$^{-1}$) |
| $a$ | Fraction of exact exchange in hybrid functionals |
Solid-State and Basis-Set Notation
| Symbol | Meaning |
|---|---|
| $\mathbf{k}$ | Crystal momentum (Bloch wavevector) in the first BZ |
| $\mathbf{G}$ | Reciprocal lattice vector |
| $\Omega$ | Unit cell volume |
| $E_{\rm cut}$ | Plane-wave kinetic energy cutoff |
| $N_1\times N_2\times N_3$ | Monkhorst–Pack $k$-point mesh |
| $r_c^a$ | PAW augmentation sphere radius for atom $a$ |
| $\hat{\mathcal{T}}$ | PAW linear transformation operator |
| $\langle\tilde p_n^a | $ |
Spin and Magnetism
| Symbol | Meaning |
|---|---|
| $\sigma$ | Spin index, $\sigma \in {\uparrow,\downarrow}$ |
| $\zeta$ | Spin polarisation $= (\rho_\uparrow-\rho_\downarrow)/\rho \in [-1,1]$ |
| $\mu_B$ | Bohr magneton $= e\hbar/2m_e = 1/2$ a.u. |
| $\mathbf{m}(\mathbf{r})$ | Local magnetisation density vector (non-collinear) |
| $J_{ij}$ | Heisenberg exchange coupling constant between sites $i,j$ |
| $\hat{\mathbf{S}}_i$ | Spin operator on site $i$ |
| $\xi$ | Spin–orbit coupling constant |
| $\hat{\mathbf{L}}$, $\hat{\mathbf{S}}$ | Orbital and spin angular momentum operators |
| $\langle L_z\rangle$ | Expectation value of orbital moment along $z$ |
| MAE | Magnetic anisotropy energy $= E(\hat{\mathbf{n}}{\rm hard}) - E(\hat{\mathbf{n}}{\rm easy})$ |
DFT+U Notation
| Symbol | Meaning |
|---|---|
| $n_{mm’}^{I\sigma}$ | Occupation matrix of the correlated subspace on site $I$, spin $\sigma$ |
| $N^{I\sigma}$ | Total occupation $= \mathrm{Tr}[\mathbf{n}^{I\sigma}]$ |
| $U$ | Average on-site Coulomb repulsion |
| $J$ | Average on-site exchange interaction |
| $U_{\rm eff}$ | Dudarev effective parameter $= U - J$ |
| $F^0, F^2, F^4$ | Slater integrals (radial Coulomb matrix elements) |
| $E_{\rm dc}$ | Double-counting correction |
Dirac Notation
We use Dirac (bra-ket) notation throughout:
$$ \langle\phi_i|\hat{O}|\phi_j\rangle = \int \phi_i^*(\mathbf{r})\,\hat{O}\,\phi_j(\mathbf{r})\,d\mathbf{r}. $$For the many-body wavefunction:
$$ \langle\Psi|\hat{H}|\Psi\rangle = \int \Psi^*(\mathbf{r}_1,\ldots,\mathbf{r}_N)\,\hat{H}\,\Psi(\mathbf{r}_1,\ldots,\mathbf{r}_N)\,d\mathbf{r}_1\cdots d\mathbf{r}_N. $$Functional Derivatives
A functional $F[\rho]$ maps a function $\rho(\mathbf{r})$ to a scalar. Its functional derivative $\delta F/\delta\rho(\mathbf{r})$ is defined by:
$$ F[\rho + \delta\rho] - F[\rho] = \int \frac{\delta F}{\delta\rho(\mathbf{r})}\,\delta\rho(\mathbf{r})\,d\mathbf{r} + \mathcal{O}(\delta\rho^2). $$This generalises the ordinary derivative to function spaces. Key examples in this course:
$$ \frac{\delta E_{\rm H}[\rho]}{\delta\rho(\mathbf{r})} = V_{\rm H}(\mathbf{r}), \qquad \frac{\delta E_{\rm xc}[\rho]}{\delta\rho(\mathbf{r})} = V_{\rm xc}(\mathbf{r}). $$Index Conventions
- Lowercase Latin $i, j, k, \ldots$: electron or orbital indices.
- Lowercase Latin $a, b$: unoccupied (virtual) orbitals in perturbation theory contexts.
- Uppercase Latin $I, J$: atomic site indices.
- Lowercase Greek $\alpha, \beta$: nuclear site indices (Chapters 1–2); Cartesian tensor indices (Chapter 6 onward); context disambiguates.
- Uppercase Greek $\Omega$: unit cell volume.
- Lowercase Greek $\sigma$: spin index $\in{\uparrow,\downarrow}$; also used for stress tensor components — context disambiguates.
- Summation over repeated spin indices is not implied (explicit sums are always written).
- $\sum_i$ without explicit limits runs over all occupied KS orbitals unless otherwise stated.
Abbreviations
| Abbreviation | Full term |
|---|---|
| DFT | Density Functional Theory |
| HK | Hohenberg–Kohn |
| KS | Kohn–Sham |
| SCF | Self-Consistent Field |
| XC | Exchange-Correlation |
| LDA | Local Density Approximation |
| LSDA | Local Spin-Density Approximation |
| GGA | Generalised Gradient Approximation |
| GEA | Gradient Expansion Approximation |
| UEG | Uniform Electron Gas |
| PAW | Projector Augmented Wave |
| PP | Pseudopotential |
| NCPP | Norm-Conserving Pseudopotential |
| USPP | Ultrasoft Pseudopotential |
| BZ | Brillouin Zone |
| MP | Monkhorst–Pack |
| SDFT | Spin-Polarised DFT |
| SOC | Spin–Orbit Coupling |
| MAE | Magnetic Anisotropy Energy |
| AIMD | Ab Initio Molecular Dynamics |
| BOMD | Born–Oppenheimer MD |
| CPMD | Car–Parrinello MD |
| BO | Born–Oppenheimer |
| HF | Hartree–Fock |
| SIE | Self-Interaction Error |
| GL | Görling–Levy |
| FLL | Fully Localised Limit (double-counting) |
| AMF | Around Mean Field (double-counting) |
| cRPA | Constrained Random Phase Approximation |
| DMFT | Dynamical Mean-Field Theory |
| NEB | Nudged Elastic Band |
| BFGS | Broyden–Fletcher–Goldfarb–Shanno |
| DIIS | Direct Inversion in the Iterative Subspace |
A Note on Sign Conventions
Several sign conventions vary between texts and codes; we follow the most common physics convention throughout:
- Exchange energy $E_{\rm x} < 0$ always (exchange is stabilising).
- Correlation energy $E_{\rm c} < 0$ for the UEG (correlation is stabilising).
- XC hole $n_{\rm xc}(\mathbf{r},\mathbf{r}’) \leq 0$ (depletion, not accumulation).
- MAE defined as $E_{\rm hard} - E_{\rm easy} > 0$: positive MAE means the hard axis costs more energy.
- Heisenberg exchange $J > 0$ favours ferromagnetic alignment; $J < 0$ favours antiferromagnetic. (Some texts use the opposite sign; we always state the convention explicitly when quoting numerical values.)
- Hubbard $U$ is always positive (repulsive on-site interaction).