Atomistic Computer Simulations

A Practical Guide

Veronika Brázdová, David R. Bowler

• Publisher: Wiley-VCH
• Year: 2013
• ISBN:
  9783527671830
• Edition: 1
• Topic: Naturwissenschaften
• Subtopic: Mathematische & Computerphysik

Many books explain the theory of atomistic computer simulations; this book teaches you how to run them

This introductory "how to" title enables readers to understand, plan, run, and analyze their own independent atomistic simulations, and decide which method to use and which questions to ask in their research project. It is written in a clear and precise language, focusing on a thorough understanding of the concepts behind the equations and how these are used in the simulations. As a result, readers will learn how to design the computational model and which parameters of the simulations are essential, as well as being able to assess whether the results are correct, find and correct errors, and extract the relevant information from the results. Finally, they will know which information needs to be included in their publications.

This book includes checklists for planning projects, analyzing output files, and for troubleshooting, as well as pseudo keywords and case studies.

The authors provide an accompanying blog for the book with worked examples, and additional material and references: http://www.atomisticsimulations.org/.

Part One The World at the Atomic Scale

Many of the macroscopic properties of matter can be predicted successfully by performing atomistic computer simulations. Indeed, the atomic theory of matter and its outworking through statistical mechanics can be seen as one of the most important scientific theories ever proposed. In this part of the book, we establish a basic foundation, considering the appropriate length and timescales, the influence of electrons and the environment on atoms, and bonding and chemical reactions. We also discuss what goes into an atomistic computer simulation, what happens during a simulation, and present an example of how atomistic simulations can be used to understand large scale problems.

1

Atoms, Molecules and Crystals

The world around us is composed of atoms in continual vibrational motion. Each atom consists of a positive nucleus and negative electrons. From here on, we will refer to an atom to mean both the electrons and the nucleus. Different elements and their interactions give rise to everything in the material world. Unless the energies involved are high enough to even break atoms apart, the interactions happen through electrons and nuclei. An investigation of material properties, chemical reactions, or diffusion on surfaces, to name but a few, must therefore involve a description of atoms and their interactions which are mediated by electrons. These interactions will contain the electrons either explicitly or implicitly.

Atoms and electrons cannot be comprehended with our normal senses, though there are many analogies which are used to help us understand them, such as spheres joined with springs or rigid bonds. We must not confuse these analogies with the real objects. In atomistic simulations, we use mathematical models to describe the behavior of atoms and how they interact. In experiments, we can only observe the response of electrons or nuclei to probes, which give characteristic signals. These signals can be calculated from atomistic simulations, and from these simulations we can draw conclusions about the material world.

In this chapter, we will look at the time- and lengthscales involved and the basic concepts forming the foundation of any atomistic simulation. We will also consider the connections between the mathematical approach and the real world.

1.1 Length- and Timescales

The length- and timescales in atomistic simulations are, in short, much shorter than those in the everyday world. The choice of units is dictated by convenience and tradition within a discipline (chemists and physicists in particular rarely agree on units) rather than by Système International d’Unités (SI units): a unit that keeps the relevant numerical values in single digits is always preferred. In atomic units, this is achieved by setting the radius of the electron in the hydrogen atom, the Bohr radius, to one¹⁾. One Bohr is 0.529 × 10⁻¹⁰ m and the hydrogen–hydrogen bond in the H₂ molecule is 1.40 Bohr (or a.u.). Another natural unit at the atomic scale is the Ångstrøm (Å), 10⁻¹⁰ m or 0.1 nanometers. Ångstrøms have the advantage of being easily recalculated to SI units. A hydrogen–hydrogen bond in the H₂ molecule is 0.74 Å long, a carbon–hydrogen bond in a methane molecule (CH₄) is 1.09 Å long and an Au–Au bond distance in bulk gold is 2.88 Å. Even though Bohrs and Ångstrøms do keep the relevant distances usefully in single digits, picometers (100 pm = 1 Å) are also sometimes used. The advantage of these is that one avoids the decimal point, as bond lengths in all but the most exact atomistic simulations are usually given to 10⁻¹² m and thus a 1.09 Å length can be written as 109 pm.

In surface and materials science, the systems of interest tend to be on the order of nanometers (1 nm = 10 Å). The semiconductor device industry is now also operating in the nm size range; because an integrated circuit’s physical size is constrained by practical limits of profitable manufacturability, performance is increased by increasing the number of components (e. g., transistors) and profitability by decreasing the size of the components. The critical dimension, defined as half the distance between identical features on the chip, is a useful measure of decreasing component sizes. The critical dimension of typical CPUs decreased from 350 nm in 1997 to 22 nm in 2012, enabling the number of transistors to increase from 7.5 × 10⁶ to 1.4 × 10⁹ (with physical sizes of 203 and 160 mm² respectively).

In biochemistry, Ångstrøms are used for the local structure: adenylate kinase, a small monomeric enzyme, isolated from E. coli is about 45 Å × 45 Å × 43 Å [1]. More commonly used, however, for the overall size of biological macromolecules is the total atomic mass, given in atomic mass units (amu, or Dalton, Da). An adenylate kinase would weigh 20–26 kDa.

Crystals are macroscopic objects, but they are composed of small sets of atoms repeated periodically in all three dimensions. In a silicon crystal, this set of atoms fits within a cube with a 5.431 Å side. In complex crystals such as zeolites dimensions of the repeated unit can be on the order of a few nm.

While length is a direct variable in atomistic simulations, time may not be. This will depend on what we are trying to simulate. If we need the direct time evolution of a system, or more often simply to sample many states of a system, then time will be important and we will use a technique such as Molecular Dynamics (see Chapter 6).

In many simulations, by contrast, the details of the dynamical process are not as important as the energies and structures involved. It is often possible to save computational time and resources by calculating a series of static “snapshots” of the atomic and electronic structure rather than modeling the whole process, for example, the initial and final state of a chemical reaction and the transition state (Chapter 3). Moreover, the range of timescales relevant in one system may be too great to allow simulations of the time evolution of all events because the sampling of time in the simulation is set by the fastest process. Atoms vibrate on a timescale of 10–100 fs (fs is 10⁻¹⁵ s). It takes about 100 fs to break an atomic bond [2, 3]; the larger a system is, the more involved atomic movements can take place, and the longer the relevant timescales are. For instance, pico- and nanosecond local fluctuations of adenylate kinase facilitate large-scale micro- to millisecond movements that have been linked to its enzymatic function [4]. In general, different functional motions of proteins range from femtoseconds to hours [5].

Where do atomistic simulations fit among other computational methods? The length- and timescales accessible to them will depend on the available computational resources, but also, critically, on the method chosen. In general, the more accurate the method, the more computationally demanding it is, leading to smaller system sizes and shorter timescales. In particular, methods that do not include electrons explicitly, such as forcefield methods, allow much larger system sizes and longer timescales than electronic structure methods, which do include electrons. Highly accurate quantum chemical methods, for example, include excitations and mixed spin states, but the system size is limited to a few dozen atoms. Electronic structure methods used in materials science, in particular, the density functional theory (DFT), can routinely handle hundreds of atoms. DFT has been applied to two million atoms [6], however, such calculations are not yet routine and the approach requires significant modifications of the standard DFT algorithms. Classical molecular dynamics (MD) simulations, which approximate the effect of electrons, are routinely applied to systems of hundreds of thousands of atoms and are performed over nanosecond and even picosecond timescales. MD simulations on millions of atoms (see, e. g., [7]) and even on 10⁸ atoms have also been performed [8].

Using classical molecular dynamics, it is also possible to run simulations up to microsecond range. However, there is no guarantee that even on such a timescale the simulated event will occur, as was demonstrated, for example, in [9], where despite a heroic effort, the native state of the protein was not found. It is even more complicated to simulate events with differing characteristic timescales in the same system because the time sampling would have to be fine enough to sample the fastest event. Multiscale methods are used to deal with these difficulties, but they are beyond the scope of this book.

Developments in atomistic computational methods and in computer power over the last few decades have increased the length- and timescales accessible to simulation. On the other side of the theory-experiment divide, developments in experimental techniques and sample preparation have allowed access to much smaller length- and timescales, as well as well-defined, small, model systems. The gap between experiment and simulation is therefore narrowing and it is often possible to investigate the same atomic system by experimental probes and by computer simulations.

1.2 Electrons in an Atom

The electronic structure of atoms determines their isolated properties and how they interact to form molecules and solids, as well as determining the structure of the periodic table of the elements. In this section, we will outline how electrons are arranged in an isolated atom.

The negatively charged electrons are distributed around the positively charged nucleus, also called the ionic core. Note the careful avoidance of the word “orbit” in the previous sentence. Electrons are quantum objects and we cannot follow their movement from place to place as we would with larger objects. The closest we can get to describing their movement is to calculate the probability that an electron will occupy a particular region of space a given distance from the nucleus and from other electrons. It is worth repeating: electrons do not behave as “normal” objects.

A full description of the electronic structure of an atom would involve a many-body function of the coordinates of all the electrons, the wavefunction. The wavefunction is a solution of the Schrödinger equation, the equation that describes a quantum system. However, it is impossible to solve this for more than two particles analytically, so an independent electron picture is often used. The solution is a set of discrete (i. e., quantum) states the electron can occupy in an atom. While electrons are indistinguishable particles, the quantum states in an atom they occupy do have different properties. Moreover, if two electrons are swapped in a system, the total wavefunction changes sign; we say that electrons are fermions. This antisymmetry of the wavefunction leads to the Pauli exclusion principle: two electrons cannot occupy the same state in one system. How can we describe an electronic state and what does it have to do with bonds between atoms?

The square of the wavefunction at a point in space gives the probability of finding the electron there. The sum over all electrons gives the total electronic charge density at that point. The electron is most likely to be found in a region of space called the atomic orbital. Again, the atomic orbi...

Table of contents

• Cover
• Contents
• Title Page
• The Authors
• Copyright
• Dedication
• Preface
• Part One The World at the Atomic Scale
• Part Two Introducing Equations to Describe the System
• Part Three Describing Interactions Between Atoms
• Part Four Setting Up and Running the Calculation
• Part Five Analyzing Results
• Appendix A UNIX
• Appendix B Scientific Computing
• Index
• Color Plates