Theoretical developments on the dynamics of a dense liquid using a statistical-mechanical approach primarily involve a small set of slow collective densities termed hydrodynamic modes. The time scales of relaxation of these modes are much longer than those for the microscopic modes of the system. The basic approach adopted here is the analysis of the time correlation functions (introduced earlier in Chapter 1) of the slow modes. In the present chapter and the next two chapters we discuss microscopic methods for calculating the correlation functions involving the fluctuation or hydrodynamic approach. We focus primarily on the simplest type of correlation functions involving fluctuations at two different spatial and time coordinates. Owing to time translation invariance, equilibrium two-point correlation functions of hydrodynamic modes at the same time over different spatial points are time-independent and provide us with information on the thermodynamic behavior of the system. On the other hand, the dynamic behavior of the system is linked to the correlation of physically observable quantities at two different times. The time correlation function of density fluctuations is particularly important for our discussion of the slow dynamics in a liquid. In the simplest of the theoretical models, the decay of the correlation with time is exponential. We discuss here how such exponential relaxation behavior can be understood using linear dynamics of the fluctuations. The formalism developed in the later parts of this chapter allows in a natural way the extension of the macroscopic hydrodynamics to intermediate length and time scales, and is referred to as generalized hydrodynamics.

Computer simulation is an indispensible research tool for modeling, understanding, and predicting nanoscale phenomena. There is a huge gap between the complexity of the programs and algorithms used in computational physics courses and and those used in research for computer simulations of nanoscale systems. The advanced computer codes used by researchers are often too complicated for students who want to develop their own codes, want to understand the essential details of computer simulations, or want to improve existing programs.

The aim of this book is to provide a comprehensive program library and description of advanced algorithms to help students and researchers learn novel methods and develop their own approaches. An important contribution of this book is that it is accompanied by an algorithm library in Fortran 90 that implements the computational approaches described in the text.

The physical problems are solved at various levels of sophistication using methods based on classical molecular dynamics, tight binding, density functional approaches, or fully correlated wave functions. Various basis functions including finite differences, Lagrange functions, plane waves, and Gaussians are introduced to solve bound state and scattering problems and to describe electronic structure and transport properties of materials. Different methods of solving the same problem are introduced and compared.

The book is divided into two parts. In the first part we concentrate on one-dimensional problems.

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.