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.

Introduction

Two-dimensional few-electron systems have been the focus of extensive theoretical and experimental investigation. Recent advances in nanofabrication techniques have enabled experiments with 2D quantum dots having highly controlled parameters such as electron number, size, shape, confinement strength, and magnetic field. The possibility of fabricating these “artificial atoms” with tunable properties is a fascinating new development in nanotechnology. The principal motivations for these investigations are the variety of possible applications in quantum computing [298], spintronics [261], information storage [199], and nanoelectronics [15, 159, 315].

Theoretical calculations of quantum dot systems are based on the effective mass approximation [42, 131, 130, 350, 205, 101, 233, 356, 184, 139, 35, 127]. In these models the electrons move in an external confining potential and interact via the Coulomb interaction. The apparent similarity of “natural” atoms and quantum dots have motivated the application of sophisticated theoretical methods borrowed from atomic physics and quantum chemistry to calculate the properties of quantum dots. Parabolically confined 2D quantum dots have been studied by several different well-established methods: exact diagonalization techniques [131, 205], Hartree–Fock approximations [101, 233, 356], and density functional approaches [184, 139]. Quantum Monte Carlo (QMC) techniques have also been used for 2D [35, 127, 255, 85] as well as 3D structures. The strongly correlated low-electronicdensity regime has received much attention owing to the intriguing possibility of the formation of Wigner molecules [356, 85].