1;Foreword;6 1.1;Acknowledgments;9 1.2;References;9 2;Contents;10 3;List of Contributors;18 4;1 Introduction;20 4.1;1.1 Historical Backdrop;20 4.2;1.2 The Density of States;33 4.3;1.3 Free Energy;37 4.4;1.4 Ergodicity, Quasi-nonergodicity and Enhanced Sampling;40 4.5;References;43 5;2 Calculating Free Energy Differences Using Perturbation Theory;51 5.1;2.1 Introduction;51 5.2;2.2 The Perturbation Formalism;52 5.3;2.3 Interpretation of the Free Energy Perturbation Equation;55 5.4;2.4 Cumulant Expansion of the Free Energy;58 5.5;2.5 Two Simple Applications of Perturbation Theory;60 5.6;2.6 How to Deal with Large Perturbations;64 5.7;2.7 A Pictorial Representation of Free Energy Perturbation;66 5.8;2.8 'Alchemical Transformations';68 5.9;2.9 Improving the Efficiency of FEP;78 5.10;2.10 Calculating Free Energy Contributions;84 5.11;2.11 Summary;89 5.12;References;90 6;3 Methods Based on Probability Distributions and Histograms;94 6.1;3.1 Introduction;94 6.2;3.2 Histogram Reweighting;95 6.3;3.3 Basic Stratification and Importance Sampling;101 6.4;3.4 Flat-Histogram Methods;109 6.5;3.5 Order Parameters, Reaction Coordinates, and Extended Ensembles;130 6.6;References;133 7;4 Thermodynamic Integration Using Constrained and Unconstrained Dynamics;136 7.1;4.1 Introduction;136 7.2;4.2 Methods for Constrained and Unconstrained Simulations;138 7.3;4.3 Generalized Coordinates and Lagrangian Formulation;140 7.4;4.4 The Derivative of the Free Energy;145 7.5;4.5 The Potential of Mean Constraint Force;148 7.6;4.6 The Adaptive Biasing Force Method;155 7.7;4.7 Discussion of Other Techniques;166 7.8;4.8 Examples of Application of ABF;167 7.9;4.9 Glycophorin A;170 7.10;4.10 Alchemical Transformations;172 7.11;4.11 Conclusion;177 7.12;Appendix;178 7.13;References;184 8;5 Nonequilibrium Methods for Equilibrium Free Energy Calculations;188 8.1;5.1 Introduction and Background;188 8.2;5.2 Jarzynski's Identity;191 8.3;5.3 Derivation of Jarzynski's Identity;192 8.4;5.4 Forward and Backward Averages: Crooks Relation;197 8.5;5.5 Derivation of the Crooks Relation (and Jarzynski's Identity);198 8.6;5.6 Implementation;199 8.7;5.7 Analysis of Nonequilibrium Free Energy Calculations;201 8.8;5.8 Illustrating Example;204 8.9;5.9 Calculating Potentials of Mean Force;208 8.10;5.10 Applications;211 8.11;5.11 Summary;211 8.12;References;212 9;6 Understanding and Improving Free Energy Calculations in Molecular Simulations: Error Analysis and Reduction Methods;216 9.1;6.1 Introduction;216 9.2;6.2 Overview of the FEP and NEW Methods;220 9.3;6.3 Understanding Free Energy Calculations;222 9.4;6.4 Modeling Free Energy Errors;232 9.5;6.5 Optimal Staging Design;243 9.6;6.6 Overlap Sampling Techniques;245 9.7;6.7 Extrapolation Methods;256 9.8;6.8 Concluding Remarks;260 9.9;References;261 10;7 Transition Path Sampling and the Calculation of Free Energies;265 10.1;7.1 Rare Events and Free Energy Landscapes;265 10.2;7.2 Transition Path Ensemble;268 10.3;7.3 Sampling the Transition Path Ensemble;271 10.4;7.4 Free Energies from Transition Path Sampling Simulations;278 10.5;7.5 The Jarzynski Identity: Path Sampling of Nonequilibrium Trajectories;280 10.6;7.6 Rare Event Kinetics and Free Energies in Path Space;286 10.7;7.7 Summary;290 10.8;References;290 11;8 Specialized Methods for Improving Ergodic Sampling Using Molecular Dynamics and Monte Carlo Simulations;293 11.1;8.1 Background;293 11.2;8.2 Measuring Ergodicity;294 11.3;8.3 Introduction to Enhanced Sampling Strategies;295 11.4;8.4 Modifying the Configurational Distribution: Non- Boltzmann Sampling;296 11.5;8.5 Methods Based on Exchanging Configurations: Parallel Tempering and Related Strategies;302 11.6;8.6 Smart Darting and Basin Hopping Monte Carlo;307 11.7;8.7 Momentum-Enhanced HMC;309 11.8;8.8 Skewing Momenta Distributions to Enhance Free Energy Calculations from Trajectory Space Methods;314 11.9;8.9 Quantum Free Energy Calculations;325 11.10;8.10 Summary;330 11.11;References;331 12;9 Potential Distribution Methods and Fre