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\usepackage{floatrow} |
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\oddsidemargin 0.0cm |
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\topmargin -21pt |
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\usepackage[square, comma, sort&compress]{natbib} |
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\bibpunct{[}{]}{,}{n}{}{;} |
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\DeclareFloatFont{tiny}{\scriptsize}% "scriptsize" is defined by floatrow, "tiny" not |
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\floatsetup[table]{font=tiny} |
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|
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+ |
|
| 28 |
|
%\renewcommand\citemid{\ } % no comma in optional reference note |
| 29 |
< |
\lstset{language=C,frame=TB,basicstyle=\footnotesize,basicstyle=\ttfamily, % |
| 29 |
> |
\lstset{language=C,frame=TB,basicstyle=\footnotesize\ttfamily, % |
| 30 |
|
xleftmargin=0.25in, xrightmargin=0.25in,captionpos=b, % |
| 31 |
|
abovecaptionskip=0.5cm, belowcaptionskip=0.5cm, escapeinside={~}{~}} |
| 32 |
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\renewcommand{\lstlistingname}{Scheme} |
| 51 |
|
\newcolumntype{M}{p{1.55in}} |
| 52 |
|
|
| 53 |
|
|
| 54 |
< |
\title{{\sc OpenMD-2.1}: Molecular Dynamics in the Open} |
| 54 |
> |
\title{{\sc OpenMD-2.2}: Molecular Dynamics in the Open} |
| 55 |
|
|
| 56 |
|
\author{Joseph Michalka, James Marr, Kelsey Stocker, Madan Lamichhane, |
| 57 |
|
Patrick Louden, \\ |
| 779 |
|
allowing the user to gauge the stability of the integrator. The |
| 780 |
|
statistics file is denoted with the \texttt{.stat} file extension. |
| 781 |
|
|
| 782 |
< |
\chapter{\label{section:forceFields}Force Fields} |
| 782 |
> |
\chapter{\label{chapter:forceFields}Force Fields} |
| 783 |
|
|
| 784 |
|
Like many molecular simulation packages, {\sc OpenMD} splits the |
| 785 |
|
potential energy into the short-ranged (bonded) portion and a |
| 793 |
|
are defined by the force field which is selected in the MetaData |
| 794 |
|
section. |
| 795 |
|
|
| 796 |
< |
\section{\label{section:shortRange}The basic interactions} |
| 796 |
> |
\section{\label{section:divisionOfLabor}Separation into Internal and |
| 797 |
> |
Cross interactions} |
| 798 |
|
|
| 799 |
< |
The energy function for a system composed of $N$ molecules is |
| 800 |
< |
traditionally written |
| 799 |
> |
The classical potential energy function for a system composed of $N$ |
| 800 |
> |
molecules is traditionally written |
| 801 |
|
\begin{equation} |
| 802 |
|
V = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
| 803 |
|
+ \sum^{N-1}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}}, |
| 804 |
|
\label{eq:totalPotential} |
| 805 |
|
\end{equation} |
| 806 |
< |
where $V^{IJ}_{\text{Cross}}$ contains all intermolecular interactions |
| 807 |
< |
between molecules $I$ and $J$, and $V^{I}_{\text{Internal}}$ is the |
| 808 |
< |
internal potential of molecule $I$: |
| 809 |
< |
\begin{align*} |
| 810 |
< |
V^{I}_{\text{Internal}} = & |
| 811 |
< |
\sum_{r_{ij} \in I} V_{\text{bond}}(r_{ij}) |
| 807 |
< |
+ \sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
| 808 |
< |
+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{ijkl}) |
| 809 |
< |
+ \sum_{\omega_{ijkl} \in I} V_{\text{inversion}}(\omega_{ijkl}) \\ |
| 810 |
< |
& + \sum_{i \in I} \sum_{(j>i+4) \in I} |
| 811 |
< |
\biggl[ V_{\text{dispersion}}(r_{ij}) + V_{\text{electrostatic}} |
| 812 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 813 |
< |
\biggr]. |
| 814 |
< |
\label{eq:internalPotential} |
| 815 |
< |
\end{align*} |
| 816 |
< |
Here $V_{\text{bond}}, V_{\text{bend}}, |
| 817 |
< |
V_{\text{torsion}},\mathrm{~and~} V_{\text{inversion}}$ represent the |
| 818 |
< |
bond, bend, torsion, and inversion potentials for all |
| 819 |
< |
topologically-connected sets of atoms within the molecule. Bonds are |
| 820 |
< |
the primary way of specifying how the atoms are connected together to |
| 821 |
< |
form the molecule (i.e. they define the molecular topology). The |
| 822 |
< |
other short-range interactions may be specified explicitly in the |
| 823 |
< |
molecule definition, or they may be deduced from bonding information. |
| 824 |
< |
For example, bends can be implicitly deduced from two bonds which |
| 825 |
< |
share an atom. Torsions can be deduced from two bends that share a |
| 826 |
< |
bond. Inversion potentials are utilized primarily to enforce |
| 827 |
< |
planarity around $sp^2$-hybridized sites, and these are specified with |
| 828 |
< |
central atoms and satellites (or an atom with bonds to exactly three |
| 829 |
< |
satellites). The pairwise portions of the non-bonded interactions are |
| 830 |
< |
usually excluded for atom pairs that are involved in the same bond, |
| 831 |
< |
bend, or torsion. All other atom pairs within a molecule are subject |
| 832 |
< |
to non-bonded pair potentials. |
| 806 |
> |
where $V^{I}_{\text{Internal}}$ contains all of the terms internal to |
| 807 |
> |
molecule $I$ (e.g. bonding, bending, torsional, and inversion terms) |
| 808 |
> |
and $V^{IJ}_{\text{Cross}}$ contains all intermolecular interactions |
| 809 |
> |
between molecules $I$ and $J$. For large molecules, the internal |
| 810 |
> |
potential may also include some non-bonded terms like electrostatic or |
| 811 |
> |
van der Waals interactions. |
| 812 |
|
|
| 813 |
|
The types of atoms being simulated, as well as the specific functional |
| 814 |
|
forms and parameters of the intra-molecular functions and the |
| 818 |
|
|
| 819 |
|
\section{\label{section:frcFile}Force Field Files} |
| 820 |
|
|
| 821 |
< |
Force field files have a number of ``Blocks'' to demarkate different |
| 821 |
> |
Force field files have a number of ``Blocks'' to delineate different |
| 822 |
|
types of information. The blocks contain AtomType data, which provide |
| 823 |
|
properties belonging to a single AtomType, as well as interaction |
| 824 |
|
information which provides information about bonded or non-bonded |
| 825 |
|
interactions that cannot be deduced from AtomType information alone. |
| 826 |
|
A simple example of a forceField file is shown in scheme |
| 827 |
< |
\ref{sch:frcExample}. |
| 827 |
> |
\ref{sch:frcExample}. |
| 828 |
|
|
| 829 |
|
\begin{lstlisting}[float,caption={[An example of a complete OpenMD |
| 830 |
|
force field file for straight-chain united-atom alkanes.] An example |
| 831 |
|
showing a complete OpenMD force field for straight-chain united-atom |
| 832 |
|
alkanes.}, label={sch:frcExample}] |
| 833 |
|
begin Options |
| 834 |
< |
Name = "alkane" end |
| 835 |
< |
Options |
| 834 |
> |
Name = "alkane" |
| 835 |
> |
end Options |
| 836 |
|
|
| 837 |
|
begin BaseAtomTypes |
| 838 |
|
//name mass |
| 875 |
|
end TorsionTypes |
| 876 |
|
\end{lstlisting} |
| 877 |
|
|
| 878 |
< |
\subsection{\label{section:ffOptions}The Options block} |
| 878 |
> |
\section{\label{section:ffOptions}The Options block} |
| 879 |
|
|
| 880 |
|
The Options block defines properties governing how the force field |
| 881 |
|
interactions are carried out. This block is delineated with the text |
| 893 |
|
Name = "alkane" // any string |
| 894 |
|
vdWtype = "Lennard-Jones" |
| 895 |
|
DistanceMixingRule = "arithmetic" // can also be "geometric" or "cubic" |
| 896 |
< |
DistanceType = "sigma" // can also be Rmin |
| 896 |
> |
DistanceType = "sigma" // can also be "Rmin" |
| 897 |
|
EnergyMixingRule = "geometric" // can also be "arithmetic" or "hhg" |
| 898 |
|
EnergyUnitScaling = 1.0 |
| 899 |
|
MetallicEnergyUnitScaling = 1.0 |
| 912 |
|
end Options |
| 913 |
|
\end{lstlisting} |
| 914 |
|
|
| 915 |
< |
\subsection{\label{section:ffBase}The BaseAtomTypes block} |
| 915 |
> |
\section{\label{section:ffBase}The BaseAtomTypes block} |
| 916 |
|
|
| 917 |
|
An AtomType the primary data structure that OpenMD uses to store |
| 918 |
|
static data about an atom. Things that belong to AtomType are |
| 938 |
|
ability to print out the names of the base atom types for displaying |
| 939 |
|
simulations in Jmol or VMD. |
| 940 |
|
|
| 941 |
< |
\begin{lstlisting}[caption={[A simple example of a BaseAtomType |
| 942 |
< |
block.] A simple example of a BaseAtomType block.}, |
| 941 |
> |
\begin{lstlisting}[caption={[A simple example of a BaseAtomTypes |
| 942 |
> |
block.] A simple example of a BaseAtomTypes block.}, |
| 943 |
|
label={sch:baseAtomTypesBlock}] |
| 944 |
|
begin BaseAtomTypes |
| 945 |
|
//Name mass (amu) |
| 960 |
|
end BaseAtomTypes |
| 961 |
|
\end{lstlisting} |
| 962 |
|
|
| 963 |
< |
\subsection{\label{section:ffAtom}The AtomTypes block} |
| 963 |
> |
\section{\label{section:ffAtom}The AtomTypes block} |
| 964 |
|
|
| 965 |
|
AtomTypes inherit most properties from BaseAtomTypes, but can override |
| 966 |
|
their lower-level properties as well. Scheme \ref{sch:atomTypesBlock} |
| 968 |
|
from the oxygen base type. |
| 969 |
|
|
| 970 |
|
\begin{lstlisting}[caption={[An example of a AtomTypes block.] A |
| 971 |
< |
simple example of an AtomType block which |
| 971 |
> |
simple example of an AtomTypes block which |
| 972 |
|
shows how multiple types can inherit from the same base type.}, |
| 973 |
|
label={sch:atomTypesBlock}] |
| 974 |
|
begin AtomTypes |
| 994 |
|
end AtomTypes |
| 995 |
|
\end{lstlisting} |
| 996 |
|
|
| 997 |
< |
\subsection{\label{section:ffDirectionalAtom}The DirectionalAtomTypes |
| 997 |
> |
\section{\label{section:ffDirectionalAtom}The DirectionalAtomTypes |
| 998 |
|
block} |
| 999 |
|
DirectionalAtoms have orientational degrees of freedom as well as |
| 1000 |
< |
translation, so they have moment of inertia tensors. |
| 1000 |
> |
translation, so moving these atoms requires information about the |
| 1001 |
> |
moments of inertias in the same way that translational motion requires |
| 1002 |
> |
mass. For DirectionalAtoms, OpenMD treats the mass distribution with |
| 1003 |
> |
higher priority than electrostatic distributions; the moment of |
| 1004 |
> |
inertia tensor, $\overleftrightarrow{\mathsf I}$, should be |
| 1005 |
> |
diagonalized to obtain body-fixed axes, and the three diagonal moments |
| 1006 |
> |
should correspond to rotational motion \textit{around} each of these |
| 1007 |
> |
body-fixed axes. Charge distributions may then result in dipole |
| 1008 |
> |
vectors that are oriented along a linear combination of the body-axes, |
| 1009 |
> |
and in quadrupole tensors that are not necessarily diagonal in the |
| 1010 |
> |
body frame. |
| 1011 |
|
|
| 1012 |
|
\begin{lstlisting}[caption={[An example of a DirectionalAtomTypes block.] A |
| 1013 |
|
simple example of a DirectionalAtomTypes block.}, |
| 1015 |
|
begin DirectionalAtomTypes |
| 1016 |
|
//Name I_xx I_yy I_zz (All moments in (amu*Ang^2) |
| 1017 |
|
SSD 1.7696 0.6145 1.1550 |
| 1029 |
– |
SSD_E 1.7696 0.6145 1.1550 |
| 1018 |
|
GBC6H6 88.781 88.781 177.561 |
| 1019 |
|
GBCH3OH 4.056 20.258 20.999 |
| 1020 |
|
GBH2O 1.777 0.581 1.196 |
| 1021 |
+ |
CO2 43.06 43.06 0.0 // single-site model for CO2 |
| 1022 |
|
end DirectionalAtomTypes |
| 1023 |
|
|
| 1024 |
|
\end{lstlisting} |
| 1025 |
|
|
| 1026 |
+ |
For a DirectionalAtom that represents a linear object, it is |
| 1027 |
+ |
appropriate for one of the moments of inertia to be zero. In this |
| 1028 |
+ |
case, OpenMD identifies that DirectionalAtom as having only 5 degrees |
| 1029 |
+ |
of freedom (three translations and two rotations), and will alter |
| 1030 |
+ |
calculation of temperatures to reflect this. |
| 1031 |
|
|
| 1032 |
< |
\subsection{\label{section::ffAtomProperties}AtomType properties} |
| 1033 |
< |
\subsubsection{\label{section:ffLJ}The LennardJonesAtomTypes block} |
| 1034 |
< |
The most basic interatomic interaction implemented in {\sc OpenMD} is |
| 1035 |
< |
the Lennard-Jones potential, which mimics the van der Waals |
| 1036 |
< |
interaction at long distances and uses an empirical repulsion at short |
| 1037 |
< |
distances. The Lennard-Jones potential is given by: |
| 1032 |
> |
\section{\label{section::ffAtomProperties}AtomType properties} |
| 1033 |
> |
\subsection{\label{section:ffLJ}The LennardJonesAtomTypes block} |
| 1034 |
> |
One of the most basic interatomic interactions implemented in {\sc |
| 1035 |
> |
OpenMD} is the Lennard-Jones potential, which mimics the van der |
| 1036 |
> |
Waals interaction at long distances and uses an empirical repulsion at |
| 1037 |
> |
short distances. The Lennard-Jones potential is given by: |
| 1038 |
|
\begin{equation} |
| 1039 |
|
V_{\text{LJ}}(r_{ij}) = |
| 1040 |
|
4\epsilon_{ij} \biggl[ |
| 1049 |
|
|
| 1050 |
|
Interactions between dissimilar particles requires the generation of |
| 1051 |
|
cross term parameters for $\sigma$ and $\epsilon$. These parameters |
| 1052 |
< |
are determined using the Lorentz-Berthelot mixing |
| 1052 |
> |
are usually determined using the Lorentz-Berthelot mixing |
| 1053 |
|
rules:\cite{Allen87} |
| 1054 |
|
\begin{equation} |
| 1055 |
|
\sigma_{ij} = \frac{1}{2}[\sigma_{ii} + \sigma_{jj}], |
| 1061 |
|
\label{eq:epsilonMix} |
| 1062 |
|
\end{equation} |
| 1063 |
|
|
| 1064 |
< |
\subsubsection{\label{section:ffCharge}The ChargeAtomTypes block} |
| 1065 |
< |
\subsubsection{\label{section:ffMultipole}The MultipoleAtomTypes block} |
| 1066 |
< |
The dipole-dipole potential has the following form: |
| 1064 |
> |
The values of $\sigma_{ii}$ and $\epsilon_{ii}$ are properties of atom |
| 1065 |
> |
type $i$, and must be specified in a section of the force field file |
| 1066 |
> |
called the {\tt LennardJonesAtomTypes} block (see listing |
| 1067 |
> |
\ref{sch:LJatomTypesBlock}). Separate Lennard-Jones interactions |
| 1068 |
> |
which are not determined by the mixing rules can also be specified in |
| 1069 |
> |
the {\tt NonbondedInteractionTypes} block (see section |
| 1070 |
> |
\ref{section:ffNBinteraction}). |
| 1071 |
> |
|
| 1072 |
> |
\begin{lstlisting}[caption={[An example of a LennardJonesAtomTypes block.] A |
| 1073 |
> |
simple example of a LennardJonesAtomTypee block. Units for |
| 1074 |
> |
$\epsilon$ are kcal / mol and for $\sigma$ are \AA\ .}, |
| 1075 |
> |
label={sch:LJatomTypesBlock}] |
| 1076 |
> |
begin LennardJonesAtomTypes |
| 1077 |
> |
//Name epsilon sigma |
| 1078 |
> |
O_TIP4P 0.1550 3.15365 |
| 1079 |
> |
O_TIP4P-Ew 0.16275 3.16435 |
| 1080 |
> |
O_TIP5P 0.16 3.12 |
| 1081 |
> |
O_TIP5P-E 0.178 3.097 |
| 1082 |
> |
O_SPCE 0.15532 3.16549 |
| 1083 |
> |
O_SPC 0.15532 3.16549 |
| 1084 |
> |
CH4 0.279 3.73 |
| 1085 |
> |
CH3 0.185 3.75 |
| 1086 |
> |
CH2 0.0866 3.95 |
| 1087 |
> |
CH 0.0189 4.68 |
| 1088 |
> |
end LennardJonesAtomTypes |
| 1089 |
> |
\end{lstlisting} |
| 1090 |
> |
|
| 1091 |
> |
\subsection{\label{section:ffCharge}The ChargeAtomTypes block} |
| 1092 |
> |
|
| 1093 |
> |
In molecular simulations, proper accumulation of the electrostatic |
| 1094 |
> |
interactions is essential and is one of the most |
| 1095 |
> |
computationally-demanding tasks. Most common molecular mechanics |
| 1096 |
> |
force fields represent atomic sites with full or partial charges |
| 1097 |
> |
protected by Lennard-Jones (short range) interactions. Partial charge |
| 1098 |
> |
values, $q_i$ are empirical representations of the distribution of |
| 1099 |
> |
electronic charge in a molecule. This means that nearly every pair |
| 1100 |
> |
interaction involves a calculation of charge-charge forces. Coupled |
| 1101 |
> |
with relatively long-ranged $r^{-1}$ decay, the monopole interactions |
| 1102 |
> |
quickly become the most expensive part of molecular simulations. The |
| 1103 |
> |
interactions between point charges can be handled via a number of |
| 1104 |
> |
different algorithms, but Coulomb's law is the fundamental physical |
| 1105 |
> |
principle governing these interactions, |
| 1106 |
> |
\begin{equation} |
| 1107 |
> |
V_{\text{charge}}(r_{ij}) = \sum_{ij}\frac{q_iq_je^2}{4 \pi \epsilon_0 |
| 1108 |
> |
r_{ij}}, |
| 1109 |
> |
\end{equation} |
| 1110 |
> |
where $q$ represents the charge on particle $i$ or $j$, and $e$ is the |
| 1111 |
> |
charge of an electron in Coulombs. $\epsilon_0$ is the permittivity |
| 1112 |
> |
of free space. |
| 1113 |
> |
|
| 1114 |
> |
\begin{lstlisting}[caption={[An example of a ChargeAtomTypes block.] A |
| 1115 |
> |
simple example of a ChargeAtomTypes block. Units for |
| 1116 |
> |
charge are in multiples of electron charge.}, |
| 1117 |
> |
label={sch:ChargeAtomTypesBlock}] |
| 1118 |
> |
begin ChargeAtomTypes |
| 1119 |
> |
// Name charge |
| 1120 |
> |
O_TIP3P -0.834 |
| 1121 |
> |
O_SPCE -0.8476 |
| 1122 |
> |
H_TIP3P 0.417 |
| 1123 |
> |
H_TIP4P 0.520 |
| 1124 |
> |
H_SPCE 0.4238 |
| 1125 |
> |
EP_TIP4P -1.040 |
| 1126 |
> |
Na+ 1.0 |
| 1127 |
> |
Cl- -1.0 |
| 1128 |
> |
end ChargeAtomTypes |
| 1129 |
> |
\end{lstlisting} |
| 1130 |
> |
|
| 1131 |
> |
\subsection{\label{section:ffMultipole}The MultipoleAtomTypes |
| 1132 |
> |
block} |
| 1133 |
> |
For complex charge distributions that are centered on single sites, it |
| 1134 |
> |
is convenient to write the total electrostatic potential in terms of |
| 1135 |
> |
multipole moments, |
| 1136 |
|
\begin{equation} |
| 1137 |
+ |
U_{\bf{ab}}(r)=\hat{M}_{\bf a} \hat{M}_{\bf b} \frac{1}{r} \label{kernel}. |
| 1138 |
+ |
\end{equation} |
| 1139 |
+ |
where the multipole operator on site $\bf a$, |
| 1140 |
+ |
\begin{equation} |
| 1141 |
+ |
\hat{M}_{\bf a} = C_{\bf a} - D_{{\bf a}\alpha} \frac{\partial}{\partial r_{\alpha}} |
| 1142 |
+ |
+ Q_{{\bf a}\alpha\beta} |
| 1143 |
+ |
\frac{\partial^2}{\partial r_{\alpha} \partial r_{\beta}} + \dots |
| 1144 |
+ |
\end{equation} |
| 1145 |
+ |
Here, the point charge, dipole, and quadrupole for site $\bf a$ are |
| 1146 |
+ |
given by $C_{\bf a}$, $D_{{\bf a}\alpha}$, and $Q_{{\bf |
| 1147 |
+ |
a}\alpha\beta}$, respectively. These are the {\it primitive} |
| 1148 |
+ |
multipoles. If the site is representing a distribution of charges, |
| 1149 |
+ |
these can be expressed as, |
| 1150 |
+ |
\begin{align} |
| 1151 |
+ |
C_{\bf a} =&\sum_{k \, \text{in \bf a}} q_k , \label{eq:charge} \\ |
| 1152 |
+ |
D_{{\bf a}\alpha} =&\sum_{k \, \text{in \bf a}} q_k r_{k\alpha}, \label{eq:dipole}\\ |
| 1153 |
+ |
Q_{{\bf a}\alpha\beta} =& \frac{1}{2} \sum_{k \, \text{in \bf a}} q_k |
| 1154 |
+ |
r_{k\alpha} r_{k\beta} . \label{eq:quadrupole} |
| 1155 |
+ |
\end{align} |
| 1156 |
+ |
Note that the definition of the primitive quadrupole here differs from |
| 1157 |
+ |
the standard traceless form, and contains an additional Taylor-series |
| 1158 |
+ |
based factor of $1/2$. |
| 1159 |
+ |
|
| 1160 |
+ |
The details of the multipolar interactions will be given later, but |
| 1161 |
+ |
many readers are familiar with the dipole-dipole potential: |
| 1162 |
+ |
\begin{equation} |
| 1163 |
|
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 1164 |
< |
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
| 1164 |
> |
\boldsymbol{\Omega}_{j}) = \frac{|{\bf D}_i||{\bf D}_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
| 1165 |
|
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
| 1166 |
|
- |
| 1167 |
|
3(\boldsymbol{\hat{u}}_i \cdot \hat{\mathbf{r}}_{ij}) % |
| 1171 |
|
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 1172 |
|
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
| 1173 |
|
are the orientational degrees of freedom for atoms $i$ and $j$ |
| 1174 |
< |
respectively. The magnitude of the dipole moment of atom $i$ is |
| 1175 |
< |
$|\mu_i|$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
| 1174 |
> |
respectively. The magnitude of the dipole moment of atom $i$ is $|{\bf |
| 1175 |
> |
D}_i|$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
| 1176 |
|
vector of $\boldsymbol{\Omega}_i$, and $\boldsymbol{\hat{r}}_{ij}$ is |
| 1177 |
|
the unit vector pointing along $\mathbf{r}_{ij}$ |
| 1178 |
|
($\boldsymbol{\hat{r}}_{ij}=\mathbf{r}_{ij}/|\mathbf{r}_{ij}|$). |
| 1179 |
|
|
| 1091 |
– |
\subsubsection{\label{section:ffGB}The FluctuatingChargeAtomTypes block} |
| 1092 |
– |
\subsubsection{\label{section:ffPol}The PolarizableAtomTypes block} |
| 1093 |
– |
\subsubsection{\label{section:ffGB}The GayBerneAtomTypes block} |
| 1094 |
– |
\subsubsection{\label{section:ffSticky}The StickyAtomTypes block} |
| 1180 |
|
|
| 1181 |
< |
One of the solvents used by {\sc OpenMD} is the extended Soft Sticky |
| 1182 |
< |
Dipole (SSD/E) water model.\cite{fennell04} The original SSD was |
| 1183 |
< |
developed by Ichiye \emph{et al.}\cite{liu96:new_model} as a modified |
| 1184 |
< |
form of the hard-sphere water model proposed by Bratko, Blum, and |
| 1181 |
> |
\begin{lstlisting}[caption={[An example of a MultipoleAtomTypes block.] A |
| 1182 |
> |
simple example of a MultipoleAtomTypes block. Dipoles are given in |
| 1183 |
> |
units of Debyes, and Quadrupole moments are given in units of Debye |
| 1184 |
> |
\AA~(or $10^{-26} \mathrm{~esu~cm}^2$)}, |
| 1185 |
> |
label={sch:MultipoleAtomTypesBlock}] |
| 1186 |
> |
begin MultipoleAtomTypes |
| 1187 |
> |
// Euler angles are given in zxz convention in units of degrees. |
| 1188 |
> |
// |
| 1189 |
> |
// point dipoles: |
| 1190 |
> |
// name d phi theta psi dipole_moment |
| 1191 |
> |
DIP d 0.0 0.0 0.0 1.91 // dipole points along z-body axis |
| 1192 |
> |
// |
| 1193 |
> |
// point quadrupoles: |
| 1194 |
> |
// name q phi theta psi Qxx Qyy Qzz |
| 1195 |
> |
CO2 q 0.0 0.0 0.0 0.0 0.0 -0.430592 //quadrupole tensor has zz element |
| 1196 |
> |
// |
| 1197 |
> |
// Atoms with both dipole and quadrupole moments: |
| 1198 |
> |
// name dq phi theta psi dipole_moment Qxx Qyy Qzz |
| 1199 |
> |
SSD dq 0.0 0.0 0.0 2.35 -1.682 1.762 -0.08 |
| 1200 |
> |
end MultipoleAtomTypes |
| 1201 |
> |
\end{lstlisting} |
| 1202 |
> |
|
| 1203 |
> |
Specifying a MultipoleAtomType requires declaring how the |
| 1204 |
> |
electrostatic frame for the site is rotated relative to the body-fixed |
| 1205 |
> |
axes for that atom. The Euler angles $(\phi, \theta, \psi)$ for this |
| 1206 |
> |
rotation must be given, and then the dipole, quadrupole, or all of |
| 1207 |
> |
these moments are specified in the electrostatic frame. In OpenMD, |
| 1208 |
> |
the Euler angles are specified in the $zxz$ convention and are entered |
| 1209 |
> |
in units of degrees. Dipole moments are entered in units of Debye, |
| 1210 |
> |
and Quadrupole moments in units of Debye \AA. |
| 1211 |
> |
|
| 1212 |
> |
\subsection{\label{section:ffGB}The FluctuatingChargeAtomTypes block} |
| 1213 |
> |
%\subsubsection{\label{section:ffPol}The PolarizableAtomTypes block} |
| 1214 |
> |
|
| 1215 |
> |
\subsection{\label{section:ffGB}The GayBerneAtomTypes block} |
| 1216 |
> |
|
| 1217 |
> |
The Gay-Berne potential has been widely used in the liquid crystal |
| 1218 |
> |
community to describe anisotropic phase |
| 1219 |
> |
behavior.~\cite{Gay:1981yu,Berne:1972pb,Kushick:1976xy,Luckhurst:1990fy,Cleaver:1996rt} |
| 1220 |
> |
The form of the Gay-Berne potential implemented in OpenMD was |
| 1221 |
> |
generalized by Cleaver {\it et al.} and is appropriate for dissimilar |
| 1222 |
> |
uniaxial ellipsoids.\cite{Cleaver:1996rt} The potential is constructed |
| 1223 |
> |
in the familiar form of the Lennard-Jones function using |
| 1224 |
> |
orientation-dependent $\sigma$ and $\epsilon$ parameters, |
| 1225 |
> |
\begin{equation*} |
| 1226 |
> |
V_{ij}({{\bf \hat u}_i}, {{\bf \hat u}_j}, {{\bf \hat |
| 1227 |
> |
r}_{ij}}) = 4\epsilon ({{\bf \hat u}_i}, {{\bf \hat u}_j}, |
| 1228 |
> |
{{\bf \hat r}_{ij}})\left[\left(\frac{\sigma_0}{r_{ij}-\sigma({{\bf \hat u |
| 1229 |
> |
}_i}, |
| 1230 |
> |
{{\bf \hat u}_j}, {{\bf \hat r}_{ij}})+\sigma_0}\right)^{12} |
| 1231 |
> |
-\left(\frac{\sigma_0}{r_{ij}-\sigma({{\bf \hat u}_i}, {{\bf \hat u}_j}, |
| 1232 |
> |
{{\bf \hat r}_{ij}})+\sigma_0}\right)^6\right] |
| 1233 |
> |
\label{eq:gb} |
| 1234 |
> |
\end{equation*} |
| 1235 |
> |
|
| 1236 |
> |
The range $(\sigma({\bf \hat{u}}_{i},{\bf \hat{u}}_{j},{\bf |
| 1237 |
> |
\hat{r}}_{ij}))$, and strength $(\epsilon({\bf \hat{u}}_{i},{\bf |
| 1238 |
> |
\hat{u}}_{j},{\bf \hat{r}}_{ij}))$ parameters |
| 1239 |
> |
are dependent on the relative orientations of the two ellipsoids (${\bf |
| 1240 |
> |
\hat{u}}_{i},{\bf \hat{u}}_{j}$) as well as the direction of the |
| 1241 |
> |
inter-ellipsoid separation (${\bf \hat{r}}_{ij}$). The shape and |
| 1242 |
> |
attractiveness of each ellipsoid is governed by a relatively small set |
| 1243 |
> |
of parameters: |
| 1244 |
> |
\begin{itemize} |
| 1245 |
> |
\item $d$: range parameter for the side-by-side (S) and cross (X) configurations |
| 1246 |
> |
\item $l$: range parameter for the end-to-end (E) configuration |
| 1247 |
> |
\item $\epsilon_X$: well-depth parameter for the cross (X) configuration |
| 1248 |
> |
\item $\epsilon_S$: well-depth parameter for the side-by-side (S) configuration |
| 1249 |
> |
\item $\epsilon_E$: well depth parameter for the end-to-end (E) configuration |
| 1250 |
> |
\item $dw$: The ``softness'' of the potential |
| 1251 |
> |
\end{itemize} |
| 1252 |
> |
Additionally, there are two universal paramters to govern the overall |
| 1253 |
> |
importance of the purely orientational ($\nu$) and the mixed |
| 1254 |
> |
orientational / translational ($\mu$) parts of strength of the |
| 1255 |
> |
interactions. These parameters have default or ``canonical'' values, |
| 1256 |
> |
but may be changed as a force field option: |
| 1257 |
> |
\begin{itemize} |
| 1258 |
> |
\item $\nu$: purely orientational part : defaults to 1 |
| 1259 |
> |
\item $\mu$: mixed orientational / translational part : defaults to |
| 1260 |
> |
2 |
| 1261 |
> |
\end{itemize} |
| 1262 |
> |
Further details of the potential are given |
| 1263 |
> |
elsewhere,\cite{Luckhurst:1990fy,Golubkov06,SunX._jp0762020} and an |
| 1264 |
> |
excellent overview of the computational methods that can be used to |
| 1265 |
> |
efficiently compute forces and torques for this potential can be found |
| 1266 |
> |
in Ref. \citealp{Golubkov06} |
| 1267 |
> |
|
| 1268 |
> |
\begin{lstlisting}[caption={[An example of a GayBerneAtomTypes block.] A |
| 1269 |
> |
simple example of a GayBerneAtomTypes block. Distances ($d$ and $l$) |
| 1270 |
> |
are given in \AA\ and energies ($\epsilon_X, \epsilon_S, \epsilon_E$) |
| 1271 |
> |
are in units of kcal/mol. $dw$ is unitless.}, |
| 1272 |
> |
label={sch:GayBerneAtomTypes}] |
| 1273 |
> |
begin GayBerneAtomTypes |
| 1274 |
> |
//Name d l eps_X eps_S eps_E dw |
| 1275 |
> |
GBlinear 2.8104 9.993 0.774729 0.774729 0.116839 1.0 |
| 1276 |
> |
GBC6H6 4.65 2.03 0.540 0.540 1.9818 0.6 |
| 1277 |
> |
GBCH3OH 2.55 3.18 0.542 0.542 0.55826 1.0 |
| 1278 |
> |
end GayBerneAtomTypes |
| 1279 |
> |
\end{lstlisting} |
| 1280 |
> |
|
| 1281 |
> |
\subsection{\label{section:ffSticky}The StickyAtomTypes block} |
| 1282 |
> |
|
| 1283 |
> |
One of the solvents that can be simulated by {\sc OpenMD} is the |
| 1284 |
> |
extended Soft Sticky Dipole (SSD/E) water model.\cite{fennell04} The |
| 1285 |
> |
original SSD was developed by Ichiye \emph{et |
| 1286 |
> |
al.}\cite{liu96:new_model} as a modified form of the hard-sphere |
| 1287 |
> |
water model proposed by Bratko, Blum, and |
| 1288 |
|
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
| 1289 |
|
with a Lennard-Jones core and a sticky potential that directs the |
| 1290 |
|
particles to assume the proper hydrogen bond orientation in the first |
| 1367 |
|
|
| 1368 |
|
Recent constant pressure simulations revealed issues in the original |
| 1369 |
|
SSD model that led to lower than expected densities at all target |
| 1370 |
< |
pressures.\cite{Ichiye03,fennell04} The default model in {\sc OpenMD} |
| 1371 |
< |
is therefore SSD/E, a density corrected derivative of SSD that |
| 1372 |
< |
exhibits improved liquid structure and transport behavior. If the use |
| 1373 |
< |
of a reaction field long-range interaction correction is desired, it |
| 1374 |
< |
is recommended that the parameters be modified to those of the SSD/RF |
| 1375 |
< |
model (an SSD variant parameterized for reaction field). These solvent |
| 1188 |
< |
parameters are listed and can be easily modified in the {\sc duff} |
| 1189 |
< |
force field file ({\tt DUFF.frc}). A table of the parameter values |
| 1190 |
< |
and the drawbacks and benefits of the different density corrected SSD |
| 1191 |
< |
models can be found in reference~\cite{fennell04}. |
| 1370 |
> |
pressures,\cite{Ichiye03,fennell04} so variants on the sticky |
| 1371 |
> |
potential can be specified by using one of a number of substitute atom |
| 1372 |
> |
types (see listing \ref{sch:StickyAtomTypes}). A table of the |
| 1373 |
> |
parameter values and the drawbacks and benefits of the different |
| 1374 |
> |
density corrected SSD models can be found in |
| 1375 |
> |
reference~\citealp{fennell04}. |
| 1376 |
|
|
| 1377 |
< |
\subsection{\label{section::ffMetals}Metallic Atom Types} |
| 1378 |
< |
\subsubsection{\label{section:ffEAM}The EAMAtomTypes block} |
| 1379 |
< |
{\sc OpenMD} implements a potential that describes bonding in |
| 1380 |
< |
transition metal |
| 1381 |
< |
systems.~\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} This |
| 1382 |
< |
potential has an attractive interaction which models ``Embedding'' a |
| 1383 |
< |
positively charged pseudo-atom core in the electron density due to the |
| 1384 |
< |
free valance ``sea'' of electrons created by the surrounding atoms in |
| 1385 |
< |
the system. A pairwise part of the potential (which is primarily |
| 1386 |
< |
repulsive) describes the interaction of the positively charged metal |
| 1387 |
< |
core ions with one another. The Embedded Atom Method ({\sc |
| 1388 |
< |
eam})~\cite{Daw84,FBD86,johnson89,Lu97} has been widely adopted in the |
| 1389 |
< |
materials science community and has been included in {\sc OpenMD}. A |
| 1206 |
< |
good review of {\sc eam} and other formulations of metallic potentials |
| 1207 |
< |
was given by Voter.\cite{Voter:95} |
| 1377 |
> |
\begin{lstlisting}[caption={[An example of a StickyAtomTypes block.] A |
| 1378 |
> |
simple example of a StickyAtomTypes block. Distances ($r_l$, $r_u$, |
| 1379 |
> |
$r_{l}'$ and $r_{u}'$) are given in \AA\ and energies ($v_0, v_{0}'$) |
| 1380 |
> |
are in units of kcal/mol. $w_0$ is unitless.}, |
| 1381 |
> |
label={sch:StickyAtomTypes}] |
| 1382 |
> |
begin StickyAtomTypes |
| 1383 |
> |
//name w0 v0 (kcal/mol) v0p rl (Ang) ru rlp rup |
| 1384 |
> |
SSD_E 0.07715 3.90 3.90 2.40 3.80 2.75 3.35 |
| 1385 |
> |
SSD_RF 0.07715 3.90 3.90 2.40 3.80 2.75 3.35 |
| 1386 |
> |
SSD 0.07715 3.7284 3.7284 2.75 3.35 2.75 4.0 |
| 1387 |
> |
SSD1 0.07715 3.6613 3.6613 2.75 3.35 2.75 4.0 |
| 1388 |
> |
end StickyAtomTypes |
| 1389 |
> |
\end{lstlisting} |
| 1390 |
|
|
| 1391 |
< |
The {\sc eam} potential has the form: |
| 1391 |
> |
\section{\label{section::ffMetals}Metallic Atom Types} |
| 1392 |
> |
|
| 1393 |
> |
{\sc OpenMD} implements a number of related potentials that describe |
| 1394 |
> |
bonding in transition metals. These potentials have an attractive |
| 1395 |
> |
interaction which models ``Embedding'' a positively charged |
| 1396 |
> |
pseudo-atom core in the electron density due to the free valance |
| 1397 |
> |
``sea'' of electrons created by the surrounding atoms in the system. |
| 1398 |
> |
A pairwise part of the potential (which is primarily repulsive) |
| 1399 |
> |
describes the interaction of the positively charged metal core ions |
| 1400 |
> |
with one another. These potentials have the form: |
| 1401 |
|
\begin{equation} |
| 1402 |
|
V = \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
| 1403 |
|
\phi_{ij}({\bf r}_{ij}) |
| 1414 |
|
transition metal potentials require two loops through the atom pairs |
| 1415 |
|
to compute the inter-atomic forces. |
| 1416 |
|
|
| 1417 |
< |
The pairwise portion of the potential, $\phi_{ij}$, is a primarily |
| 1418 |
< |
repulsive interaction between atoms $i$ and $j$. In the original |
| 1228 |
< |
formulation of {\sc eam}\cite{Daw84}, $\phi_{ij}$ was an entirely |
| 1229 |
< |
repulsive term; however later refinements to {\sc eam} allowed for |
| 1230 |
< |
more general forms for $\phi$.\cite{Daw89} The effective cutoff |
| 1231 |
< |
distance, $r_{{\text cut}}$ is the distance at which the values of |
| 1232 |
< |
$f(r)$ and $\phi(r)$ drop to zero for all atoms present in the |
| 1233 |
< |
simulation. In practice, this distance is fairly small, limiting the |
| 1234 |
< |
summations in the {\sc eam} equation to the few dozen atoms |
| 1235 |
< |
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
| 1236 |
< |
interactions. |
| 1417 |
> |
The pairwise portion of the potential, $\phi_{ij}$, is usually a |
| 1418 |
> |
repulsive interaction between atoms $i$ and $j$. |
| 1419 |
|
|
| 1420 |
< |
In computing forces for alloys, mixing rules as outlined by |
| 1421 |
< |
Johnson~\cite{johnson89} are used to compute the heterogenous pair |
| 1422 |
< |
potential, |
| 1420 |
> |
\subsection{\label{section:ffEAM}The EAMAtomTypes block} |
| 1421 |
> |
The Embedded Atom Method ({\sc eam}) is one of the most widely-used |
| 1422 |
> |
potentials for transition |
| 1423 |
> |
metals.~\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02,Daw84,FBD86,johnson89,Lu97} |
| 1424 |
> |
It has been widely adopted in the materials science community and a |
| 1425 |
> |
good review of {\sc eam} and other formulations of metallic potentials |
| 1426 |
> |
was given by Voter.\cite{Voter:95} |
| 1427 |
> |
|
| 1428 |
> |
In the original formulation of {\sc eam}\cite{Daw84}, the pair |
| 1429 |
> |
potential, $\phi_{ij}$ was an entirely repulsive term; however later |
| 1430 |
> |
refinements to {\sc eam} allowed for more general forms for |
| 1431 |
> |
$\phi$.\cite{Daw89} The effective cutoff distance, $r_{{\text cut}}$ |
| 1432 |
> |
is the distance at which the values of $f(r)$ and $\phi(r)$ drop to |
| 1433 |
> |
zero for all atoms present in the simulation. In practice, this |
| 1434 |
> |
distance is fairly small, limiting the summations in the {\sc eam} |
| 1435 |
> |
equation to the few dozen atoms surrounding atom $i$ for both the |
| 1436 |
> |
density $\rho$ and pairwise $\phi$ interactions. |
| 1437 |
> |
|
| 1438 |
> |
In computing forces for alloys, OpenMD uses mixing rules outlined by |
| 1439 |
> |
Johnson~\cite{johnson89} to compute the heterogenous pair potential, |
| 1440 |
|
\begin{equation} |
| 1441 |
|
\label{eq:johnson} |
| 1442 |
|
\phi_{ab}(r)=\frac{1}{2}\left( |
| 1468 |
|
$\mbox{kcal mol}^{-1}$ as in the rest of the {\sc OpenMD} force field |
| 1469 |
|
files. |
| 1470 |
|
|
| 1471 |
< |
\subsubsection{\label{section:ffSC}The SuttonChenAtomTypes block} |
| 1471 |
> |
\begin{lstlisting}[caption={[An example of a EAMAtomTypes block.] A |
| 1472 |
> |
simple example of a EAMAtomTypes block. Here the only data provided is |
| 1473 |
> |
the name of a {\tt funcfl} file which contains the raw data for spline |
| 1474 |
> |
interpolations for the density, functional, and pair potential.}, |
| 1475 |
> |
label={sch:EAMAtomTypes}] |
| 1476 |
> |
begin EAMAtomTypes |
| 1477 |
> |
Au Au.u3.funcfl |
| 1478 |
> |
Ag Ag.u3.funcfl |
| 1479 |
> |
Cu Cu.u3.funcfl |
| 1480 |
> |
Ni Ni.u3.funcfl |
| 1481 |
> |
Pd Pd.u3.funcfl |
| 1482 |
> |
Pt Pt.u3.funcfl |
| 1483 |
> |
end EAMAtomTypes |
| 1484 |
> |
\end{lstlisting} |
| 1485 |
|
|
| 1486 |
+ |
\subsection{\label{section:ffSC}The SuttonChenAtomTypes block} |
| 1487 |
+ |
|
| 1488 |
|
The Sutton-Chen ({\sc sc})~\cite{Chen90} potential has been used to |
| 1489 |
< |
study a wide range of phenomena in metals. Although it is similar in |
| 1490 |
< |
form to the {\sc eam} potential, the Sutton-Chen model takes on a |
| 1491 |
< |
simpler form, |
| 1489 |
> |
study a wide range of phenomena in metals. Although it has the same |
| 1490 |
> |
basic form as the {\sc eam} potential, the Sutton-Chen model requires |
| 1491 |
> |
a simpler set of parameters, |
| 1492 |
|
\begin{equation} |
| 1493 |
|
\label{eq:SCP1} |
| 1494 |
|
U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq |
| 1495 |
< |
i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] , |
| 1495 |
> |
i}\epsilon_{ij}V^{pair}_{ij}(r_{ij})-c_{i}\epsilon_{ii}\sqrt{\rho_{i}}\right] , |
| 1496 |
|
\end{equation} |
| 1497 |
|
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
| 1498 |
|
\begin{equation} |
| 1499 |
|
\label{eq:SCP2} |
| 1500 |
|
V^{pair}_{ij}(r)=\left( |
| 1501 |
< |
\frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( |
| 1501 |
> |
\frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}} \hspace{1in} \rho_{i}=\sum_{j\neq i}\left( |
| 1502 |
|
\frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}} |
| 1503 |
|
\end{equation} |
| 1504 |
|
|
| 1506 |
|
interactions of the pseudo-atom cores. The $\sqrt{\rho_i}$ term in |
| 1507 |
|
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 1508 |
|
the interactions between the valence electrons and the cores of the |
| 1509 |
< |
pseudo-atoms. $D_{ij}$, $D_{ii}$, $c_i$ and $\alpha_{ij}$ are |
| 1510 |
< |
parameters used to tune the potential for different transition |
| 1511 |
< |
metals. |
| 1509 |
> |
pseudo-atoms. $\epsilon_{ij}$, $\epsilon_{ii}$, $c_i$ and |
| 1510 |
> |
$\alpha_{ij}$ are parameters used to tune the potential for different |
| 1511 |
> |
transition metals. |
| 1512 |
|
|
| 1513 |
|
The {\sc sc} potential form has also been parameterized by Qi {\it et |
| 1514 |
|
al.}\cite{Qi99} These parameters were obtained via empirical and {\it |
| 1515 |
|
ab initio} calculations to match structural features of the FCC |
| 1516 |
< |
crystal. To specify the original Sutton-Chen variant of the {\sc sc} |
| 1517 |
< |
force field, the user would add the {\tt forceFieldVariant = "SC";} |
| 1304 |
< |
line to the meta-data file, while specification of the Qi {\it et al.} |
| 1305 |
< |
quantum-adapted variant of the {\sc sc} potential, the user would add |
| 1306 |
< |
the {\tt forceFieldVariant = "QSC";} line to the meta-data file. |
| 1516 |
> |
crystal. Interested readers are encouraged to consult reference |
| 1517 |
> |
\citealp{Qi99} for further details. |
| 1518 |
|
|
| 1519 |
< |
\subsection{\label{section::ffShortRange}Short Range Interactions} |
| 1520 |
< |
\subsubsection{\label{section:ffBond}The BondTypes block} |
| 1521 |
< |
\subsubsection{\label{section:ffBend}The BendTypes block} |
| 1522 |
< |
A harmonic bend potential is represented by the following function: |
| 1523 |
< |
\begin{equation} |
| 1524 |
< |
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 |
| 1525 |
< |
)^2, \label{eq:bendPot} |
| 1526 |
< |
\end{equation} |
| 1527 |
< |
where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$, |
| 1528 |
< |
$\theta_0$ is the equilibrium bond angle, and $k_{\theta}$ is the |
| 1529 |
< |
force constant which determines the strength of the harmonic bend. |
| 1519 |
> |
\begin{lstlisting}[caption={[An example of a SCAtomTypes block.] A |
| 1520 |
> |
simple example of a SCAtomTypes block. Distances ($\alpha$) |
| 1521 |
> |
are given in \AA\ and energies ($\epsilon$) are (by convention) given in |
| 1522 |
> |
units of eV. These units must be specified in the {\tt Options} block |
| 1523 |
> |
using the keyword {\tt MetallicEnergyUnitScaling}. Without this {\tt |
| 1524 |
> |
Options} keyword, the default units for $\epsilon$ are kcal/mol. The |
| 1525 |
> |
other parameters, $m$, $n$, and $c$ are unitless.}, |
| 1526 |
> |
label={sch:SCAtomTypes}] |
| 1527 |
> |
begin SCAtomTypes |
| 1528 |
> |
// Name epsilon(eV) c m n alpha(angstroms) |
| 1529 |
> |
Ni 0.0073767 84.745 5.0 10.0 3.5157 |
| 1530 |
> |
Cu 0.0057921 84.843 5.0 10.0 3.6030 |
| 1531 |
> |
Rh 0.0024612 305.499 5.0 13.0 3.7984 |
| 1532 |
> |
Pd 0.0032864 148.205 6.0 12.0 3.8813 |
| 1533 |
> |
Ag 0.0039450 96.524 6.0 11.0 4.0691 |
| 1534 |
> |
Ir 0.0037674 224.815 6.0 13.0 3.8344 |
| 1535 |
> |
Pt 0.0097894 71.336 7.0 11.0 3.9163 |
| 1536 |
> |
Au 0.0078052 53.581 8.0 11.0 4.0651 |
| 1537 |
> |
Au2 0.0078052 53.581 8.0 11.0 4.0651 |
| 1538 |
> |
end SCAtomTypes |
| 1539 |
> |
\end{lstlisting} |
| 1540 |
|
|
| 1541 |
< |
\subsubsection{\label{section:ffTorsion}The TorsionTypes block} |
| 1542 |
< |
The torsion potential is often represented as a cosine series of the |
| 1543 |
< |
form: |
| 1541 |
> |
\section{\label{section::ffShortRange}Short Range Interactions} |
| 1542 |
> |
The internal structure of a molecule is usually specified in terms of |
| 1543 |
> |
a set of ``bonded'' terms in the potential energy function for |
| 1544 |
> |
molecule $I$, |
| 1545 |
> |
\begin{align*} |
| 1546 |
> |
V^{I}_{\text{Internal}} = & |
| 1547 |
> |
\sum_{r_{ij} \in I} V_{\text{bond}}(r_{ij}) |
| 1548 |
> |
+ \sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
| 1549 |
> |
+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{ijkl}) |
| 1550 |
> |
+ \sum_{\omega_{ijkl} \in I} V_{\text{inversion}}(\omega_{ijkl}) \\ |
| 1551 |
> |
& + \sum_{i \in I} \sum_{(j>i+4) \in I} |
| 1552 |
> |
\biggl[ V_{\text{dispersion}}(r_{ij}) + V_{\text{electrostatic}} |
| 1553 |
> |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 1554 |
> |
\biggr]. |
| 1555 |
> |
\label{eq:internalPotential} |
| 1556 |
> |
\end{align*} |
| 1557 |
> |
Here $V_{\text{bond}}, V_{\text{bend}}, |
| 1558 |
> |
V_{\text{torsion}},\mathrm{~and~} V_{\text{inversion}}$ represent the |
| 1559 |
> |
bond, bend, torsion, and inversion potentials for all |
| 1560 |
> |
topologically-connected sets of atoms within the molecule. Bonds are |
| 1561 |
> |
the primary way of specifying how the atoms are connected together to |
| 1562 |
> |
form the molecule (i.e. they define the molecular topology). The |
| 1563 |
> |
other short-range interactions may be specified explicitly in the |
| 1564 |
> |
molecule definition, or they may be deduced from bonding information. |
| 1565 |
> |
For example, bends can be implicitly deduced from two bonds which |
| 1566 |
> |
share an atom. Torsions can be deduced from two bends that share a |
| 1567 |
> |
bond. Inversion potentials are utilized primarily to enforce |
| 1568 |
> |
planarity around $sp^2$-hybridized sites, and these are specified with |
| 1569 |
> |
central atoms and satellites (or an atom with bonds to exactly three |
| 1570 |
> |
satellites). Non-bonded interactions are usually excluded for atom |
| 1571 |
> |
pairs that are involved in the same bond, bend, or torsion, but all |
| 1572 |
> |
other atom pairs are included in the standard non-bonded interactions. |
| 1573 |
> |
|
| 1574 |
> |
Bond lengths, angles, and torsions (dihedrals) are ``natural'' |
| 1575 |
> |
coordinates to treat molecular motion, as it is usually in these |
| 1576 |
> |
coordinates that most chemists understand the behavior of molecules. |
| 1577 |
> |
The bond lengths and angles are often considered ``hard'' degrees of |
| 1578 |
> |
freedom. That is, we can't deform them very much without a |
| 1579 |
> |
significant energetic penalty. On the other hand, dihedral angles or |
| 1580 |
> |
torsions are ``soft'' and typically undergo significant deformation |
| 1581 |
> |
under normal conditions. |
| 1582 |
> |
|
| 1583 |
> |
\subsection{\label{section:ffBond}The BondTypes block} |
| 1584 |
> |
|
| 1585 |
> |
Bonds are the primary way to specify how the atoms are connected |
| 1586 |
> |
together to form the molecule. In general, bonds exert strong |
| 1587 |
> |
restoring forces to keep the molecule compact. The bond energy |
| 1588 |
> |
functions are relatively simple functions of the distance between two |
| 1589 |
> |
atomic sites, |
| 1590 |
|
\begin{equation} |
| 1591 |
< |
V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi] |
| 1592 |
< |
+ c_2[1 + \cos(2\phi)] |
| 1593 |
< |
+ c_3[1 + \cos(3\phi)], |
| 1594 |
< |
\label{eq:origTorsionPot} |
| 1595 |
< |
\end{equation} |
| 1596 |
< |
where: |
| 1591 |
> |
b = \left| \vec{r}_{ij} \right| = \sqrt{(x_j - x_i)^2 + (y_j - y_i)^2 |
| 1592 |
> |
+ (z_j - z_i)^2}. |
| 1593 |
> |
\end{equation} |
| 1594 |
> |
All BondTypes must specify two AtomType names ($i$ and $j$) that |
| 1595 |
> |
describe when that bond should be applied, as well as an equilibrium |
| 1596 |
> |
bond length, $b_{ij}^0$, in units of \AA. The most common forms for |
| 1597 |
> |
bonding potentials are {\tt Harmonic} bonds, |
| 1598 |
|
\begin{equation} |
| 1599 |
< |
\cos\phi = (\hat{\mathbf{r}}_{ij} \times \hat{\mathbf{r}}_{jk}) \cdot |
| 1600 |
< |
(\hat{\mathbf{r}}_{jk} \times \hat{\mathbf{r}}_{kl}). |
| 1333 |
< |
\label{eq:torsPhi} |
| 1599 |
> |
V_{\text{bond}}(b) = \frac{k_{ij}}{2} \left(b - |
| 1600 |
> |
b_{ij}^0 \right)^2 |
| 1601 |
|
\end{equation} |
| 1602 |
< |
Here, $\hat{\mathbf{r}}_{\alpha\beta}$ are the set of unit bond |
| 1336 |
< |
vectors between atoms $i$, $j$, $k$, and $l$. For computational |
| 1337 |
< |
efficiency, the torsion potential has been recast after the method of |
| 1338 |
< |
{\sc charmm},\cite{Brooks83} in which the angle series is converted to |
| 1339 |
< |
a power series of the form: |
| 1602 |
> |
and {\tt Morse} bonds, |
| 1603 |
|
\begin{equation} |
| 1604 |
< |
V_{\text{torsion}}(\phi) = |
| 1342 |
< |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0, |
| 1343 |
< |
\label{eq:torsionPot} |
| 1604 |
> |
V_{\text{bond}}(b) = D_{ij} \left[ 1 - e^{-\beta_{ij} (b - b_{ij}^0)} \right]^2 |
| 1605 |
|
\end{equation} |
| 1345 |
– |
where: |
| 1346 |
– |
\begin{align*} |
| 1347 |
– |
k_0 &= c_1 + c_3, \\ |
| 1348 |
– |
k_1 &= c_1 - 3c_3, \\ |
| 1349 |
– |
k_2 &= 2 c_2, \\ |
| 1350 |
– |
k_3 &= 4c_3. |
| 1351 |
– |
\end{align*} |
| 1352 |
– |
By recasting the potential as a power series, repeated trigonometric |
| 1353 |
– |
evaluations are avoided during the calculation of the potential |
| 1354 |
– |
energy. |
| 1606 |
|
|
| 1607 |
< |
\subsubsection{\label{section:ffInversion}The InversionTypes block} |
| 1608 |
< |
\subsection{\label{section::ffLongRange}Long Range Interactions} |
| 1609 |
< |
\subsubsection{\label{section:ffInversion}The NonBondedInteraction block} |
| 1607 |
> |
\begin{figure}[h] |
| 1608 |
> |
\centering |
| 1609 |
> |
\includegraphics[width=2.5in]{bond.pdf} |
| 1610 |
> |
\caption[Bond coordinates]{The coordinate describing a |
| 1611 |
> |
a bond between atoms $i$ and $j$ is $|r_{ij}|$, the length of the |
| 1612 |
> |
$\vec{r}_{ij}$ vector. } |
| 1613 |
> |
\label{fig:bond} |
| 1614 |
> |
\end{figure} |
| 1615 |
|
|
| 1616 |
< |
|
| 1617 |
< |
|
| 1618 |
< |
(see Fig.~\ref{fig:lipidModel}), The parameters for $k_{\theta}$ and |
| 1363 |
< |
$\theta_0$ are borrowed from those in TraPPE.\cite{Siepmann1998} |
| 1364 |
< |
|
| 1365 |
< |
Calculating the long-range (non-bonded) potential involves a sum over |
| 1366 |
< |
all pairs of atoms (except for those atoms which are involved in a |
| 1367 |
< |
bond, bend, or torsion with each other). If done poorly, calculating |
| 1368 |
< |
the the long-range interactions for $N$ atoms would involve $N(N-1)/2$ |
| 1369 |
< |
evaluations of atomic distances. To reduce the number of distance |
| 1370 |
< |
evaluations between pairs of atoms, {\sc OpenMD} allows the use of |
| 1371 |
< |
switched cutoffs with Verlet neighbor lists.\cite{Allen87} Neutral |
| 1372 |
< |
groups which contain charges will exhibit pathological forces unless |
| 1373 |
< |
the cutoff is applied to the neutral groups evenly instead of to the |
| 1374 |
< |
individual atoms.\cite{leach01:mm} {\sc OpenMD} allows users to |
| 1375 |
< |
specify cutoff groups which may contain an arbitrary number of atoms |
| 1376 |
< |
in the molecule. Atoms in a cutoff group are treated as a single unit |
| 1377 |
< |
for the evaluation of the switching function: |
| 1616 |
> |
OpenMD can also simulate some less common types of bond potentials, |
| 1617 |
> |
including {\tt Fixed} bonds (which are constrained to be at a |
| 1618 |
> |
specified bond length), |
| 1619 |
|
\begin{equation} |
| 1620 |
< |
V_{\mathrm{long-range}} = \sum_{a} \sum_{b>a} s(r_{ab}) \sum_{i \in a} \sum_{j \in b} V_{ij}(r_{ij}), |
| 1620 |
> |
V_{\text{bond}}(b) = 0. |
| 1621 |
|
\end{equation} |
| 1622 |
< |
where $r_{ab}$ is the distance between the centers of mass of the two |
| 1623 |
< |
cutoff groups ($a$ and $b$). |
| 1622 |
> |
{\tt Cubic} bonds include terms to model anharmonicity, |
| 1623 |
> |
\begin{equation} |
| 1624 |
> |
V_{\text{bond}}(b) = K_3 (b - b_{ij}^0)^3 + K_2 (b - b_{ij}^0)^2 + K_1 (b - b_{ij}^0) + K_0, |
| 1625 |
> |
\end{equation} |
| 1626 |
> |
and {\tt Quartic} bonds, include another term in the Taylor |
| 1627 |
> |
expansion around $b_{ij}^0$, |
| 1628 |
> |
\begin{equation} |
| 1629 |
> |
V_{\text{bond}}(b) = K_4 (b - b_{ij}^0)^4 + K_3 (b - b_{ij}^0)^3 + |
| 1630 |
> |
K_2 (b - b_{ij}^0)^2 + K_1 (b - b_{ij}^0) + K_0, |
| 1631 |
> |
\end{equation} |
| 1632 |
> |
can also be simulated. Note that the factor of $1/2$ that is included |
| 1633 |
> |
in the {\tt Harmonic} bond type force constant is {\it not} present in |
| 1634 |
> |
either the {\tt Cubic} or {\tt Quartic} bond types. |
| 1635 |
|
|
| 1636 |
< |
The sums over $a$ and $b$ are over the cutoff groups that are present |
| 1385 |
< |
in the simulation. Atoms which are not explicitly defined as members |
| 1386 |
< |
of a {\tt cutoffGroup} are treated as a group consisting of only one |
| 1387 |
< |
atom. The switching function, $s(r)$ is the standard cubic switching |
| 1388 |
< |
function, |
| 1636 |
> |
{\tt Polynomial} bonds which can have any number of terms, |
| 1637 |
|
\begin{equation} |
| 1638 |
< |
S(r) = |
| 1391 |
< |
\begin{cases} |
| 1392 |
< |
1 & \text{if $r \le r_{\text{sw}}$},\\ |
| 1393 |
< |
\frac{(r_{\text{cut}} + 2r - 3r_{\text{sw}})(r_{\text{cut}} - r)^2} |
| 1394 |
< |
{(r_{\text{cut}} - r_{\text{sw}})^3} |
| 1395 |
< |
& \text{if $r_{\text{sw}} < r \le r_{\text{cut}}$}, \\ |
| 1396 |
< |
0 & \text{if $r > r_{\text{cut}}$.} |
| 1397 |
< |
\end{cases} |
| 1398 |
< |
\label{eq:dipoleSwitching} |
| 1638 |
> |
V_{\text{bond}}(b) = \sum_n K_n (b - b_{ij}^0)^n. |
| 1639 |
|
\end{equation} |
| 1640 |
< |
Here, $r_{\text{sw}}$ is the {\tt switchingRadius}, or the distance |
| 1641 |
< |
beyond which interactions are reduced, and $r_{\text{cut}}$ is the |
| 1402 |
< |
{\tt cutoffRadius}, or the distance at which interactions are |
| 1403 |
< |
truncated. |
| 1640 |
> |
can also be specified by giving a sequence of integer ($n$) and force |
| 1641 |
> |
constant ($K_n$) pairs. |
| 1642 |
|
|
| 1643 |
< |
Users of {\sc OpenMD} do not need to specify the {\tt cutoffRadius} or |
| 1644 |
< |
{\tt switchingRadius}. In simulations containing only Lennard-Jones |
| 1645 |
< |
atoms, the cutoff radius has a default value of $2.5\sigma_{ii}$, |
| 1646 |
< |
where $\sigma_{ii}$ is the largest Lennard-Jones length parameter |
| 1647 |
< |
present in the simulation. In simulations containing charged or |
| 1648 |
< |
dipolar atoms, the default cutoff radius is $15 \mbox{\AA}$. |
| 1643 |
> |
The order of terms in the BondTypes block is: |
| 1644 |
> |
\begin{itemize} |
| 1645 |
> |
\item {\tt AtomType} 1 |
| 1646 |
> |
\item {\tt AtomType} 2 |
| 1647 |
> |
\item {\tt BondType} (one of {\tt Harmonic}, {\tt Morse}, {\tt Fixed}, {\tt |
| 1648 |
> |
Cubic}, {\tt Quartic}, or {\tt Polynomial}) |
| 1649 |
> |
\item $b_{ij}^0$, the equilibrium bond length in \AA |
| 1650 |
> |
\item any other parameters required by the {\tt BondType} |
| 1651 |
> |
\end{itemize} |
| 1652 |
|
|
| 1653 |
< |
The {\tt switchingRadius} is set to a default value of 95\% of the |
| 1654 |
< |
{\tt cutoffRadius}. In the special case of a simulation containing |
| 1655 |
< |
{\it only} Lennard-Jones atoms, the default switching radius takes the |
| 1656 |
< |
same value as the cutoff radius, and {\sc OpenMD} will use a shifted |
| 1657 |
< |
potential to remove discontinuities in the potential at the cutoff. |
| 1658 |
< |
Both radii may be specified in the meta-data file. |
| 1659 |
< |
|
| 1660 |
< |
Force fields can be added to {\sc OpenMD}, although it comes with a few |
| 1661 |
< |
simple examples (Lennard-Jones, {\sc duff}, {\sc water}, and {\sc |
| 1662 |
< |
eam}) which are explained in the following sections. |
| 1663 |
< |
|
| 1664 |
< |
\section{\label{sec:LJPot}The Lennard Jones Force Field} |
| 1665 |
< |
|
| 1666 |
< |
Scheme |
| 1667 |
< |
\ref{sch:LJFF} gives an example meta-data file that |
| 1668 |
< |
sets up a system of 108 Ar particles to be simulated using the |
| 1669 |
< |
Lennard-Jones force field. |
| 1670 |
< |
|
| 1430 |
< |
\begin{lstlisting}[float,caption={[Invocation of the Lennard-Jones |
| 1431 |
< |
force field] A sample startup file for a small Lennard-Jones |
| 1432 |
< |
simulation.},label={sch:LJFF}] |
| 1433 |
< |
<OpenMD> |
| 1434 |
< |
<MetaData> |
| 1435 |
< |
#include "argon.md" |
| 1436 |
< |
|
| 1437 |
< |
component{ |
| 1438 |
< |
type = "Ar"; |
| 1439 |
< |
nMol = 108; |
| 1440 |
< |
} |
| 1441 |
< |
|
| 1442 |
< |
forceField = "LJ"; |
| 1443 |
< |
</MetaData> |
| 1444 |
< |
<Snapshot> // not shown in this scheme |
| 1445 |
< |
</Snapshot> |
| 1446 |
< |
</OpenMD> |
| 1653 |
> |
\begin{lstlisting}[caption={[An example of a BondTypes block.] A |
| 1654 |
> |
simple example of a BondTypes block. Distances ($b_0$) |
| 1655 |
> |
are given in \AA\ and force constants are given in |
| 1656 |
> |
units so that when multiplied by the correct power of distance they |
| 1657 |
> |
return energies in kcal/mol. For example $k$ for a Harmonic bond is |
| 1658 |
> |
in units of kcal/mol/\AA$^2$.}, |
| 1659 |
> |
label={sch:BondTypes}] |
| 1660 |
> |
begin BondTypes |
| 1661 |
> |
//Atom1 Atom2 Harmonic b0 k (kcal/mol/A^2) |
| 1662 |
> |
CH2 CH2 Harmonic 1.526 260 |
| 1663 |
> |
//Atom1 Atom2 Morse b0 D beta (A^-1) |
| 1664 |
> |
CN NC Morse 1.157437 212.95 2.5802 |
| 1665 |
> |
//Atom1 Atom2 Fixed b0 |
| 1666 |
> |
CT HC Fixed 1.09 |
| 1667 |
> |
//Atom1 Atom2 Cubic b0 K3 K2 K1 K0 |
| 1668 |
> |
//Atom1 Atom2 Quartic b0 K4 K3 K2 K1 K0 |
| 1669 |
> |
//Atom1 Atom2 Polynomial b0 n Kn [m Km] |
| 1670 |
> |
end BondTypes |
| 1671 |
|
\end{lstlisting} |
| 1672 |
|
|
| 1673 |
+ |
There are advantages and disadvantages of all of the different types |
| 1674 |
+ |
of bonds, but specific simulation tasks may call for specific |
| 1675 |
+ |
behaviors. |
| 1676 |
|
|
| 1677 |
< |
\section{\label{section:DUFF}Dipolar Unified-Atom Force Field} |
| 1677 |
> |
\subsection{\label{section:ffBend}The BendTypes block} |
| 1678 |
> |
The equilibrium geometries and energy functions for bending motions in |
| 1679 |
> |
a molecule are strongly dependent on the bonding environment of the |
| 1680 |
> |
central atomic site. For example, different types of hybridized |
| 1681 |
> |
carbon centers require different bending angles and force constants to |
| 1682 |
> |
describe the local geometry. |
| 1683 |
|
|
| 1684 |
< |
The dipolar unified-atom force field ({\sc duff}) was developed to |
| 1685 |
< |
simulate lipid bilayers. These types of simulations require a model |
| 1686 |
< |
capable of forming bilayers, while still being sufficiently |
| 1687 |
< |
computationally efficient to allow large systems ($\sim$100's of |
| 1688 |
< |
phospholipids, $\sim$1000's of waters) to be simulated for long times |
| 1689 |
< |
($\sim$10's of nanoseconds). With this goal in mind, {\sc duff} has no |
| 1690 |
< |
point charges. Charge-neutral distributions are replaced with dipoles, |
| 1691 |
< |
while most atoms and groups of atoms are reduced to Lennard-Jones |
| 1692 |
< |
interaction sites. This simplification reduces the length scale of |
| 1461 |
< |
long range interactions from $\frac{1}{r}$ to $\frac{1}{r^3}$, |
| 1462 |
< |
removing the need for the computationally expensive Ewald |
| 1463 |
< |
sum. Instead, Verlet neighbor-lists and cutoff radii are used for the |
| 1464 |
< |
dipolar interactions, and, if desired, a reaction field may be added |
| 1465 |
< |
to mimic longer range interactions. |
| 1684 |
> |
The bending potential energy functions used in most force fields are |
| 1685 |
> |
often simple functions of the angle between two bonds, |
| 1686 |
> |
\begin{equation} |
| 1687 |
> |
\theta_{ijk} = \cos^{-1} \left(\frac{\vec{r}_{ji} \cdot |
| 1688 |
> |
\vec{r}_{jk}}{\left| \vec{r}_{ji} \right| \left| \vec{r}_{jk} |
| 1689 |
> |
\right|} \right) |
| 1690 |
> |
\end{equation} |
| 1691 |
> |
Here atom $j$ is the central atom that is bonded to two partners $i$ |
| 1692 |
> |
and $k$. |
| 1693 |
|
|
| 1694 |
< |
As an example, lipid head-groups in {\sc duff} are represented as |
| 1468 |
< |
point dipole interaction sites. Placing a dipole at the head group's |
| 1469 |
< |
center of mass mimics the charge separation found in common |
| 1470 |
< |
phospholipid head groups such as phosphatidylcholine.\cite{Cevc87} |
| 1471 |
< |
Additionally, a large Lennard-Jones site is located at the |
| 1472 |
< |
pseudoatom's center of mass. The model is illustrated by the red atom |
| 1473 |
< |
in Fig.~\ref{fig:lipidModel}. The water model we use to |
| 1474 |
< |
complement the dipoles of the lipids is a |
| 1475 |
< |
reparameterization\cite{fennell04} of the soft sticky dipole (SSD) |
| 1476 |
< |
model of Ichiye |
| 1477 |
< |
\emph{et al.}\cite{liu96:new_model} |
| 1478 |
< |
|
| 1479 |
< |
\begin{figure} |
| 1694 |
> |
\begin{figure}[h] |
| 1695 |
|
\centering |
| 1696 |
< |
\includegraphics[width=\linewidth]{lipidModel.pdf} |
| 1697 |
< |
\caption[A representation of a lipid model in {\sc duff}]{A |
| 1698 |
< |
representation of the lipid model. $\phi$ is the torsion angle, |
| 1699 |
< |
$\theta$ is the bend angle, and $\mu$ is the dipole moment of the head |
| 1700 |
< |
group.} |
| 1701 |
< |
\label{fig:lipidModel} |
| 1696 |
> |
\includegraphics[width=3.5in]{bend.pdf} |
| 1697 |
> |
\caption[Bend angle coordinates]{The coordinate describing a bend |
| 1698 |
> |
between atoms $i$, $j$, and $k$ is the angle $\theta_{ijk} = |
| 1699 |
> |
\cos^{-1} \left(\hat{r}_{ji} \cdot \hat{r}_{jk}\right)$ where $\hat{r}_{ji}$ is |
| 1700 |
> |
the unit vector between atoms $j$ and $i$. } |
| 1701 |
> |
\label{fig:bend} |
| 1702 |
|
\end{figure} |
| 1703 |
|
|
| 1489 |
– |
A set of scalable parameters has been used to model the alkyl groups |
| 1490 |
– |
with Lennard-Jones sites. For this, parameters from the TraPPE force |
| 1491 |
– |
field of Siepmann \emph{et al.}\cite{Siepmann1998} have been |
| 1492 |
– |
utilized. TraPPE is a unified-atom representation of n-alkanes which |
| 1493 |
– |
is parametrized against phase equilibria using Gibbs ensemble Monte |
| 1494 |
– |
Carlo simulation techniques.\cite{Siepmann1998} One of the advantages |
| 1495 |
– |
of TraPPE is that it generalizes the types of atoms in an alkyl chain |
| 1496 |
– |
to keep the number of pseudoatoms to a minimum; thus, the parameters |
| 1497 |
– |
for a unified atom such as $\text{CH}_2$ do not change depending on |
| 1498 |
– |
what species are bonded to it. |
| 1704 |
|
|
| 1705 |
< |
As is required by TraPPE, {\sc duff} also constrains all bonds to be |
| 1706 |
< |
of fixed length. Typically, bond vibrations are the fastest motions in |
| 1707 |
< |
a molecular dynamic simulation. With these vibrations present, small |
| 1708 |
< |
time steps between force evaluations must be used to ensure adequate |
| 1709 |
< |
energy conservation in the bond degrees of freedom. By constraining |
| 1710 |
< |
the bond lengths, larger time steps may be used when integrating the |
| 1711 |
< |
equations of motion. A simulation using {\sc duff} is illustrated in |
| 1712 |
< |
Scheme \ref{sch:DUFF}. |
| 1705 |
> |
All BendTypes must specify three AtomType names ($i$, $j$ and $k$) |
| 1706 |
> |
that describe when that bend potential should be applied, as well as |
| 1707 |
> |
an equilibrium bending angle, $\theta_{ijk}^0$, in units of |
| 1708 |
> |
degrees. The most common forms for bending potentials are {\tt |
| 1709 |
> |
Harmonic} bends, |
| 1710 |
> |
\begin{equation} |
| 1711 |
> |
V_{\text{bend}}(\theta_{ijk}) = \frac{k_{ijk}}{2}( \theta_{ijk} - \theta_{ijk}^0 |
| 1712 |
> |
)^2, \label{eq:bendPot} |
| 1713 |
> |
\end{equation} |
| 1714 |
> |
where $k_{ijk}$ is the force constant which determines the strength of |
| 1715 |
> |
the harmonic bend. {\tt UreyBradley} bends utilize an additional 1-3 |
| 1716 |
> |
bond-type interaction in addition to the harmonic bending potential: |
| 1717 |
> |
\begin{equation} |
| 1718 |
> |
V_{\text{bend}}(\vec{r}_i , \vec{r}_j, \vec{r}_k) |
| 1719 |
> |
=\frac{k_{ijk}}{2}( \theta_{ijk} - \theta_{ijk}^0)^2 |
| 1720 |
> |
+ \frac{k_{ub}}{2}( r_{ik} - s_0 )^2. \label{eq:ubBend} |
| 1721 |
> |
\end{equation} |
| 1722 |
|
|
| 1723 |
< |
\begin{lstlisting}[float,caption={[Invocation of {\sc duff}]A portion |
| 1724 |
< |
of a startup file showing a simulation utilizing {\sc |
| 1725 |
< |
duff}},label={sch:DUFF}] |
| 1726 |
< |
<OpenMD> |
| 1727 |
< |
<MetaData> |
| 1728 |
< |
#include "water.md" |
| 1515 |
< |
#include "lipid.md" |
| 1723 |
> |
A {\tt Cosine} bend is a variant on the harmonic bend which utilizes |
| 1724 |
> |
the cosine of the angle instead of the angle itself, |
| 1725 |
> |
\begin{equation} |
| 1726 |
> |
V_{\text{bend}}(\theta_{ijk}) = \frac{k_{ijk}}{2}( \cos\theta_{ijk} - |
| 1727 |
> |
\cos \theta_{ijk}^0 )^2. \label{eq:cosBend} |
| 1728 |
> |
\end{equation} |
| 1729 |
|
|
| 1730 |
< |
component{ |
| 1731 |
< |
type = "simpleLipid_16"; |
| 1732 |
< |
nMol = 60; |
| 1733 |
< |
} |
| 1730 |
> |
OpenMD can also simulate some less common types of bend potentials, |
| 1731 |
> |
including {\tt Cubic} bends, which include terms to model anharmonicity, |
| 1732 |
> |
\begin{equation} |
| 1733 |
> |
V_{\text{bend}}(\theta_{ijk}) = K_3 (\theta_{ijk} - \theta_{ijk}^0)^3 + K_2 (\theta_{ijk} - \theta_{ijk}^0)^2 + K_1 (\theta_{ijk} - \theta_{ijk}^0) + K_0, |
| 1734 |
> |
\end{equation} |
| 1735 |
> |
and {\tt Quartic} bends, which include another term in the Taylor |
| 1736 |
> |
expansion around $\theta_{ijk}^0$, |
| 1737 |
> |
\begin{equation} |
| 1738 |
> |
V_{\text{bend}}(\theta_{ijk}) = K_4 (\theta_{ijk} - \theta_{ijk}^0)^4 + K_3 (\theta_{ijk} - \theta_{ijk}^0)^3 + |
| 1739 |
> |
K_2 (\theta_{ijk} - \theta_{ijk}^0)^2 + K_1 (\theta_{ijk} - |
| 1740 |
> |
\theta_{ijk}^0) + K_0, |
| 1741 |
> |
\end{equation} |
| 1742 |
> |
can also be simulated. Note that the factor of $1/2$ that is included |
| 1743 |
> |
in the {\tt Harmonic} bend type force constant is {\it not} present in |
| 1744 |
> |
either the {\tt Cubic} or {\tt Quartic} bend types. |
| 1745 |
|
|
| 1746 |
< |
component{ |
| 1747 |
< |
type = "SSD_water"; |
| 1748 |
< |
nMol = 1936; |
| 1749 |
< |
} |
| 1746 |
> |
{\tt Polynomial} bends which can have any number of terms, |
| 1747 |
> |
\begin{equation} |
| 1748 |
> |
V_{\text{bend}}(\theta_{ijk}) = \sum_n K_n (\theta_{ijk} - \theta_{ijk}^0)^n. |
| 1749 |
> |
\end{equation} |
| 1750 |
> |
can also be specified by giving a sequence of integer ($n$) and force |
| 1751 |
> |
constant ($K_n$) pairs. |
| 1752 |
|
|
| 1753 |
< |
forceField = "DUFF"; |
| 1754 |
< |
</MetaData> |
| 1755 |
< |
<Snapshot> // not shown in this scheme |
| 1756 |
< |
</Snapshot> |
| 1757 |
< |
</OpenMD> |
| 1753 |
> |
The order of terms in the BendTypes block is: |
| 1754 |
> |
\begin{itemize} |
| 1755 |
> |
\item {\tt AtomType} 1 |
| 1756 |
> |
\item {\tt AtomType} 2 (this is the central atom) |
| 1757 |
> |
\item {\tt AtomType} 3 |
| 1758 |
> |
\item {\tt BendType} (one of {\tt Harmonic}, {\tt UreyBradley}, {\tt |
| 1759 |
> |
Cosine}, {\tt Cubic}, {\tt Quartic}, or {\tt Polynomial}) |
| 1760 |
> |
\item $\theta_{ijk}^0$, the equilibrium bending angle in degrees. |
| 1761 |
> |
\item any other parameters required by the {\tt BendType} |
| 1762 |
> |
\end{itemize} |
| 1763 |
> |
|
| 1764 |
> |
\begin{lstlisting}[caption={[An example of a BendTypes block.] A |
| 1765 |
> |
simple example of a BendTypes block. By convention, equilibrium angles |
| 1766 |
> |
($\theta_0$) are given in degrees but force constants are given in |
| 1767 |
> |
units so that when multiplied by the correct power of angle (in |
| 1768 |
> |
radians) they return energies in kcal/mol. For example $k$ for a |
| 1769 |
> |
Harmonic bend is in units of kcal/mol/radians$^2$.}, |
| 1770 |
> |
label={sch:BendTypes}] |
| 1771 |
> |
begin BendTypes |
| 1772 |
> |
//Atom1 Atom2 Atom3 Harmonic theta0(deg) Ktheta(kcal/mol/radians^2) |
| 1773 |
> |
CT CT CT Harmonic 109.5 80.000000 |
| 1774 |
> |
CH2 CH CH2 Harmonic 112.0 117.68 |
| 1775 |
> |
CH3 CH2 SH Harmonic 96.0 67.220 |
| 1776 |
> |
//UreyBradley |
| 1777 |
> |
//Atom1 Atom2 Atom3 UreyBradley theta0 Ktheta s0 Kub |
| 1778 |
> |
//Cosine |
| 1779 |
> |
//Atom1 Atom2 Atom3 Cosine theta0 Ktheta(kcal/mol) |
| 1780 |
> |
//Cubic |
| 1781 |
> |
//Atom1 Atom2 Atom3 Cubic theta0 K3 K2 K1 K0 |
| 1782 |
> |
//Quartic |
| 1783 |
> |
//Atom1 Atom2 Atom3 Quartic theta0 K4 K3 K2 K1 K0 |
| 1784 |
> |
//Polynomial |
| 1785 |
> |
//Atom1 Atom2 Atom3 Polynomial theta0 n Kn [m Km] |
| 1786 |
> |
end BendTypes |
| 1787 |
|
\end{lstlisting} |
| 1788 |
|
|
| 1789 |
+ |
Note that the parameters for a particular bend type are the same for |
| 1790 |
+ |
any bending triplet of the same atomic types (in the same or reversed |
| 1791 |
+ |
order). Changing the AtomType in the Atom2 position will change the |
| 1792 |
+ |
matched bend types in the force field. |
| 1793 |
|
|
| 1794 |
+ |
\subsection{\label{section:ffTorsion}The TorsionTypes block} |
| 1795 |
+ |
The torsion potential for rotation of groups around a central bond can |
| 1796 |
+ |
often be represented with various cosine functions. For two |
| 1797 |
+ |
tetrahedral ($sp^3$) carbons connected by a single bond, the torsion |
| 1798 |
+ |
potential might be |
| 1799 |
+ |
\begin{equation*} |
| 1800 |
+ |
V_{\text{torsion}} = \frac{v}{2} \left[ 1 + \cos( 3 \phi ) \right] |
| 1801 |
+ |
\end{equation*} |
| 1802 |
+ |
where $v$ is the barrier for going from {\em staggered} $\rightarrow$ |
| 1803 |
+ |
{\em eclipsed} conformations, while for $sp^2$ carbons connected by a |
| 1804 |
+ |
double bond, the potential might be |
| 1805 |
+ |
\begin{equation*} |
| 1806 |
+ |
V_{\text{torsion}} = \frac{w}{2} \left[ 1 - \cos( 2 \phi ) \right] |
| 1807 |
+ |
\end{equation*} |
| 1808 |
+ |
where $w$ is the barrier for going from {\em cis} $\rightarrow$ {\em |
| 1809 |
+ |
trans} conformations. |
| 1810 |
|
|
| 1811 |
< |
The cross potential between molecules $I$ and $J$, |
| 1812 |
< |
$V^{IJ}_{\text{Cross}}$, is as follows: |
| 1811 |
> |
A general torsion potentials can be represented as a cosine series of |
| 1812 |
> |
the form: |
| 1813 |
|
\begin{equation} |
| 1814 |
< |
V^{IJ}_{\text{Cross}} = |
| 1815 |
< |
\sum_{i \in I} \sum_{j \in J} |
| 1816 |
< |
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 1817 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 1543 |
< |
+ V_{\text{sticky}} |
| 1544 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 1545 |
< |
\biggr], |
| 1546 |
< |
\label{eq:crossPotentail} |
| 1814 |
> |
V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi_{ijkl}] |
| 1815 |
> |
+ c_2[1 - \cos(2\phi_{ijkl})] |
| 1816 |
> |
+ c_3[1 + \cos(3\phi_{ijkl})], |
| 1817 |
> |
\label{eq:origTorsionPot} |
| 1818 |
|
\end{equation} |
| 1819 |
< |
where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 1549 |
< |
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 1550 |
< |
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model |
| 1551 |
< |
(Sec.~\ref{section:SSD}). Note that not all atom types include all |
| 1552 |
< |
interactions. |
| 1553 |
< |
|
| 1554 |
< |
|
| 1555 |
< |
\section{\label{section:WATER}The {\sc water} Force Field} |
| 1556 |
< |
|
| 1557 |
< |
In addition to the {\sc duff} force field's solvent description, a |
| 1558 |
< |
separate {\sc water} force field has been included for simulating most |
| 1559 |
< |
of the common rigid-body water models. This force field includes the |
| 1560 |
< |
simple and point-dipolar models (SSD, SSD1, SSD/E, SSD/RF, and DPD |
| 1561 |
< |
water), as well as the common charge-based models (SPC, SPC/E, TIP3P, |
| 1562 |
< |
TIP4P, and |
| 1563 |
< |
TIP5P).\cite{liu96:new_model,Ichiye03,fennell04,Marrink01,Berendsen81,Berendsen87,Jorgensen83,Mahoney00} |
| 1564 |
< |
In order to handle these models, charge-charge interactions were |
| 1565 |
< |
included in the force-loop: |
| 1819 |
> |
where the angle $\phi_{ijkl}$ is defined |
| 1820 |
|
\begin{equation} |
| 1821 |
< |
V_{\text{charge}}(r_{ij}) = \sum_{ij}\frac{q_iq_je^2}{r_{ij}}, |
| 1821 |
> |
\cos\phi_{ijkl} = (\hat{\mathbf{r}}_{ij} \times \hat{\mathbf{r}}_{jk}) \cdot |
| 1822 |
> |
(\hat{\mathbf{r}}_{jk} \times \hat{\mathbf{r}}_{kl}). |
| 1823 |
> |
\label{eq:torsPhi} |
| 1824 |
|
\end{equation} |
| 1825 |
< |
where $q$ represents the charge on particle $i$ or $j$, and $e$ is the |
| 1826 |
< |
charge of an electron in Coulombs. The charge-charge interaction |
| 1827 |
< |
support is rudimentary in the current version of {\sc OpenMD}. As with |
| 1828 |
< |
the other pair interactions, charges can be simulated with a pure |
| 1829 |
< |
cutoff or a reaction field. The various methods for performing the |
| 1830 |
< |
Ewald summation have not yet been included. The {\sc water} force |
| 1831 |
< |
field can be easily expanded through modification of the {\sc water} |
| 1832 |
< |
force field file ({\tt WATER.frc}). By adding atom types and inserting |
| 1833 |
< |
the appropriate parameters, it is possible to extend the force field |
| 1578 |
< |
to handle rigid molecules other than water. |
| 1825 |
> |
Here, $\hat{\mathbf{r}}_{\alpha\beta}$ are the set of unit bond |
| 1826 |
> |
vectors between atoms $i$, $j$, $k$, and $l$. Note that some force |
| 1827 |
> |
fields define the zero of the $\phi_{ijkl}$ angle when atoms $i$ and |
| 1828 |
> |
$l$ are in the {\em trans} configuration, while most define the zero |
| 1829 |
> |
angle for when $i$ and $l$ are in the fully eclipsed orientation. The |
| 1830 |
> |
behavior of the torsion parser can be altered with the {\tt |
| 1831 |
> |
TorsionAngleConvention} keyword in the Options block. The default |
| 1832 |
> |
behavior is {\tt "180\_is\_trans"} while the opposite behavior can be |
| 1833 |
> |
invoked by setting this keyword to {\tt "0\_is\_trans"}. |
| 1834 |
|
|
| 1835 |
+ |
\begin{figure}[h] |
| 1836 |
+ |
\centering |
| 1837 |
+ |
\includegraphics[width=4.5in]{torsion.pdf} |
| 1838 |
+ |
\caption[Torsion or dihedral angle coordinates]{The coordinate |
| 1839 |
+ |
describing a torsion between atoms $i$, $j$, $k$, and $l$ is the |
| 1840 |
+ |
dihedral angle $\phi_{ijkl}$ which measures the relative rotation of |
| 1841 |
+ |
the two terminal atoms around the axis defined by the middle bond. } |
| 1842 |
+ |
\label{fig:torsion} |
| 1843 |
+ |
\end{figure} |
| 1844 |
|
|
| 1845 |
< |
\section{\label{section:sc}The Sutton-Chen Force Field} |
| 1845 |
> |
For computational efficiency, OpenMD recasts torsion potential in the |
| 1846 |
> |
method of {\sc charmm},\cite{Brooks83} in which the angle series is |
| 1847 |
> |
converted to a power series of the form: |
| 1848 |
> |
\begin{equation} |
| 1849 |
> |
V_{\text{torsion}}(\phi_{ijkl}) = |
| 1850 |
> |
k_3 \cos^3 \phi_{ijkl} + k_2 \cos^2 \phi_{ijkl} + k_1 \cos \phi_{ijkl} + k_0, |
| 1851 |
> |
\label{eq:torsionPot} |
| 1852 |
> |
\end{equation} |
| 1853 |
> |
where: |
| 1854 |
> |
\begin{align*} |
| 1855 |
> |
k_0 &= c_1 + 2 c_2 + c_3, \\ |
| 1856 |
> |
k_1 &= c_1 - 3c_3, \\ |
| 1857 |
> |
k_2 &= - 2 c_2, \\ |
| 1858 |
> |
k_3 &= 4 c_3. |
| 1859 |
> |
\end{align*} |
| 1860 |
> |
By recasting the potential as a power series, repeated trigonometric |
| 1861 |
> |
evaluations are avoided during the calculation of the potential |
| 1862 |
> |
energy. |
| 1863 |
|
|
| 1864 |
+ |
Using this framework, OpenMD implements a variety of different |
| 1865 |
+ |
potential energy functions for torsions: |
| 1866 |
+ |
\begin{itemize} |
| 1867 |
+ |
\item {\tt Cubic}: |
| 1868 |
+ |
\begin{equation*} |
| 1869 |
+ |
V_{\text{torsion}}(\phi) = |
| 1870 |
+ |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0, |
| 1871 |
+ |
\end{equation*} |
| 1872 |
+ |
\item {\tt Quartic}: |
| 1873 |
+ |
\begin{equation*} |
| 1874 |
+ |
V_{\text{torsion}}(\phi) = k_4 \cos^4 \phi + |
| 1875 |
+ |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0, |
| 1876 |
+ |
\end{equation*} |
| 1877 |
+ |
\item {\tt Polynomial}: |
| 1878 |
+ |
\begin{equation*} |
| 1879 |
+ |
V_{\text{torsion}}(\phi) = \sum_n k_n \cos^n \phi , |
| 1880 |
+ |
\end{equation*} |
| 1881 |
+ |
\item {\tt Charmm}: |
| 1882 |
+ |
\begin{equation*} |
| 1883 |
+ |
V_{\text{torsion}}(\phi) = \sum_n K_n \left( 1 + cos(n |
| 1884 |
+ |
\phi - \delta_n) \right), |
| 1885 |
+ |
\end{equation*} |
| 1886 |
+ |
\item {\tt Opls}: |
| 1887 |
+ |
\begin{equation*} |
| 1888 |
+ |
V_{\text{torsion}}(\phi) = \frac{1}{2} \left(v_1 (1 + \cos \phi) \right) |
| 1889 |
+ |
+ v_2 (1 - \cos 2 \phi) + v_3 (1 + \cos 3 \phi), |
| 1890 |
+ |
\end{equation*} |
| 1891 |
+ |
\item {\tt Trappe}:\cite{Siepmann1998} |
| 1892 |
+ |
\begin{equation*} |
| 1893 |
+ |
V_{\text{torsion}}(\phi) = c_0 + c_1 (1 + \cos \phi) + c_2 (1 - \cos 2 \phi) + |
| 1894 |
+ |
c_3 (1 + \cos 3 \phi), |
| 1895 |
+ |
\end{equation*} |
| 1896 |
+ |
\item {\tt Harmonic}: |
| 1897 |
+ |
\begin{equation*} |
| 1898 |
+ |
V_{\text{torsion}}(\phi) = \frac{d_0}{2} \left(\phi - \phi^0\right). |
| 1899 |
+ |
\end{equation*} |
| 1900 |
+ |
\end{itemize} |
| 1901 |
|
|
| 1902 |
< |
\section{\label{section:clay}The CLAY force field} |
| 1902 |
> |
Most torsion types don't require specific angle information in the |
| 1903 |
> |
parameters since they are typically expressed in cosine polynomials. |
| 1904 |
> |
{\tt Charmm} and {\tt Harmonic} torsions are a bit different. {\tt |
| 1905 |
> |
Charmm} torsion types require a set of phase angles, $\delta_n$ that |
| 1906 |
> |
are expressed in degrees, and associated periodicity numbers, $n$. |
| 1907 |
> |
{\tt Harmonic} torsions have an equilibrium torsion angle, $\phi_0$ |
| 1908 |
> |
that is measured in degrees, while $d_0$ has units of |
| 1909 |
> |
kcal/mol/degrees$^2$. All other torsion parameters are measured in |
| 1910 |
> |
units of kcal/mol. |
| 1911 |
|
|
| 1912 |
< |
The {\sc clay} force field is based on an ionic (nonbonded) |
| 1913 |
< |
description of the metal-oxygen interactions associated with hydrated |
| 1914 |
< |
phases. All atoms are represented as point charges and are allowed |
| 1915 |
< |
complete translational freedom. Metal-oxygen interactions are based on |
| 1916 |
< |
a simple Lennard-Jones potential combined with electrostatics. The |
| 1917 |
< |
empirical parameters were optimized by Cygan {\it et |
| 1918 |
< |
al.}\cite{Cygan04} on the basis of known mineral structures, and |
| 1919 |
< |
partial atomic charges were derived from periodic DFT quantum chemical |
| 1920 |
< |
calculations of simple oxide, hydroxide, and oxyhydroxide model |
| 1921 |
< |
compounds with well-defined structures. |
| 1912 |
> |
\begin{lstlisting}[caption={[An example of a TorsionTypes block.] A |
| 1913 |
> |
simple example of a TorsionTypes block. Energy constants are given in |
| 1914 |
> |
kcal / mol, and when required by the form, $\delta$ angles are given |
| 1915 |
> |
in degrees.}, |
| 1916 |
> |
label={sch:TorsionTypes}] |
| 1917 |
> |
begin TorsionTypes |
| 1918 |
> |
//Cubic |
| 1919 |
> |
//Atom1 Atom2 Atom3 Atom4 Cubic k3 k2 k1 k0 |
| 1920 |
> |
CH2 CH2 CH2 CH2 Cubic 5.9602 -0.2568 -3.802 2.1586 |
| 1921 |
> |
CH2 CH CH CH2 Cubic 3.3254 -0.4215 -1.686 1.1661 |
| 1922 |
> |
//Trappe |
| 1923 |
> |
//Atom1 Atom2 Atom3 Atom4 Trappe c0 c1 c2 c3 |
| 1924 |
> |
CH3 CH2 CH2 SH Trappe 0.10507 -0.10342 0.03668 0.60874 |
| 1925 |
> |
//Charmm |
| 1926 |
> |
//Atom1 Atom2 Atom3 Atom4 Charmm Kchi n delta [Kchi n delta] |
| 1927 |
> |
CT CT CT C Charmm 0.156 3 0.00 |
| 1928 |
> |
OH CT CT OH Charmm 0.144 3 0.00 1.175 2 0 |
| 1929 |
> |
HC CT CM CM Charmm 1.150 1 0.00 0.38 3 180 |
| 1930 |
> |
//Quartic |
| 1931 |
> |
//Atom1 Atom2 Atom3 Atom4 Quartic k4 k3 k2 k1 k0 |
| 1932 |
> |
//Polynomial |
| 1933 |
> |
//Atom1 Atom2 Atom3 Atom4 Polynomial n Kn [m Km] |
| 1934 |
> |
S CH2 CH2 C Polynomial 0 2.218 1 2.905 2 -3.136 3 -0.7313 4 6.272 5 -7.528 |
| 1935 |
> |
end TorsionTypes |
| 1936 |
> |
\end{lstlisting} |
| 1937 |
> |
|
| 1938 |
> |
Note that the parameters for a particular torsion type are the same |
| 1939 |
> |
for any torsional quartet of the same atomic types (in the same or |
| 1940 |
> |
reversed order). |
| 1941 |
> |
|
| 1942 |
> |
\subsection{\label{section:ffInversion}The InversionTypes block} |
| 1943 |
> |
|
| 1944 |
> |
Inversion potentials are often added to force fields to enforce |
| 1945 |
> |
planarity around $sp^2$-hybridized carbons or to correct vibrational |
| 1946 |
> |
frequencies for umbrella-like vibrational modes for central atoms |
| 1947 |
> |
bonded to exactly three satellite groups. |
| 1948 |
> |
|
| 1949 |
> |
In OpenMD's version of an inversion, the central atom is entered {\it |
| 1950 |
> |
first} in each line in the {\tt InversionTypes} block. Note that |
| 1951 |
> |
this is quite different than how other codes treat Improper torsional |
| 1952 |
> |
potentials to mimic inversion behavior. In some other widely-used |
| 1953 |
> |
simulation packages, the central atom is treated as atom 3 in a |
| 1954 |
> |
standard torsion form: |
| 1955 |
> |
\begin{itemize} |
| 1956 |
> |
\item OpenMD: I - (J - K - L) (e.g. I is $sp^2$ hybridized carbon) |
| 1957 |
> |
\item AMBER: I - J - K - L (e.g. K is $sp^2$ hybridized carbon) |
| 1958 |
> |
\end{itemize} |
| 1959 |
> |
|
| 1960 |
> |
The inversion angle itself is defined as: |
| 1961 |
> |
\begin{equation} |
| 1962 |
> |
\cos\omega_{i-jkl} = \left(\hat{\mathbf{r}}_{il} \times |
| 1963 |
> |
\hat{\mathbf{r}}_{ij}\right)\cdot\left( \hat{\mathbf{r}}_{il} \times |
| 1964 |
> |
\hat{\mathbf{r}}_{ik}\right) |
| 1965 |
> |
\end{equation} |
| 1966 |
> |
Here, $\hat{\mathbf{r}}_{\alpha\beta}$ are the set of unit bond |
| 1967 |
> |
vectors between the central atoms $i$, and the satellite atoms $j$, |
| 1968 |
> |
$k$, and $l$. Note that other definitions of inversion angles are |
| 1969 |
> |
possible, so users are encouraged to be particularly careful when |
| 1970 |
> |
converting other force field files for use with OpenMD. |
| 1971 |
> |
|
| 1972 |
> |
There are many common ways to create planarity or umbrella behavior in |
| 1973 |
> |
a potential energy function, and OpenMD implements a number of the |
| 1974 |
> |
more common functions: |
| 1975 |
> |
\begin{itemize} |
| 1976 |
> |
\item {\tt ImproperCosine}: |
| 1977 |
> |
\begin{equation*} |
| 1978 |
> |
V_{\text{torsion}}(\omega) = \sum_n \frac{K_n}{2} \left( 1 + cos(n |
| 1979 |
> |
\omega - \delta_n) \right), |
| 1980 |
> |
\end{equation*} |
| 1981 |
> |
\item {\tt AmberImproper}: |
| 1982 |
> |
\begin{equation*} |
| 1983 |
> |
V_{\text{torsion}}(\omega) = \frac{v}{2} (1 - \cos\left(2 \left(\omega - \omega_0\right)\right), |
| 1984 |
> |
\end{equation*} |
| 1985 |
> |
\item {\tt Harmonic}: |
| 1986 |
> |
\begin{equation*} |
| 1987 |
> |
V_{\text{torsion}}(\omega) = \frac{d}{2} \left(\omega - \omega_0\right). |
| 1988 |
> |
\end{equation*} |
| 1989 |
> |
\end{itemize} |
| 1990 |
> |
\begin{lstlisting}[caption={[An example of an InversionTypes block.] A |
| 1991 |
> |
simple example of a InversionTypes block. Angles ($\delta_n$ and |
| 1992 |
> |
$\omega_0$) angles are given in degrees, while energy parameters ($v, |
| 1993 |
> |
K_n$) are given in kcal / mol. The Harmonic Inversion type has a |
| 1994 |
> |
force constant that must be given in kcal/mol/degrees$^2$.}, |
| 1995 |
> |
label={sch:InversionTypes}] |
| 1996 |
> |
begin InversionTypes |
| 1997 |
> |
//Harmonic |
| 1998 |
> |
//Atom1 Atom2 Atom3 Atom4 Harmonic d(kcal/mol/deg^2) omega0 |
| 1999 |
> |
RCHar3 X X X Harmonic 1.21876e-2 180.0 |
| 2000 |
> |
//AmberImproper |
| 2001 |
> |
//Atom1 Atom2 Atom3 Atom4 AmberImproper v(kcal/mol) |
| 2002 |
> |
C CT N O AmberImproper 10.500000 |
| 2003 |
> |
CA CA CA CT AmberImproper 1.100000 |
| 2004 |
> |
//ImproperCosine |
| 2005 |
> |
//Atom1 Atom2 Atom3 Atom4 ImproperCosine Kn n delta_n [Kn n delta_n] |
| 2006 |
> |
end InversionTypes |
| 2007 |
> |
\end{lstlisting} |
| 2008 |
> |
|
| 2009 |
> |
\section{\label{section::ffLongRange}Long Range Interactions} |
| 2010 |
> |
|
| 2011 |
> |
Calculating the long-range (non-bonded) potential involves a sum over |
| 2012 |
> |
all pairs of atoms (except for those atoms which are involved in a |
| 2013 |
> |
bond, bend, or torsion with each other). Many of these interactions |
| 2014 |
> |
can be inferred from the AtomTypes, |
| 2015 |
|
|
| 2016 |
+ |
\subsection{\label{section:ffNBinteraction}The NonBondedInteractions |
| 2017 |
+ |
block} |
| 2018 |
|
|
| 2019 |
+ |
The user might want like to specify explicit or special interactions |
| 2020 |
+ |
that override the default non-bodned interactions that are inferred |
| 2021 |
+ |
from the AtomTypes. To do this, OpenMD implements a |
| 2022 |
+ |
NonBondedInteractions block to allow the user to specify the following |
| 2023 |
+ |
(pair-wise) non-bonded interactions: |
| 2024 |
+ |
|
| 2025 |
+ |
\begin{itemize} |
| 2026 |
+ |
\item {\tt LennardJones}: |
| 2027 |
+ |
\begin{equation*} |
| 2028 |
+ |
V_{\text{NB}}(r) = 4 \epsilon_{ij} \left( |
| 2029 |
+ |
\left(\frac{\sigma_{ij}}{r} \right)^{12} - |
| 2030 |
+ |
\left(\frac{\sigma_{ij}}{r} \right)^{6} \right), |
| 2031 |
+ |
\end{equation*} |
| 2032 |
+ |
\item {\tt ShiftedMorse}: |
| 2033 |
+ |
\begin{equation*} |
| 2034 |
+ |
V_{\text{NB}}(r) = D_{ij} \left( e^{-2 \beta_{ij} (r - |
| 2035 |
+ |
r^0)} - 2 e^{- \beta_{ij} (r - |
| 2036 |
+ |
r^0)} \right), |
| 2037 |
+ |
\end{equation*} |
| 2038 |
+ |
\item {\tt RepulsiveMorse}: |
| 2039 |
+ |
\begin{equation*} |
| 2040 |
+ |
V_{\text{NB}}(r) = D_{ij} \left( e^{-2 \beta_{ij} (r - |
| 2041 |
+ |
r^0)} \right), |
| 2042 |
+ |
\end{equation*} |
| 2043 |
+ |
\item {\tt RepulsivePower}: |
| 2044 |
+ |
\begin{equation*} |
| 2045 |
+ |
V_{\text{NB}}(r) = \epsilon_{ij} |
| 2046 |
+ |
\left(\frac{\sigma_{ij}}{r} \right)^{n_{ij}}. |
| 2047 |
+ |
\end{equation*} |
| 2048 |
+ |
\end{itemize} |
| 2049 |
+ |
|
| 2050 |
+ |
\begin{lstlisting}[caption={[An example of a NonBondedInteractions block.] A |
| 2051 |
+ |
simple example of a NonBondedInteractions block. Distances ($\sigma, |
| 2052 |
+ |
r_0$) are given in \AA, while energies ($\epsilon, D0$) are in |
| 2053 |
+ |
kcal/mol. The Morse potentials have an additional parameter $\beta_0$ |
| 2054 |
+ |
which is in units of \AA$^{-1}$.}, |
| 2055 |
+ |
label={sch:InversionTypes}] |
| 2056 |
+ |
begin NonBondedInteractions |
| 2057 |
+ |
|
| 2058 |
+ |
//Lennard-Jones |
| 2059 |
+ |
//Atom1 Atom2 LennardJones sigma epsilon |
| 2060 |
+ |
Au CH3 LennardJones 3.54 0.2146 |
| 2061 |
+ |
Au CH2 LennardJones 3.54 0.1749 |
| 2062 |
+ |
Au CH LennardJones 3.54 0.1749 |
| 2063 |
+ |
Au S LennardJones 2.40 8.465 |
| 2064 |
+ |
|
| 2065 |
+ |
//Shifted Morse |
| 2066 |
+ |
//Atom1 Atom2 ShiftedMorse r0 D0 beta0 |
| 2067 |
+ |
Au O_SPCE ShiftedMorse 3.70 0.0424 0.769 |
| 2068 |
+ |
|
| 2069 |
+ |
//Repulsive Morse |
| 2070 |
+ |
//Atom1 Atom2 RepulsiveMorse r0 D0 beta0 |
| 2071 |
+ |
Au H_SPCE RepulsiveMorse -1.00 0.00850 0.769 |
| 2072 |
+ |
|
| 2073 |
+ |
//Repulsive Power |
| 2074 |
+ |
//Atom1 Atom2 RepulsivePower sigma epsilon n |
| 2075 |
+ |
Au ON RepulsivePower 3.47005 0.186208 11 |
| 2076 |
+ |
Au NO RepulsivePower 3.53955 0.168629 11 |
| 2077 |
+ |
end NonBondedInteractions |
| 2078 |
+ |
\end{lstlisting} |
| 2079 |
+ |
|
| 2080 |
|
\section{\label{section:electrostatics}Electrostatics} |
| 2081 |
|
|
| 2082 |
< |
To aid in performing simulations in more traditional force fields, we |
| 2083 |
< |
have added routines to carry out electrostatic interactions using a |
| 2084 |
< |
number of different electrostatic summation methods. These methods |
| 2085 |
< |
are extended from the damped and cutoff-neutralized Coulombic sum |
| 2086 |
< |
originally proposed by Wolf, {\it et al.}\cite{Wolf99} One of these, |
| 2087 |
< |
the damped shifted force method, shows a remarkable ability to |
| 2088 |
< |
reproduce the energetic and dynamic characteristics exhibited by |
| 2089 |
< |
simulations employing lattice summation techniques. The basic idea is |
| 2090 |
< |
to construct well-behaved real-space summation methods using two tricks: |
| 2082 |
> |
Because nearly all force fields involve electrostatic interactions in |
| 2083 |
> |
one form or another, OpenMD implements a number of different |
| 2084 |
> |
electrostatic summation methods. These methods are extended from the |
| 2085 |
> |
damped and cutoff-neutralized Coulombic sum originally proposed by |
| 2086 |
> |
Wolf, {\it et al.}\cite{Wolf99} One of these, the damped shifted force |
| 2087 |
> |
method, shows a remarkable ability to reproduce the energetic and |
| 2088 |
> |
dynamic characteristics exhibited by simulations employing lattice |
| 2089 |
> |
summation techniques. The basic idea is to construct well-behaved |
| 2090 |
> |
real-space summation methods using two tricks: |
| 2091 |
|
\begin{enumerate} |
| 2092 |
|
\item shifting through the use of image charges, and |
| 2093 |
|
\item damping the electrostatic interaction. |
| 2206 |
|
\end{equation} |
| 2207 |
|
the shifted potential (eq. (\ref{eq:SPPot})) becomes |
| 2208 |
|
\begin{equation} |
| 2209 |
< |
V_{\textrm{DSP}}(r) = q_iq_j\left(\frac{\textrm{erfc}\left(\alpha r\right)}{r}-\ |
| 1728 |
< |
frac{\textrm{erfc}\left(\alpha R_\textrm{c}\right)}{R_\textrm{c}}\right) \quad r |
| 2209 |
> |
V_{\textrm{DSP}}(r) = q_iq_j\left(\frac{\textrm{erfc}\left(\alpha r\right)}{r}-\frac{\textrm{erfc}\left(\alpha R_\textrm{c}\right)}{R_\textrm{c}}\right) \quad r |
| 2210 |
|
\leqslant R_\textrm{c}, |
| 2211 |
|
\label{eq:DSPPot} |
| 2212 |
|
\end{equation} |
| 2250 |
|
this reason, the default electrostatic summation method utilized by |
| 2251 |
|
{\sc OpenMD} is the DSF (Eq. \ref{eq:DSFPot}) with a damping parameter |
| 2252 |
|
($\alpha$) that is set algorithmically from the cutoff radius. |
| 2253 |
+ |
|
| 2254 |
+ |
|
| 2255 |
+ |
\section{\label{section:cutoffGroups}Switching Functions} |
| 2256 |
+ |
|
| 2257 |
+ |
Calculating the the long-range interactions for $N$ atoms involves |
| 2258 |
+ |
$N(N-1)/2$ evaluations of atomic distances if it is done in a brute |
| 2259 |
+ |
force manner. To reduce the number of distance evaluations between |
| 2260 |
+ |
pairs of atoms, {\sc OpenMD} allows the use of hard or switched |
| 2261 |
+ |
cutoffs with Verlet neighbor lists.\cite{Allen87} Neutral groups which |
| 2262 |
+ |
contain charges can exhibit pathological forces unless the cutoff is |
| 2263 |
+ |
applied to the neutral groups evenly instead of to the individual |
| 2264 |
+ |
atoms.\cite{leach01:mm} {\sc OpenMD} allows users to specify cutoff |
| 2265 |
+ |
groups which may contain an arbitrary number of atoms in the molecule. |
| 2266 |
+ |
Atoms in a cutoff group are treated as a single unit for the |
| 2267 |
+ |
evaluation of the switching function: |
| 2268 |
+ |
\begin{equation} |
| 2269 |
+ |
V_{\mathrm{long-range}} = \sum_{a} \sum_{b>a} s(r_{ab}) \sum_{i \in a} \sum_{j \in b} V_{ij}(r_{ij}), |
| 2270 |
+ |
\end{equation} |
| 2271 |
+ |
where $r_{ab}$ is the distance between the centers of mass of the two |
| 2272 |
+ |
cutoff groups ($a$ and $b$). |
| 2273 |
+ |
|
| 2274 |
+ |
The sums over $a$ and $b$ are over the cutoff groups that are present |
| 2275 |
+ |
in the simulation. Atoms which are not explicitly defined as members |
| 2276 |
+ |
of a {\tt cutoffGroup} are treated as a group consisting of only one |
| 2277 |
+ |
atom. The switching function, $s(r)$ is the standard cubic switching |
| 2278 |
+ |
function, |
| 2279 |
+ |
\begin{equation} |
| 2280 |
+ |
S(r) = |
| 2281 |
+ |
\begin{cases} |
| 2282 |
+ |
1 & \text{if $r \le r_{\text{sw}}$},\\ |
| 2283 |
+ |
\frac{(r_{\text{cut}} + 2r - 3r_{\text{sw}})(r_{\text{cut}} - r)^2} |
| 2284 |
+ |
{(r_{\text{cut}} - r_{\text{sw}})^3} |
| 2285 |
+ |
& \text{if $r_{\text{sw}} < r \le r_{\text{cut}}$}, \\ |
| 2286 |
+ |
0 & \text{if $r > r_{\text{cut}}$.} |
| 2287 |
+ |
\end{cases} |
| 2288 |
+ |
\label{eq:dipoleSwitching} |
| 2289 |
+ |
\end{equation} |
| 2290 |
+ |
Here, $r_{\text{sw}}$ is the {\tt switchingRadius}, or the distance |
| 2291 |
+ |
beyond which interactions are reduced, and $r_{\text{cut}}$ is the |
| 2292 |
+ |
{\tt cutoffRadius}, or the distance at which interactions are |
| 2293 |
+ |
truncated. |
| 2294 |
+ |
|
| 2295 |
+ |
Users of {\sc OpenMD} do not need to specify the {\tt cutoffRadius} or |
| 2296 |
+ |
{\tt switchingRadius}. |
| 2297 |
+ |
If the {\tt cutoffRadius} was not explicitly set, OpenMD will attempt |
| 2298 |
+ |
to guess an appropriate choice. If the system contains electrostatic |
| 2299 |
+ |
atoms, the default cutoff radius is 12 \AA. In systems without |
| 2300 |
+ |
electrostatic (charge or multipolar) atoms, the atom types present in the simulation will be |
| 2301 |
+ |
polled for suggested cutoff values (e.g. $2.5 max(\left\{ \sigma |
| 2302 |
+ |
\right\})$ for Lennard-Jones atoms. The largest suggested value |
| 2303 |
+ |
that was found will be used. |
| 2304 |
+ |
|
| 2305 |
+ |
By default, OpenMD uses shifted force potentials to force the |
| 2306 |
+ |
potential energy and forces to smoothly approach zero at the cutoff |
| 2307 |
+ |
radius. If the user would like to use another cutoff method |
| 2308 |
+ |
they may do so by setting the {\tt cutoffMethod} parameter to: |
| 2309 |
+ |
\begin{itemize} |
| 2310 |
+ |
\item {\tt HARD} |
| 2311 |
+ |
\item {\tt SWITCHED} |
| 2312 |
+ |
\item {\tt SHIFTED\_FORCE} (default): |
| 2313 |
+ |
\item {\tt TAYLOR\_SHIFTED} |
| 2314 |
+ |
\item {\tt SHIFTED\_POTENTIAL} |
| 2315 |
+ |
\end{itemize} |
| 2316 |
+ |
|
| 2317 |
+ |
The {\tt switchingRadius} is set to a default value of 95\% of the |
| 2318 |
+ |
{\tt cutoffRadius}. In the special case of a simulation containing |
| 2319 |
+ |
{\it only} Lennard-Jones atoms, the default switching radius takes the |
| 2320 |
+ |
same value as the cutoff radius, and {\sc OpenMD} will use a shifted |
| 2321 |
+ |
potential to remove discontinuities in the potential at the cutoff. |
| 2322 |
+ |
Both radii may be specified in the meta-data file. |
| 2323 |
|
|
| 2324 |
+ |
|
| 2325 |
|
\section{\label{section:pbc}Periodic Boundary Conditions} |
| 2326 |
|
|
| 2327 |
|
\newcommand{\roundme}{\operatorname{round}} |
