LAPW+LO: A Variational All-Electron Basis

• LAPW+LO is a variational, all-electron framework that partitions space into muffin-tin spheres and an interstitial region to accurately represent electron behavior.
• It incorporates local orbitals to systematically eliminate linearization errors and capture semicore and high-energy conduction states for improved spectral and energetic precision.
• The method enables systematic convergence in DFT, GW, and ab initio spectroscopy, ensuring benchmark-quality calculations for diverse material classes.

The linearized augmented planewave plus local orbital (LAPW+LO) basis is a variational, all-electron framework for first-principles electronic structure calculations, especially within density functional theory (DFT) and many-body perturbation theory. It partitions space into non-overlapping, atom-centered muffin-tin (MT) spheres and an interstitial region, and constructs basis functions that combine interstitial plane waves with atom-centered solutions of the radial Schrödinger or Dirac equation. Local orbitals, strictly confined to the MT spheres, are included to capture semicore and high-lying conduction states, systematically eliminating the linearization error intrinsic to the basic LAPW approach. This method underpins state-of-the-art implementations such as the full-potential all-electron code exciting, and is considered the gold standard for calculations of ground-state, excited-state, and spectroscopic properties in solids and molecules (Tillack et al., 2019, Nabok et al., 2016, Raya-Moreno et al., 16 Jan 2026).

1. Formal Construction of the LAPW+LO Basis

The LAPW+LO basis exploits the distinct behavior of the Kohn-Sham (or quasiparticle) wavefunctions in the MT and interstitial regions:

• Interstitial region (I): Each basis function is a plane wave, $$\phi_{\mathbf{k}+\mathbf{G}}(\mathbf{r}) = (1/\sqrt{\Omega})\,e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}$$, indexed by crystalline momentum $\mathbf{k}$ and reciprocal lattice vector $\mathbf{G}$.
• MT sphere $\alpha$: The basis function is expanded in spherical harmonics $Y_{\ell m}(\widehat{\mathbf{r}}_{\alpha})$ and two radial functions, $u_{\ell}^{\alpha}(r_{\alpha};E_{\ell}^{\alpha})$ and its energy derivative $$\dot u_{\ell}^{\alpha}(r_{\alpha};E_{\ell}^{\alpha})$$, both solutions to the radial Schrödinger (or Dirac) equation for the spherically averaged potential $V_{\alpha}(r)$:

$$\left[-\frac{1}{2}\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{2r^2} + V_{\alpha}(r)\right]u_{\ell}^{\alpha}(r;E) = E\,u_{\ell}^{\alpha}(r;E)$$

The APW augmentation coefficients $A_{l m}$ and $B_{l m}$ are fixed by imposing continuity of the value and the first radial derivative at the MT boundary:

$$\psi^{\text{I}}_{\mathbf{k}+\mathbf{G}}(R_{\mathrm{MT}}) = \psi^{\text{MT}}_{\mathbf{k}+\mathbf{G}}(R_{\mathrm{MT}}),\quad \frac{d}{dr}\psi^{\text{I}}_{\mathbf{k}+\mathbf{G}}(R_{\mathrm{MT}}) = \frac{d}{dr}\psi^{\text{MT}}_{\mathbf{k}+\mathbf{G}}(R_{\mathrm{MT}})$$

• Local orbitals (LO): For each atom $\alpha$ and channel $(l, m)$, functions of the form

$$\phi_{\mathsf L}(\mathbf{r}) = \Theta(R_{\mathrm{MT}}^{\alpha}-r_{\alpha})\left[a\,u_{l}^{\alpha}(r_{\alpha}; E_{l}^{\alpha}) + b\,\dot u_{l}^{\alpha}(r_{\alpha};E_{l}^{\alpha}) + c\,u_{l}^{\alpha}(r_{\alpha}; E_{l,\mathrm{LO}}^{\alpha})\right]Y_{l m}(\widehat{\mathbf{r}}_{\alpha})$$

vanish at the sphere boundary together with their radial derivatives, where $E_{l,\mathrm{LO}}^{\alpha}$ typically targets semicore or high-lying states (Tillack et al., 2019).

2. Linearization and Systematic Extension

The canonical LAPW basis expands radial functions via a first-order Taylor expansion about $E_\ell$:

$$u_{l}^{\alpha}(r;E) \approx u_{l}^{\alpha}(r;E_{l}^{\alpha}) + (E-E_{l}^{\alpha})\dot u_{l}^{\alpha}(r;E_{l}^{\alpha})$$

Truncation after the linear term introduces the "linearization error," which grows quadratically with $|E-E_l|$ and impacts the representation of states far from $E_{l}^{\alpha}$, particularly high-lying conduction bands and deep semicore levels (Michalicek et al., 2013, Nabok et al., 2016). The addition of LOs at several energies $E_{l,\mathrm{LO}}$ systematically enlarges the energy window over which the basis is accurate, thereby eliminating this error to leading order. Advanced variants employ high-derivative local orbitals (HDLOs) built from $\ddot u_{l}^{\alpha}(r;E_{l}^{\alpha})$, which further reduce the energy scaling of the representation error from quadratic to cubic or beyond (Michalicek et al., 2013, Kutepov, 2020).

3. Role and Construction of Local Orbitals

Local orbitals are strictly confined to individual MT spheres and constructed as linear combinations of $u_{l}(r;E_{l})$, $\dot u_{l}(r;E_{l})$, and either $\ddot u_{l}(r;E_{l})$ or $u_{l}(r;E_{l,\mathrm{LO}})$, with coefficients chosen to satisfy normalization and to enforce vanishing value and derivative at the MT boundary. Two principal classes of LOs are prevalent (Michalicek et al., 2013, Kutepov, 2020):

• Higher-energy LOs (HELOs): Use $u_{l}(r;E^{\mathrm{LO}}_{l})$ at energies well above (or below) $E_{l}$, enhancing completeness for GW and high-energy conduction bands.
• Higher-derivative LOs (HDLOs): Employ $\ddot u_{l}(r;E_{l})$ for maximal suppression of the linearization error within the energy window pertinent to DFT.

The presence of LOs allows the basis to remain compact in the interstitial plane waves while capturing semicore, tightly bound, or very diffuse states—without requiring a prohibitive increase in $G_{\max}$ or the numerical basis set size (Nabok et al., 2016).

4. Variational Properties, Completeness, and Basis Optimization

The LAPW+LO basis possesses a blockwise completeness: interstitial plane waves span smooth, delocalized electron behavior, while atom-centered (augmented and local) components capture rapid MT variations and localized correlation. This enables systematic, variational convergence of total energies, bandgaps, quasiparticle corrections, and derived properties. Explicit formulations, such as the three-component formalism in mixed APW+MTO+LO algorithms, provide a transparent decomposition of the overlap and Hamiltonian matrices, facilitating both analytic and numerical implementation and enabling direct assessment of the relative error of different components (Kotani et al., 2014).

Optimal use involves:

• Choice of $R_{\mathrm{MT}}K_{\max} \approx 8-12$ for plane waves.
• Angular momentum expansion to $\ell_{\max} \approx 7-8$ or until convergence in target properties.
• Multiple LOs per $(\ell, \alpha)$, spanning the relevant energy window (e.g., using centers of semicore, valence, and high-energy ranges).
• Pruning and orthogonalization to avoid near-linear dependence among LOs (Nabok et al., 2016, Raya-Moreno et al., 16 Jan 2026).

5. Numerical Impact: Convergence, Accuracy, and Comparison

Numerical benchmarks for semiconductors (e.g., ZnO) and metals exhibit the following consequences of LO inclusion (Nabok et al., 2016, Michalicek et al., 2013):

• DFT total energy: Adding HDLOs or HELOs accelerates convergence in $R_{\mathrm{MT}}K_{\max}$, reduces spurious dependence on internal parameters (e.g., $E_{\ell}$), and yields energies lower or comparable to those attained with APW+lo or legacy LAPW.
• Bandstructure and QP corrections: Converged GW gaps require LOs spanning high-energy unoccupied states; absence of sufficient LO channels causes artificial saturation or underestimation of QP gaps. For example, in ZnO the direct gap at $\Gamma$ increases from 2.42 eV (default LO basis) to 2.94 eV (optimized LO basis), in agreement with high-quality PAW benchmarks (Nabok et al., 2016).
• Semicore description: LOs enable accurate variational treatment of semicore levels by targeting $E_{l,\mathrm{LO}}$ near semicore binding energies, eliminating "ghost" or spurious eigenstates and allowing energetic and spectroscopic convergence without inflating the global plane wave basis (Tillack et al., 2019).

A representative table, summarizing accuracy gains as a function of basis extension:

Basis Type	$\Delta E_{\text{tot}}$ (mHtr)	QP Gap Error (meV)	Main LO Feature
LAPW	$\sim$10	$>$500	None
LAPW+HELO	$\sim$2	$\sim$60	$u(E\!\gg\!E_\ell)$ in LO
LAPW+HDLO	$\lesssim$1	$\lesssim$5	$\ddot u(E_\ell)$ in LO

Data condensed from (Michalicek et al., 2013, Nabok et al., 2016)

6. Extensions, Alternatives, and Algorithmic Features

Recent advancements and methodological extensions include:

• Multi-radius and multi-derivative bases: The SLAPWMR formalism enables distinct matching radii for each $G$, $l$, and site, minimizing linearization and low-$G$ errors without enlarging the derivative subspace. This enables higher-order matching (vanishing third derivatives at the boundary), yielding $O\left[(E-E_l)^6\right]$ scaling in linearization error and further reducing the basis set size for given accuracy (Goldstein, 2024).
• Supplemented tight-binding functions: Two-energy linearizations yield two independent orthonormal LOs per $(l,m)$, extending the LAPW+LO approach into a broader, systematically complete basis (Nikolaev et al., 2015).
• Hybrid and mixed basis strategies: The PMT approach, combining APWs, muffin-tin orbitals (MTOs), and LOs in a unified three-component formalism, attains similar accuracy with a reduced number of plane waves by supplementing plane wave and atom-centered behavior (Kotani et al., 2014).

Implementationally, symmetry-adapted basis construction, dual-basis self-validation, automated LO selection, and efficient real-arithmetic diagonalization are routine in modern codes (e.g., exciting), and are essential for microhartree-level numerical accuracy (Raya-Moreno et al., 16 Jan 2026).

7. Physical and Computational Significance

The LAPW+LO method offers maximal variational flexibility for all-electron, full-potential calculations:

• All-electron representation: No pseudization or shape approximation is required for core or valence states; true radial and angular behavior is retained within each sphere.
• High-fidelity property calculation: Enables benchmark-quality evaluation of Kohn-Sham and quasiparticle band structures, effective masses, densities of states, and spectroscopic matrix elements down to errors of a few meV or less, across diverse material classes, including metals, semiconductors, insulators, and hybrid organic-inorganic interfaces (Tillack et al., 2019, Raya-Moreno et al., 16 Jan 2026).
• Systematic improvability: By judicious addition of LOs and optimization of basis parameters, both linearization and basis-incompleteness errors can be eliminated to within arbitrary precision, independent of internal orbital energies or sphere radii.
• Robustness: The method is numerically well-conditioned, avoids ill-posedness arising from overcompleteness or ghost states, and can be faithfully extended to relativistic (Dirac–Kohn–Sham) frameworks with clear separation between essential relativistic corrections and variational flexibility (Kutepov, 2020).

The LAPW+LO formalism is foundational for modern ab initio spectroscopy and materials modeling, enabling direct, transparent, and reproducible connection between theoretical frameworks and experimentally measurable quantities.