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Planewave Basis SetEdit

Planewave Basis Set is a foundational tool in computational physics and chemistry used to represent electronic wavefunctions as sums of plane waves. Its natural fit for periodic systems—such as crystals and surfaces—stems from Bloch’s theorem, which ensures that electronic states in a periodic potential can be decomposed into combinations of plane waves modulated by a crystal momentum. This framework is central to many modern ab initio methods and serves as a clean, systematic way to approach electronic structure problems.

In a planewave basis set, the wavefunctions are expanded as psi_k(r) = sum_G c_G(k) e^{i (k+G)·r}, where k is a wavevector in the Brillouin zone and G runs over reciprocal lattice vectors. The expansion is controlled by an energy cutoff, E_cut, which limits the magnitude of G included in the sum. Increasing E_cut adds plane waves and hence enhances accuracy in a predictable, controllable manner. This makes the planewave approach particularly attractive for convergent studies and for materials where translational symmetry plays a crucial role. See also plane-wave basis set for a broader treatment and context.

Overview

  • Basis and convergence: The planewave basis is systematic and unbiased, because there is a clear parameter (the energy cutoff) that governs completeness. The number of plane waves grows as a function of E_cut and the size of the reciprocal lattice, so computational cost increases with the desired precision. See energy cutoff for details on how this parameter influences efficiency and accuracy.
  • Periodic systems and Bloch’s theorem: The method is especially powerful for periodic solids, where electronic states are labeled by a crystal momentum k and the wavefunctions are periodic modulo a phase factor. This leads naturally to calculations over the Brillouin zone and to concepts like the band structure. See Bloch theorem and Brillouin zone for related ideas.
  • Pseudopotentials and all-electron variants: To avoid explicitly treating tightly bound core electrons, many planewave calculations employ pseudopotentials, which replace the all-electron potential in the core region with a smoother representation. This reduces required E_cut and accelerates convergence. See pseudopotential and ultrasoft pseudopotential for common approaches; for all-electron treatments in a planewave framework, see PAW and APW methods that combine plane waves with localized augmentations.
  • Software and practice: Numerous software packages implement planewave methods, often in the context of density functional theory, and provide ready-made pseudopotentials, k-point grids, and convergence testing tools. Prominent examples include Quantum ESPRESSO, VASP, ABINIT, and CASTEP projects, each with its own strengths and community practices. See also density functional theory for the broader theoretical framework underpinning these codes.

Mathematical formulation and practicalities

  • Wavefunction expansion: The electronic wavefunction in a periodic potential is expanded in a basis of plane waves with coefficients that depend on the Bloch wavevector k. The choice of G-vectors is dictated by the crystal structure and the imposed energy cutoff.
  • Reciprocal space and Fourier methods: Because plane waves are eigenfunctions of the translation operators in a lattice, many operations (such as kinetic energy evaluation) become diagonal or near-diagonal in reciprocal space, enabling efficient Fast Fourier Transform (FFT) algorithms. See Fourier transform for foundational mathematics and reciprocal lattice for lattice terminology.
  • Pseudopotentials and core-valence separation: Pseudopotentials replace the strong oscillations of core electron wavefunctions near nuclei with smoother, effective potentials, allowing a smaller basis set to achieve the same accuracy for valence properties. See pseudopotential and norm-conserving pseudopotential and consider alternatives like ultrasoft pseudopotential and PAW when higher accuracy or transferability is needed.
  • All-electron variants and augmentation schemes: In some contexts, plane waves are combined with augmentation regions to recover core-like behavior, leading to methods such as augmented plane waves and linearized augmented plane wave techniques. See APW and LAPW for augments of plane waves with localized basis functions.
  • k-point sampling and Brillouin zone integration: Calculations for solids require sampling of k-points in the Brillouin zone to approximate integrals over electronic states. The density of k-point grids affects both accuracy and cost. See k-point sampling and Brillouin zone.

Strengths and limitations

  • Strengths:
    • Systematic improvability via E_cut, offering a straightforward path to higher accuracy.
    • Uniform error control across a wide range of properties for periodic systems.
    • Mathematical simplicity and ease of integration with many algorithms for solving the Kohn–Sham equations in density functional theory.
    • Compatibility with a wide class of materials problems, including metals, semiconductors, and insulators, as well as surfaces and interfaces.
  • Limitations:
    • Computational cost scales with the number of plane waves, making plane-wave methods less economical for very large systems or for materials with localized, tightly bound states unless pseudopotentials or augmentation schemes are used.
    • Not always ideal for isolated molecules or systems with strong localization, where localized basis sets (such as Gaussian-type orbitals) can be more efficient.
    • Requires careful treatment of convergence with respect to E_cut and k-point density, and careful selection of pseudopotentials or augmentation schemes for reliable results.

Variants and related methods

  • Pseudopotentials and alternatives: Norm-conserving and ultrasoft pseudopotentials are common choices to reduce the plane-wave basis size. See pseudopotential and ultrasoft pseudopotential.
  • Projector augmented-wave methods: PAW provides a way to recover near all-electron accuracy while retaining the efficiency of a planewave-like representation. See also APW for related augmentation strategies.
  • Augmented plane waves and LAPW: In all-electron schemes, plane waves are augmented in the vicinity of atomic cores, leading to methods such as LAPW and APW that combine the strengths of plane waves with localized basis functions.
  • Software ecosystems: Popular platforms implementing planewave DFT include Quantum ESPRESSO, VASP, ABINIT, and CASTEP, each with documentation on using E_cut, k-point grids, and pseudopotentials effectively.

Applications

  • Solid-state properties: Band structures, densities of states, total energies, and structural optimizations for crystalline materials.
  • Surfaces and defects: Calculations of surface energies, work functions, adsorption energies, and defect formation energies in periodic cells.
  • Phonons and finite-temperature properties: Some planewave implementations support phonon calculations via perturbation theory, enabling insights into vibrational properties and thermodynamics.
  • Materials discovery: High-throughput studies and screening of candidate materials often rely on planewave-based DFT for reliable comparison across many compositions.

See also