892 |
|
simulations. For instance, instantaneous temperature of an |
893 |
|
Hamiltonian system of $N$ particle can be measured by |
894 |
|
\[ |
895 |
< |
T(t) = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
895 |
> |
T = \sum\limits_{i = 1}^N {\frac{{m_i v_i^2 }}{{fk_B }}} |
896 |
|
\] |
897 |
|
where $m_i$ and $v_i$ are the mass and velocity of $i$th particle |
898 |
|
respectively, $f$ is the number of degrees of freedom, and $k_B$ is |
913 |
|
These three individual steps will be covered in the following |
914 |
|
sections. Sec.~\ref{introSec:initialSystemSettings} deals with the |
915 |
|
initialization of a simulation. Sec.~\ref{introSec:production} will |
916 |
< |
discusses issues in production run, including the force evaluation |
917 |
< |
and the numerical integration schemes of the equations of motion . |
918 |
< |
Sec.~\ref{introSection:Analysis} provides the theoretical tools for |
919 |
< |
trajectory analysis. |
916 |
> |
discusses issues in production run. Sec.~\ref{introSection:Analysis} |
917 |
> |
provides the theoretical tools for trajectory analysis. |
918 |
|
|
919 |
|
\subsection{\label{introSec:initialSystemSettings}Initialization} |
920 |
|
|
984 |
|
|
985 |
|
\subsection{\label{introSection:production}Production} |
986 |
|
|
987 |
< |
\subsubsection{\label{introSec:forceCalculation}The Force Calculation} |
987 |
> |
Production run is the most important steps of the simulation, in |
988 |
> |
which the equilibrated structure is used as a starting point and the |
989 |
> |
motions of the molecules are collected for later analysis. In order |
990 |
> |
to capture the macroscopic properties of the system, the molecular |
991 |
> |
dynamics simulation must be performed in correct and efficient way. |
992 |
|
|
993 |
< |
\subsubsection{\label{introSection:integrationSchemes} Integration |
994 |
< |
Schemes} |
993 |
> |
The most expensive part of a molecular dynamics simulation is the |
994 |
> |
calculation of non-bonded forces, such as van der Waals force and |
995 |
> |
Coulombic forces \textit{etc}. For a system of $N$ particles, the |
996 |
> |
complexity of the algorithm for pair-wise interactions is $O(N^2 )$, |
997 |
> |
which making large simulations prohibitive in the absence of any |
998 |
> |
computation saving techniques. |
999 |
> |
|
1000 |
> |
A natural approach to avoid system size issue is to represent the |
1001 |
> |
bulk behavior by a finite number of the particles. However, this |
1002 |
> |
approach will suffer from the surface effect. To offset this, |
1003 |
> |
\textit{Periodic boundary condition} is developed to simulate bulk |
1004 |
> |
properties with a relatively small number of particles. In this |
1005 |
> |
method, the simulation box is replicated throughout space to form an |
1006 |
> |
infinite lattice. During the simulation, when a particle moves in |
1007 |
> |
the primary cell, its image in other cells move in exactly the same |
1008 |
> |
direction with exactly the same orientation. Thus, as a particle |
1009 |
> |
leaves the primary cell, one of its images will enter through the |
1010 |
> |
opposite face. |
1011 |
> |
%\begin{figure} |
1012 |
> |
%\centering |
1013 |
> |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
1014 |
> |
%\caption[An illustration of periodic boundary conditions]{A 2-D |
1015 |
> |
%illustration of periodic boundary conditions. As one particle leaves |
1016 |
> |
%the right of the simulation box, an image of it enters the left.} |
1017 |
> |
%\label{introFig:pbc} |
1018 |
> |
%\end{figure} |
1019 |
|
|
1020 |
+ |
%cutoff and minimum image convention |
1021 |
+ |
Another important technique to improve the efficiency of force |
1022 |
+ |
evaluation is to apply cutoff where particles farther than a |
1023 |
+ |
predetermined distance, are not included in the calculation |
1024 |
+ |
\cite{Frenkel1996}. The use of a cutoff radius will cause a |
1025 |
+ |
discontinuity in the potential energy curve |
1026 |
+ |
(Fig.~\ref{introFig:shiftPot}). Fortunately, one can shift the |
1027 |
+ |
potential to ensure the potential curve go smoothly to zero at the |
1028 |
+ |
cutoff radius. Cutoff strategy works pretty well for Lennard-Jones |
1029 |
+ |
interaction because of its short range nature. However, simply |
1030 |
+ |
truncating the electrostatic interaction with the use of cutoff has |
1031 |
+ |
been shown to lead to severe artifacts in simulations. Ewald |
1032 |
+ |
summation, in which the slowly conditionally convergent Coulomb |
1033 |
+ |
potential is transformed into direct and reciprocal sums with rapid |
1034 |
+ |
and absolute convergence, has proved to minimize the periodicity |
1035 |
+ |
artifacts in liquid simulations. Taking the advantages of the fast |
1036 |
+ |
Fourier transform (FFT) for calculating discrete Fourier transforms, |
1037 |
+ |
the particle mesh-based methods are accelerated from $O(N^{3/2})$ to |
1038 |
+ |
$O(N logN)$. An alternative approach is \emph{fast multipole |
1039 |
+ |
method}, which treats Coulombic interaction exactly at short range, |
1040 |
+ |
and approximate the potential at long range through multipolar |
1041 |
+ |
expansion. In spite of their wide acceptances at the molecular |
1042 |
+ |
simulation community, these two methods are hard to be implemented |
1043 |
+ |
correctly and efficiently. Instead, we use a damped and |
1044 |
+ |
charge-neutralized Coulomb potential method developed by Wolf and |
1045 |
+ |
his coworkers. The shifted Coulomb potential for particle $i$ and |
1046 |
+ |
particle $j$ at distance $r_{rj}$ is given by: |
1047 |
+ |
\begin{equation} |
1048 |
+ |
V(r_{ij})= \frac{q_i q_j \textrm{erfc}(\alpha |
1049 |
+ |
r_{ij})}{r_{ij}}-\lim_{r_{ij}\rightarrow |
1050 |
+ |
R_\textrm{c}}\left\{\frac{q_iq_j \textrm{erfc}(\alpha |
1051 |
+ |
r_{ij})}{r_{ij}}\right\}. \label{introEquation:shiftedCoulomb} |
1052 |
+ |
\end{equation} |
1053 |
+ |
where $\alpha$ is the convergence parameter. Due to the lack of |
1054 |
+ |
inherent periodicity and rapid convergence,this method is extremely |
1055 |
+ |
efficient and easy to implement. |
1056 |
+ |
%\begin{figure} |
1057 |
+ |
%\centering |
1058 |
+ |
%\includegraphics[width=\linewidth]{pbcFig.eps} |
1059 |
+ |
%\caption[An illustration of shifted Coulomb potential]{An illustration of shifted Coulomb potential.} |
1060 |
+ |
%\label{introFigure:shiftedCoulomb} |
1061 |
+ |
%\end{figure} |
1062 |
+ |
|
1063 |
+ |
%multiple time step |
1064 |
+ |
|
1065 |
|
\subsection{\label{introSection:Analysis} Analysis} |
1066 |
|
|
1067 |
+ |
Recently, advanced visualization technique are widely applied to |
1068 |
+ |
monitor the motions of molecules. Although the dynamics of the |
1069 |
+ |
system can be described qualitatively from animation, quantitative |
1070 |
+ |
trajectory analysis are more appreciable. According to the |
1071 |
+ |
principles of Statistical Mechanics, |
1072 |
+ |
Sec.~\ref{introSection:statisticalMechanics}, one can compute |
1073 |
+ |
thermodynamics properties, analyze fluctuations of structural |
1074 |
+ |
parameters, and investigate time-dependent processes of the molecule |
1075 |
+ |
from the trajectories. |
1076 |
+ |
|
1077 |
+ |
\subsubsection{\label{introSection:thermodynamicsProperties}Thermodynamics Properties} |
1078 |
+ |
|
1079 |
+ |
Thermodynamics properties, which can be expressed in terms of some |
1080 |
+ |
function of the coordinates and momenta of all particles in the |
1081 |
+ |
system, can be directly computed from molecular dynamics. The usual |
1082 |
+ |
way to measure the pressure is based on virial theorem of Clausius |
1083 |
+ |
which states that the virial is equal to $-3Nk_BT$. For a system |
1084 |
+ |
with forces between particles, the total virial, $W$, contains the |
1085 |
+ |
contribution from external pressure and interaction between the |
1086 |
+ |
particles: |
1087 |
+ |
\[ |
1088 |
+ |
W = - 3PV + \left\langle {\sum\limits_{i < j} {r{}_{ij} \cdot |
1089 |
+ |
f_{ij} } } \right\rangle |
1090 |
+ |
\] |
1091 |
+ |
where $f_{ij}$ is the force between particle $i$ and $j$ at a |
1092 |
+ |
distance $r_{ij}$. Thus, the expression for the pressure is given |
1093 |
+ |
by: |
1094 |
+ |
\begin{equation} |
1095 |
+ |
P = \frac{{Nk_B T}}{V} - \frac{1}{{3V}}\left\langle {\sum\limits_{i |
1096 |
+ |
< j} {r{}_{ij} \cdot f_{ij} } } \right\rangle |
1097 |
+ |
\end{equation} |
1098 |
+ |
|
1099 |
+ |
\subsubsection{\label{introSection:structuralProperties}Structural Properties} |
1100 |
+ |
|
1101 |
+ |
Structural Properties of a simple fluid can be described by a set of |
1102 |
+ |
distribution functions. Among these functions,\emph{pair |
1103 |
+ |
distribution function}, also known as \emph{radial distribution |
1104 |
+ |
function}, is of most fundamental importance to liquid-state theory. |
1105 |
+ |
Pair distribution function can be gathered by Fourier transforming |
1106 |
+ |
raw data from a series of neutron diffraction experiments and |
1107 |
+ |
integrating over the surface factor \cite{Powles73}. The experiment |
1108 |
+ |
result can serve as a criterion to justify the correctness of the |
1109 |
+ |
theory. Moreover, various equilibrium thermodynamic and structural |
1110 |
+ |
properties can also be expressed in terms of radial distribution |
1111 |
+ |
function \cite{allen87:csl}. |
1112 |
+ |
|
1113 |
+ |
A pair distribution functions $g(r)$ gives the probability that a |
1114 |
+ |
particle $i$ will be located at a distance $r$ from a another |
1115 |
+ |
particle $j$ in the system |
1116 |
+ |
\[ |
1117 |
+ |
g(r) = \frac{V}{{N^2 }}\left\langle {\sum\limits_i {\sum\limits_{j |
1118 |
+ |
\ne i} {\delta (r - r_{ij} )} } } \right\rangle. |
1119 |
+ |
\] |
1120 |
+ |
Note that the delta function can be replaced by a histogram in |
1121 |
+ |
computer simulation. Figure |
1122 |
+ |
\ref{introFigure:pairDistributionFunction} shows a typical pair |
1123 |
+ |
distribution function for the liquid argon system. The occurrence of |
1124 |
+ |
several peaks in the plot of $g(r)$ suggests that it is more likely |
1125 |
+ |
to find particles at certain radial values than at others. This is a |
1126 |
+ |
result of the attractive interaction at such distances. Because of |
1127 |
+ |
the strong repulsive forces at short distance, the probability of |
1128 |
+ |
locating particles at distances less than about 2.5{\AA} from each |
1129 |
+ |
other is essentially zero. |
1130 |
+ |
|
1131 |
+ |
%\begin{figure} |
1132 |
+ |
%\centering |
1133 |
+ |
%\includegraphics[width=\linewidth]{pdf.eps} |
1134 |
+ |
%\caption[Pair distribution function for the liquid argon |
1135 |
+ |
%]{Pair distribution function for the liquid argon} |
1136 |
+ |
%\label{introFigure:pairDistributionFunction} |
1137 |
+ |
%\end{figure} |
1138 |
+ |
|
1139 |
+ |
\subsubsection{\label{introSection:timeDependentProperties}Time-dependent |
1140 |
+ |
Properties} |
1141 |
+ |
|
1142 |
+ |
Time-dependent properties are usually calculated using \emph{time |
1143 |
+ |
correlation function}, which correlates random variables $A$ and $B$ |
1144 |
+ |
at two different time |
1145 |
+ |
\begin{equation} |
1146 |
+ |
C_{AB} (t) = \left\langle {A(t)B(0)} \right\rangle. |
1147 |
+ |
\label{introEquation:timeCorrelationFunction} |
1148 |
+ |
\end{equation} |
1149 |
+ |
If $A$ and $B$ refer to same variable, this kind of correlation |
1150 |
+ |
function is called \emph{auto correlation function}. One example of |
1151 |
+ |
auto correlation function is velocity auto-correlation function |
1152 |
+ |
which is directly related to transport properties of molecular |
1153 |
+ |
liquids: |
1154 |
+ |
\[ |
1155 |
+ |
D = \frac{1}{3}\int\limits_0^\infty {\left\langle {v(t) \cdot v(0)} |
1156 |
+ |
\right\rangle } dt |
1157 |
+ |
\] |
1158 |
+ |
where $D$ is diffusion constant. Unlike velocity autocorrelation |
1159 |
+ |
function which is averaging over time origins and over all the |
1160 |
+ |
atoms, dipole autocorrelation are calculated for the entire system. |
1161 |
+ |
The dipole autocorrelation function is given by: |
1162 |
+ |
\[ |
1163 |
+ |
c_{dipole} = \left\langle {u_{tot} (t) \cdot u_{tot} (t)} |
1164 |
+ |
\right\rangle |
1165 |
+ |
\] |
1166 |
+ |
Here $u_{tot}$ is the net dipole of the entire system and is given |
1167 |
+ |
by |
1168 |
+ |
\[ |
1169 |
+ |
u_{tot} (t) = \sum\limits_i {u_i (t)} |
1170 |
+ |
\] |
1171 |
+ |
In principle, many time correlation functions can be related with |
1172 |
+ |
Fourier transforms of the infrared, Raman, and inelastic neutron |
1173 |
+ |
scattering spectra of molecular liquids. In practice, one can |
1174 |
+ |
extract the IR spectrum from the intensity of dipole fluctuation at |
1175 |
+ |
each frequency using the following relationship: |
1176 |
+ |
\[ |
1177 |
+ |
\hat c_{dipole} (v) = \int_{ - \infty }^\infty {c_{dipole} (t)e^{ - |
1178 |
+ |
i2\pi vt} dt} |
1179 |
+ |
\] |
1180 |
+ |
|
1181 |
|
\section{\label{introSection:rigidBody}Dynamics of Rigid Bodies} |
1182 |
|
|
1183 |
|
Rigid bodies are frequently involved in the modeling of different |
2039 |
|
\] |
2040 |
|
where $x_OR$, $y_OR$, $z_OR$ are the components of the vector |
2041 |
|
joining center of resistance $R$ and origin $O$. |
1857 |
– |
|
1858 |
– |
%\section{\label{introSection:correlationFunctions}Correlation Functions} |