Book contents
- Frontmatter
- Contents
- Preface
- Part I One-dimensional problems
- Part II Two- and three-dimensional systems
- 7 Three-dimensional real-space approach: from quantum dots to Bose–Einstein condensates
- 8 Variational calculations in two dimensions: quantum dots
- 9 Variational calculations in three dimensions: atoms and molecules
- 10 Monte Carlo calculations
- 11 Molecular dynamics simulations
- 12 Tight-binding approach to electronic structure calculations
- 13 Plane wave density functional calculations
- 14 Density functional calculations with atomic orbitals
- 15 Real-space density functional calculations
- 16 Time-dependent density functional calculations
- 17 Scattering and transport in nanostructures
- 18 Numerical linear algebra
- Appendix: Code descriptions
- References
- Index
15 - Real-space density functional calculations
from Part II - Two- and three-dimensional systems
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface
- Part I One-dimensional problems
- Part II Two- and three-dimensional systems
- 7 Three-dimensional real-space approach: from quantum dots to Bose–Einstein condensates
- 8 Variational calculations in two dimensions: quantum dots
- 9 Variational calculations in three dimensions: atoms and molecules
- 10 Monte Carlo calculations
- 11 Molecular dynamics simulations
- 12 Tight-binding approach to electronic structure calculations
- 13 Plane wave density functional calculations
- 14 Density functional calculations with atomic orbitals
- 15 Real-space density functional calculations
- 16 Time-dependent density functional calculations
- 17 Scattering and transport in nanostructures
- 18 Numerical linear algebra
- Appendix: Code descriptions
- References
- Index
Summary
In this chapter we present a real-space approach to density functional calculations. Real-space calculations [28, 134, 207, 291, 4, 202, 118, 39, 116, 76, 123, 325, 122, 117, 326, 361, 124, 223, 257, 244, 125, 138, 245] are being rapidly developed as alternatives to plane wave calculations. In this chapter we will use a real-space grid with a finite difference representation for the kinetic energy operator. The advantage of real-space grid calculations is their simplicity and versatility (e.g., there are no matrix elements to be calculated and the boundary conditions are more easy imposed). As with plane wave basis sets, the accuracy can be improved easily and systematically. In fact, there exists a rigorous cutoff for the plane waves, which can be represented in a given grid without aliasing, that provides a convenient connection between the two schemes. Pseudopotentials, developed in the plane wave context, can be applied equally well in grid-based methods, resulting in an accurate and efficient evaluation of the electron–ion potential.
Unlike in the case of plane waves, the evaluation of the kinetic energy using finite differences is approximate, but it can be significantly improved by using high-order representations of the Laplacian operator. However, an important difference between finite difference schemes and basis set approaches is the lack of a Rayleigh–Ritz variational principle in the finite difference case.
- Type
- Chapter
- Information
- Computational NanoscienceApplications for Molecules, Clusters, and Solids, pp. 332 - 338Publisher: Cambridge University PressPrint publication year: 2011