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Localized Atomic-Orbital Basis

A localized atomic-orbital (LCAO) basis represents the one-electron problem directly in terms of atom-centered functions. The same algebra applies to finite and periodic systems; lattice translations and Bloch sums are additional structure introduced only in the periodic case. The basis is generally nonorthogonal, so its overlap matrix must be retained. Hartree atomic units are used unless stated otherwise.

Atomic Orbitals and Indices

Let

\[ \mu\equiv(i,n,\ell,m) \]

denote a spatial atomic orbital: \(i\) labels its center, \(n\) labels the radial function, including repeated functions with the same \(\ell\), and \((\ell,m)\) are angular-momentum indices. A typical numerical atomic orbital centered at \(\boldsymbol R_i\) has the form

\[ \phi_\mu(\boldsymbol r) = R_{i n\ell} \left( \lvert\boldsymbol r-\boldsymbol R_i\rvert \right) Y_{\ell m} \left( \widehat{\boldsymbol r-\boldsymbol R_i} \right). \]

Here \(R_{i n\ell}\) is the radial factor and the hat denotes the unit direction of \(\boldsymbol r-\boldsymbol R_i\). The radial functions may be numerical or analytic and need not be mutually orthogonal. The symbol \(Y_{\ell m}\) denotes a chosen spherical-harmonic basis. This theory does not fix its real or complex form, phase convention, normalization, or component ordering; all basis-dependent quantities are understood to use one consistent choice.

Spin is appended explicitly,

\[ \lvert\phi_\mu;s\rangle = \lvert\phi_\mu\rangle\otimes\lvert s\rangle, \qquad s\in\{\uparrow,\downarrow\}. \]

The spatial basis is not assumed to depend on spin.

Nonorthogonal Basis Matrices

For a finite system, define the spatial basis-overlap matrix by

\[ S_{\mu\nu} = \langle\phi_\mu\vert\phi_\nu\rangle, \]

and the spinor overlap and generic operator matrix elements by

\[ S_{\mu s,\nu s'} = \langle\phi_\mu;s\vert\phi_\nu;s'\rangle = S_{\mu\nu}\delta_{ss'}, \]
\[ X_{\mu s,\nu s'} = \langle\phi_\mu;s\vert\hat X\vert\phi_\nu;s'\rangle. \]

Thus \(\mathsf S=[S_{\mu\nu}]\) is the ordinary spatial LCAO overlap matrix, \(\mathbf S=[S_{\mu s,\nu s'}]\) is its complete spinor form, and \(\mathbf X=[X_{\mu s,\nu s'}]\) is the spinor basis matrix of a generic one-electron operator \(\hat X\). Taking \(\hat X=\hat H_{\mathrm{KS}}\) gives the Kohn–Sham matrix \(\mathbf H\); the operator itself is defined in Kohn–Sham Density Functional Theory. The eigenproblems below use this ordinary overlap and therefore apply directly to all-electron or norm-conserving problems. When LCAO auxiliary spinors are used in PAW, the right-hand side instead contains the complete PAW overlap matrix defined in Projector-Augmented-Wave Method.

A Kohn–Sham spinor orbital is expanded as

\[ \lvert\psi_a\rangle = \sum_{\mu s} c_{\mu s,a} \lvert\phi_\mu;s\rangle. \]

Collecting the coefficients into \(\mathbf C\), the Kohn–Sham problem is

\[ \mathbf H\mathbf C = \mathbf S\mathbf C\mathbf E, \qquad \mathbf C^\dagger\mathbf S\mathbf C = \mathbf I. \]

The columns of \(\mathbf C\) are the coefficient vectors \(c_{\mu s,a}\), and \(\mathbf E=\operatorname{diag}(\varepsilon_a)\) is the diagonal eigenvalue matrix.

The overlap matrix must be positive definite on the retained basis. Very small eigenvalues of \(\mathsf S\) indicate linear dependencies and make the coefficient problem ill-conditioned.

The density matrix is

\[ \mathbf D = \mathbf C\mathbf f\mathbf C^\dagger, \]

where \(\mathbf f=\operatorname{diag}(f_a)\) is the diagonal occupation matrix. In components,

\[ D_{\mu s,\nu s'} = \sum_a f_a c_{\mu s,a}c_{\nu s',a}^*. \]

It reconstructs the density operator as

\[ \hat\rho = \sum_{\mu s,\nu s'} \lvert\phi_\mu;s\rangle D_{\mu s,\nu s'} \langle\phi_\nu;s'\rvert. \]

In a nonorthogonal basis, \(\mathbf D\) is not the matrix of covariant operator matrix elements:

\[ \langle\phi_\mu;s\vert\hat\rho\vert\phi_\nu;s'\rangle = \left( \mathbf S\mathbf D\mathbf S \right)_{\mu s,\nu s'}. \]

Consequently,

\[ N_{\mathrm e} = \operatorname{tr}(\mathbf D\mathbf S), \qquad \langle\hat X\rangle = \operatorname{tr}(\mathbf D\mathbf X). \]

Here \(\operatorname{tr}\) is the ordinary matrix trace over both orbital and spin indices.

For integer occupations, idempotency of the density operator becomes

\[ \mathbf D\mathbf S\mathbf D = \mathbf D. \]

The local spin-density matrix follows directly:

\[ n_{ss'}(\boldsymbol r) = \sum_{\mu\nu} \phi_\mu(\boldsymbol r) D_{\mu s,\nu s'} \phi_\nu^*(\boldsymbol r). \]

The components \(n_{ss'}\) form the \(2\times2\) local spin-density matrix \(\mathbf n(\boldsymbol r)\). Its scalar electron and spin-polarization densities are

\[ n(\boldsymbol r) = \operatorname{tr}_s\mathbf n(\boldsymbol r), \qquad m^\alpha(\boldsymbol r) = \operatorname{tr}_s \left[ \sigma_\alpha\mathbf n(\boldsymbol r) \right]. \]

These relations fix the meaning of \(\mathbf D\) independently of any storage ordering used by a particular implementation.

Projection of \(\mathbf D\) or a real-space scalar field into an auxiliary basis is derived in Auxiliary-Basis Expansion and Projection of Density.

Spin Structure

Because the spatial basis is spin independent, the full overlap matrix factorizes as

\[ \mathbf S = \mathsf S\otimes\sigma_0, \qquad \left(\mathsf S\right)_{\mu\nu} \equiv S_{\mu\nu} \equiv \langle \phi_\mu \vert \phi_\nu \rangle. \]

The Hamiltonian need not factorize in spin. Applying the Pauli-trace convention of Kohn–Sham Density Functional Theory to its spin blocks gives

\[ \mathbf H = \mathsf H^0\otimes\sigma_0 + \sum_{\alpha=x,y,z} \mathsf H^\alpha\otimes\sigma_\alpha. \]

Here \(\mathsf H^0\) and \(\mathsf H^\alpha\) are matrices over the spatial AO indices, while \(\mathbf H\) is the complete matrix over both orbital and spin indices. Their italic matrix elements are

\[ \left(\mathsf H^A\right)_{\mu\nu} \equiv H_{\mu\nu}^A, \qquad A\in\{0,x,y,z\}. \]

The spin-vector components admit an additional real–imaginary split when the spatial AOs are real and there is no external orbital magnetic field. For the minimal Hamiltonian containing real spin-independent terms, magnetic exchange, and spin–orbit coupling, write

\[ \mathsf H^\alpha = \mathsf B^\alpha + \mathsf H^{\mathrm{SO},\alpha}, \qquad B_{\mu\nu}^\alpha \equiv \operatorname{Re}H_{\mu\nu}^\alpha, \qquad H_{\mu\nu}^{\mathrm{SO},\alpha} \equiv \mathrm i\operatorname{Im}H_{\mu\nu}^\alpha. \]

Hermiticity makes the exchange sector real symmetric and the spin–orbit sector purely imaginary antisymmetric. For a finite system,

\[ \left(\mathsf B^\alpha\right)^T = \mathsf B^\alpha, \qquad \left(\mathsf H^{\mathrm{SO},\alpha}\right)^T = -\mathsf H^{\mathrm{SO},\alpha}. \]

For the periodic cell-shift-indexed matrix blocks introduced below, the corresponding relations are

\[ B_{\nu\mu}^\alpha(-\boldsymbol R) = B_{\mu\nu}^\alpha(\boldsymbol R), \qquad H_{\nu\mu}^{\mathrm{SO},\alpha}(-\boldsymbol R) = -H_{\mu\nu}^{\mathrm{SO},\alpha}(\boldsymbol R). \]

Under time reversal, the real spin-vector sector \(\mathsf B^\alpha\) is odd, whereas the purely imaginary sector \(\mathsf H^{\mathrm{SO},\alpha}\) is even. Their identification with magnetic exchange and spin–orbit coupling therefore depends on the minimal Hamiltonian content stated above.

A spin-independent Hamiltonian has only \(\mathsf H^0\). A collinear formulation with quantization axis \(z\) retains \(\mathsf H^0\) and \(\mathsf H^z\), while a general noncollinear or spin–orbit-coupled Hamiltonian may require all four Pauli components. The same algebraic Pauli decomposition produces \(\mathsf D^0\) and \(\mathsf D^\alpha\) from the complete density matrix \(\mathbf D\), but the exchange–SOC interpretation is specific to the Hamiltonian. No additional factor of two is included when occupations refer to explicit spinor states.

The density-independent spin-traceless operator supplied by a fully relativistic norm-conserving pseudopotential, including its projector-centered LCAO matrix, is derived in Fixed Spin–Orbit Operators in an LCAO Basis.

Periodic LCAO Basis Matrices

For a periodic system, let \(\lvert\phi_{\mu\boldsymbol R};s\rangle\) be the translate of orbital \(\mu\) in cell \(\boldsymbol R\), so that its spatial part satisfies

\[ \phi_{\mu\boldsymbol R}(\boldsymbol r) = \phi_{\mu\boldsymbol 0}(\boldsymbol r-\boldsymbol R). \]

The corresponding Bloch sum at \(\boldsymbol k\) follows the cell-inner-product convention of Periodic Systems:

\[ \lvert\phi_{\mu\boldsymbol k};s\rangle = \sum_{\boldsymbol R} \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol R } \lvert\phi_{\mu\boldsymbol R};s\rangle. \]

This sum has no global normalization prefactor. Its overlap is the cell inner product

\[ S_{\mu s,\nu s'}(\boldsymbol k) = \langle \phi_{\mu\boldsymbol k};s \vert \phi_{\nu\boldsymbol k};s' \rangle_\Omega. \]

The Bloch function need not be constructed on an extended real-space domain. Implementations normally obtain \(\mathbf H(\boldsymbol k)\) and \(\mathbf S(\boldsymbol k)\) directly by folding their localized cell-shift-indexed matrix blocks.

For a lattice-translation-invariant operator \(\hat X\), define its cell-shift-indexed matrix blocks by displacing the right state relative to the left,

\[ X_{\mu s,\nu s'}(\boldsymbol R) \equiv \langle \phi_{\mu\boldsymbol 0};s \lvert\hat X\rvert \phi_{\nu\boldsymbol R};s' \rangle. \]

Let \(\mathbf X(\boldsymbol R)\) collect all orbital and spin components \(X_{\mu s,\nu s'}(\boldsymbol R)\). Its Bloch matrix is

\[ \mathbf X(\boldsymbol k) = \sum_{\boldsymbol R} \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol R } \mathbf X(\boldsymbol R). \]

The Kohn–Sham spinors are expanded in this Bloch basis as

\[ \lvert\psi_{a\boldsymbol k}\rangle = \sum_{\mu s} c_{\mu s,a}(\boldsymbol k) \lvert\phi_{\mu\boldsymbol k};s\rangle. \]

Substitution into the Kohn–Sham equation gives the generalized eigenproblem

\[ \mathbf H(\boldsymbol k) \mathbf C(\boldsymbol k) = \mathbf S(\boldsymbol k) \mathbf C(\boldsymbol k) \mathbf E(\boldsymbol k). \]

Cell normalization of the Kohn–Sham spinors requires

\[ \mathbf C(\boldsymbol k)^\dagger \mathbf S(\boldsymbol k) \mathbf C(\boldsymbol k) = \mathbf I. \]

The density matrix at each \(\boldsymbol k\) point is

\[ \mathbf D(\boldsymbol k) = \mathbf C(\boldsymbol k) \mathbf f(\boldsymbol k) \mathbf C(\boldsymbol k)^\dagger. \]

Here \(\mathbf f(\boldsymbol k)=\operatorname{diag}(f_{a\boldsymbol k})\) contains the occupations of the cell-normalized Bloch spinors.

Traces per primitive cell are

\[ N_{\mathrm e} = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \operatorname{tr} \left[ \mathbf D(\boldsymbol k) \mathbf S(\boldsymbol k) \right], \]
\[ \langle\hat X\rangle_{\mathrm{cell}} = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \operatorname{tr} \left[ \mathbf D(\boldsymbol k) \mathbf X(\boldsymbol k) \right]. \]

The occupations are those of explicit spinor states.

Cell-Shift-Indexed Density-Matrix Blocks

At the continuum level, the cell-shift-indexed density-matrix blocks are the Brillouin-zone transform

\[ \mathbf D(\boldsymbol R) = \frac{1}{\Omega_{\mathrm{BZ}}} \int_{\mathrm{BZ}} \mathrm d\boldsymbol k\, \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol R } \mathbf D(\boldsymbol k). \]

In this definition, \(\boldsymbol R\) is the cell of the unstarred, row orbital relative to the cell of the starred, column orbital. To reconstruct the periodic density, let \(\boldsymbol R_0\) be the cell of the starred orbital; the unstarred orbital is then in \(\boldsymbol R_0+\boldsymbol R\). The spin-density matrix is

\[ n_{ss'}(\boldsymbol r) = \sum_{\boldsymbol R_0} \sum_{\boldsymbol R} \sum_{\mu\nu} \phi_{\mu,\boldsymbol R_0+\boldsymbol R}(\boldsymbol r) D_{\mu s,\nu s'}(\boldsymbol R) \phi_{\nu,\boldsymbol R_0}^*(\boldsymbol r). \]

The lattice sums are formally infinite; an exact cutoff or numerical screening makes only finitely many terms relevant at a given point. Hermiticity gives

\[ \mathbf D(\boldsymbol R)^\dagger = \mathbf D(-\boldsymbol R). \]

For a complete, unreduced uniform mesh \(\mathcal K\), the sampled values and the dual finite translation set form the discrete pair

\[ \mathbf D_{\mathcal K}(\boldsymbol R) = \frac{1}{N_{\boldsymbol k}} \sum_{\boldsymbol k\in\mathcal K} \mathrm e^{ +\mathrm i\boldsymbol k\cdot\boldsymbol R } \mathbf D(\boldsymbol k), \qquad \boldsymbol R\in\mathcal R_{\mathcal K}, \]
\[ \mathbf D(\boldsymbol k) = \sum_{\boldsymbol R\in\mathcal R_{\mathcal K}} \mathrm e^{ -\mathrm i\boldsymbol k\cdot\boldsymbol R } \mathbf D_{\mathcal K}(\boldsymbol R), \qquad \boldsymbol k\in\mathcal K, \]

where \(\mathcal R_{\mathcal K}\) contains \(N_{\boldsymbol k}=|\mathcal K|\) translations. This pair applies to the finite sampled data; a general weighted or symmetry-reduced quadrature does not define an exact inverse transform.

The same positive phase appears in \(\mathbf X(\boldsymbol k)\) and \(\mathbf D(\boldsymbol R)\), but the displacement has a different role: \(\mathbf X(\boldsymbol R)\) is an operator matrix element with the right ket displaced from the left bra, whereas \(\mathbf D(\boldsymbol R)\) is a density matrix block with the unstarred row orbital displaced from the starred column orbital. The displacement convention must therefore accompany the phase convention.

Scope and Numerical Character

Localized bases can represent both isolated and periodic systems without changing their local building blocks. Their finite spatial extent often makes \(\mathbf H(\boldsymbol R)\) and \(\mathbf S(\boldsymbol R)\) sparse. The decay of \(\mathbf D(\boldsymbol R)\) is a property of the electronic state and is generally slower in metals than in insulators.

Basis quality is controlled by the number and shapes of radial functions, angular-momentum channels, polarization functions, and confinement. There is no single monotone parameter equivalent to a plane-wave energy cutoff. Convergence must therefore be established for the observables of interest, together with the conditioning of \(\mathsf S\).

Because atom-centered basis functions move and may change with the nuclei, energy derivatives contain basis-response terms customarily grouped under Pulay contributions.

References