This text provides a detailed description of the techniques employed in molecular modelling and computational chemistry. The first part of the book covers the two major methods used to describe the interactions within a system (quantum mechanics and molecular mechanics). The second part then deals with techniques that use such energy models, including energy minimisation, molecular dynamics, Monte Carlo simulations and conformational analysis. The author also discusses the use of more advanced modelling techniques such as the calculation of free energies and the simulation of chemical reactions. In addition he considers methods such as database searching that can be used to design new molecules with specific properties. Many of the topics are treated in considerable depth but the author assumes that the reader has only a basic knowledge of the relevant physical and chemical principles.
Most of the theoretical sections are accompanied by simple calculations together with examples drawn from the literature. The book is well illustrated and a colour plate section highlights the impact of computer molecular graphics. The book will prove a valuable text for postgraduate students and professionals and many sections will be useful to final-year undergraduates taking courses in molecular modelling or computational chemistry.
This book is intended to provide an introduction to some of the techniques used in molecular modelling and computational chemistry, and to illustrate how these techniques can be used to study physical, chemical and biological phenomena. A major objective is to provide, in one volume, some of the theoretical background to the vast array of methods available to the molecular modeller. I also hope that the book will help the reader to select the most appropriate method for a problem and so make the most of his or her modelling hardware and software. Many modelling programs are extremely simple to use and are often supplied with seductive graphical interfaces which obviously helps to make modelling techniques more accessible, but it can also be very easy to select a wholly inappropriate technique or method.
Most molecular modelling studies involve three stages. In the first stage a model is selected to describe the intra- and inter- molecular interactions in the system. The two most common models that are used in molecular modelling are quantum mechanics and molecular mechanics. These models enable the energy of any arrangement of the atoms and molecules in the system to be calculated, and allow the modeller to determine how the energy of the system varies as the positions of the atoms and molecules change. The second stage of a molecular modelling study is the calculation itself, such as an energy minimisation, a molecular dynamics or Monte Carlo simulation, or a conformational search. Finally, the calculation must be analysed, not only to calculate properties but also to check that it has been performed properly.
The book is organised so that some of the techniques discussed in later chapters refer to material discussed earlier, though I have tried to make each chapter as independent of the others as possible. Some readers may therefore be pleased to know that it is not essential to completely digest the chapters on quantum mechanics and molecular mechanics in order to read about methods for searching conformational space! Readers with experience in one or more areas may of course wish to be more selective.
I have tried to provide as much of the underlying theory as seems appropriate to enable the reader to understand the fundamentals of each method. In doing so I have assumed some background knowledge of quantum mechanics, statistical mechanics, conformational analysis and mathematics. A reader with an undergraduate degree in chemistry should have covered this material, which should also be familiar to many undergraduates in the final year of their degree course. Full discussions can be found in the suggestions for further reading at the end of each chapter. I have also attempted to provide a reasonable selection of original references, though in a book of this scope it is obviously impossible to provide a comprehensive coverage of the literature. In this context, I apologise in advance if any technique is inappropriately inattributed.
In Chapter 1 we consider some of the historical background to molecular modelling and discuss a number of important general principles that are common to many modelling methods. We also examine the use of computer graphics, the Internet and the World-Wide Web and the molecular modelling literature. Chapter 1 concludes with a brief summary of some relevant mathematical concepts. Chapters 2 and 3 describe quantum mechanics and molecular mechanics , which are the two major methods used to model the interactions within a molecular system. These methods can be used to calculate the energy of a given arrangement of the atoms as well as certain other properties. In chapters 4-8 we examine energy minimisation, molecular dynamics, Monte Carlo simulations and conformational analysis. These techniques use an appropriate energy model to determine a wide range of structural and thermodynamic properties. The final two chapters describe various techniques that combine concepts from previous chapters. In Chapter 8 we discuss the calculation of free energies using computer simulation , continuum solvent models, and methods for simulating chemical reactions. Chapter 9 is concerned with computational methods for discovering and designing new molecules, such as database searching , de novo design and quantitative structure-activity relationships.
The range of systems that can be considered in molecular modelling is extremely broad, from isolated molecules through simple atomic and molecular liquids to polymers, biological macromolecules such as proteins and DNA and solids. Many of the techniques are illustrated with examples chosen to reflect the breadth of applications. It is inevitable that for reasons of space some techniques must be dealt with in a rudimentary fashion (or not at all), and that many interesting and important applications cannot be described. Molecular modelling is a rapidly developing discipline, and has benefitted from the dramatic improvements in computer hardware and software of recent years. Calculations that were major undertakings only a few years ago can now be performed using personal computing facilities. Thus, examples used to indicate the 'state of the art' at the time of writing will invariably be routine within a short time.
Chapter 1. Useful Concepts in Molecular Modelling
1.1 Introduction 1.2 Coordinate systems 1.3 Potential Energy surfaces 1.4 Molecular Graphics 1.5 Surfaces 1.6 Computer hardware and sofware 1.7 Units of length and energy 1.8 The molecular modelling literature 1.9 The Internet 1.10 Mathematical Concepts 1.10.1 Series expansions 1.10.2 Vectors 1.10.3 Matrices, eigenvalues and eigenvectors 1.10.4 Complex numbers 1.10.5 Lagrange multipliers 1.10.6 Multiple integrals 1.10.7 Some basic elements of statistics
Chapter 2. Quantum mechanical models
2.1 Introduction 2.1.1 Operators 2.1.2 Atomic units 2.1.3 Exact solutions to the Schrödinger equation 2.2 One-electron atoms 2.3 Polyelectronic atoms and molecules 2.3.1 The Born-Oppenheimer approximation 2.3.2 The helium atom 2.3.3 General polyelectronic systems and Slater Determinants 2.4 Molecular orbital calculations 2.4.1 Calculating the energy from the wavefunction: the hydrogen molecule 2.4.2 The energy of a general polyelectronic system 2.4.3 Shorthand representations of the 1 and 2 electron integrals 2.4.4 The energy of a closed-shell system 2.5 The Hartree-Fock equations 2.5.1 The Hartree-Fock equations for atoms and Slater's rules 2.5.2 The linear combination of atomic orbitals (LCAO) in Hartree-Fock theory 2.5.3 Closed-shell systems and the Roothaan-Hall equations 2.5.4 Solving the Roothaan-Hall equations 2.5.5 A simple illustration of the Roothaan-Hall approach 2.5.6 The application of the Hartree-Fock equations to molecular systems 2.6 Basis sets 2.6.1 Creating a basis set 2.7 Open-shell systems 2.8 Electron correlation 2.8.1 Configuration interaction 2.8.2 Many body perturbation theory 2.9 Practical considerations when running ab initio calculations 2.9.1 Convergence of Self-consistent field calculations 2.9.2 The Direct SCF method 2.9.3 Setting up the calculation and the choice of coordinates 2.9.4 Calculating derivatives of the energy 2.9.5 Basis set superposition error 2.10 Approximate molecular orbital theories 2.11 Semi-empirical methods 2.11.1 Zero-differential overlap 2.11.2 CNDO 2.11.3 INDO 2.11.4 NDDO 2.11.5 MINDO/3 2.11.6 MNDO 2.11.7 AM1 2.11.8 PM3 2.11.9 SAM1 2.11.10 Programs for peforming semi-empirical quantum mechanical calculations 2.12 Hückel theory 2.12.1 Extended Hückel theory 2.13 Valence bond theories 2.14 The calculation of molecular properties using quantum mechanics 2.14.1 Energies, Koopman's theorem and ionisation potentials 2.14.2 Calculation of Electric multipoles 2.14.3 The total electric density distribution and molecular orbitals 2.14.4 Population analysis 2.14.5 Mulliken and Löwdin population analysis 2.14.6 Partitioning the electron density using the theory of atoms in molecules 2.14.7 Bond orders 2.14.8 Electrostatic potentials 2.14.9 Thermodynamic and structural properties 2.15 Performance of semi-empirical methods 2.16 Energy component analysis 2.16.1 The water dimer
Chapter 3 Empirical Force field models: molecular mechanics
3.1 Introduction 3.1.1 A simple molecular mechanics force field 3.2 Some general features of molecular mechanics force fields 3.3 Bond stretching 3.4 Angle bending 3.5 Torsional terms 3.6 Improper torsions and out-of-plane bending motions 3.7 Cross terms Non-bonded interactions 3.8 Electrostatic interactions 3.8.1 The central multipole expansion 3.8.2 Point-charge electrostatic models 3.8.3 Calculating partial atomic charges 3.8.4 Charges derived from the molecular electrostatic potential 3.8.5 Deriving charge models for large systems 3.8.6 Rapid methods for calculating atomic charges 3.8.7 Beyond partial atomic charge models 3.8.8 Distributed multipole models 3.8.9 Applications of charge schemes to the study of aromatic-aromatic interactions 3.8.10 Polarization 3.8.11 Solvent dielectric models 3.9 van der Waals interactions 3.9.1 Dispersive interactions 3.9.2 The repulsive contribution 3.9.3 Modelling van der Waals interactions 3.9.4 van der Waals interactions in polyatomic systems 3.9.5 Reduced units 3.10 Many body effects in empirical potentials 3.11 Effective pair potentials 3.12 Hydrogen bonding in molecular mechanics 3.13 Force field models for the simulation of liquid water 3.13.1 Simple water models 3.13.2 Polarisable water models 3.13.3 Ab initio potentials for water 3.14 United atom force fields and reduced representations 3.14.1 Other simplified models 3.15 Derivatives of the molecular mechanics energy function 3.16 Calculating thermodynamic properties using a force field 3.17 Force field parameterisation 3.18 Transferrability of force field parameters 3.19 The treatment of delocalised [[pi]]-systems 3.20 Force fields for metals and inorganic systems 3.20.1 Force fields for zeolites Appendix 3.1 The Interaction between two Drude molecules
Chapter 4 Energy minimization and other methods for exploring the energy surface
4.1 Introduction 4.1.1 Energy minimisation: statement of the problem 4.1.2 Derivatives 4.2 Non-derivative minimisation methods 4.2.1 The Simplex method 4.2.2 The sequential univariate method 4.3 Introduction to derivative minimisation methods 4.4 First-order minimisation methods 4.4.1 The steepest descents method 4.4.2 Line search in one dimension 4.4.3 Arbitrary step approach 4.4.4 Conjugate gradients minimisation 4.5 Second derivative methods: the Newton-Raphson method 4.5.1 Variants on the Newton-Raphson method 4.6 Quasi-Newton methods 4.7 Which minimisation method should I use? 4.7.1 Distinguishing between minima, maxima and transition points 4.7.2 Convergence criteria 4.8 Applications of energy minimisation 4.8.1 Normal mode analysis 4.8.2 The study of intermolecular processes using energy minimisation and normal mode analysis 4.9 The determination of transition points and reaction pathways 4.9.1 Methods to locate saddle points 4.9.2 Reaction path following 4.9.3 Locating transition points and elucidating reaction pathways for large systems 4.9.4 The transition structures of pericyclic reactions
Chapter 5 An introduction to computer simulation methods
5.1 Introduction 5.1.1 Time averages, ensemble averages and some historical background 5.1.2 A brief description of the molecular dynamics method 5.1.3 The basic elements of the Monte Carlo method 5.1.4 Differences between the molecular dynamics and Monte Carlo methods 5.2 Calculation of simple thermodynamic properties 5.2.1 Energy 5.2.2 Heat capacity 5.2.3 Pressure 5.2.4 Temperature 5.2.5 Radial disribution functions 5.3 The concept of phase space 5.4 Practical aspects of computer simulation 5.4.1 Setting up and running a simulation 5.4.2 Choosing the initial configuration 5.5 Boundaries 5.5.1 Periodic boundary conditions 5.5.2 Non-periodic boundary conditions 5.6 Monitoring the equilibration 5.7 Truncating the potential and the minimum image convention 5.7.1 Non-bonded neighbour lists 5.7.2 Group-based cutoffs 5.7.3 Problems with cutoffs and how to avoid them 5.8 Long-range forces 5.8.1 The Ewald summation method 5.8.2 The reaction field and image charge methods 5.9 The cell-multipole method for calculating non-bonded interactions 5.10 Analysing the results of a simulation and estimating errors Appendix 5.1 Basic statistical mechanics Appendix 5.2 Relationship between heat capacity and energy fluctuations Appendix 5.3 Calculation of the real gas contribution to the virial Appendix 5.4 Fomulae to translate particle back into central box for various periodic shapes
Chapter 6 Molecular Dynamics
6.1 Introduction 6.2 Molecular dynamics using simple models 6.3 Molecular dynamics with continuous potentials 6.3.1 Finite difference methods 6.3.2 Predictor-corrector integration methods 6.3.3 Which integration algorithm is most appropriate? 6.3.4 Choosing the time step 6.4 Setting up and running a molecular dynamics simulation 6.4.1 Calculating the temperature 6.5 Constraint dynamics 6.6 Time-dependent properties 6.6.1 Correlation functions 6.6.2 Orientational correlation functions 6.6.3 Transport properties 6.7 Constant temperature and constant pressure molecular dynamics 6.7.1 Constant temperature dynamics 6.7.2 Constant pressure dynamics 6.8 Incorporating solvent effects into molecular dynamics: Potentials of Mean Force and Stochastic dynamics 6.8.1 Practical aspects of stochastic dynamics simulations 6.9 Conformational changes from molecular dynamics simulations 6.10 Molecular Dynamics simulations of chain amphiphiles 6.10.1 Simulations of lipids 6.10.2 Simulations of Langmuir-Blodgett films Appendix 6.1 Energy conservation in molecular dynamics Appendix 6.2 Fourier series and fourier analysis
Chapter 7 Monte Carlo simulation methods
7.1 Introduction 7.2 Calculating properties by integration 7.3 Some theoretical background to the Metropolis method 7.4 Implementation of the Metropolis Monte Carlo method 7.4.1 Random number generators 7.5 Monte Carlo simulation of molecules 7.5.1 Rigid molecules 7.5.2 Monte Carlo simulations of flexible molecules 7.6 Models used in Monte Carlo simulations of polymers 7.6.1 Lattice models of polymers 7.6.2 'Continuous' polymer models 7.7 `Biased' Monte Carlo methods 7.8 Monte Carlo sampling from different ensembles 7.8.1 Grand-canonical Monte Carlo simulations 7.8.2 Grand-canonical Monte Carlo simulations of adsorption processes 7.9 Calculating the Chemical Potential 7.10 The Configurational Bias Monte Carlo method 7.10.1 Applications of the Configurational bias Monte Carlo method 7.11 Simulating phase equilibria by the Gibbs Ensemble Monte Carlo method 7.12 Monte Carlo or molecular dynamics? Appendix 7.1 The Marsaglia random number generator
Chapter 8 Conformational analysis
8.1 Introduction 8.2 Systematic methods for exploring conformational space 8.3 Model-building approaches 8.4 Random search methods 8.5 Genetic algorithms 8.6 Distance geometry 8.6.1 The use of distance geometry in NMR 8.7 Exploring conformational space using simulation methods 8.7.1 Simulated annealing 8.7.2 Solving protein structures by restrained molecular dynamics refinement 8.7.3 X-ray crystallographic refinement 8.7.4 Molecular dynamics refinement of NMR data 8.7.5 Time-averaged NMR refinement 8.8 Which conformational search method should I use? 8.9 Structural databases 8.10 Molecular fitting 8.11 Clustering algorithms and pattern recognition techniques in molecular modelling 8.12 Reducing the dimensionality of a data set 8.12.1 Principal components analysis 8.13 The role of conformational analysis in predicting the structure of peptides and proteins 8.13.1 Some basic principles of protein structure 8.13.2 The hydrophobic effect 8.13.3 First-principles methods for predicting protein structure 8.13.4 Lattice models for invesigating protein structure 8.13.5 Rule-based approaches 8.13.6 Homology and comparative modelling methods 8.13.7 Aligning protein sequences 8.13.8 Constructing and evaluating an homology model 8.13.9 Predicting protein structures by 'threading' 8.13.10 A comparison of comparative modelling stategies
Chapter 9 Three challenges in molecular modelling: free energies, solvation and simulating reactions
9.1 The difficulty of calculating free energies from a computer simulation 9.2 The calculation of free energy differences 9.2.1 Thermodynamic perturbation 9.2.2 Implementation of free-energy perturbation 9.2.3 Thermodynamic integration 9.2.4 The 'Slow growth' method 9.3 Applications of methods for calculating free energy differences 9.3.1 Thermodynamic cycles 9.3.2 Applications of the thermodynamic cycle perturbation method 9.3.3 The calculation of absolute free energies 9.4 The calculation of enthalpy and entropy differences 9.5 Partitioning the free energy 9.6 Potential pitfalls with free energy calculations 9.6.1 Implementation aspects 9.7 Potentials of mean force 9.7.1 Umbrella sampling 9.7.2 Calculating the potential of mean force for flexible molecules 9.8 Continuum representations of the solvent 9.8.1 Thermodynamic background 9.9 The electrostatic contribution to the free energy of solvation: the Born and Onsager models 9.9.1 Calculating the electrostatic contribution via quantum mechanics 9.9.2 Continuum models for molecular mechanics 9.9.3 The Langevin dipole model 9.9.4 Methods based upon the Poisson-Boltzmann equation 9.9.5 Applications of finite difference Poisson-Boltzmann calculations 9.6 Non-electrostatic contributions to the solvation free energy 9.7 Very simple solvation models 9.8 Modelling chemical reactions 9.8.1 Empirical approaches to simulating reactions 9.8.2 The potential of mean force of a reaction 9.8.3 Combined quantum mechanical/molecular mechanical approaches 9.9 Density functional theory 9.9.1 Density Functional Methods in the study of processes on solids 9.9.2 The Car-Parinello method 9.9.3 The Application of ab initio molecular dynamics to a chemisorption process Appendix 9.1 Calculating free energy differences using thermodynamic integration Appendix 9.2 The Slow growth method for calculating free energy differences
Chapter 10 The use of molecular modelling to discover and design new molecules
10.1 Molecular modelling in drug discovery 10.2 Deriving and using three-dimensional pharmacophores 10.2.1 Constrained systematic search 10.2.2 Ensemble distance geometry and ensemble molecular dynamics 10.2.3 Clique detection methods for finding pharmacophores 10.2.4 Incorporating geometric features in a 3D pharmacophore 10.3 Molecular docking 10.4 Structure-basd methods to identify lead compounds 10.4.1 Finding lead compounds by searching 3D databases 10.4.2 Sources of 3D data 10.5 de novo ligand design 10.5.1 Finding favourable positions of molecular fragments within a binding site 10.5.2 Connecting molecular fragments in a binding site 10.5.3 The use of structure-based design methods to design HIV-1 Protease inhibitors 10.6 Molecular similarity 10.7 Quantitative Structure-activity relationships 10.7.1 The calculation of partition coefficients 10.7.2 Deriving the QSAR equation 10.7.3 Non-linear models: neural networks and genetic algorithms 10.7.4 Interpreting a QSAR equation 10.7.5 Partial least squares 10.7.6 Partial least squares and molecular field analysis
Computational Chemistry List (a discussion forum for all those interested in computational chemistry)
Imperial College Chemistry home page (contains links to a host of useful modelling- related pages)
Network Science (an on-line journal of science and computers)