Notation and Conventions
This page fixes notation shared by the theory documents. It is a reference for symbols and object types, not a substitute for their physical definitions. Method-specific normalization, Fourier, and augmentation conventions remain in the corresponding electronic-structure chapters.
Typographic Conventions
| Notation | Meaning |
|---|---|
| \(\hat X\) | Quantum operator. |
| \(\mathbf X\) | Complete spinor matrix over spatial and spin indices, or a complete spin-matrix field such as \(\mathbf n(\boldsymbol r)\). |
| \(\mathsf X\) | Matrix over spatial basis indices only. With no Pauli superscript it denotes a spinless or explicitly reduced spatial matrix. |
| \(\boldsymbol x\) | Cartesian vector or pseudovector, including positions, reciprocal vectors, and spin-polarization vectors. |
| \(X_{\mu s,\nu s'}\) | Matrix element of \(\mathbf X\), with spatial and spin indices shown explicitly. |
| \(X_{\mu\nu}\) | Matrix element of \(\mathsf X\). Indexed matrix elements are italic even when their parent matrix is bold or sans serif. |
| \(\mathsf X^0\), \(\mathsf X^\alpha\) | Spatial matrices of the scalar and spin-vector Pauli components, with \(\alpha=x,y,z\). |
Roman subscripts and superscripts label components or sectors; they do not by themselves imply tensor transformation rules. A dagger denotes Hermitian conjugation, \(T\) denotes transpose, and \(*\) denotes complex conjugation. Integration measures are written immediately after the integral sign, as in \(\int_\Omega\mathrm d\boldsymbol r\,f(\boldsymbol r)\).
Indices
| Index | Meaning |
|---|---|
| \(i,j\) | Atoms; in a periodic system, atoms in the unit cell unless stated otherwise. |
| \(\mu,\nu\) | Global spatial atomic-orbital basis indices. For LCAO, \(\mu\) may resolve as \((i,n,\ell,m)\). |
| \(\lambda,\lambda'\) | Global auxiliary-basis indices, with basis functions written as \(\chi_\lambda\) and \(\chi_{\lambda'}\). |
| \(\ell,m,m'\) | Angular-momentum shell and components. |
| \(s,s'\) | Explicit spin indices, \(s\in\{\uparrow,\downarrow\}\). |
| \(a,b\) | One-electron states or bands. |
| \(A\) | Pauli component \(A\in\{0,x,y,z\}\). |
| \(\alpha,\beta\) | Cartesian or Pauli-vector components \(x,y,z\). |
| \(\boldsymbol R\) | Cell-shift vector. |
| \(\boldsymbol k\) | Crystal momentum. |
| \(\boldsymbol G\) | Reciprocal-lattice vector. |
Indices not listed here are scoped to the chapter in which they are introduced and may be reused for different method-specific channels. The projector and partial-wave chapters explicitly scope \(\mu,\nu\) to their local atom-centered channels.
Spin remains explicit in component notation: for example, \(H_{\mu s,\nu s'}\), rather than introducing a separate symbol for the composite index \((\mu,s)\).
Core Electronic Objects
| Symbol | Meaning |
|---|---|
| \(\hat H\) | One-electron Hamiltonian operator. |
| \(\mathbf H\) | Complete spinor Hamiltonian matrix in a stated basis. |
| \(\mathsf H\) | Spinless or explicitly reduced Hamiltonian matrix over spatial basis indices. |
| \(\hat S\) | Operator whose basis matrix is the overlap matrix in a generalized eigenproblem. For an ordinary orthonormal problem, \(\hat S=\hat I\). |
| \(\mathbf S\) | Complete spinor overlap matrix in the chosen basis. |
| \(\mathsf S\) | Spatial overlap matrix. |
| \(\hat\rho\) | One-particle density operator. |
| \(\rho_{ss'}(\boldsymbol r,\boldsymbol r')\) | Spin components of the real-space kernel of \(\hat\rho\). |
| \(\mathbf D\) | Complete spinor density matrix in a chosen basis, \(\mathbf D=\mathbf C\mathbf f\mathbf C^\dagger\). |
| \(\mathsf D\) | Spinless or explicitly reduced density matrix over spatial basis indices; \(\mathsf D^0\) and \(\mathsf D^\alpha\) denote Pauli-component matrices of \(\mathbf D\). |
| \(\mathbf n(\boldsymbol r)\) | Local spin-density matrix with components \(n_{ss'}(\boldsymbol r)=\rho_{ss'}(\boldsymbol r,\boldsymbol r)\). |
| \(n(\boldsymbol r)\) | Scalar electron density, obtained by tracing \(\mathbf n\) over spin. |
| \(\boldsymbol m(\boldsymbol r)\) | Pauli spin-polarization density. It is not a magnetic-moment density unless the corresponding factor and sign convention are applied. |
The operator, its real-space kernel, and its basis matrix are distinct objects. In particular, \(\mathbf D\) is reserved for density matrices in a chosen basis; in a nonorthogonal basis it is not the covariant matrix of \(\hat\rho\). The detailed relation is given in Localized Atomic-Orbital Basis.
Spin and Traces
The spinor formulation is the default. Occupations therefore satisfy \(0\le f_a\le1\), with no implicit spin-degeneracy factor. A reduced spinless formulation may use occupations up to two only when stated explicitly.
\(\sigma_0\) denotes the spin identity and \(\boldsymbol\sigma=(\sigma_x,\sigma_y,\sigma_z)\) the Pauli matrices. The Pauli decomposition and the definitions of the scalar and spin-vector components are fixed in Kohn–Sham Density Functional Theory. In a spatial basis, the corresponding matrix decomposition is
Thus \(\mathbf X\) and \(\{\mathsf X^0,\mathsf X^x,\mathsf X^y,\mathsf X^z\}\) are equivalent forms of the same spinor matrix. By contrast, a superscript-free \(\mathsf X\) denotes a genuinely spinless or explicitly reduced spatial matrix. A spin-independent spinor matrix has only \(\mathsf X^0\); this is a property of \(\mathbf X\), not another meaning of “spinless.”
\(\operatorname{tr}_s\) traces only the explicit spin indices. \(\operatorname{tr}\) without a subscript traces the complete one-electron space or its complete finite matrix, as determined by context. All theory documents use the lowercase operator name \(\operatorname{tr}\).
Units and Periodic Normalization
Hartree atomic units are used unless stated otherwise. Energies are therefore in hartree and lengths in bohr.
No translation symmetry is assumed unless it is introduced explicitly. For periodic systems:
- electronic states are normalized using the primitive-cell inner product;
- Brillouin-zone averages use \(\Omega_{\mathrm{BZ}}^{-1}\int_{\mathrm{BZ}}\mathrm d\boldsymbol k\);
- weights in a numerical Brillouin-zone quadrature satisfy \(\sum_{\boldsymbol k}w_{\boldsymbol k}=1\);
- electron counts, traces, energies, and densities are per unit cell;
- \(N_{\boldsymbol k}\) denotes the cardinality of a complete unreduced uniform mesh only when a finite discrete Fourier pair is explicitly introduced;
- real-space matrix blocks use the displacement of the right ket relative to the left bra.
The Bloch phases, reciprocal-lattice convention, and complete Fourier pairs are defined in Periodic Systems. They should not be reconstructed from a storage format or code-specific array ordering.