Atomic orbitals

Besides plane waves and real-space grids, atomic orbitals [59, 332, 158, 133, 258, 319] are also a popular choice as basis states in electronic structure calculations. Each choice of basis states has its own advantages and disadvantages. The most important advantages of plane waves and real space grids are their straightforward formalism and simple control of accuracy (an energy cutoff and a grid spacing, respectively). However, methods based on the linear combination of atomic orbitals (LCAO) are more efficient in terms of basis size, because atomic orbitals are much better suited to represent molecular or Bloch wave functions. Another advantage of localized atomic orbitals is that the Hamiltonian matrix becomes sparse as the system size increases. This has recently renewed interest in LCAO bases because the sparsity makes them suitable for order-N methods [246, 215, 200, 104], in which computational effort scales linearly with system size. Local-atomic-orbital bases also offer a natural way of quantifying the magnitudes of atomic charge, orbital population, bond charge, charge transfer, etc.

The disadvantages of LCAO include the facts that (i) the functions can become overcomplete (linear dependence can occur in a calculation if two similar functions are centered at the same atom), (ii) they are difficult to program (especially if high-angular-momentum functions are needed), and (iii) it is difficult to test or demonstrate absolute convergence since there are many more parameters than the energy cutoff of the plane wave approach.

Molecular dynamics simulation is one of the most fundamental tools of materials modeling. Such simulations are used to study chemical reactions, fluid flow, phase transitions, droplet formation, and many other physical and chemical phenomena. Many textbook and review articles [119, 263, 96, 5, 113] exist in the literature, and in this chapter we restrict ourselves to a basic introduction.

Classical molecular dynamics uses Newton's equations of motion to describe the time development of a system. These calculations involve a long series of time steps, at each of which Newton's laws are used to determine the new positions and velocities from the old positions and velocities. The computation is simple but has to be repeated many times. For accurate simulation the time step is very small and the calculation takes a long time to simulate a real time interval. The force calculation in an N-particle system may scale as O(N2), thus the calculation time can be quite long. In the last few decades sophisticated computational algorithms have been developed to address these problems. In this chapter we study two prototypical examples of MD simulations: the Lennard–Jones system and structure of Si described by the Stillinger–Weber potential [299].

Introduction

Classical molecular dynamics (MD) uses potentials based on empirical data or on independent electronic structure calculations. It is a powerful tool for investigating many-body condensed matter systems.

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].