| 9 |
|
\usepackage{longtable} |
| 10 |
|
\pagestyle{plain} |
| 11 |
|
\pagenumbering{arabic} |
| 12 |
+ |
\usepackage{floatrow} |
| 13 |
|
\oddsidemargin 0.0cm |
| 14 |
|
\evensidemargin 0.0cm |
| 15 |
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\topmargin -21pt |
| 18 |
|
\textwidth 6.5in |
| 19 |
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\brokenpenalty=10000 |
| 20 |
|
\renewcommand{\baselinestretch}{1.2} |
| 21 |
+ |
\usepackage[square, comma, sort&compress]{natbib} |
| 22 |
+ |
\bibpunct{[}{]}{,}{n}{}{;} |
| 23 |
|
|
| 24 |
+ |
\DeclareFloatFont{tiny}{\scriptsize}% "scriptsize" is defined by floatrow, "tiny" not |
| 25 |
+ |
\floatsetup[table]{font=tiny} |
| 26 |
+ |
|
| 27 |
+ |
|
| 28 |
|
%\renewcommand\citemid{\ } % no comma in optional reference note |
| 29 |
< |
\lstset{language=C,frame=TB,basicstyle=\tiny,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 |
|
\renewcommand{\lstlistingname}{Scheme} |
| 45 |
|
\newcolumntype{H}{p{0.75in}} |
| 46 |
|
\newcolumntype{I}{p{5in}} |
| 47 |
|
|
| 48 |
+ |
\newcolumntype{J}{p{1.5in}} |
| 49 |
+ |
\newcolumntype{K}{p{1.2in}} |
| 50 |
+ |
\newcolumntype{L}{p{1.5in}} |
| 51 |
+ |
\newcolumntype{M}{p{1.55in}} |
| 52 |
|
|
| 42 |
– |
\title{{\sc OpenMD}: Molecular Dynamics in the Open} |
| 53 |
|
|
| 54 |
< |
\author{Shenyu Kuang, Chunlei Li, Charles F. Vardeman II, \\ |
| 45 |
< |
Teng Lin, Christopher J. Fennell, Xiuquan Sun, \\ |
| 46 |
< |
Kyle Daily, Yang Zheng, Matthew A. Meineke, and J. Daniel Gezelter\\ |
| 47 |
< |
Department of Chemistry and Biochemistry\\ |
| 48 |
< |
University of Notre Dame\\ |
| 49 |
< |
Notre Dame, Indiana 46556} |
| 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, \\ |
| 58 |
+ |
Teng Lin, Charles F. Vardeman II, Christopher J. Fennell, Shenyu |
| 59 |
+ |
Kuang, Xiuquan Sun, \\ |
| 60 |
+ |
Chunlei Li, Kyle Daily, Yang Zheng, Matthew A. Meineke, and \\ |
| 61 |
+ |
J. Daniel Gezelter \\ |
| 62 |
+ |
Department of Chemistry and Biochemistry\\ |
| 63 |
+ |
University of Notre Dame\\ |
| 64 |
+ |
Notre Dame, Indiana 46556} |
| 65 |
+ |
|
| 66 |
|
\maketitle |
| 67 |
|
|
| 68 |
|
\section*{Preface} |
| 79 |
|
that is easy to learn. |
| 80 |
|
|
| 81 |
|
\tableofcontents |
| 82 |
< |
%\listoffigures |
| 83 |
< |
%\listoftables |
| 82 |
> |
\listoffigures |
| 83 |
> |
\listoftables |
| 84 |
|
|
| 85 |
|
\mainmatter |
| 86 |
|
|
| 147 |
|
leave an interaction region. |
| 148 |
|
|
| 149 |
|
{\tt Atoms} may also be grouped in more traditional ways into {\tt |
| 150 |
< |
bonds}, {\tt bends}, and {\tt torsions}. These groupings allow the |
| 151 |
< |
correct choice of interaction parameters for short-range interactions |
| 152 |
< |
to be chosen from the definitions in the {\tt forceField}. |
| 150 |
> |
bonds}, {\tt bends}, {\tt torsions}, and {\tt inversions}. These |
| 151 |
> |
groupings allow the correct choice of interaction parameters for |
| 152 |
> |
short-range interactions to be chosen from the definitions in the {\tt |
| 153 |
> |
forceField}. |
| 154 |
|
|
| 155 |
|
All of these groups of {\tt atoms} are brought together in the {\tt |
| 156 |
|
molecule}, which is the fundamental structure for setting up and {\sc |
| 506 |
|
\endhead |
| 507 |
|
\hline |
| 508 |
|
\endfoot |
| 509 |
< |
{\tt forceField} & string & Sets the force field. & Possible force |
| 510 |
< |
fields are DUFF, WATER, LJ, EAM, SC, and CLAY. \\ |
| 509 |
> |
{\tt forceField} & string & Sets the base name for the force field file & |
| 510 |
> |
OpenMD appends a {\tt .frc} to the end of this to look for a force |
| 511 |
> |
field file.\\ |
| 512 |
|
{\tt component} & & Defines the molecular components of the system & |
| 513 |
|
Every {\tt $<$MetaData$>$} block must have a component statement. \\ |
| 514 |
|
{\tt minimizer} & string & Chooses a minimizer & Possible minimizers |
| 515 |
|
are SD and CG. Either {\tt ensemble} or {\tt minimizer} must be specified. \\ |
| 516 |
|
{\tt ensemble} & string & Sets the ensemble. & Possible ensembles are |
| 517 |
< |
NVE, NVT, NPTi, NPAT, NPTf, NPTxyz, and LD. Either {\tt ensemble} |
| 517 |
> |
NVE, NVT, NPTi, NPAT, NPTf, NPTxyz, LD and LangevinHull. Either {\tt ensemble} |
| 518 |
|
or {\tt minimizer} must be specified. \\ |
| 519 |
|
{\tt dt} & fs & Sets the time step. & Selection of {\tt dt} should be |
| 520 |
|
small enough to sample the fastest motion of the simulation. ({\tt |
| 578 |
|
default is the first eight of these columns in order.) \\ |
| 579 |
|
& & \multicolumn{2}{p{3.5in}}{Allowed |
| 580 |
|
column names are: {\sc time, total\_energy, potential\_energy, kinetic\_energy, |
| 581 |
< |
temperature, pressure, volume, conserved\_quantity, |
| 581 |
> |
temperature, pressure, volume, conserved\_quantity, hullvolume, gyrvolume, |
| 582 |
|
translational\_kinetic, rotational\_kinetic, long\_range\_potential, |
| 583 |
|
short\_range\_potential, vanderwaals\_potential, |
| 584 |
< |
electrostatic\_potential, bond\_potential, bend\_potential, |
| 585 |
< |
dihedral\_potential, improper\_potential, vraw, vharm, |
| 586 |
< |
pressure\_tensor\_x, pressure\_tensor\_y, pressure\_tensor\_z}} \\ |
| 584 |
> |
electrostatic\_potential, metallic\_potential, |
| 585 |
> |
hydrogen\_bonding\_potential, bond\_potential, bend\_potential, |
| 586 |
> |
dihedral\_potential, inversion\_potential, raw\_potential, restraint\_potential, |
| 587 |
> |
pressure\_tensor, system\_dipole, heatflux, electronic\_temperature}} \\ |
| 588 |
|
{\tt printPressureTensor} & logical & sets whether {\sc OpenMD} will print |
| 589 |
|
out the pressure tensor & can be useful for calculations of the bulk |
| 590 |
|
modulus \\ |
| 725 |
|
<MetaData> |
| 726 |
|
molecule{ |
| 727 |
|
name = "I2"; |
| 728 |
< |
atom[0]{ |
| 729 |
< |
type = "I"; |
| 730 |
< |
} |
| 713 |
< |
atom[1]{ |
| 714 |
< |
type = "I"; |
| 715 |
< |
} |
| 716 |
< |
bond{ |
| 717 |
< |
members( 0, 1); |
| 718 |
< |
} |
| 728 |
> |
atom[0]{ type = "I"; } |
| 729 |
> |
atom[1]{ type = "I"; } |
| 730 |
> |
bond{ members( 0, 1); } |
| 731 |
|
} |
| 732 |
|
molecule{ |
| 733 |
|
name = "HCl" |
| 734 |
< |
atom[0]{ |
| 735 |
< |
type = "H"; |
| 736 |
< |
} |
| 725 |
< |
atom[1]{ |
| 726 |
< |
type = "Cl"; |
| 727 |
< |
} |
| 728 |
< |
bond{ |
| 729 |
< |
members( 0, 1); |
| 730 |
< |
} |
| 734 |
> |
atom[0]{ type = "H";} |
| 735 |
> |
atom[1]{ type = "Cl";} |
| 736 |
> |
bond{ members( 0, 1); } |
| 737 |
|
} |
| 738 |
|
component{ |
| 739 |
|
type = "HCl"; |
| 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:empiricalEnergy}The Empirical Energy |
| 777 |
< |
Functions} |
| 782 |
> |
\chapter{\label{chapter:forceFields}Force Fields} |
| 783 |
|
|
| 784 |
< |
Like many simulation packages, {\sc OpenMD} splits the potential energy |
| 785 |
< |
into the short-ranged (bonded) portion and a long-range (non-bonded) |
| 786 |
< |
potential, |
| 784 |
> |
Like many molecular simulation packages, {\sc OpenMD} splits the |
| 785 |
> |
potential energy into the short-ranged (bonded) portion and a |
| 786 |
> |
long-range (non-bonded) potential, |
| 787 |
|
\begin{equation} |
| 788 |
|
V = V_{\mathrm{short-range}} + V_{\mathrm{long-range}}. |
| 789 |
|
\end{equation} |
| 790 |
< |
The short-ranged portion includes the explicit bonds, bends, and |
| 791 |
< |
torsions which have been defined in the meta-data file for the |
| 792 |
< |
molecules which are present in the simulation. The functional forms and |
| 793 |
< |
parameters for these interactions are defined by the force field which |
| 794 |
< |
is chosen. |
| 790 |
> |
The short-ranged portion includes the bonds, bends, torsions, and |
| 791 |
> |
inversions which have been defined in the meta-data file for the |
| 792 |
> |
molecules. The functional forms and parameters for these interactions |
| 793 |
> |
are defined by the force field which is selected in the MetaData |
| 794 |
> |
section. |
| 795 |
|
|
| 796 |
< |
Calculating the long-range (non-bonded) potential involves a sum over |
| 797 |
< |
all pairs of atoms (except for those atoms which are involved in a |
| 793 |
< |
bond, bend, or torsion with each other). If done poorly, calculating |
| 794 |
< |
the the long-range interactions for $N$ atoms would involve $N(N-1)/2$ |
| 795 |
< |
evaluations of atomic distances. To reduce the number of distance |
| 796 |
< |
evaluations between pairs of atoms, {\sc OpenMD} uses a switched cutoff |
| 797 |
< |
with Verlet neighbor lists.\cite{Allen87} It is well known that |
| 798 |
< |
neutral groups which contain charges will exhibit pathological forces |
| 799 |
< |
unless the cutoff is applied to the neutral groups evenly instead of |
| 800 |
< |
to the individual atoms.\cite{leach01:mm} {\sc OpenMD} allows users to |
| 801 |
< |
specify cutoff groups which may contain an arbitrary number of atoms |
| 802 |
< |
in the molecule. Atoms in a cutoff group are treated as a single unit |
| 803 |
< |
for the evaluation of the switching function: |
| 804 |
< |
\begin{equation} |
| 805 |
< |
V_{\mathrm{long-range}} = \sum_{a} \sum_{b>a} s(r_{ab}) \sum_{i \in a} \sum_{j \in b} V_{ij}(r_{ij}), |
| 806 |
< |
\end{equation} |
| 807 |
< |
where $r_{ab}$ is the distance between the centers of mass of the two |
| 808 |
< |
cutoff groups ($a$ and $b$). |
| 796 |
> |
\section{\label{section:divisionOfLabor}Separation into Internal and |
| 797 |
> |
Cross interactions} |
| 798 |
|
|
| 799 |
< |
The sums over $a$ and $b$ are over the cutoff groups that are present |
| 800 |
< |
in the simulation. Atoms which are not explicitly defined as members |
| 812 |
< |
of a {\tt cutoffGroup} are treated as a group consisting of only one |
| 813 |
< |
atom. The switching function, $s(r)$ is the standard cubic switching |
| 814 |
< |
function, |
| 799 |
> |
The classical potential energy function for a system composed of $N$ |
| 800 |
> |
molecules is traditionally written |
| 801 |
|
\begin{equation} |
| 802 |
< |
S(r) = |
| 803 |
< |
\begin{cases} |
| 804 |
< |
1 & \text{if $r \le r_{\text{sw}}$},\\ |
| 819 |
< |
\frac{(r_{\text{cut}} + 2r - 3r_{\text{sw}})(r_{\text{cut}} - r)^2} |
| 820 |
< |
{(r_{\text{cut}} - r_{\text{sw}})^3} |
| 821 |
< |
& \text{if $r_{\text{sw}} < r \le r_{\text{cut}}$}, \\ |
| 822 |
< |
0 & \text{if $r > r_{\text{cut}}$.} |
| 823 |
< |
\end{cases} |
| 824 |
< |
\label{eq:dipoleSwitching} |
| 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 |
< |
Here, $r_{\text{sw}}$ is the {\tt switchingRadius}, or the distance |
| 807 |
< |
beyond which interactions are reduced, and $r_{\text{cut}}$ is the |
| 808 |
< |
{\tt cutoffRadius}, or the distance at which interactions are |
| 809 |
< |
truncated. |
| 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 |
< |
Users of {\sc OpenMD} do not need to specify the {\tt cutoffRadius} or |
| 814 |
< |
{\tt switchingRadius}. In simulations containing only Lennard-Jones |
| 815 |
< |
atoms, the cutoff radius has a default value of $2.5\sigma_{ii}$, |
| 816 |
< |
where $\sigma_{ii}$ is the largest Lennard-Jones length parameter |
| 817 |
< |
present in the simulation. In simulations containing charged or |
| 836 |
< |
dipolar atoms, the default cutoff radius is $15 \mbox{\AA}$. |
| 813 |
> |
The types of atoms being simulated, as well as the specific functional |
| 814 |
> |
forms and parameters of the intra-molecular functions and the |
| 815 |
> |
long-range potentials are defined by the force field. In the following |
| 816 |
> |
sections we discuss the stucture of an OpenMD force field file and the |
| 817 |
> |
specification of blocks that may be present within these files. |
| 818 |
|
|
| 819 |
< |
The {\tt switchingRadius} is set to a default value of 95\% of the |
| 839 |
< |
{\tt cutoffRadius}. In the special case of a simulation containing |
| 840 |
< |
{\it only} Lennard-Jones atoms, the default switching radius takes the |
| 841 |
< |
same value as the cutoff radius, and {\sc OpenMD} will use a shifted |
| 842 |
< |
potential to remove discontinuities in the potential at the cutoff. |
| 843 |
< |
Both radii may be specified in the meta-data file. |
| 819 |
> |
\section{\label{section:frcFile}Force Field Files} |
| 820 |
|
|
| 821 |
< |
Force fields can be added to {\sc OpenMD}, although it comes with a few |
| 822 |
< |
simple examples (Lennard-Jones, {\sc duff}, {\sc water}, and {\sc |
| 823 |
< |
eam}) which are explained in the following sections. |
| 824 |
< |
|
| 825 |
< |
\section{\label{sec:LJPot}The Lennard Jones Force Field} |
| 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}. |
| 828 |
|
|
| 829 |
< |
The most basic force field implemented in {\sc OpenMD} is the |
| 830 |
< |
Lennard-Jones force field, which mimics the van der Waals interaction |
| 831 |
< |
at long distances and uses an empirical repulsion at short |
| 832 |
< |
distances. The Lennard-Jones potential is given by: |
| 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" |
| 835 |
> |
end Options |
| 836 |
> |
|
| 837 |
> |
begin BaseAtomTypes |
| 838 |
> |
//name mass |
| 839 |
> |
C 12.0107 |
| 840 |
> |
end BaseAtomTypes |
| 841 |
> |
|
| 842 |
> |
begin AtomTypes |
| 843 |
> |
//name base mass |
| 844 |
> |
CH4 C 16.05 |
| 845 |
> |
CH3 C 15.04 |
| 846 |
> |
CH2 C 14.03 |
| 847 |
> |
end AtomTypes |
| 848 |
> |
|
| 849 |
> |
begin LennardJonesAtomTypes |
| 850 |
> |
//name epsilon sigma |
| 851 |
> |
CH4 0.2941 3.73 |
| 852 |
> |
CH3 0.1947 3.75 |
| 853 |
> |
CH2 0.09140 3.95 |
| 854 |
> |
end LennardJonesAtomTypes |
| 855 |
> |
|
| 856 |
> |
begin BondTypes |
| 857 |
> |
//AT1 AT2 Type r0 k |
| 858 |
> |
CH3 CH3 Harmonic 1.526 260 |
| 859 |
> |
CH3 CH2 Harmonic 1.526 260 |
| 860 |
> |
CH2 CH2 Harmonic 1.526 260 |
| 861 |
> |
end BondTypes |
| 862 |
> |
|
| 863 |
> |
begin BendTypes |
| 864 |
> |
//AT1 AT2 AT3 Type theta0 k |
| 865 |
> |
CH3 CH2 CH3 Harmonic 114.0 124.19 |
| 866 |
> |
CH3 CH2 CH2 Harmonic 114.0 124.19 |
| 867 |
> |
CH2 CH2 CH2 Harmonic 114.0 124.19 |
| 868 |
> |
end BendTypes |
| 869 |
> |
|
| 870 |
> |
begin TorsionTypes |
| 871 |
> |
//AT1 AT2 AT3 AT4 Type |
| 872 |
> |
CH3 CH2 CH2 CH3 Trappe 0.0 0.70544 -0.13549 1.5723 |
| 873 |
> |
CH3 CH2 CH2 CH2 Trappe 0.0 0.70544 -0.13549 1.5723 |
| 874 |
> |
CH2 CH2 CH2 CH2 Trappe 0.0 0.70544 -0.13549 1.5723 |
| 875 |
> |
end TorsionTypes |
| 876 |
> |
\end{lstlisting} |
| 877 |
> |
|
| 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 |
| 882 |
> |
tags {\tt begin Options} and {\tt end Options}. Most options don't |
| 883 |
> |
need to be set as they come with fairly sensible default values, but |
| 884 |
> |
the various keywords and their possible values are given in Scheme |
| 885 |
> |
\ref{sch:optionsBlock}. |
| 886 |
> |
|
| 887 |
> |
\begin{lstlisting}[caption={[A force field Options block showing default values |
| 888 |
> |
for many force field options.] A force field Options block showing default values |
| 889 |
> |
for many force field options. Most of these options do not need to be |
| 890 |
> |
specified if the default values are working.}, |
| 891 |
> |
label={sch:optionsBlock}] |
| 892 |
> |
begin Options |
| 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" |
| 897 |
> |
EnergyMixingRule = "geometric" // can also be "arithmetic" or "hhg" |
| 898 |
> |
EnergyUnitScaling = 1.0 |
| 899 |
> |
MetallicEnergyUnitScaling = 1.0 |
| 900 |
> |
DistanceUnitScaling = 1.0 |
| 901 |
> |
AngleUnitScaling = 1.0 |
| 902 |
> |
TorsionAngleConvention = "180_is_trans" // can also be "0_is_trans" |
| 903 |
> |
vdW-12-scale = 0.0 |
| 904 |
> |
vdW-13-scale = 0.0 |
| 905 |
> |
vdW-14-scale = 0.0 |
| 906 |
> |
electrostatic-12-scale = 0.0 |
| 907 |
> |
electrostatic-13-scale = 0.0 |
| 908 |
> |
electrostatic-14-scale = 0.0 |
| 909 |
> |
GayBerneMu = 2.0 |
| 910 |
> |
GayBerneNu = 1.0 |
| 911 |
> |
EAMMixingMethod = "Johnson" // can also be "Daw" |
| 912 |
> |
end Options |
| 913 |
> |
\end{lstlisting} |
| 914 |
> |
|
| 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 |
| 919 |
> |
universal properties (i.e. does this atom have a fixed charge? What |
| 920 |
> |
is its mass?) Dynamic properties of an atom are not intended to be |
| 921 |
> |
properties of an atom type. A BaseAtomType can be used to build |
| 922 |
> |
extended sets of related atom types that all fall back to one |
| 923 |
> |
particular type. For example, one might want a series of atomTypes |
| 924 |
> |
that inherit from more basic types: |
| 925 |
> |
\begin{displaymath} |
| 926 |
> |
\mathtt{ALA-CA} \rightarrow \mathtt{CT} \rightarrow \mathtt{CSP3} \rightarrow \mathtt{C} |
| 927 |
> |
\end{displaymath} |
| 928 |
> |
where for each step to the right, the atomType falls back to more |
| 929 |
> |
primitive data. That is, the mass could be a property of the {\tt C} |
| 930 |
> |
type, while Lennard-Jones parameters could be properties of the {\tt |
| 931 |
> |
CSP3} type. {\tt CT} could have charge information and its own set |
| 932 |
> |
of Lennard-Jones parameter that override the CSP3 parameters. And the |
| 933 |
> |
{\tt ALA-CA} type might have specific torsion or charge information |
| 934 |
> |
that override the lower level types. A BaseAtomType contains only |
| 935 |
> |
information a primitive name and the mass of this atom type. |
| 936 |
> |
BaseAtomTypes can also be useful in creating files that can be easily |
| 937 |
> |
viewed in visualization programs. The {\tt Dump2XYZ} utility has the |
| 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 BaseAtomTypes |
| 942 |
> |
block.] A simple example of a BaseAtomTypes block.}, |
| 943 |
> |
label={sch:baseAtomTypesBlock}] |
| 944 |
> |
begin BaseAtomTypes |
| 945 |
> |
//Name mass (amu) |
| 946 |
> |
H 1.0079 |
| 947 |
> |
O 15.9994 |
| 948 |
> |
Si 28.0855 |
| 949 |
> |
Al 26.981538 |
| 950 |
> |
Mg 24.3050 |
| 951 |
> |
Ca 40.078 |
| 952 |
> |
Fe 55.845 |
| 953 |
> |
Li 6.941 |
| 954 |
> |
Na 22.98977 |
| 955 |
> |
K 39.0983 |
| 956 |
> |
Cs 132.90545 |
| 957 |
> |
Ca 40.078 |
| 958 |
> |
Ba 137.327 |
| 959 |
> |
Cl 35.453 |
| 960 |
> |
end BaseAtomTypes |
| 961 |
> |
\end{lstlisting} |
| 962 |
> |
|
| 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} |
| 967 |
> |
shows an example where multiple types of oxygen atoms can inherit mass |
| 968 |
> |
from the oxygen base type. |
| 969 |
> |
|
| 970 |
> |
\begin{lstlisting}[caption={[An example of a AtomTypes block.] A |
| 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 |
| 975 |
> |
//Name baseatomtype |
| 976 |
> |
h* H |
| 977 |
> |
ho H |
| 978 |
> |
o* O |
| 979 |
> |
oh O |
| 980 |
> |
ob O |
| 981 |
> |
obos O |
| 982 |
> |
obts O |
| 983 |
> |
obss O |
| 984 |
> |
ohs O |
| 985 |
> |
st Si |
| 986 |
> |
ao Al |
| 987 |
> |
at Al |
| 988 |
> |
mgo Mg |
| 989 |
> |
mgh Mg |
| 990 |
> |
cao Ca |
| 991 |
> |
cah Ca |
| 992 |
> |
feo Fe |
| 993 |
> |
lio Li |
| 994 |
> |
end AtomTypes |
| 995 |
> |
\end{lstlisting} |
| 996 |
> |
|
| 997 |
> |
\section{\label{section:ffDirectionalAtom}The DirectionalAtomTypes |
| 998 |
> |
block} |
| 999 |
> |
DirectionalAtoms have orientational degrees of freedom as well as |
| 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.}, |
| 1014 |
> |
label={sch:datomTypesBlock}] |
| 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 |
| 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 |
> |
\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[ |
| 1045 |
|
\end{equation} |
| 1046 |
|
where $r_{ij}$ is the distance between particles $i$ and $j$, |
| 1047 |
|
$\sigma_{ij}$ scales the length of the interaction, and |
| 1048 |
< |
$\epsilon_{ij}$ scales the well depth of the potential. Scheme |
| 866 |
< |
\ref{sch:LJFF} gives an example meta-data file that |
| 867 |
< |
sets up a system of 108 Ar particles to be simulated using the |
| 868 |
< |
Lennard-Jones force field. |
| 1048 |
> |
$\epsilon_{ij}$ scales the well depth of the potential. |
| 1049 |
|
|
| 870 |
– |
\begin{lstlisting}[float,caption={[Invocation of the Lennard-Jones |
| 871 |
– |
force field] A sample startup file for a small Lennard-Jones |
| 872 |
– |
simulation.},label={sch:LJFF}] |
| 873 |
– |
<OpenMD> |
| 874 |
– |
<MetaData> |
| 875 |
– |
#include "argon.md" |
| 876 |
– |
|
| 877 |
– |
component{ |
| 878 |
– |
type = "Ar"; |
| 879 |
– |
nMol = 108; |
| 880 |
– |
} |
| 881 |
– |
|
| 882 |
– |
forceField = "LJ"; |
| 883 |
– |
</MetaData> |
| 884 |
– |
<Snapshot> // not shown in this scheme |
| 885 |
– |
</Snapshot> |
| 886 |
– |
</OpenMD> |
| 887 |
– |
\end{lstlisting} |
| 888 |
– |
|
| 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 |
< |
\section{\label{section:DUFF}Dipolar Unified-Atom Force Field} |
| 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 |
< |
The dipolar unified-atom force field ({\sc duff}) was developed to |
| 1073 |
< |
simulate lipid bilayers. These types of simulations require a model |
| 1074 |
< |
capable of forming bilayers, while still being sufficiently |
| 1075 |
< |
computationally efficient to allow large systems ($\sim$100's of |
| 1076 |
< |
phospholipids, $\sim$1000's of waters) to be simulated for long times |
| 1077 |
< |
($\sim$10's of nanoseconds). With this goal in mind, {\sc duff} has no |
| 1078 |
< |
point charges. Charge-neutral distributions are replaced with dipoles, |
| 1079 |
< |
while most atoms and groups of atoms are reduced to Lennard-Jones |
| 1080 |
< |
interaction sites. This simplification reduces the length scale of |
| 1081 |
< |
long range interactions from $\frac{1}{r}$ to $\frac{1}{r^3}$, |
| 1082 |
< |
removing the need for the computationally expensive Ewald |
| 1083 |
< |
sum. Instead, Verlet neighbor-lists and cutoff radii are used for the |
| 1084 |
< |
dipolar interactions, and, if desired, a reaction field may be added |
| 1085 |
< |
to mimic longer range interactions. |
| 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 |
< |
As an example, lipid head-groups in {\sc duff} are represented as |
| 921 |
< |
point dipole interaction sites. Placing a dipole at the head group's |
| 922 |
< |
center of mass mimics the charge separation found in common |
| 923 |
< |
phospholipid head groups such as phosphatidylcholine.\cite{Cevc87} |
| 924 |
< |
Additionally, a large Lennard-Jones site is located at the |
| 925 |
< |
pseudoatom's center of mass. The model is illustrated by the red atom |
| 926 |
< |
in Fig.~\ref{fig:lipidModel}. The water model we use to |
| 927 |
< |
complement the dipoles of the lipids is a |
| 928 |
< |
reparameterization\cite{fennell04} of the soft sticky dipole (SSD) |
| 929 |
< |
model of Ichiye |
| 930 |
< |
\emph{et al.}\cite{liu96:new_model} |
| 1091 |
> |
\subsection{\label{section:ffCharge}The ChargeAtomTypes block} |
| 1092 |
|
|
| 1093 |
< |
\begin{figure} |
| 1094 |
< |
\centering |
| 1095 |
< |
\includegraphics[width=\linewidth]{lipidModel.pdf} |
| 1096 |
< |
\caption[A representation of a lipid model in {\sc duff}]{A |
| 1097 |
< |
representation of the lipid model. $\phi$ is the torsion angle, |
| 1098 |
< |
$\theta$ is the bend angle, and $\mu$ is the dipole moment of the head |
| 1099 |
< |
group.} |
| 1100 |
< |
\label{fig:lipidModel} |
| 1101 |
< |
\end{figure} |
| 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 |
< |
A set of scalable parameters has been used to model the alkyl groups |
| 1115 |
< |
with Lennard-Jones sites. For this, parameters from the TraPPE force |
| 1116 |
< |
field of Siepmann \emph{et al.}\cite{Siepmann1998} have been |
| 1117 |
< |
utilized. TraPPE is a unified-atom representation of n-alkanes which |
| 1118 |
< |
is parametrized against phase equilibria using Gibbs ensemble Monte |
| 1119 |
< |
Carlo simulation techniques.\cite{Siepmann1998} One of the advantages |
| 1120 |
< |
of TraPPE is that it generalizes the types of atoms in an alkyl chain |
| 1121 |
< |
to keep the number of pseudoatoms to a minimum; thus, the parameters |
| 1122 |
< |
for a unified atom such as $\text{CH}_2$ do not change depending on |
| 1123 |
< |
what species are bonded to it. |
| 1124 |
< |
|
| 1125 |
< |
As is required by TraPPE, {\sc duff} also constrains all bonds to be |
| 1126 |
< |
of fixed length. Typically, bond vibrations are the fastest motions in |
| 1127 |
< |
a molecular dynamic simulation. With these vibrations present, small |
| 1128 |
< |
time steps between force evaluations must be used to ensure adequate |
| 957 |
< |
energy conservation in the bond degrees of freedom. By constraining |
| 958 |
< |
the bond lengths, larger time steps may be used when integrating the |
| 959 |
< |
equations of motion. A simulation using {\sc duff} is illustrated in |
| 960 |
< |
Scheme \ref{sch:DUFF}. |
| 961 |
< |
|
| 962 |
< |
\begin{lstlisting}[float,caption={[Invocation of {\sc duff}]A portion |
| 963 |
< |
of a startup file showing a simulation utilizing {\sc |
| 964 |
< |
duff}},label={sch:DUFF}] |
| 965 |
< |
<OpenMD> |
| 966 |
< |
<MetaData> |
| 967 |
< |
#include "water.md" |
| 968 |
< |
#include "lipid.md" |
| 969 |
< |
|
| 970 |
< |
component{ |
| 971 |
< |
type = "simpleLipid_16"; |
| 972 |
< |
nMol = 60; |
| 973 |
< |
} |
| 974 |
< |
|
| 975 |
< |
component{ |
| 976 |
< |
type = "SSD_water"; |
| 977 |
< |
nMol = 1936; |
| 978 |
< |
} |
| 979 |
< |
|
| 980 |
< |
forceField = "DUFF"; |
| 981 |
< |
</MetaData> |
| 982 |
< |
<Snapshot> // not shown in this scheme |
| 983 |
< |
</Snapshot> |
| 984 |
< |
</OpenMD> |
| 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:energyFunctions}{\sc duff} Energy Functions} |
| 1132 |
< |
|
| 1133 |
< |
The total potential energy function in {\sc duff} is |
| 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 |
< |
V = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
| 992 |
< |
+ \sum^{N-1}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}}, |
| 993 |
< |
\label{eq:totalPotential} |
| 1137 |
> |
U_{\bf{ab}}(r)=\hat{M}_{\bf a} \hat{M}_{\bf b} \frac{1}{r} \label{kernel}. |
| 1138 |
|
\end{equation} |
| 1139 |
< |
where $V^{I}_{\text{Internal}}$ is the internal potential of molecule $I$: |
| 1139 |
> |
where the multipole operator on site $\bf a$, |
| 1140 |
|
\begin{equation} |
| 1141 |
< |
V^{I}_{\text{Internal}} = |
| 1142 |
< |
\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
| 1143 |
< |
+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{ijkl}) |
| 1000 |
< |
+ \sum_{i \in I} \sum_{(j>i+4) \in I} |
| 1001 |
< |
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 1002 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 1003 |
< |
\biggr]. |
| 1004 |
< |
\label{eq:internalPotential} |
| 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 $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
| 1146 |
< |
within the molecule $I$, and $V_{\text{torsion}}$ is the torsion |
| 1147 |
< |
potential for all 1, 4 bonded pairs. The pairwise portions of the |
| 1148 |
< |
non-bonded interactions are excluded for atom pairs that are involved |
| 1149 |
< |
in the smae bond, bend, or torsion. All other atom pairs within a |
| 1150 |
< |
molecule are subject to the LJ pair potential. |
| 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 bend potential of a molecule is represented by the following function: |
| 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{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 |
| 1164 |
< |
)^2, \label{eq:bendPot} |
| 1163 |
> |
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 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}) % |
| 1168 |
> |
(\boldsymbol{\hat{u}}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr]. |
| 1169 |
> |
\label{eq:dipolePot} |
| 1170 |
|
\end{equation} |
| 1171 |
< |
where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
| 1172 |
< |
(see Fig.~\ref{fig:lipidModel}), $\theta_0$ is the equilibrium |
| 1173 |
< |
bond angle, and $k_{\theta}$ is the force constant which determines the |
| 1174 |
< |
strength of the harmonic bend. The parameters for $k_{\theta}$ and |
| 1175 |
< |
$\theta_0$ are borrowed from those in TraPPE.\cite{Siepmann1998} |
| 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 $|{\bf |
| 1175 |
> |
D}_i|$, $\boldsymbol{\hat{u}}_i$ 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 |
|
|
| 1180 |
< |
The torsion potential and parameters are also borrowed from TraPPE. It is |
| 1181 |
< |
of the form: |
| 1182 |
< |
\begin{equation} |
| 1183 |
< |
V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi] |
| 1184 |
< |
+ c_2[1 + \cos(2\phi)] |
| 1185 |
< |
+ c_3[1 + \cos(3\phi)], |
| 1180 |
> |
|
| 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 |
| 1291 |
> |
solvation shell. Thus, the interaction between two SSD water molecules |
| 1292 |
> |
\emph{i} and \emph{j} is given by the potential |
| 1293 |
> |
\begin{equation} |
| 1294 |
> |
V_{ij} = |
| 1295 |
> |
V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp} |
| 1296 |
> |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
| 1297 |
> |
V_{ij}^{sp} |
| 1298 |
> |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
| 1299 |
> |
\label{eq:ssdPot} |
| 1300 |
> |
\end{equation} |
| 1301 |
> |
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
| 1302 |
> |
\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and |
| 1303 |
> |
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
| 1304 |
> |
orientations of the respective molecules. The Lennard-Jones and dipole |
| 1305 |
> |
parts of the potential are given by equations \ref{eq:lennardJonesPot} |
| 1306 |
> |
and \ref{eq:dipolePot} respectively. The sticky part is described by |
| 1307 |
> |
the following, |
| 1308 |
> |
\begin{equation} |
| 1309 |
> |
u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)= |
| 1310 |
> |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij}, |
| 1311 |
> |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + |
| 1312 |
> |
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij}, |
| 1313 |
> |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
| 1314 |
> |
\label{eq:stickyPot} |
| 1315 |
> |
\end{equation} |
| 1316 |
> |
where $\nu_0$ is a strength parameter for the sticky potential, and |
| 1317 |
> |
$s$ and $s^\prime$ are cubic switching functions which turn off the |
| 1318 |
> |
sticky interaction beyond the first solvation shell. The $w$ function |
| 1319 |
> |
can be thought of as an attractive potential with tetrahedral |
| 1320 |
> |
geometry: |
| 1321 |
> |
\begin{equation} |
| 1322 |
> |
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
| 1323 |
> |
\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
| 1324 |
> |
\label{eq:stickyW} |
| 1325 |
> |
\end{equation} |
| 1326 |
> |
while the $w^\prime$ function counters the normal aligned and |
| 1327 |
> |
anti-aligned structures favored by point dipoles: |
| 1328 |
> |
\begin{equation} |
| 1329 |
> |
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
| 1330 |
> |
(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
| 1331 |
> |
\label{eq:stickyWprime} |
| 1332 |
> |
\end{equation} |
| 1333 |
> |
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
| 1334 |
> |
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
| 1335 |
> |
enhances the tetrahedral geometry for hydrogen bonded structures), |
| 1336 |
> |
while $w^\prime$ is a purely empirical function. A more detailed |
| 1337 |
> |
description of the functional parts and variables in this potential |
| 1338 |
> |
can be found in the original SSD |
| 1339 |
> |
articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03} |
| 1340 |
> |
|
| 1341 |
> |
\begin{figure} |
| 1342 |
> |
\centering |
| 1343 |
> |
\includegraphics[width=\linewidth]{waterAngle.pdf} |
| 1344 |
> |
\caption[Coordinate definition for the SSD/E water model]{Coordinates |
| 1345 |
> |
for the interaction between two SSD/E water molecules. $\theta_{ij}$ |
| 1346 |
> |
is the angle that $r_{ij}$ makes with the $\hat{z}$ vector in the |
| 1347 |
> |
body-fixed frame for molecule $i$. The $\hat{z}$ vector bisects the |
| 1348 |
> |
HOH angle in each water molecule. } |
| 1349 |
> |
\label{fig:ssd} |
| 1350 |
> |
\end{figure} |
| 1351 |
> |
|
| 1352 |
> |
Since SSD/E is a single-point {\it dipolar} model, the force |
| 1353 |
> |
calculations are simplified significantly relative to the standard |
| 1354 |
> |
{\it charged} multi-point models. In the original Monte Carlo |
| 1355 |
> |
simulations using this model, Ichiye {\it et al.} reported that using |
| 1356 |
> |
SSD decreased computer time by a factor of 6-7 compared to other |
| 1357 |
> |
models.\cite{liu96:new_model} What is most impressive is that these |
| 1358 |
> |
savings did not come at the expense of accurate depiction of the |
| 1359 |
> |
liquid state properties. Indeed, SSD/E maintains reasonable agreement |
| 1360 |
> |
with the Head-Gordon diffraction data for the structural features of |
| 1361 |
> |
liquid water.\cite{hura00,liu96:new_model} Additionally, the dynamical |
| 1362 |
> |
properties exhibited by SSD/E agree with experiment better than those |
| 1363 |
> |
of more computationally expensive models (like TIP3P and |
| 1364 |
> |
SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate |
| 1365 |
> |
depiction of solvent properties makes SSD/E a very attractive model |
| 1366 |
> |
for the simulation of large scale biochemical simulations. |
| 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} 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 |
> |
\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 |
> |
\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}) |
| 1404 |
> |
\end{equation} |
| 1405 |
> |
where $F_{i} $ is an embedding functional that approximates the energy |
| 1406 |
> |
required to embed a positively-charged core ion $i$ into a linear |
| 1407 |
> |
superposition of spherically averaged atomic electron densities given |
| 1408 |
> |
by $\rho_{i}$, |
| 1409 |
> |
\begin{equation} |
| 1410 |
> |
\rho_{i} = \sum_{j \neq i} f_{j}({\bf r}_{ij}), |
| 1411 |
> |
\end{equation} |
| 1412 |
> |
Since the density at site $i$ ($\rho_i$) must be computed before the |
| 1413 |
> |
embedding functional can be evaluated, {\sc eam} and the related |
| 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 usually a |
| 1418 |
> |
repulsive interaction between atoms $i$ and $j$. |
| 1419 |
> |
|
| 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( |
| 1443 |
> |
\frac{f_{b}(r)}{f_{a}(r)}\phi_{aa}(r)+ |
| 1444 |
> |
\frac{f_{a}(r)}{f_{b}(r)}\phi_{bb}(r) |
| 1445 |
> |
\right). |
| 1446 |
> |
\end{equation} |
| 1447 |
> |
No mixing rule is needed for the densities, since the density at site |
| 1448 |
> |
$i$ is simply the linear sum of density contributions of all the other |
| 1449 |
> |
atoms. |
| 1450 |
> |
|
| 1451 |
> |
The {\sc eam} force field illustrates an additional feature of {\sc |
| 1452 |
> |
OpenMD}. Foiles, Baskes and Daw fit {\sc eam} potentials for Cu, Ag, |
| 1453 |
> |
Au, Ni, Pd, Pt and alloys of these metals.\cite{FBD86} These fits are |
| 1454 |
> |
included in {\sc OpenMD} as the {\tt u3} variant of the {\sc eam} force |
| 1455 |
> |
field. Voter and Chen reparamaterized a set of {\sc eam} functions |
| 1456 |
> |
which do a better job of predicting melting points.\cite{Voter:87} |
| 1457 |
> |
These functions are included in {\sc OpenMD} as the {\tt VC} variant of |
| 1458 |
> |
the {\sc eam} force field. An additional set of functions (the |
| 1459 |
> |
``Universal 6'' functions) are included in {\sc OpenMD} as the {\tt u6} |
| 1460 |
> |
variant of {\sc eam}. For example, to specify the Voter-Chen variant |
| 1461 |
> |
of the {\sc eam} force field, the user would add the {\tt |
| 1462 |
> |
forceFieldVariant = "VC";} line to the meta-data file. |
| 1463 |
> |
|
| 1464 |
> |
The potential files used by the {\sc eam} force field are in the |
| 1465 |
> |
standard {\tt funcfl} format, which is the format utilized by a number |
| 1466 |
> |
of other codes (e.g. ParaDyn~\cite{Paradyn}, {\sc dynamo 86}). It |
| 1467 |
> |
should be noted that the energy units in these files are in eV, not |
| 1468 |
> |
$\mbox{kcal mol}^{-1}$ as in the rest of the {\sc OpenMD} force field |
| 1469 |
> |
files. |
| 1470 |
> |
|
| 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 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}\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}} \hspace{1in} \rho_{i}=\sum_{j\neq i}\left( |
| 1502 |
> |
\frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}} |
| 1503 |
> |
\end{equation} |
| 1504 |
> |
|
| 1505 |
> |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
| 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. $\epsilon_{ij}$, $\epsilon_{ii}$, $c_i$ 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. Interested readers are encouraged to consult reference |
| 1517 |
> |
\citealp{Qi99} for further details. |
| 1518 |
> |
|
| 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 |
> |
\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 |
> |
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 |
> |
V_{\text{bond}}(b) = \frac{k_{ij}}{2} \left(b - |
| 1600 |
> |
b_{ij}^0 \right)^2 |
| 1601 |
> |
\end{equation} |
| 1602 |
> |
and {\tt Morse} bonds, |
| 1603 |
> |
\begin{equation} |
| 1604 |
> |
V_{\text{bond}}(b) = D_{ij} \left[ 1 - e^{-\beta_{ij} (b - b_{ij}^0)} \right]^2 |
| 1605 |
> |
\end{equation} |
| 1606 |
> |
|
| 1607 |
> |
OpenMD can also simulate some less common types of bond potentials, |
| 1608 |
> |
including {\tt Fixed} bonds (which are constrained to be at a |
| 1609 |
> |
specified bond length), |
| 1610 |
> |
\begin{equation} |
| 1611 |
> |
V_{\text{bond}}(b) = 0. |
| 1612 |
> |
\end{equation} |
| 1613 |
> |
{\tt Cubic} bonds include terms to model anharmonicity, |
| 1614 |
> |
\begin{equation} |
| 1615 |
> |
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, |
| 1616 |
> |
\end{equation} |
| 1617 |
> |
and {\tt Quartic} bonds, include another term in the Taylor |
| 1618 |
> |
expansion around $b_{ij}^0$, |
| 1619 |
> |
\begin{equation} |
| 1620 |
> |
V_{\text{bond}}(b) = K_4 (b - b_{ij}^0)^4 + K_3 (b - b_{ij}^0)^3 + |
| 1621 |
> |
K_2 (b - b_{ij}^0)^2 + K_1 (b - b_{ij}^0) + K_0, |
| 1622 |
> |
\end{equation} |
| 1623 |
> |
can also be simulated. Note that the factor of $1/2$ that is included |
| 1624 |
> |
in the {\tt Harmonic} bond type force constant is {\it not} present in |
| 1625 |
> |
either the {\tt Cubic} or {\tt Quartic} bond types. |
| 1626 |
> |
|
| 1627 |
> |
{\tt Polynomial} bonds which can have any number of terms, |
| 1628 |
> |
\begin{equation} |
| 1629 |
> |
V_{\text{bond}}(b) = \sum_n K_n (b - b_{ij}^0)^n. |
| 1630 |
> |
\end{equation} |
| 1631 |
> |
can also be specified by giving a sequence of integer ($n$) and force |
| 1632 |
> |
constant ($K_n$) pairs. |
| 1633 |
> |
|
| 1634 |
> |
The order of terms in the BondTypes block is: |
| 1635 |
> |
\begin{itemize} |
| 1636 |
> |
\item {\tt AtomType} 1 |
| 1637 |
> |
\item {\tt AtomType} 2 |
| 1638 |
> |
\item {\tt BondType} (one of {\tt Harmonic}, {\tt Morse}, {\tt Fixed}, {\tt |
| 1639 |
> |
Cubic}, {\tt Quartic}, or {\tt Polynomial}) |
| 1640 |
> |
\item $b_{ij}^0$, the equilibrium bond length in \AA |
| 1641 |
> |
\item any other parameters required by the {\tt BondType} |
| 1642 |
> |
\end{itemize} |
| 1643 |
> |
|
| 1644 |
> |
\begin{lstlisting}[caption={[An example of a BondTypes block.] A |
| 1645 |
> |
simple example of a BondTypes block. Distances ($b_0$) |
| 1646 |
> |
are given in \AA\ and force constants are given in |
| 1647 |
> |
units so that when multiplied by the correct power of distance they |
| 1648 |
> |
return energies in kcal/mol. For example $k$ for a Harmonic bond is |
| 1649 |
> |
in units of kcal/mol/\AA$^2$.}, |
| 1650 |
> |
label={sch:BondTypes}] |
| 1651 |
> |
begin BondTypes |
| 1652 |
> |
//Atom1 Atom2 Harmonic b0 k (kcal/mol/A^2) |
| 1653 |
> |
CH2 CH2 Harmonic 1.526 260 |
| 1654 |
> |
//Atom1 Atom2 Morse b0 D beta (A^-1) |
| 1655 |
> |
CN NC Morse 1.157437 212.95 2.5802 |
| 1656 |
> |
//Atom1 Atom2 Fixed b0 |
| 1657 |
> |
CT HC Fixed 1.09 |
| 1658 |
> |
//Atom1 Atom2 Cubic b0 K3 K2 K1 K0 |
| 1659 |
> |
//Atom1 Atom2 Quartic b0 K4 K3 K2 K1 K0 |
| 1660 |
> |
//Atom1 Atom2 Polynomial b0 n Kn [m Km] |
| 1661 |
> |
end BondTypes |
| 1662 |
> |
\end{lstlisting} |
| 1663 |
> |
|
| 1664 |
> |
There are advantages and disadvantages of all of the different types |
| 1665 |
> |
of bonds, but specific simulation tasks may call for specific |
| 1666 |
> |
behaviors. |
| 1667 |
> |
|
| 1668 |
> |
\subsection{\label{section:ffBend}The BendTypes block} |
| 1669 |
> |
The equilibrium geometries and energy functions for bending motions in |
| 1670 |
> |
a molecule are strongly dependent on the bonding environment of the |
| 1671 |
> |
central atomic site. For example, different types of hybridized |
| 1672 |
> |
carbon centers require different bending angles and force constants to |
| 1673 |
> |
describe the local geometry. |
| 1674 |
> |
|
| 1675 |
> |
The bending potential energy functions used in most force fields are |
| 1676 |
> |
often simple functions of the angle between two bonds, |
| 1677 |
> |
\begin{equation} |
| 1678 |
> |
\theta_{ijk} = \cos^{-1} \left(\frac{\vec{r}_{ij} \cdot |
| 1679 |
> |
\vec{r}_{jk}}{\left| \vec{r}_{ij} \right| \left| \vec{r}_{ij} |
| 1680 |
> |
\right|} \right) |
| 1681 |
> |
\end{equation} |
| 1682 |
> |
Here atom $j$ is the central atom that is bonded to two partners $i$ |
| 1683 |
> |
and $k$. |
| 1684 |
> |
|
| 1685 |
> |
All BendTypes must specify three AtomType names ($i$, $j$ and $k$) |
| 1686 |
> |
that describe when that bend potential should be applied, as well as |
| 1687 |
> |
an equilibrium bending angle, $\theta_{ijk}^0$, in units of |
| 1688 |
> |
degrees. The most common forms for bending potentials are {\tt |
| 1689 |
> |
Harmonic} bends, |
| 1690 |
> |
\begin{equation} |
| 1691 |
> |
V_{\text{bend}}(\theta_{ijk}) = \frac{k_{ijk}}{2}( \theta_{ijk} - \theta_{ijk}^0 |
| 1692 |
> |
)^2, \label{eq:bendPot} |
| 1693 |
> |
\end{equation} |
| 1694 |
> |
where $k_{ijk}$ is the force constant which determines the strength of |
| 1695 |
> |
the harmonic bend. {\tt UreyBradley} bends utilize an additional 1-3 |
| 1696 |
> |
bond-type interaction in addition to the harmonic bending potential: |
| 1697 |
> |
\begin{equation} |
| 1698 |
> |
V_{\text{bend}}(\vec{r}_i , \vec{r}_j, \vec{r}_k) |
| 1699 |
> |
=\frac{k_{ijk}}{2}( \theta_{ijk} - \theta_{ijk}^0)^2 |
| 1700 |
> |
+ \frac{k_{ub}}{2}( r_{ik} - s_0 )^2. \label{eq:ubBend} |
| 1701 |
> |
\end{equation} |
| 1702 |
> |
|
| 1703 |
> |
A {\tt Cosine} bend is a variant on the harmonic bend which utilizes |
| 1704 |
> |
the cosine of the angle instead of the angle itself, |
| 1705 |
> |
\begin{equation} |
| 1706 |
> |
V_{\text{bend}}(\theta_{ijk}) = \frac{k_{ijk}}{2}( \cos\theta_{ijk} - |
| 1707 |
> |
\cos \theta_{ijk}^0 )^2. \label{eq:cosBend} |
| 1708 |
> |
\end{equation} |
| 1709 |
> |
|
| 1710 |
> |
OpenMD can also simulate some less common types of bend potentials, |
| 1711 |
> |
including {\tt Cubic} bends, which include terms to model anharmonicity, |
| 1712 |
> |
\begin{equation} |
| 1713 |
> |
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, |
| 1714 |
> |
\end{equation} |
| 1715 |
> |
and {\tt Quartic} bends, which include another term in the Taylor |
| 1716 |
> |
expansion around $\theta_{ijk}^0$, |
| 1717 |
> |
\begin{equation} |
| 1718 |
> |
V_{\text{bend}}(\theta_{ijk}) = K_4 (\theta_{ijk} - \theta_{ijk}^0)^4 + K_3 (\theta_{ijk} - \theta_{ijk}^0)^3 + |
| 1719 |
> |
K_2 (\theta_{ijk} - \theta_{ijk}^0)^2 + K_1 (\theta_{ijk} - |
| 1720 |
> |
\theta_{ijk}^0) + K_0, |
| 1721 |
> |
\end{equation} |
| 1722 |
> |
can also be simulated. Note that the factor of $1/2$ that is included |
| 1723 |
> |
in the {\tt Harmonic} bend type force constant is {\it not} present in |
| 1724 |
> |
either the {\tt Cubic} or {\tt Quartic} bend types. |
| 1725 |
> |
|
| 1726 |
> |
{\tt Polynomial} bends which can have any number of terms, |
| 1727 |
> |
\begin{equation} |
| 1728 |
> |
V_{\text{bend}}(\theta_{ijk}) = \sum_n K_n (\theta_{ijk} - \theta_{ijk}^0)^n. |
| 1729 |
> |
\end{equation} |
| 1730 |
> |
can also be specified by giving a sequence of integer ($n$) and force |
| 1731 |
> |
constant ($K_n$) pairs. |
| 1732 |
> |
|
| 1733 |
> |
The order of terms in the BendTypes block is: |
| 1734 |
> |
\begin{itemize} |
| 1735 |
> |
\item {\tt AtomType} 1 |
| 1736 |
> |
\item {\tt AtomType} 2 (this is the central atom) |
| 1737 |
> |
\item {\tt AtomType} 3 |
| 1738 |
> |
\item {\tt BendType} (one of {\tt Harmonic}, {\tt UreyBradley}, {\tt |
| 1739 |
> |
Cosine}, {\tt Cubic}, {\tt Quartic}, or {\tt Polynomial}) |
| 1740 |
> |
\item $\theta_{ijk}^0$, the equilibrium bending angle in degrees. |
| 1741 |
> |
\item any other parameters required by the {\tt BendType} |
| 1742 |
> |
\end{itemize} |
| 1743 |
> |
|
| 1744 |
> |
\begin{lstlisting}[caption={[An example of a BendTypes block.] A |
| 1745 |
> |
simple example of a BendTypes block. By convention, equilibrium angles |
| 1746 |
> |
($\theta_0$) are given in degrees but force constants are given in |
| 1747 |
> |
units so that when multiplied by the correct power of angle (in |
| 1748 |
> |
radians) they return energies in kcal/mol. For example $k$ for a |
| 1749 |
> |
Harmonic bend is in units of kcal/mol/radians$^2$.}, |
| 1750 |
> |
label={sch:BendTypes}] |
| 1751 |
> |
begin BendTypes |
| 1752 |
> |
//Atom1 Atom2 Atom3 Harmonic theta0(deg) Ktheta(kcal/mol/radians^2) |
| 1753 |
> |
CT CT CT Harmonic 109.5 80.000000 |
| 1754 |
> |
CH2 CH CH2 Harmonic 112.0 117.68 |
| 1755 |
> |
CH3 CH2 SH Harmonic 96.0 67.220 |
| 1756 |
> |
//UreyBradley |
| 1757 |
> |
//Atom1 Atom2 Atom3 UreyBradley theta0 Ktheta s0 Kub |
| 1758 |
> |
//Cosine |
| 1759 |
> |
//Atom1 Atom2 Atom3 Cosine theta0 Ktheta(kcal/mol) |
| 1760 |
> |
//Cubic |
| 1761 |
> |
//Atom1 Atom2 Atom3 Cubic theta0 K3 K2 K1 K0 |
| 1762 |
> |
//Quartic |
| 1763 |
> |
//Atom1 Atom2 Atom3 Quartic theta0 K4 K3 K2 K1 K0 |
| 1764 |
> |
//Polynomial |
| 1765 |
> |
//Atom1 Atom2 Atom3 Polynomial theta0 n Kn [m Km] |
| 1766 |
> |
end BendTypes |
| 1767 |
> |
\end{lstlisting} |
| 1768 |
> |
|
| 1769 |
> |
Note that the parameters for a particular bend type are the same for |
| 1770 |
> |
any bending triplet of the same atomic types (in the same or reversed |
| 1771 |
> |
order). Changing the AtomType in the Atom2 position will change the |
| 1772 |
> |
matched bend types in the force field. |
| 1773 |
> |
|
| 1774 |
> |
\subsection{\label{section:ffTorsion}The TorsionTypes block} |
| 1775 |
> |
The torsion potential for rotation of groups around a central bond can |
| 1776 |
> |
often be represented with various cosine functions. For two |
| 1777 |
> |
tetrahedral ($sp^3$) carbons connected by a single bond, the torsion |
| 1778 |
> |
potential might be |
| 1779 |
> |
\begin{equation*} |
| 1780 |
> |
V_{\text{torsion}} = \frac{v}{2} \left[ 1 + \cos( 3 \phi ) \right] |
| 1781 |
> |
\end{equation*} |
| 1782 |
> |
where $v$ is the barrier for going from {\em staggered} $\rightarrow$ |
| 1783 |
> |
{\em eclipsed} conformations, while for $sp^2$ carbons connected by a |
| 1784 |
> |
double bond, the potential might be |
| 1785 |
> |
\begin{equation*} |
| 1786 |
> |
V_{\text{torsion}} = \frac{w}{2} \left[ 1 - \cos( 2 \phi ) \right] |
| 1787 |
> |
\end{equation*} |
| 1788 |
> |
where $w$ is the barrier for going from {\em cis} $\rightarrow$ {\em |
| 1789 |
> |
trans} conformations. |
| 1790 |
> |
|
| 1791 |
> |
A general torsion potentials can be represented as a cosine series of |
| 1792 |
> |
the form: |
| 1793 |
> |
\begin{equation} |
| 1794 |
> |
V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi_{ijkl}] |
| 1795 |
> |
+ c_2[1 - \cos(2\phi_{ijkl})] |
| 1796 |
> |
+ c_3[1 + \cos(3\phi_{ijkl})], |
| 1797 |
|
\label{eq:origTorsionPot} |
| 1798 |
|
\end{equation} |
| 1799 |
+ |
where the angle $\phi_{ijkl}$ is defined |
| 1800 |
+ |
\begin{equation} |
| 1801 |
+ |
\cos\phi_{ijkl} = (\hat{\mathbf{r}}_{ij} \times \hat{\mathbf{r}}_{jk}) \cdot |
| 1802 |
+ |
(\hat{\mathbf{r}}_{jk} \times \hat{\mathbf{r}}_{kl}). |
| 1803 |
+ |
\label{eq:torsPhi} |
| 1804 |
+ |
\end{equation} |
| 1805 |
+ |
Here, $\hat{\mathbf{r}}_{\alpha\beta}$ are the set of unit bond |
| 1806 |
+ |
vectors between atoms $i$, $j$, $k$, and $l$. |
| 1807 |
+ |
|
| 1808 |
+ |
For computational efficiency, OpenMD recasts torsion potential in the |
| 1809 |
+ |
method of {\sc charmm},\cite{Brooks83} in which the angle series is |
| 1810 |
+ |
converted to a power series of the form: |
| 1811 |
+ |
\begin{equation} |
| 1812 |
+ |
V_{\text{torsion}}(\phi_{ijkl}) = |
| 1813 |
+ |
k_3 \cos^3 \phi_{ijkl} + k_2 \cos^2 \phi_{ijkl} + k_1 \cos \phi_{ijkl} + k_0, |
| 1814 |
+ |
\label{eq:torsionPot} |
| 1815 |
+ |
\end{equation} |
| 1816 |
|
where: |
| 1817 |
+ |
\begin{align*} |
| 1818 |
+ |
k_0 &= c_1 + 2 c_2 + c_3, \\ |
| 1819 |
+ |
k_1 &= c_1 - 3c_3, \\ |
| 1820 |
+ |
k_2 &= - 2 c_2, \\ |
| 1821 |
+ |
k_3 &= 4 c_3. |
| 1822 |
+ |
\end{align*} |
| 1823 |
+ |
By recasting the potential as a power series, repeated trigonometric |
| 1824 |
+ |
evaluations are avoided during the calculation of the potential |
| 1825 |
+ |
energy. |
| 1826 |
+ |
|
| 1827 |
+ |
Using this framework, OpenMD implements a variety of different |
| 1828 |
+ |
potential energy functions for torsions: |
| 1829 |
+ |
\begin{itemize} |
| 1830 |
+ |
\item {\tt Cubic}: |
| 1831 |
+ |
\begin{equation*} |
| 1832 |
+ |
V_{\text{torsion}}(\phi) = |
| 1833 |
+ |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0, |
| 1834 |
+ |
\end{equation*} |
| 1835 |
+ |
\item {\tt Quartic}: |
| 1836 |
+ |
\begin{equation*} |
| 1837 |
+ |
V_{\text{torsion}}(\phi) = k_4 \cos^4 \phi + |
| 1838 |
+ |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0, |
| 1839 |
+ |
\end{equation*} |
| 1840 |
+ |
\item {\tt Polynomial}: |
| 1841 |
+ |
\begin{equation*} |
| 1842 |
+ |
V_{\text{torsion}}(\phi) = \sum_n k_n \cos^n \phi , |
| 1843 |
+ |
\end{equation*} |
| 1844 |
+ |
\item {\tt Charmm}: |
| 1845 |
+ |
\begin{equation*} |
| 1846 |
+ |
V_{\text{torsion}}(\phi) = \sum_n K_n \left( 1 + cos(n |
| 1847 |
+ |
\phi - \delta_n) \right), |
| 1848 |
+ |
\end{equation*} |
| 1849 |
+ |
\item {\tt Opls}: |
| 1850 |
+ |
\begin{equation*} |
| 1851 |
+ |
V_{\text{torsion}}(\phi) = \frac{1}{2} \left(v_1 (1 + \cos \phi) \right) |
| 1852 |
+ |
+ v_2 (1 - \cos 2 \phi) + v_3 (1 + \cos 3 \phi), |
| 1853 |
+ |
\end{equation*} |
| 1854 |
+ |
\item {\tt Trappe}:\cite{Siepmann1998} |
| 1855 |
+ |
\begin{equation*} |
| 1856 |
+ |
V_{\text{torsion}}(\phi) = c_0 + c_1 (1 + \cos \phi) + c_2 (1 - \cos 2 \phi) + |
| 1857 |
+ |
c_3 (1 + \cos 3 \phi), |
| 1858 |
+ |
\end{equation*} |
| 1859 |
+ |
\item {\tt Harmonic}: |
| 1860 |
+ |
\begin{equation*} |
| 1861 |
+ |
V_{\text{torsion}}(\phi) = \frac{d_0}{2} \left(\phi - \phi^0\right). |
| 1862 |
+ |
\end{equation*} |
| 1863 |
+ |
\end{itemize} |
| 1864 |
+ |
|
| 1865 |
+ |
Most torsion types don't require specific angle information in the |
| 1866 |
+ |
parameters since they are typically expressed in cosine polynomials. |
| 1867 |
+ |
{\tt Charmm} and {\tt Harmonic} torsions are a bit different. {\tt |
| 1868 |
+ |
Charmm} torsion types require a set of phase angles, $\delta_n$ that |
| 1869 |
+ |
are expressed in degrees, and associated periodicity numbers, $n$. |
| 1870 |
+ |
{\tt Harmonic} torsions have an equilibrium torsion angle, $\phi_0$ |
| 1871 |
+ |
that is measured in degrees, while $d_0$ has units of |
| 1872 |
+ |
kcal/mol/degrees$^2$. All other torsion parameters are measured in |
| 1873 |
+ |
units of kcal/mol. |
| 1874 |
+ |
|
| 1875 |
+ |
\begin{lstlisting}[caption={[An example of a TorsionTypes block.] A |
| 1876 |
+ |
simple example of a TorsionTypes block. Energy constants are given in |
| 1877 |
+ |
kcal / mol, and when required by the form, $\delta$ angles are given |
| 1878 |
+ |
in degrees.}, |
| 1879 |
+ |
label={sch:TorsionTypes}] |
| 1880 |
+ |
begin TorsionTypes |
| 1881 |
+ |
//Cubic |
| 1882 |
+ |
//Atom1 Atom2 Atom3 Atom4 Cubic k3 k2 k1 k0 |
| 1883 |
+ |
CH2 CH2 CH2 CH2 Cubic 5.9602 -0.2568 -3.802 2.1586 |
| 1884 |
+ |
CH2 CH CH CH2 Cubic 3.3254 -0.4215 -1.686 1.1661 |
| 1885 |
+ |
//Trappe |
| 1886 |
+ |
//Atom1 Atom2 Atom3 Atom4 Trappe c0 c1 c2 c3 |
| 1887 |
+ |
CH3 CH2 CH2 SH Trappe 0.10507 -0.10342 0.03668 0.60874 |
| 1888 |
+ |
//Charmm |
| 1889 |
+ |
//Atom1 Atom2 Atom3 Atom4 Charmm Kchi n delta [Kchi n delta] |
| 1890 |
+ |
CT CT CT C Charmm 0.156 3 0.00 |
| 1891 |
+ |
OH CT CT OH Charmm 0.144 3 0.00 1.175 2 0 |
| 1892 |
+ |
HC CT CM CM Charmm 1.150 1 0.00 0.38 3 180 |
| 1893 |
+ |
//Quartic |
| 1894 |
+ |
//Atom1 Atom2 Atom3 Atom4 Quartic k4 k3 k2 k1 k0 |
| 1895 |
+ |
//Polynomial |
| 1896 |
+ |
//Atom1 Atom2 Atom3 Atom4 Polynomial n Kn [m Km] |
| 1897 |
+ |
S CH2 CH2 C Polynomial 0 2.218 1 2.905 2 -3.136 3 -0.7313 4 6.272 5 -7.528 |
| 1898 |
+ |
end TorsionTypes |
| 1899 |
+ |
\end{lstlisting} |
| 1900 |
+ |
|
| 1901 |
+ |
Note that the parameters for a particular torsion type are the same |
| 1902 |
+ |
for any torsional quartet of the same atomic types (in the same or |
| 1903 |
+ |
reversed order). |
| 1904 |
+ |
|
| 1905 |
+ |
\subsection{\label{section:ffInversion}The InversionTypes block} |
| 1906 |
+ |
|
| 1907 |
+ |
Inversion potentials are often added to force fields to enforce |
| 1908 |
+ |
planarity around $sp^2$-hybridized carbons or to correct vibrational |
| 1909 |
+ |
frequencies for umbrella-like vibrational modes for central atoms |
| 1910 |
+ |
bonded to exactly three satellite groups. |
| 1911 |
+ |
|
| 1912 |
+ |
In OpenMD's version of an inversion, the central atom is entered {\it |
| 1913 |
+ |
first} in each line in the {\tt InversionTypes} block. Note that |
| 1914 |
+ |
this is quite different than how other codes treat Improper torsional |
| 1915 |
+ |
potentials to mimic inversion behavior. In some other widely-used |
| 1916 |
+ |
simulation packages, the central atom is treated as atom 3 in a |
| 1917 |
+ |
standard torsion form: |
| 1918 |
+ |
\begin{itemize} |
| 1919 |
+ |
\item OpenMD: I - (J - K - L) (e.g. I is $sp^2$ hybridized carbon) |
| 1920 |
+ |
\item AMBER: I - J - K - L (e.g. K is $sp^2$ hybridized carbon) |
| 1921 |
+ |
\end{itemize} |
| 1922 |
+ |
|
| 1923 |
+ |
The inversion angle itself is defined as: |
| 1924 |
|
\begin{equation} |
| 1925 |
< |
\cos\phi = (\hat{\mathbf{r}}_{ij} \times \hat{\mathbf{r}}_{jk}) \cdot |
| 1926 |
< |
(\hat{\mathbf{r}}_{jk} \times \hat{\mathbf{r}}_{kl}). |
| 1927 |
< |
\label{eq:torsPhi} |
| 1925 |
> |
\cos\omega_{i-jkl} = \left(\hat{\mathbf{r}}_{il} \times |
| 1926 |
> |
\hat{\mathbf{r}}_{ij}\right)\cdot\left( \hat{\mathbf{r}}_{il} \times |
| 1927 |
> |
\hat{\mathbf{r}}_{ik}\right) |
| 1928 |
|
\end{equation} |
| 1929 |
|
Here, $\hat{\mathbf{r}}_{\alpha\beta}$ are the set of unit bond |
| 1930 |
< |
vectors between atoms $i$, $j$, $k$, and $l$. For computational |
| 1931 |
< |
efficiency, the torsion potential has been recast after the method of |
| 1932 |
< |
{\sc charmm},\cite{Brooks83} in which the angle series is converted to |
| 1933 |
< |
a power series of the form: |
| 1043 |
< |
\begin{equation} |
| 1044 |
< |
V_{\text{torsion}}(\phi) = |
| 1045 |
< |
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0, |
| 1046 |
< |
\label{eq:torsionPot} |
| 1047 |
< |
\end{equation} |
| 1048 |
< |
where: |
| 1049 |
< |
\begin{align*} |
| 1050 |
< |
k_0 &= c_1 + c_3, \\ |
| 1051 |
< |
k_1 &= c_1 - 3c_3, \\ |
| 1052 |
< |
k_2 &= 2 c_2, \\ |
| 1053 |
< |
k_3 &= 4c_3. |
| 1054 |
< |
\end{align*} |
| 1055 |
< |
By recasting the potential as a power series, repeated trigonometric |
| 1056 |
< |
evaluations are avoided during the calculation of the potential |
| 1057 |
< |
energy. |
| 1930 |
> |
vectors between the central atoms $i$, and the satellite atoms $j$, |
| 1931 |
> |
$k$, and $l$. Note that other definitions of inversion angles are |
| 1932 |
> |
possible, so users are encouraged to be particularly careful when |
| 1933 |
> |
converting other force field files for use with OpenMD. |
| 1934 |
|
|
| 1935 |
+ |
There are many common ways to create planarity or umbrella behavior in |
| 1936 |
+ |
a potential energy function, and OpenMD implements a number of the |
| 1937 |
+ |
more common functions: |
| 1938 |
+ |
\begin{itemize} |
| 1939 |
+ |
\item {\tt ImproperCosine}: |
| 1940 |
+ |
\begin{equation*} |
| 1941 |
+ |
V_{\text{torsion}}(\omega) = \sum_n \frac{K_n}{2} \left( 1 + cos(n |
| 1942 |
+ |
\omega - \delta_n) \right), |
| 1943 |
+ |
\end{equation*} |
| 1944 |
+ |
\item {\tt AmberImproper}: |
| 1945 |
+ |
\begin{equation*} |
| 1946 |
+ |
V_{\text{torsion}}(\omega) = \frac{v}{2} (1 - \cos\left(2 \left(\omega - \omega_0\right)\right), |
| 1947 |
+ |
\end{equation*} |
| 1948 |
+ |
\item {\tt Harmonic}: |
| 1949 |
+ |
\begin{equation*} |
| 1950 |
+ |
V_{\text{torsion}}(\omega) = \frac{d}{2} \left(\omega - \omega_0\right). |
| 1951 |
+ |
\end{equation*} |
| 1952 |
+ |
\end{itemize} |
| 1953 |
+ |
\begin{lstlisting}[caption={[An example of an InversionTypes block.] A |
| 1954 |
+ |
simple example of a InversionTypes block. Angles ($\delta_n$ and |
| 1955 |
+ |
$\omega_0$) angles are given in degrees, while energy parameters ($v, |
| 1956 |
+ |
K_n$) are given in kcal / mol. The Harmonic Inversion type has a |
| 1957 |
+ |
force constant that must be given in kcal/mol/degrees$^2$.}, |
| 1958 |
+ |
label={sch:InversionTypes}] |
| 1959 |
+ |
begin InversionTypes |
| 1960 |
+ |
//Harmonic |
| 1961 |
+ |
//Atom1 Atom2 Atom3 Atom4 Harmonic d(kcal/mol/deg^2) omega0 |
| 1962 |
+ |
RCHar3 X X X Harmonic 1.21876e-2 180.0 |
| 1963 |
+ |
//AmberImproper |
| 1964 |
+ |
//Atom1 Atom2 Atom3 Atom4 AmberImproper v(kcal/mol) |
| 1965 |
+ |
C CT N O AmberImproper 10.500000 |
| 1966 |
+ |
CA CA CA CT AmberImproper 1.100000 |
| 1967 |
+ |
//ImproperCosine |
| 1968 |
+ |
//Atom1 Atom2 Atom3 Atom4 ImproperCosine Kn n delta_n [Kn n delta_n] |
| 1969 |
+ |
end InversionTypes |
| 1970 |
+ |
\end{lstlisting} |
| 1971 |
|
|
| 1972 |
< |
The cross potential between molecules $I$ and $J$, |
| 1061 |
< |
$V^{IJ}_{\text{Cross}}$, is as follows: |
| 1062 |
< |
\begin{equation} |
| 1063 |
< |
V^{IJ}_{\text{Cross}} = |
| 1064 |
< |
\sum_{i \in I} \sum_{j \in J} |
| 1065 |
< |
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 1066 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 1067 |
< |
+ V_{\text{sticky}} |
| 1068 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 1069 |
< |
\biggr], |
| 1070 |
< |
\label{eq:crossPotentail} |
| 1071 |
< |
\end{equation} |
| 1072 |
< |
where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 1073 |
< |
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 1074 |
< |
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model |
| 1075 |
< |
(Sec.~\ref{section:SSD}). Note that not all atom types include all |
| 1076 |
< |
interactions. |
| 1972 |
> |
\section{\label{section::ffLongRange}Long Range Interactions} |
| 1973 |
|
|
| 1974 |
< |
The dipole-dipole potential has the following form: |
| 1975 |
< |
\begin{equation} |
| 1976 |
< |
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 1977 |
< |
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
| 1082 |
< |
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
| 1083 |
< |
- |
| 1084 |
< |
3(\boldsymbol{\hat{u}}_i \cdot \hat{\mathbf{r}}_{ij}) % |
| 1085 |
< |
(\boldsymbol{\hat{u}}_j \cdot \hat{\mathbf{r}}_{ij}) \biggr]. |
| 1086 |
< |
\label{eq:dipolePot} |
| 1087 |
< |
\end{equation} |
| 1088 |
< |
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 1089 |
< |
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
| 1090 |
< |
are the orientational degrees of freedom for atoms $i$ and $j$ |
| 1091 |
< |
respectively. The magnitude of the dipole moment of atom $i$ is |
| 1092 |
< |
$|\mu_i|$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
| 1093 |
< |
vector of $\boldsymbol{\Omega}_i$, and $\boldsymbol{\hat{r}}_{ij}$ is |
| 1094 |
< |
the unit vector pointing along $\mathbf{r}_{ij}$ |
| 1095 |
< |
($\boldsymbol{\hat{r}}_{ij}=\mathbf{r}_{ij}/|\mathbf{r}_{ij}|$). |
| 1974 |
> |
Calculating the long-range (non-bonded) potential involves a sum over |
| 1975 |
> |
all pairs of atoms (except for those atoms which are involved in a |
| 1976 |
> |
bond, bend, or torsion with each other). Many of these interactions |
| 1977 |
> |
can be inferred from the AtomTypes, |
| 1978 |
|
|
| 1979 |
< |
\subsection{\label{section:SSD}The {\sc duff} Water Models: SSD/E |
| 1980 |
< |
and SSD/RF} |
| 1979 |
> |
\subsection{\label{section:ffNBinteraction}The NonBondedInteractions |
| 1980 |
> |
block} |
| 1981 |
|
|
| 1982 |
< |
In the interest of computational efficiency, the default solvent used |
| 1983 |
< |
by {\sc OpenMD} is the extended Soft Sticky Dipole (SSD/E) water |
| 1984 |
< |
model.\cite{fennell04} The original SSD was developed by Ichiye |
| 1985 |
< |
\emph{et al.}\cite{liu96:new_model} as a modified form of the hard-sphere |
| 1986 |
< |
water model proposed by Bratko, Blum, and |
| 1105 |
< |
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
| 1106 |
< |
with a Lennard-Jones core and a sticky potential that directs the |
| 1107 |
< |
particles to assume the proper hydrogen bond orientation in the first |
| 1108 |
< |
solvation shell. Thus, the interaction between two SSD water molecules |
| 1109 |
< |
\emph{i} and \emph{j} is given by the potential |
| 1110 |
< |
\begin{equation} |
| 1111 |
< |
V_{ij} = |
| 1112 |
< |
V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp} |
| 1113 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
| 1114 |
< |
V_{ij}^{sp} |
| 1115 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
| 1116 |
< |
\label{eq:ssdPot} |
| 1117 |
< |
\end{equation} |
| 1118 |
< |
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
| 1119 |
< |
\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and |
| 1120 |
< |
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
| 1121 |
< |
orientations of the respective molecules. The Lennard-Jones and dipole |
| 1122 |
< |
parts of the potential are given by equations \ref{eq:lennardJonesPot} |
| 1123 |
< |
and \ref{eq:dipolePot} respectively. The sticky part is described by |
| 1124 |
< |
the following, |
| 1125 |
< |
\begin{equation} |
| 1126 |
< |
u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)= |
| 1127 |
< |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij}, |
| 1128 |
< |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + |
| 1129 |
< |
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij}, |
| 1130 |
< |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
| 1131 |
< |
\label{eq:stickyPot} |
| 1132 |
< |
\end{equation} |
| 1133 |
< |
where $\nu_0$ is a strength parameter for the sticky potential, and |
| 1134 |
< |
$s$ and $s^\prime$ are cubic switching functions which turn off the |
| 1135 |
< |
sticky interaction beyond the first solvation shell. The $w$ function |
| 1136 |
< |
can be thought of as an attractive potential with tetrahedral |
| 1137 |
< |
geometry: |
| 1138 |
< |
\begin{equation} |
| 1139 |
< |
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
| 1140 |
< |
\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
| 1141 |
< |
\label{eq:stickyW} |
| 1142 |
< |
\end{equation} |
| 1143 |
< |
while the $w^\prime$ function counters the normal aligned and |
| 1144 |
< |
anti-aligned structures favored by point dipoles: |
| 1145 |
< |
\begin{equation} |
| 1146 |
< |
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
| 1147 |
< |
(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
| 1148 |
< |
\label{eq:stickyWprime} |
| 1149 |
< |
\end{equation} |
| 1150 |
< |
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
| 1151 |
< |
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
| 1152 |
< |
enhances the tetrahedral geometry for hydrogen bonded structures), |
| 1153 |
< |
while $w^\prime$ is a purely empirical function. A more detailed |
| 1154 |
< |
description of the functional parts and variables in this potential |
| 1155 |
< |
can be found in the original SSD |
| 1156 |
< |
articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03} |
| 1982 |
> |
The user might want like to specify explicit or special interactions |
| 1983 |
> |
that override the default non-bodned interactions that are inferred |
| 1984 |
> |
from the AtomTypes. To do this, OpenMD implements a |
| 1985 |
> |
NonBondedInteractions block to allow the user to specify the following |
| 1986 |
> |
(pair-wise) non-bonded interactions: |
| 1987 |
|
|
| 1988 |
< |
\begin{figure} |
| 1989 |
< |
\centering |
| 1990 |
< |
\includegraphics[width=\linewidth]{waterAngle.pdf} |
| 1991 |
< |
\caption[Coordinate definition for the SSD/E water model]{Coordinates |
| 1992 |
< |
for the interaction between two SSD/E water molecules. $\theta_{ij}$ |
| 1993 |
< |
is the angle that $r_{ij}$ makes with the $\hat{z}$ vector in the |
| 1994 |
< |
body-fixed frame for molecule $i$. The $\hat{z}$ vector bisects the |
| 1995 |
< |
HOH angle in each water molecule. } |
| 1996 |
< |
\label{fig:ssd} |
| 1997 |
< |
\end{figure} |
| 1988 |
> |
\begin{itemize} |
| 1989 |
> |
\item {\tt LennardJones}: |
| 1990 |
> |
\begin{equation*} |
| 1991 |
> |
V_{\text{NB}}(r) = 4 \epsilon_{ij} \left( |
| 1992 |
> |
\left(\frac{\sigma_{ij}}{r} \right)^{12} - |
| 1993 |
> |
\left(\frac{\sigma_{ij}}{r} \right)^{6} \right), |
| 1994 |
> |
\end{equation*} |
| 1995 |
> |
\item {\tt ShiftedMorse}: |
| 1996 |
> |
\begin{equation*} |
| 1997 |
> |
V_{\text{NB}}(r) = D_{ij} \left( e^{-2 \beta_{ij} (r - |
| 1998 |
> |
r^0)} - 2 e^{- \beta_{ij} (r - |
| 1999 |
> |
r^0)} \right), |
| 2000 |
> |
\end{equation*} |
| 2001 |
> |
\item {\tt RepulsiveMorse}: |
| 2002 |
> |
\begin{equation*} |
| 2003 |
> |
V_{\text{NB}}(r) = D_{ij} \left( e^{-2 \beta_{ij} (r - |
| 2004 |
> |
r^0)} \right), |
| 2005 |
> |
\end{equation*} |
| 2006 |
> |
\item {\tt RepulsivePower}: |
| 2007 |
> |
\begin{equation*} |
| 2008 |
> |
V_{\text{NB}}(r) = \epsilon_{ij} |
| 2009 |
> |
\left(\frac{\sigma_{ij}}{r} \right)^{n_{ij}}. |
| 2010 |
> |
\end{equation*} |
| 2011 |
> |
\end{itemize} |
| 2012 |
|
|
| 2013 |
+ |
\begin{lstlisting}[caption={[An example of a NonBondedInteractions block.] A |
| 2014 |
+ |
simple example of a NonBondedInteractions block. Distances ($\sigma, |
| 2015 |
+ |
r_0$) are given in \AA, while energies ($\epsilon, D0$) are in |
| 2016 |
+ |
kcal/mol. The Morse potentials have an additional parameter $\beta_0$ |
| 2017 |
+ |
which is in units of \AA$^{-1}$.}, |
| 2018 |
+ |
label={sch:InversionTypes}] |
| 2019 |
+ |
begin NonBondedInteractions |
| 2020 |
|
|
| 2021 |
< |
Since SSD/E is a single-point {\it dipolar} model, the force |
| 2022 |
< |
calculations are simplified significantly relative to the standard |
| 2023 |
< |
{\it charged} multi-point models. In the original Monte Carlo |
| 2024 |
< |
simulations using this model, Ichiye {\it et al.} reported that using |
| 2025 |
< |
SSD decreased computer time by a factor of 6-7 compared to other |
| 2026 |
< |
models.\cite{liu96:new_model} What is most impressive is that these |
| 1176 |
< |
savings did not come at the expense of accurate depiction of the |
| 1177 |
< |
liquid state properties. Indeed, SSD/E maintains reasonable agreement |
| 1178 |
< |
with the Head-Gordon diffraction data for the structural features of |
| 1179 |
< |
liquid water.\cite{hura00,liu96:new_model} Additionally, the dynamical |
| 1180 |
< |
properties exhibited by SSD/E agree with experiment better than those |
| 1181 |
< |
of more computationally expensive models (like TIP3P and |
| 1182 |
< |
SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate |
| 1183 |
< |
depiction of solvent properties makes SSD/E a very attractive model |
| 1184 |
< |
for the simulation of large scale biochemical simulations. |
| 2021 |
> |
//Lennard-Jones |
| 2022 |
> |
//Atom1 Atom2 LennardJones sigma epsilon |
| 2023 |
> |
Au CH3 LennardJones 3.54 0.2146 |
| 2024 |
> |
Au CH2 LennardJones 3.54 0.1749 |
| 2025 |
> |
Au CH LennardJones 3.54 0.1749 |
| 2026 |
> |
Au S LennardJones 2.40 8.465 |
| 2027 |
|
|
| 2028 |
< |
Recent constant pressure simulations revealed issues in the original |
| 2029 |
< |
SSD model that led to lower than expected densities at all target |
| 2030 |
< |
pressures.\cite{Ichiye03,fennell04} The default model in {\sc OpenMD} |
| 1189 |
< |
is therefore SSD/E, a density corrected derivative of SSD that |
| 1190 |
< |
exhibits improved liquid structure and transport behavior. If the use |
| 1191 |
< |
of a reaction field long-range interaction correction is desired, it |
| 1192 |
< |
is recommended that the parameters be modified to those of the SSD/RF |
| 1193 |
< |
model (an SSD variant parameterized for reaction field). These solvent |
| 1194 |
< |
parameters are listed and can be easily modified in the {\sc duff} |
| 1195 |
< |
force field file ({\tt DUFF.frc}). A table of the parameter values |
| 1196 |
< |
and the drawbacks and benefits of the different density corrected SSD |
| 1197 |
< |
models can be found in reference~\cite{fennell04}. |
| 2028 |
> |
//Shifted Morse |
| 2029 |
> |
//Atom1 Atom2 ShiftedMorse r0 D0 beta0 |
| 2030 |
> |
Au O_SPCE ShiftedMorse 3.70 0.0424 0.769 |
| 2031 |
|
|
| 2032 |
< |
\section{\label{section:WATER}The {\sc water} Force Field} |
| 2032 |
> |
//Repulsive Morse |
| 2033 |
> |
//Atom1 Atom2 RepulsiveMorse r0 D0 beta0 |
| 2034 |
> |
Au H_SPCE RepulsiveMorse -1.00 0.00850 0.769 |
| 2035 |
|
|
| 2036 |
< |
In addition to the {\sc duff} force field's solvent description, a |
| 2037 |
< |
separate {\sc water} force field has been included for simulating most |
| 2038 |
< |
of the common rigid-body water models. This force field includes the |
| 2039 |
< |
simple and point-dipolar models (SSD, SSD1, SSD/E, SSD/RF, and DPD |
| 2040 |
< |
water), as well as the common charge-based models (SPC, SPC/E, TIP3P, |
| 2041 |
< |
TIP4P, and |
| 1207 |
< |
TIP5P).\cite{liu96:new_model,Ichiye03,fennell04,Marrink01,Berendsen81,Berendsen87,Jorgensen83,Mahoney00} |
| 1208 |
< |
In order to handle these models, charge-charge interactions were |
| 1209 |
< |
included in the force-loop: |
| 1210 |
< |
\begin{equation} |
| 1211 |
< |
V_{\text{charge}}(r_{ij}) = \sum_{ij}\frac{q_iq_je^2}{r_{ij}}, |
| 1212 |
< |
\end{equation} |
| 1213 |
< |
where $q$ represents the charge on particle $i$ or $j$, and $e$ is the |
| 1214 |
< |
charge of an electron in Coulombs. The charge-charge interaction |
| 1215 |
< |
support is rudimentary in the current version of {\sc OpenMD}. As with |
| 1216 |
< |
the other pair interactions, charges can be simulated with a pure |
| 1217 |
< |
cutoff or a reaction field. The various methods for performing the |
| 1218 |
< |
Ewald summation have not yet been included. The {\sc water} force |
| 1219 |
< |
field can be easily expanded through modification of the {\sc water} |
| 1220 |
< |
force field file ({\tt WATER.frc}). By adding atom types and inserting |
| 1221 |
< |
the appropriate parameters, it is possible to extend the force field |
| 1222 |
< |
to handle rigid molecules other than water. |
| 1223 |
< |
|
| 1224 |
< |
\section{\label{section:eam}Embedded Atom Method} |
| 1225 |
< |
|
| 1226 |
< |
{\sc OpenMD} implements a potential that describes bonding in |
| 1227 |
< |
transition metal |
| 1228 |
< |
systems.~\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} This |
| 1229 |
< |
potential has an attractive interaction which models ``Embedding'' a |
| 1230 |
< |
positively charged pseudo-atom core in the electron density due to the |
| 1231 |
< |
free valance ``sea'' of electrons created by the surrounding atoms in |
| 1232 |
< |
the system. A pairwise part of the potential (which is primarily |
| 1233 |
< |
repulsive) describes the interaction of the positively charged metal |
| 1234 |
< |
core ions with one another. The Embedded Atom Method ({\sc |
| 1235 |
< |
eam})~\cite{Daw84,FBD86,johnson89,Lu97} has been widely adopted in the |
| 1236 |
< |
materials science community and has been included in {\sc OpenMD}. A |
| 1237 |
< |
good review of {\sc eam} and other formulations of metallic potentials |
| 1238 |
< |
was given by Voter.\cite{Voter:95} |
| 1239 |
< |
|
| 1240 |
< |
The {\sc eam} potential has the form: |
| 1241 |
< |
\begin{equation} |
| 1242 |
< |
V = \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
| 1243 |
< |
\phi_{ij}({\bf r}_{ij}) |
| 1244 |
< |
\end{equation} |
| 1245 |
< |
where $F_{i} $ is an embedding functional that approximates the energy |
| 1246 |
< |
required to embed a positively-charged core ion $i$ into a linear |
| 1247 |
< |
superposition of spherically averaged atomic electron densities given |
| 1248 |
< |
by $\rho_{i}$, |
| 1249 |
< |
\begin{equation} |
| 1250 |
< |
\rho_{i} = \sum_{j \neq i} f_{j}({\bf r}_{ij}), |
| 1251 |
< |
\end{equation} |
| 1252 |
< |
Since the density at site $i$ ($\rho_i$) must be computed before the |
| 1253 |
< |
embedding functional can be evaluated, {\sc eam} and the related |
| 1254 |
< |
transition metal potentials require two loops through the atom pairs |
| 1255 |
< |
to compute the inter-atomic forces. |
| 1256 |
< |
|
| 1257 |
< |
The pairwise portion of the potential, $\phi_{ij}$, is a primarily |
| 1258 |
< |
repulsive interaction between atoms $i$ and $j$. In the original |
| 1259 |
< |
formulation of {\sc eam}\cite{Daw84}, $\phi_{ij}$ was an entirely |
| 1260 |
< |
repulsive term; however later refinements to {\sc eam} allowed for |
| 1261 |
< |
more general forms for $\phi$.\cite{Daw89} The effective cutoff |
| 1262 |
< |
distance, $r_{{\text cut}}$ is the distance at which the values of |
| 1263 |
< |
$f(r)$ and $\phi(r)$ drop to zero for all atoms present in the |
| 1264 |
< |
simulation. In practice, this distance is fairly small, limiting the |
| 1265 |
< |
summations in the {\sc eam} equation to the few dozen atoms |
| 1266 |
< |
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
| 1267 |
< |
interactions. |
| 1268 |
< |
|
| 1269 |
< |
In computing forces for alloys, mixing rules as outlined by |
| 1270 |
< |
Johnson~\cite{johnson89} are used to compute the heterogenous pair |
| 1271 |
< |
potential, |
| 1272 |
< |
\begin{equation} |
| 1273 |
< |
\label{eq:johnson} |
| 1274 |
< |
\phi_{ab}(r)=\frac{1}{2}\left( |
| 1275 |
< |
\frac{f_{b}(r)}{f_{a}(r)}\phi_{aa}(r)+ |
| 1276 |
< |
\frac{f_{a}(r)}{f_{b}(r)}\phi_{bb}(r) |
| 1277 |
< |
\right). |
| 1278 |
< |
\end{equation} |
| 1279 |
< |
No mixing rule is needed for the densities, since the density at site |
| 1280 |
< |
$i$ is simply the linear sum of density contributions of all the other |
| 1281 |
< |
atoms. |
| 1282 |
< |
|
| 1283 |
< |
The {\sc eam} force field illustrates an additional feature of {\sc |
| 1284 |
< |
OpenMD}. Foiles, Baskes and Daw fit {\sc eam} potentials for Cu, Ag, |
| 1285 |
< |
Au, Ni, Pd, Pt and alloys of these metals.\cite{FBD86} These fits are |
| 1286 |
< |
included in {\sc OpenMD} as the {\tt u3} variant of the {\sc eam} force |
| 1287 |
< |
field. Voter and Chen reparamaterized a set of {\sc eam} functions |
| 1288 |
< |
which do a better job of predicting melting points.\cite{Voter:87} |
| 1289 |
< |
These functions are included in {\sc OpenMD} as the {\tt VC} variant of |
| 1290 |
< |
the {\sc eam} force field. An additional set of functions (the |
| 1291 |
< |
``Universal 6'' functions) are included in {\sc OpenMD} as the {\tt u6} |
| 1292 |
< |
variant of {\sc eam}. For example, to specify the Voter-Chen variant |
| 1293 |
< |
of the {\sc eam} force field, the user would add the {\tt |
| 1294 |
< |
forceFieldVariant = "VC";} line to the meta-data file. |
| 1295 |
< |
|
| 1296 |
< |
The potential files used by the {\sc eam} force field are in the |
| 1297 |
< |
standard {\tt funcfl} format, which is the format utilized by a number |
| 1298 |
< |
of other codes (e.g. ParaDyn~\cite{Paradyn}, {\sc dynamo 86}). It |
| 1299 |
< |
should be noted that the energy units in these files are in eV, not |
| 1300 |
< |
$\mbox{kcal mol}^{-1}$ as in the rest of the {\sc OpenMD} force field |
| 1301 |
< |
files. |
| 1302 |
< |
|
| 1303 |
< |
\section{\label{section:sc}The Sutton-Chen Force Field} |
| 1304 |
< |
|
| 1305 |
< |
The Sutton-Chen ({\sc sc})~\cite{Chen90} potential has been used to |
| 1306 |
< |
study a wide range of phenomena in metals. Although it is similar in |
| 1307 |
< |
form to the {\sc eam} potential, the Sutton-Chen model takes on a |
| 1308 |
< |
simpler form, |
| 1309 |
< |
\begin{equation} |
| 1310 |
< |
\label{eq:SCP1} |
| 1311 |
< |
U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq |
| 1312 |
< |
i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] , |
| 1313 |
< |
\end{equation} |
| 1314 |
< |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
| 1315 |
< |
\begin{equation} |
| 1316 |
< |
\label{eq:SCP2} |
| 1317 |
< |
V^{pair}_{ij}(r)=\left( |
| 1318 |
< |
\frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( |
| 1319 |
< |
\frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}} |
| 1320 |
< |
\end{equation} |
| 1321 |
< |
|
| 1322 |
< |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
| 1323 |
< |
interactions of the pseudo-atom cores. The $\sqrt{\rho_i}$ term in |
| 1324 |
< |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 1325 |
< |
the interactions between the valence electrons and the cores of the |
| 1326 |
< |
pseudo-atoms. $D_{ij}$, $D_{ii}$, $c_i$ and $\alpha_{ij}$ are |
| 1327 |
< |
parameters used to tune the potential for different transition |
| 1328 |
< |
metals. |
| 2036 |
> |
//Repulsive Power |
| 2037 |
> |
//Atom1 Atom2 RepulsivePower sigma epsilon n |
| 2038 |
> |
Au ON RepulsivePower 3.47005 0.186208 11 |
| 2039 |
> |
Au NO RepulsivePower 3.53955 0.168629 11 |
| 2040 |
> |
end NonBondedInteractions |
| 2041 |
> |
\end{lstlisting} |
| 2042 |
|
|
| 1330 |
– |
The {\sc sc} potential form has also been parameterized by Qi {\it et |
| 1331 |
– |
al.}\cite{Qi99} These parameters were obtained via empirical and {\it |
| 1332 |
– |
ab initio} calculations to match structural features of the FCC |
| 1333 |
– |
crystal. To specify the original Sutton-Chen variant of the {\sc sc} |
| 1334 |
– |
force field, the user would add the {\tt forceFieldVariant = "SC";} |
| 1335 |
– |
line to the meta-data file, while specification of the Qi {\it et al.} |
| 1336 |
– |
quantum-adapted variant of the {\sc sc} potential, the user would add |
| 1337 |
– |
the {\tt forceFieldVariant = "QSC";} line to the meta-data file. |
| 1338 |
– |
|
| 1339 |
– |
\section{\label{section:clay}The CLAY force field} |
| 1340 |
– |
|
| 1341 |
– |
The {\sc clay} force field is based on an ionic (nonbonded) |
| 1342 |
– |
description of the metal-oxygen interactions associated with hydrated |
| 1343 |
– |
phases. All atoms are represented as point charges and are allowed |
| 1344 |
– |
complete translational freedom. Metal-oxygen interactions are based on |
| 1345 |
– |
a simple Lennard-Jones potential combined with electrostatics. The |
| 1346 |
– |
empirical parameters were optimized by Cygan {\it et |
| 1347 |
– |
al.}\cite{Cygan04} on the basis of known mineral structures, and |
| 1348 |
– |
partial atomic charges were derived from periodic DFT quantum chemical |
| 1349 |
– |
calculations of simple oxide, hydroxide, and oxyhydroxide model |
| 1350 |
– |
compounds with well-defined structures. |
| 1351 |
– |
|
| 1352 |
– |
|
| 2043 |
|
\section{\label{section:electrostatics}Electrostatics} |
| 2044 |
|
|
| 2045 |
|
To aid in performing simulations in more traditional force fields, we |
| 2169 |
|
\end{equation} |
| 2170 |
|
the shifted potential (eq. (\ref{eq:SPPot})) becomes |
| 2171 |
|
\begin{equation} |
| 2172 |
< |
V_{\textrm{DSP}}(r) = q_iq_j\left(\frac{\textrm{erfc}\left(\alpha r\right)}{r}-\ |
| 1483 |
< |
frac{\textrm{erfc}\left(\alpha R_\textrm{c}\right)}{R_\textrm{c}}\right) \quad r |
| 2172 |
> |
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 |
| 2173 |
|
\leqslant R_\textrm{c}, |
| 2174 |
|
\label{eq:DSPPot} |
| 2175 |
|
\end{equation} |
| 2213 |
|
this reason, the default electrostatic summation method utilized by |
| 2214 |
|
{\sc OpenMD} is the DSF (Eq. \ref{eq:DSFPot}) with a damping parameter |
| 2215 |
|
($\alpha$) that is set algorithmically from the cutoff radius. |
| 2216 |
+ |
|
| 2217 |
+ |
|
| 2218 |
+ |
\section{\label{section:cutoffGroups}Switching Functions} |
| 2219 |
+ |
|
| 2220 |
+ |
If done poorly, calculating the the long-range interactions for $N$ |
| 2221 |
+ |
atoms would involve $N(N-1)/2$ evaluations of atomic distances. To |
| 2222 |
+ |
reduce the number of distance evaluations between pairs of atoms, {\sc |
| 2223 |
+ |
OpenMD} allows the use of switched cutoffs with Verlet neighbor |
| 2224 |
+ |
lists.\cite{Allen87} Neutral groups which contain charges will exhibit |
| 2225 |
+ |
pathological forces unless the cutoff is applied to the neutral groups |
| 2226 |
+ |
evenly instead of to the individual atoms.\cite{leach01:mm} {\sc |
| 2227 |
+ |
OpenMD} allows users to specify cutoff groups which may contain an |
| 2228 |
+ |
arbitrary number of atoms in the molecule. Atoms in a cutoff group |
| 2229 |
+ |
are treated as a single unit for the evaluation of the switching |
| 2230 |
+ |
function: |
| 2231 |
+ |
\begin{equation} |
| 2232 |
+ |
V_{\mathrm{long-range}} = \sum_{a} \sum_{b>a} s(r_{ab}) \sum_{i \in a} \sum_{j \in b} V_{ij}(r_{ij}), |
| 2233 |
+ |
\end{equation} |
| 2234 |
+ |
where $r_{ab}$ is the distance between the centers of mass of the two |
| 2235 |
+ |
cutoff groups ($a$ and $b$). |
| 2236 |
+ |
|
| 2237 |
+ |
The sums over $a$ and $b$ are over the cutoff groups that are present |
| 2238 |
+ |
in the simulation. Atoms which are not explicitly defined as members |
| 2239 |
+ |
of a {\tt cutoffGroup} are treated as a group consisting of only one |
| 2240 |
+ |
atom. The switching function, $s(r)$ is the standard cubic switching |
| 2241 |
+ |
function, |
| 2242 |
+ |
\begin{equation} |
| 2243 |
+ |
S(r) = |
| 2244 |
+ |
\begin{cases} |
| 2245 |
+ |
1 & \text{if $r \le r_{\text{sw}}$},\\ |
| 2246 |
+ |
\frac{(r_{\text{cut}} + 2r - 3r_{\text{sw}})(r_{\text{cut}} - r)^2} |
| 2247 |
+ |
{(r_{\text{cut}} - r_{\text{sw}})^3} |
| 2248 |
+ |
& \text{if $r_{\text{sw}} < r \le r_{\text{cut}}$}, \\ |
| 2249 |
+ |
0 & \text{if $r > r_{\text{cut}}$.} |
| 2250 |
+ |
\end{cases} |
| 2251 |
+ |
\label{eq:dipoleSwitching} |
| 2252 |
+ |
\end{equation} |
| 2253 |
+ |
Here, $r_{\text{sw}}$ is the {\tt switchingRadius}, or the distance |
| 2254 |
+ |
beyond which interactions are reduced, and $r_{\text{cut}}$ is the |
| 2255 |
+ |
{\tt cutoffRadius}, or the distance at which interactions are |
| 2256 |
+ |
truncated. |
| 2257 |
|
|
| 2258 |
+ |
Users of {\sc OpenMD} do not need to specify the {\tt cutoffRadius} or |
| 2259 |
+ |
{\tt switchingRadius}. In simulations containing only Lennard-Jones |
| 2260 |
+ |
atoms, the cutoff radius has a default value of $2.5\sigma_{ii}$, |
| 2261 |
+ |
where $\sigma_{ii}$ is the largest Lennard-Jones length parameter |
| 2262 |
+ |
present in the simulation. In simulations containing charged or |
| 2263 |
+ |
dipolar atoms, the default cutoff radius is $15 \mbox{\AA}$. |
| 2264 |
+ |
|
| 2265 |
+ |
The {\tt switchingRadius} is set to a default value of 95\% of the |
| 2266 |
+ |
{\tt cutoffRadius}. In the special case of a simulation containing |
| 2267 |
+ |
{\it only} Lennard-Jones atoms, the default switching radius takes the |
| 2268 |
+ |
same value as the cutoff radius, and {\sc OpenMD} will use a shifted |
| 2269 |
+ |
potential to remove discontinuities in the potential at the cutoff. |
| 2270 |
+ |
Both radii may be specified in the meta-data file. |
| 2271 |
+ |
|
| 2272 |
+ |
|
| 2273 |
+ |
\section{\label{section:WATER}The {\sc water} Force Field} |
| 2274 |
+ |
|
| 2275 |
+ |
In addition to the {\sc duff} force field's solvent description, a |
| 2276 |
+ |
separate {\sc water} force field has been included for simulating most |
| 2277 |
+ |
of the common rigid-body water models. This force field includes the |
| 2278 |
+ |
simple and point-dipolar models (SSD, SSD1, SSD/E, SSD/RF, and DPD |
| 2279 |
+ |
water), as well as the common charge-based models (SPC, SPC/E, TIP3P, |
| 2280 |
+ |
TIP4P, and |
| 2281 |
+ |
TIP5P).\cite{liu96:new_model,Ichiye03,fennell04,Marrink01,Berendsen81,Berendsen87,Jorgensen83,Mahoney00} |
| 2282 |
+ |
In order to handle these models, charge-charge interactions were |
| 2283 |
+ |
included in the force-loop: |
| 2284 |
+ |
\begin{equation} |
| 2285 |
+ |
V_{\text{charge}}(r_{ij}) = \sum_{ij}\frac{q_iq_je^2}{r_{ij}}, |
| 2286 |
+ |
\end{equation} |
| 2287 |
+ |
where $q$ represents the charge on particle $i$ or $j$, and $e$ is the |
| 2288 |
+ |
charge of an electron in Coulombs. The charge-charge interaction |
| 2289 |
+ |
support is rudimentary in the current version of {\sc OpenMD}. As with |
| 2290 |
+ |
the other pair interactions, charges can be simulated with a pure |
| 2291 |
+ |
cutoff or a reaction field. The various methods for performing the |
| 2292 |
+ |
Ewald summation have not yet been included. The {\sc water} force |
| 2293 |
+ |
field can be easily expanded through modification of the {\sc water} |
| 2294 |
+ |
force field file ({\tt WATER.frc}). By adding atom types and inserting |
| 2295 |
+ |
the appropriate parameters, it is possible to extend the force field |
| 2296 |
+ |
to handle rigid molecules other than water. |
| 2297 |
+ |
|
| 2298 |
+ |
|
| 2299 |
+ |
|
| 2300 |
|
\section{\label{section:pbc}Periodic Boundary Conditions} |
| 2301 |
|
|
| 2302 |
|
\newcommand{\roundme}{\operatorname{round}} |
| 2678 |
|
& (with separate barostats on each box dimension) & \\ |
| 2679 |
|
LD & Langevin Dynamics & {\tt ensemble = LD;} \\ |
| 2680 |
|
& (approximates the effects of an implicit solvent) & \\ |
| 2681 |
+ |
LangevinHull & Non-periodic Langevin Dynamics & {\tt ensemble = LangevinHull;} \\ |
| 2682 |
+ |
& (Langevin Dynamics for molecules on convex hull;\\ |
| 2683 |
+ |
& Newtonian for interior molecules) & \\ |
| 2684 |
|
\end{tabular} |
| 2685 |
|
\end{center} |
| 2686 |
|
|
| 3166 |
|
in the body-fixed frame) and ${\bf V}$ is a generalized velocity, |
| 3167 |
|
${\bf V} = |
| 3168 |
|
\left\{{\bf v},{\bf \omega}\right\}$. The right side of |
| 3169 |
< |
Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a |
| 3169 |
> |
Eq. \ref{LDGeneralizedForm} consists of three generalized forces: a |
| 3170 |
|
system force (${\bf F}_{s}$), a frictional or dissipative force (${\bf |
| 3171 |
|
F}_{f}$) and a stochastic force (${\bf F}_{r}$). While the evolution |
| 3172 |
|
of the system in Newtonian mechanics is typically done in the lab |
| 3388 |
|
\endhead |
| 3389 |
|
\hline |
| 3390 |
|
\endfoot |
| 3391 |
< |
{\tt viscosity} & centipoise & Sets the value of viscosity of the implicit |
| 3391 |
> |
{\tt viscosity} & poise & Sets the value of viscosity of the implicit |
| 3392 |
|
solvent \\ |
| 3393 |
|
{\tt targetTemp} & K & Sets the target temperature of the system. |
| 3394 |
|
This parameter must be specified to use Langevin dynamics. \\ |
| 3395 |
|
{\tt HydroPropFile} & string & Specifies the name of the resistance |
| 3396 |
|
tensor (usually a {\tt .diff} file) which is precalculated by {\tt |
| 3397 |
< |
Hydro}. This keyworkd is not necessary if the simulation contains only |
| 3397 |
> |
Hydro}. This keyword is not necessary if the simulation contains only |
| 3398 |
|
simple bodies (spheres and ellipsoids). \\ |
| 3399 |
|
{\tt beadSize} & $\mbox{\AA}$ & Sets the diameter of the beads to use |
| 3400 |
|
when the {\tt RoughShell} model is used to approximate the resistance |
| 3401 |
|
tensor. |
| 3402 |
|
\label{table:ldParameters} |
| 3403 |
+ |
\end{longtable} |
| 3404 |
+ |
|
| 3405 |
+ |
\section{Constant Pressure without periodic boundary conditions (The LangevinHull)} |
| 3406 |
+ |
|
| 3407 |
+ |
The Langevin Hull\cite{Vardeman2011} uses an external bath at a fixed constant pressure |
| 3408 |
+ |
($P$) and temperature ($T$) with an effective solvent viscosity |
| 3409 |
+ |
($\eta$). This bath interacts only with the objects on the exterior |
| 3410 |
+ |
hull of the system. Defining the hull of the atoms in a simulation is |
| 3411 |
+ |
done in a manner similar to the approach of Kohanoff, Caro and |
| 3412 |
+ |
Finnis.\cite{Kohanoff:2005qm} That is, any instantaneous configuration |
| 3413 |
+ |
of the atoms in the system is considered as a point cloud in three |
| 3414 |
+ |
dimensional space. Delaunay triangulation is used to find all facets |
| 3415 |
+ |
between coplanar |
| 3416 |
+ |
neighbors.\cite{delaunay,springerlink:10.1007/BF00977785} In highly |
| 3417 |
+ |
symmetric point clouds, facets can contain many atoms, but in all but |
| 3418 |
+ |
the most symmetric of cases, the facets are simple triangles in |
| 3419 |
+ |
3-space which contain exactly three atoms. |
| 3420 |
+ |
|
| 3421 |
+ |
The convex hull is the set of facets that have {\it no concave |
| 3422 |
+ |
corners} at an atomic site.\cite{Barber96,EDELSBRUNNER:1994oq} This |
| 3423 |
+ |
eliminates all facets on the interior of the point cloud, leaving only |
| 3424 |
+ |
those exposed to the bath. Sites on the convex hull are dynamic; as |
| 3425 |
+ |
molecules re-enter the cluster, all interactions between atoms on that |
| 3426 |
+ |
molecule and the external bath are removed. Since the edge is |
| 3427 |
+ |
determined dynamically as the simulation progresses, no {\it a priori} |
| 3428 |
+ |
geometry is defined. The pressure and temperature bath interacts only |
| 3429 |
+ |
with the atoms on the edge and not with atoms interior to the |
| 3430 |
+ |
simulation. |
| 3431 |
+ |
|
| 3432 |
+ |
Atomic sites in the interior of the simulation move under standard |
| 3433 |
+ |
Newtonian dynamics, |
| 3434 |
+ |
\begin{equation} |
| 3435 |
+ |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U, |
| 3436 |
+ |
\label{eq:Newton} |
| 3437 |
+ |
\end{equation} |
| 3438 |
+ |
where $m_i$ is the mass of site $i$, ${\mathbf v}_i(t)$ is the |
| 3439 |
+ |
instantaneous velocity of site $i$ at time $t$, and $U$ is the total |
| 3440 |
+ |
potential energy. For atoms on the exterior of the cluster |
| 3441 |
+ |
(i.e. those that occupy one of the vertices of the convex hull), the |
| 3442 |
+ |
equation of motion is modified with an external force, ${\mathbf |
| 3443 |
+ |
F}_i^{\mathrm ext}$: |
| 3444 |
+ |
\begin{equation} |
| 3445 |
+ |
m_i \dot{\mathbf v}_i(t)=-{\mathbf \nabla}_i U + {\mathbf F}_i^{\mathrm ext}. |
| 3446 |
+ |
\end{equation} |
| 3447 |
+ |
|
| 3448 |
+ |
The external bath interacts indirectly with the atomic sites through |
| 3449 |
+ |
the intermediary of the hull facets. Since each vertex (or atom) |
| 3450 |
+ |
provides one corner of a triangular facet, the force on the facets are |
| 3451 |
+ |
divided equally to each vertex. However, each vertex can participate |
| 3452 |
+ |
in multiple facets, so the resultant force is a sum over all facets |
| 3453 |
+ |
$f$ containing vertex $i$: |
| 3454 |
+ |
\begin{equation} |
| 3455 |
+ |
{\mathbf F}_{i}^{\mathrm ext} = \sum_{\begin{array}{c}\mathrm{facets\ |
| 3456 |
+ |
} f \\ \mathrm{containing\ } i\end{array}} \frac{1}{3}\ {\mathbf |
| 3457 |
+ |
F}_f^{\mathrm ext} |
| 3458 |
+ |
\end{equation} |
| 3459 |
+ |
|
| 3460 |
+ |
The external pressure bath applies a force to the facets of the convex |
| 3461 |
+ |
hull in direct proportion to the area of the facet, while the thermal |
| 3462 |
+ |
coupling depends on the solvent temperature, viscosity and the size |
| 3463 |
+ |
and shape of each facet. The thermal interactions are expressed as a |
| 3464 |
+ |
standard Langevin description of the forces, |
| 3465 |
+ |
\begin{equation} |
| 3466 |
+ |
\begin{array}{rclclcl} |
| 3467 |
+ |
{\mathbf F}_f^{\text{ext}} & = & \text{external pressure} & + & \text{drag force} & + & \text{random force} \\ |
| 3468 |
+ |
& = & -\hat{n}_f P A_f & - & \Xi_f(t) {\mathbf v}_f(t) & + & {\mathbf R}_f(t) |
| 3469 |
+ |
\end{array} |
| 3470 |
+ |
\end{equation} |
| 3471 |
+ |
Here, $A_f$ and $\hat{n}_f$ are the area and (outward-facing) normal |
| 3472 |
+ |
vectors for facet $f$, respectively. ${\mathbf v}_f(t)$ is the |
| 3473 |
+ |
velocity of the facet centroid, |
| 3474 |
+ |
\begin{equation} |
| 3475 |
+ |
{\mathbf v}_f(t) = \frac{1}{3} \sum_{i=1}^{3} {\mathbf v}_i, |
| 3476 |
+ |
\end{equation} |
| 3477 |
+ |
and $\Xi_f(t)$ is an approximate ($3 \times 3$) resistance tensor that |
| 3478 |
+ |
depends on the geometry and surface area of facet $f$ and the |
| 3479 |
+ |
viscosity of the bath. The resistance tensor is related to the |
| 3480 |
+ |
fluctuations of the random force, $\mathbf{R}(t)$, by the |
| 3481 |
+ |
fluctuation-dissipation theorem (see Eq. \ref{eq:randomForce}). |
| 3482 |
+ |
|
| 3483 |
+ |
Once the resistance tensor is known for a given facet, a stochastic |
| 3484 |
+ |
vector that has the properties in Eq. (\ref{eq:randomForce}) can be |
| 3485 |
+ |
calculated efficiently by carrying out a Cholesky decomposition to |
| 3486 |
+ |
obtain the square root matrix of the resistance tensor (see |
| 3487 |
+ |
Eq. \ref{eq:Cholesky}). |
| 3488 |
+ |
|
| 3489 |
+ |
Our treatment of the resistance tensor for the Langevin Hull facets is |
| 3490 |
+ |
approximate. $\Xi_f$ for a rigid triangular plate would normally be |
| 3491 |
+ |
treated as a $6 \times 6$ tensor that includes translational and |
| 3492 |
+ |
rotational drag as well as translational-rotational coupling. The |
| 3493 |
+ |
computation of resistance tensors for rigid bodies has been detailed |
| 3494 |
+ |
elsewhere,\cite{JoseGarciadelaTorre02012000,Garcia-de-la-Torre:2001wd,GarciadelaTorreJ2002,Sun:2008fk} |
| 3495 |
+ |
but the standard approach involving bead approximations would be |
| 3496 |
+ |
prohibitively expensive if it were recomputed at each step in a |
| 3497 |
+ |
molecular dynamics simulation. |
| 3498 |
+ |
|
| 3499 |
+ |
Instead, we are utilizing an approximate resistance tensor obtained by |
| 3500 |
+ |
first constructing the Oseen tensor for the interaction of the |
| 3501 |
+ |
centroid of the facet ($f$) with each of the subfacets $\ell=1,2,3$, |
| 3502 |
+ |
\begin{equation} |
| 3503 |
+ |
T_{\ell f}=\frac{A_\ell}{8\pi\eta R_{\ell f}}\left(I + |
| 3504 |
+ |
\frac{\mathbf{R}_{\ell f}\mathbf{R}_{\ell f}^T}{R_{\ell f}^2}\right) |
| 3505 |
+ |
\end{equation} |
| 3506 |
+ |
Here, $A_\ell$ is the area of subfacet $\ell$ which is a triangle |
| 3507 |
+ |
containing two of the vertices of the facet along with the centroid. |
| 3508 |
+ |
$\mathbf{R}_{\ell f}$ is the vector between the centroid of facet $f$ |
| 3509 |
+ |
and the centroid of sub-facet $\ell$, and $I$ is the ($3 \times 3$) |
| 3510 |
+ |
identity matrix. $\eta$ is the viscosity of the external bath. |
| 3511 |
+ |
|
| 3512 |
+ |
The tensors for each of the sub-facets are added together, and the |
| 3513 |
+ |
resulting matrix is inverted to give a $3 \times 3$ resistance tensor |
| 3514 |
+ |
for translations of the triangular facet, |
| 3515 |
+ |
\begin{equation} |
| 3516 |
+ |
\Xi_f(t) =\left[\sum_{i=1}^3 T_{if}\right]^{-1}. |
| 3517 |
+ |
\end{equation} |
| 3518 |
+ |
Note that this treatment ignores rotations (and |
| 3519 |
+ |
translational-rotational coupling) of the facet. In compact systems, |
| 3520 |
+ |
the facets stay relatively fixed in orientation between |
| 3521 |
+ |
configurations, so this appears to be a reasonably good approximation. |
| 3522 |
+ |
|
| 3523 |
+ |
At each |
| 3524 |
+ |
molecular dynamics time step, the following process is carried out: |
| 3525 |
+ |
\begin{enumerate} |
| 3526 |
+ |
\item The standard inter-atomic forces ($\nabla_iU$) are computed. |
| 3527 |
+ |
\item Delaunay triangulation is carried out using the current atomic |
| 3528 |
+ |
configuration. |
| 3529 |
+ |
\item The convex hull is computed and facets are identified. |
| 3530 |
+ |
\item For each facet: |
| 3531 |
+ |
\begin{itemize} |
| 3532 |
+ |
\item[a.] The force from the pressure bath ($-\hat{n}_fPA_f$) is |
| 3533 |
+ |
computed. |
| 3534 |
+ |
\item[b.] The resistance tensor ($\Xi_f(t)$) is computed using the |
| 3535 |
+ |
viscosity ($\eta$) of the bath. |
| 3536 |
+ |
\item[c.] Facet drag ($-\Xi_f(t) \mathbf{v}_f(t)$) forces are |
| 3537 |
+ |
computed. |
| 3538 |
+ |
\item[d.] Random forces ($\mathbf{R}_f(t)$) are computed using the |
| 3539 |
+ |
resistance tensor and the temperature ($T$) of the bath. |
| 3540 |
+ |
\end{itemize} |
| 3541 |
+ |
\item The facet forces are divided equally among the vertex atoms. |
| 3542 |
+ |
\item Atomic positions and velocities are propagated. |
| 3543 |
+ |
\end{enumerate} |
| 3544 |
+ |
The Delaunay triangulation and computation of the convex hull are done |
| 3545 |
+ |
using calls to the qhull library,\cite{Qhull} and for this reason, if |
| 3546 |
+ |
qhull is not detected during the build, the Langevin Hull integrator |
| 3547 |
+ |
will not be available. There is a minimal penalty for computing the |
| 3548 |
+ |
convex hull and resistance tensors at each step in the molecular |
| 3549 |
+ |
dynamics simulation (roughly 0.02 $\times$ cost of a single force |
| 3550 |
+ |
evaluation). |
| 3551 |
+ |
|
| 3552 |
+ |
\begin{longtable}[c]{GBF} |
| 3553 |
+ |
\caption{Meta-data Keywords: Required parameters for the Langevin Hull integrator} |
| 3554 |
+ |
\\ |
| 3555 |
+ |
{\bf keyword} & {\bf units} & {\bf use} \\ \hline |
| 3556 |
+ |
\endhead |
| 3557 |
+ |
\hline |
| 3558 |
+ |
\endfoot |
| 3559 |
+ |
{\tt viscosity} & poise & Sets the value of viscosity of the implicit |
| 3560 |
+ |
solven . \\ |
| 3561 |
+ |
{\tt targetTemp} & K & Sets the target temperature of the system. |
| 3562 |
+ |
This parameter must be specified to use Langevin Hull dynamics. \\ |
| 3563 |
+ |
{\tt targetPressure} & atm & Sets the target pressure of the system. |
| 3564 |
+ |
This parameter must be specified to use Langevin Hull dynamics. \\ |
| 3565 |
+ |
{\tt usePeriodicBoundaryConditions} & logical & Turns off periodic boundary conditions. |
| 3566 |
+ |
This parameter must be set to \tt false \\ |
| 3567 |
+ |
\label{table:lhullParameters} |
| 3568 |
|
\end{longtable} |
| 3569 |
|
|
| 3570 |
+ |
|
| 3571 |
|
\section{\label{sec:constraints}Constraint Methods} |
| 3572 |
|
|
| 3573 |
|
\subsection{\label{section:rattle}The {\sc rattle} Method for Bond |
| 3716 |
|
\label{table:zconParams} |
| 3717 |
|
\end{longtable} |
| 3718 |
|
|
| 3719 |
< |
\chapter{\label{section:restraints}Restraints} |
| 3720 |
< |
Restraints are external potentials that are added to a system to keep |
| 3721 |
< |
particular molecules or collections of particles close to some |
| 3722 |
< |
reference structure. A restraint can be a collective |
| 3719 |
> |
% \chapter{\label{section:restraints}Restraints} |
| 3720 |
> |
% Restraints are external potentials that are added to a system to |
| 3721 |
> |
% keep particular molecules or collections of particles close to some |
| 3722 |
> |
% reference structure. A restraint can be a collective |
| 3723 |
|
|
| 3724 |
|
\chapter{\label{section:thermInt}Thermodynamic Integration} |
| 3725 |
|
|
| 3858 |
|
Einstein crystal |
| 3859 |
|
\label{table:thermIntParams} |
| 3860 |
|
\end{longtable} |
| 3861 |
+ |
|
| 3862 |
+ |
\chapter{\label{section:rnemd}Reverse Non-Equilibrium Molecular Dynamics (RNEMD)} |
| 3863 |
+ |
|
| 3864 |
+ |
There are many ways to compute transport properties from molecular |
| 3865 |
+ |
dynamics simulations. Equilibrium Molecular Dynamics (EMD) |
| 3866 |
+ |
simulations can be used by computing relevant time correlation |
| 3867 |
+ |
functions and assuming linear response theory holds. For some transport properties (notably thermal conductivity), EMD approaches |
| 3868 |
+ |
are subject to noise and poor convergence of the relevant |
| 3869 |
+ |
correlation functions. Traditional Non-equilibrium Molecular Dynamics |
| 3870 |
+ |
(NEMD) methods impose a gradient (e.g. thermal or momentum) on a |
| 3871 |
+ |
simulation. However, the resulting flux is often difficult to |
| 3872 |
+ |
measure. Furthermore, problems arise for NEMD simulations of |
| 3873 |
+ |
heterogeneous systems, such as phase-phase boundaries or interfaces, |
| 3874 |
+ |
where the type of gradient to enforce at the boundary between |
| 3875 |
+ |
materials is unclear. |
| 3876 |
|
|
| 3877 |
+ |
{\it Reverse} Non-Equilibrium Molecular Dynamics (RNEMD) methods adopt |
| 3878 |
+ |
a different approach in that an unphysical {\it flux} is imposed |
| 3879 |
+ |
between different regions or ``slabs'' of the simulation box. The |
| 3880 |
+ |
response of the system is to develop a temperature or momentum {\it |
| 3881 |
+ |
gradient} between the two regions. Since the amount of the applied |
| 3882 |
+ |
flux is known exactly, and the measurement of gradient is generally |
| 3883 |
+ |
less complicated, imposed-flux methods typically take shorter |
| 3884 |
+ |
simulation times to obtain converged results for transport properties. |
| 3885 |
|
|
| 3886 |
+ |
\begin{figure} |
| 3887 |
+ |
\includegraphics[width=\linewidth]{rnemdDemo} |
| 3888 |
+ |
\caption{The (VSS) RNEMD approach imposes unphysical transfer of both |
| 3889 |
+ |
linear momentum and kinetic energy between a ``hot'' slab and a |
| 3890 |
+ |
``cold'' slab in the simulation box. The system responds to this |
| 3891 |
+ |
imposed flux by generating both momentum and temperature gradients. |
| 3892 |
+ |
The slope of the gradients can then be used to compute transport |
| 3893 |
+ |
properties (e.g. shear viscosity and thermal conductivity).} |
| 3894 |
+ |
\label{rnemdDemo} |
| 3895 |
+ |
\end{figure} |
| 3896 |
+ |
|
| 3897 |
+ |
\section{\label{section:algo}Three algorithms for carrying out RNEMD simulations} |
| 3898 |
+ |
\subsection{\label{subsection:swapping}The swapping algorithm} |
| 3899 |
+ |
The original ``swapping'' approaches by M\"{u}ller-Plathe {\it et |
| 3900 |
+ |
al.}\cite{ISI:000080382700030,MullerPlathe:1997xw} can be understood |
| 3901 |
+ |
as a sequence of imaginary elastic collisions between particles in |
| 3902 |
+ |
opposite slabs. In each collision, the entire momentum vectors of |
| 3903 |
+ |
both particles may be exchanged to generate a thermal |
| 3904 |
+ |
flux. Alternatively, a single component of the momentum vectors may be |
| 3905 |
+ |
exchanged to generate a shear flux. This algorithm turns out to be |
| 3906 |
+ |
quite useful in many simulations. However, the M\"{u}ller-Plathe |
| 3907 |
+ |
swapping approach perturbs the system away from ideal |
| 3908 |
+ |
Maxwell-Boltzmann distributions, and this may leads to undesirable |
| 3909 |
+ |
side-effects when the applied flux becomes large.\cite{Maginn:2010} |
| 3910 |
+ |
This limits the applicability of the swapping algorithm, so in OpenMD, |
| 3911 |
+ |
we have implemented two additional algorithms for RNEMD in addition to the |
| 3912 |
+ |
original swapping approach. |
| 3913 |
+ |
|
| 3914 |
+ |
\subsection{\label{subsection:nivs}Non-Isotropic Velocity Scaling (NIVS)} |
| 3915 |
+ |
Instead of having momentum exchange between {\it individual particles} |
| 3916 |
+ |
in each slab, the NIVS algorithm applies velocity scaling to all of |
| 3917 |
+ |
the selected particles in both slabs.\cite{kuang:164101} A combination of linear |
| 3918 |
+ |
momentum, kinetic energy, and flux constraint equations governs the |
| 3919 |
+ |
amount of velocity scaling performed at each step. Interested readers |
| 3920 |
+ |
should consult ref. \citealp{kuang:164101} for further details on the |
| 3921 |
+ |
methodology. |
| 3922 |
+ |
|
| 3923 |
+ |
NIVS has been shown to be very effective at producing thermal |
| 3924 |
+ |
gradients and for computing thermal conductivities, particularly for |
| 3925 |
+ |
heterogeneous interfaces. Although the NIVS algorithm can also be |
| 3926 |
+ |
applied to impose a directional momentum flux, thermal anisotropy was |
| 3927 |
+ |
observed in relatively high flux simulations, and the method is not |
| 3928 |
+ |
suitable for imposing a shear flux or for computing shear viscosities. |
| 3929 |
+ |
|
| 3930 |
+ |
\subsection{\label{subsection:vss}Velocity Shearing and Scaling (VSS)} |
| 3931 |
+ |
The third RNEMD algorithm implemented in OpenMD utilizes a series of |
| 3932 |
+ |
simultaneous velocity shearing and scaling exchanges between the two |
| 3933 |
+ |
slabs.\cite{2012MolPh.110..691K} This method results in a set of simpler equations to satisfy |
| 3934 |
+ |
the conservation constraints while creating a desired flux between the |
| 3935 |
+ |
two slabs. |
| 3936 |
+ |
|
| 3937 |
+ |
The VSS approach is versatile in that it may be used to implement both |
| 3938 |
+ |
thermal and shear transport either separately or simultaneously. |
| 3939 |
+ |
Perturbations of velocities away from the ideal Maxwell-Boltzmann |
| 3940 |
+ |
distributions are minimal, and thermal anisotropy is kept to a |
| 3941 |
+ |
minimum. This ability to generate simultaneous thermal and shear |
| 3942 |
+ |
fluxes has been utilized to map out the shear viscosity of SPC/E water |
| 3943 |
+ |
over a wide range of temperatures (90~K) just with a single simulation. |
| 3944 |
+ |
VSS-RNEMD also allows the directional momentum flux to have |
| 3945 |
+ |
arbitary directions, which could aid in the study of anisotropic solid |
| 3946 |
+ |
surfaces in contact with liquid environments. |
| 3947 |
+ |
|
| 3948 |
+ |
\section{\label{section:usingRNEMD}Using OpenMD to perform a RNEMD simulation} |
| 3949 |
+ |
\subsection{\label{section:rnemdParams} What the user needs to specify} |
| 3950 |
+ |
To carry out a RNEMD simulation, |
| 3951 |
+ |
a user must specify a number of parameters in the MetaData (.md) file. |
| 3952 |
+ |
Because the RNEMD methods have a large number of parameters, these |
| 3953 |
+ |
must be enclosed in a {\it separate} {\tt RNEMD\{...\}} block. The most important |
| 3954 |
+ |
parameters to specify are the {\tt useRNEMD}, {\tt fluxType} and flux |
| 3955 |
+ |
parameters. Most other parameters (summarized in table |
| 3956 |
+ |
\ref{table:rnemd}) have reasonable default values. {\tt fluxType} |
| 3957 |
+ |
sets up the kind of exchange that will be carried out between the two |
| 3958 |
+ |
slabs (either Kinetic Energy ({\tt KE}) or momentum ({\tt Px, Py, Pz, |
| 3959 |
+ |
Pvector}), or some combination of these). The flux is specified |
| 3960 |
+ |
with the use of three possible parameters: {\tt kineticFlux} for |
| 3961 |
+ |
kinetic energy exchange, as well as {\tt momentumFlux} or {\tt |
| 3962 |
+ |
momentumFluxVector} for simulations with directional exchange. |
| 3963 |
+ |
|
| 3964 |
+ |
\subsection{\label{section:rnemdResults} Processing the results} |
| 3965 |
+ |
OpenMD will generate a {\tt .rnemd} |
| 3966 |
+ |
file with the same prefix as the original {\tt .md} file. This file |
| 3967 |
+ |
contains a running average of properties of interest computed within a |
| 3968 |
+ |
set of bins that divide the simulation cell along the $z$-axis. The |
| 3969 |
+ |
first column of the {\tt .rnemd} file is the $z$ coordinate of the |
| 3970 |
+ |
center of each bin, while following columns may contain the average |
| 3971 |
+ |
temperature, velocity, or density within each bin. The output format |
| 3972 |
+ |
in the {\tt .rnemd} file can be altered with the {\tt outputFields}, |
| 3973 |
+ |
{\tt outputBins}, and {\tt outputFileName} parameters. A report at |
| 3974 |
+ |
the top of the {\tt .rnemd} file contains the current exchange totals |
| 3975 |
+ |
as well as the average flux applied during the simulation. Using the |
| 3976 |
+ |
slope of the temperature or velocity gradient obtaine from the {\tt |
| 3977 |
+ |
.rnemd} file along with the applied flux, the user can very simply |
| 3978 |
+ |
arrive at estimates of thermal conductivities ($\lambda$), |
| 3979 |
+ |
\begin{equation} |
| 3980 |
+ |
J_z = -\lambda \frac{\partial T}{\partial z}, |
| 3981 |
+ |
\end{equation} |
| 3982 |
+ |
and shear viscosities ($\eta$), |
| 3983 |
+ |
\begin{equation} |
| 3984 |
+ |
j_z(p_x) = -\eta \frac{\partial \langle v_x \rangle}{\partial z}. |
| 3985 |
+ |
\end{equation} |
| 3986 |
+ |
Here, the quantities on the left hand side are the actual flux values |
| 3987 |
+ |
(in the header of the {\tt .rnemd} file), while the slopes are |
| 3988 |
+ |
obtained from linear fits to the gradients observed in the {\tt |
| 3989 |
+ |
.rnemd} file. |
| 3990 |
+ |
|
| 3991 |
+ |
More complicated simulations (including interfaces) require a bit more |
| 3992 |
+ |
care. Here the second derivative may be required to compute the |
| 3993 |
+ |
interfacial thermal conductance, |
| 3994 |
+ |
\begin{align} |
| 3995 |
+ |
G^\prime &= \left(\nabla\lambda \cdot \mathbf{\hat{n}}\right)_{z_0} \\ |
| 3996 |
+ |
&= \frac{\partial}{\partial z}\left(-\frac{J_z}{ |
| 3997 |
+ |
\left(\frac{\partial T}{\partial z}\right)}\right)_{z_0} \\ |
| 3998 |
+ |
&= J_z\left(\frac{\partial^2 T}{\partial z^2}\right)_{z_0} \Big/ |
| 3999 |
+ |
\left(\frac{\partial T}{\partial z}\right)_{z_0}^2. |
| 4000 |
+ |
\label{derivativeG} |
| 4001 |
+ |
\end{align} |
| 4002 |
+ |
where $z_0$ is the location of the interface between two materials and |
| 4003 |
+ |
$\mathbf{\hat{n}}$ is a unit vector normal to the interface. We |
| 4004 |
+ |
suggest that users interested in interfacial conductance consult |
| 4005 |
+ |
reference \citealp{kuang:AuThl} for other approaches to computing $G$. |
| 4006 |
+ |
Users interested in {\it friction coefficients} at heterogeneous |
| 4007 |
+ |
interfaces may also find reference \citealp{2012MolPh.110..691K} |
| 4008 |
+ |
useful. |
| 4009 |
+ |
|
| 4010 |
+ |
\newpage |
| 4011 |
+ |
|
| 4012 |
+ |
\begin{longtable}[c]{JKLM} |
| 4013 |
+ |
\caption{Meta-data Keywords: Parameters for RNEMD simulations}\\ |
| 4014 |
+ |
\multicolumn{4}{c}{The following keywords must be enclosed inside a {\tt RNEMD\{...\}} block.} |
| 4015 |
+ |
\\ \hline |
| 4016 |
+ |
{\bf keyword} & {\bf units} & {\bf use} & {\bf remarks} \\ \hline |
| 4017 |
+ |
\endhead |
| 4018 |
+ |
\hline |
| 4019 |
+ |
\endfoot |
| 4020 |
+ |
{\tt useRNEMD} & logical & perform RNEMD? & default is ``false'' \\ |
| 4021 |
+ |
{\tt objectSelection} & string & see section \ref{section:syntax} |
| 4022 |
+ |
for selection syntax & default is ``select all'' \\ |
| 4023 |
+ |
{\tt method} & string & exchange method & one of the following: |
| 4024 |
+ |
{\tt Swap, NIVS,} or {\tt VSS} (default is {\tt VSS}) \\ |
| 4025 |
+ |
{\tt fluxType} & string & what is being exchanged between slabs? & one |
| 4026 |
+ |
of the following: {\tt KE, Px, Py, Pz, Pvector, KE+Px, KE+Py, KE+Pvector} \\ |
| 4027 |
+ |
{\tt kineticFlux} & kcal mol$^{-1}$ \AA$^{-2}$ fs$^{-1}$ & specify the kinetic energy flux & \\ |
| 4028 |
+ |
{\tt momentumFlux} & amu \AA$^{-1}$ fs$^{-2}$ & specify the momentum flux & \\ |
| 4029 |
+ |
{\tt momentumFluxVector} & amu \AA$^{-1}$ fs$^{-2}$ & specify the momentum flux when |
| 4030 |
+ |
{\tt Pvector} is part of the exchange & Vector3d input\\ |
| 4031 |
+ |
{\tt exchangeTime} & fs & how often to perform the exchange & default is 100 fs\\ |
| 4032 |
+ |
|
| 4033 |
+ |
{\tt slabWidth} & $\mbox{\AA}$ & width of the two exchange slabs & default is $\mathsf{H}_{zz} / 10.0$ \\ |
| 4034 |
+ |
{\tt slabAcenter} & $\mbox{\AA}$ & center of the end slab & default is 0 \\ |
| 4035 |
+ |
{\tt slabBcenter} & $\mbox{\AA}$ & center of the middle slab & default is $\mathsf{H}_{zz} / 2$ \\ |
| 4036 |
+ |
{\tt outputFileName} & string & file name for output histograms & default is the same prefix as the |
| 4037 |
+ |
.md file, but with the {\tt .rnemd} extension \\ |
| 4038 |
+ |
{\tt outputBins} & int & number of $z$-bins in the output histogram & |
| 4039 |
+ |
default is 20 \\ |
| 4040 |
+ |
{\tt outputFields} & string & columns to print in the {\tt .rnemd} |
| 4041 |
+ |
file where each column is separated by a pipe ($\mid$) symbol. & Allowed column names are: {\sc z, temperature, velocity, density} \\ |
| 4042 |
+ |
\label{table:rnemd} |
| 4043 |
+ |
\end{longtable} |
| 4044 |
+ |
|
| 4045 |
|
\chapter{\label{section:minimizer}Energy Minimization} |
| 4046 |
|
|
| 4047 |
< |
As one of the basic procedures of molecular modeling, energy |
| 2925 |
< |
minimization is used to identify local configurations that are stable |
| 4047 |
> |
Energy minimization is used to identify local configurations that are stable |
| 4048 |
|
points on the potential energy surface. There is a vast literature on |
| 4049 |
|
energy minimization algorithms have been developed to search for the |
| 4050 |
|
global energy minimum as well as to find local structures which are |
| 4171 |
|
\begin{figure} |
| 4172 |
|
\centering |
| 4173 |
|
\includegraphics[width=3in]{heirarchy.pdf} |
| 4174 |
< |
\caption[Class heirarchy for StuntDoubles in {\sc OpenMD}-4]{ \\ The |
| 4175 |
< |
class heirarchy of StuntDoubles in {\sc OpenMD}-4. The selection |
| 4174 |
> |
\caption[Class heirarchy for StuntDoubles in {\sc OpenMD}]{ \\ The |
| 4175 |
> |
class heirarchy of StuntDoubles in {\sc OpenMD}. The selection |
| 4176 |
|
syntax allows the user to select any of the objects that are descended |
| 4177 |
|
from a StuntDouble.} |
| 4178 |
|
\label{fig:heirarchy} |
| 4312 |
|
\end{center} |
| 4313 |
|
|
| 4314 |
|
For example, the phrase {\tt select mass > 16.0 and charge < -2} |
| 4315 |
< |
wouldselect StuntDoubles which have mass greater than 16.0 and charges |
| 4315 |
> |
would select StuntDoubles which have mass greater than 16.0 and charges |
| 4316 |
|
less than -2. |
| 4317 |
|
|
| 4318 |
|
\subsection{\label{section:within}Within expressions} |
| 4352 |
|
-z & {\tt -{}-zconstraint} & replace the atom types of zconstraint molecules (default=off) \\ |
| 4353 |
|
-r & {\tt -{}-rigidbody} & add a pseudo COM atom to rigidbody (default=off) \\ |
| 4354 |
|
-t & {\tt -{}-watertype} & replace the atom type of water model (default=on) \\ |
| 4355 |
< |
-b & {\tt -{}-basetype} & using base atom type (default=off) \\ |
| 4355 |
> |
-b & {\tt -{}-basetype} & using base atom type |
| 4356 |
> |
(default=off) \\ |
| 4357 |
> |
-v& {\tt -{}-velocities} & Print velocities in xyz file (default=off)\\ |
| 4358 |
> |
-f& {\tt -{}-forces} & Print forces xyz file (default=off)\\ |
| 4359 |
> |
-u& {\tt -{}-vectors} & Print vectors (dipoles, etc) in xyz file |
| 4360 |
> |
(default=off)\\ |
| 4361 |
> |
-c& {\tt -{}-charges} & Print charges in xyz file (default=off)\\ |
| 4362 |
> |
-e& {\tt -{}-efield} & Print electric field vector in xyz file |
| 4363 |
> |
(default=off)\\ |
| 4364 |
|
& {\tt -{}-repeatX=INT} & The number of images to repeat in the x direction (default=`0') \\ |
| 4365 |
|
& {\tt -{}-repeatY=INT} & The number of images to repeat in the y direction (default=`0') \\ |
| 4366 |
|
& {\tt -{}-repeatZ=INT} & The number of images to repeat in the z direction (default=`0') \\ |
| 4452 |
|
& {\tt -{}-sele1=selection script} & select the first StuntDouble set \\ |
| 4453 |
|
& {\tt -{}-sele2=selection script} & select the second StuntDouble set \\ |
| 4454 |
|
& {\tt -{}-sele3=selection script} & select the third StuntDouble set \\ |
| 4455 |
< |
& {\tt -{}-refsele=selection script} & select reference (can only be used with {\tt -{}-gxyz}) \\ |
| 4455 |
> |
& {\tt -{}-refsele=selection script} & select reference (can only |
| 4456 |
> |
be used with {\tt -{}-gxyz}) \\ |
| 4457 |
> |
& {\tt -{}-comsele=selection script} |
| 4458 |
> |
& select stunt doubles for center-of-mass |
| 4459 |
> |
reference point\\ |
| 4460 |
> |
& {\tt -{}-seleoffset=INT} & global index offset for a second object (used |
| 4461 |
> |
to define a vector between sites in molecule)\\ |
| 4462 |
> |
|
| 4463 |
|
& {\tt -{}-molname=STRING} & molecule name \\ |
| 4464 |
|
& {\tt -{}-begin=INT} & begin internal index \\ |
| 4465 |
|
& {\tt -{}-end=INT} & end internal index \\ |
| 4466 |
+ |
& {\tt -{}-radius=DOUBLE} & nanoparticle radius\\ |
| 4467 |
|
\hline |
| 4468 |
|
\multicolumn{3}{|l|}{One option from the following group of options is required:} \\ |
| 4469 |
|
\hline |
| 4470 |
< |
& {\tt -{}-gofr} & $g(r)$ \\ |
| 4471 |
< |
& {\tt -{}-r\_theta} & $g(r, \cos(\theta))$ \\ |
| 4472 |
< |
& {\tt -{}-r\_omega} & $g(r, \cos(\omega))$ \\ |
| 4473 |
< |
& {\tt -{}-theta\_omega} & $g(\cos(\theta), \cos(\omega))$ \\ |
| 4470 |
> |
& {\tt -{}-bo} & bond order parameter ({\tt -{}-rcut} must be specified) \\ |
| 4471 |
> |
& {\tt -{}-bor} & bond order parameter as a function of |
| 4472 |
> |
radius ({\tt -{}-rcut} must be specified) \\ |
| 4473 |
> |
& {\tt -{}-bad} & $N(\theta)$ bond angle density within ({\tt -{}-rcut} must be specified) \\ |
| 4474 |
> |
& {\tt -{}-count} & count of molecules matching selection |
| 4475 |
> |
criteria (and associated statistics) \\ |
| 4476 |
> |
-g& {\tt -{}-gofr} & $g(r)$ \\ |
| 4477 |
> |
& {\tt -{}-gofz} & $g(z)$ \\ |
| 4478 |
> |
& {\tt -{}-r\_theta} & $g(r, \cos(\theta))$ \\ |
| 4479 |
> |
& {\tt -{}-r\_omega} & $g(r, \cos(\omega))$ \\ |
| 4480 |
> |
& {\tt -{}-r\_z} & $g(r, z)$ \\ |
| 4481 |
> |
& {\tt -{}-theta\_omega} & $g(\cos(\theta), \cos(\omega))$ \\ |
| 4482 |
|
& {\tt -{}-gxyz} & $g(x, y, z)$ \\ |
| 4483 |
< |
& {\tt -{}-p2} & $P_2$ order parameter ({\tt -{}-sele1} and {\tt -{}-sele2} must be specified) \\ |
| 4483 |
> |
& {\tt -{}-twodgofr} & 2D $g(r)$ (Slab width {\tt -{}-dz} must be specified)\\ |
| 4484 |
> |
-p& {\tt -{}-p2} & $P_2$ order parameter ({\tt -{}-sele1} must be specified, {\tt -{}-sele2} is optional) \\ |
| 4485 |
> |
& {\tt -{}-rp2} & Ripple order parameter ({\tt -{}-sele1} and {\tt -{}-sele2} must be specified) \\ |
| 4486 |
|
& {\tt -{}-scd} & $S_{CD}$ order parameter(either {\tt -{}-sele1}, {\tt -{}-sele2}, {\tt -{}-sele3} are specified or {\tt -{}-molname}, {\tt -{}-begin}, {\tt -{}-end} are specified) \\ |
| 4487 |
< |
& {\tt -{}-density} & density plot ({\tt -{}-sele1} must be specified) \\ |
| 4488 |
< |
& {\tt -{}-slab\_density} & slab density ({\tt -{}-sele1} must be specified) |
| 4487 |
> |
-d& {\tt -{}-density} & density plot \\ |
| 4488 |
> |
& {\tt -{}-slab\_density} & slab density \\ |
| 4489 |
> |
& {\tt -{}-p\_angle} & $p(\cos(\theta))$ ($\theta$ |
| 4490 |
> |
is the angle between molecular axis and radial vector from origin\\ |
| 4491 |
> |
& {\tt -{}-hxy} & Calculates the undulation spectrum, $h(x,y)$, of an interface \\ |
| 4492 |
> |
& {\tt -{}-rho\_r} & $\rho(r)$\\ |
| 4493 |
> |
& {\tt -{}-angle\_r} & $\theta(r)$ (spatially resolves the |
| 4494 |
> |
angle between the molecular axis and the radial vector from the |
| 4495 |
> |
origin)\\ |
| 4496 |
> |
& {\tt -{}-hullvol} & hull volume of nanoparticle\\ |
| 4497 |
> |
& {\tt -{}-rodlength} & length of nanorod\\ |
| 4498 |
> |
-Q& {\tt -{}-tet\_param} & tetrahedrality order parameter ($Q$)\\ |
| 4499 |
> |
& {\tt -{}-tet\_param\_z} & spatially-resolved tetrahedrality order |
| 4500 |
> |
parameter $Q(z)$\\ |
| 4501 |
> |
& {\tt -{}-rnemdz} & slab-resolved RNEMD statistics (temperature, |
| 4502 |
> |
density, velocity)\\ |
| 4503 |
> |
& {\tt -{}-rnemdr} & shell-resolved RNEMD statistics (temperature, |
| 4504 |
> |
density, angular velocity) |
| 4505 |
|
\end{longtable} |
| 4506 |
|
|
| 4507 |
|
\subsection{\label{section:DynamicProps}DynamicProps} |
| 4542 |
|
-o& {\tt -{}-output=filename} & output file name \\ |
| 4543 |
|
& {\tt -{}-sele1=selection script} & select first StuntDouble set \\ |
| 4544 |
|
& {\tt -{}-sele2=selection script} & select second StuntDouble set (if sele2 is not set, use script from sele1) \\ |
| 4545 |
+ |
& {\tt -{}-order=INT} & Lengendre Polynomial Order\\ |
| 4546 |
+ |
-z& {\tt -{}-nzbins=INT} & Number of $z$ bins (default=`100`)\\ |
| 4547 |
+ |
-m& {\tt -{}-memory=memory specification} |
| 4548 |
+ |
&Available memory |
| 4549 |
+ |
(default=`2G`)\\ |
| 4550 |
|
\hline |
| 4551 |
|
\multicolumn{3}{|l|}{One option from the following group of options is required:} \\ |
| 4552 |
|
\hline |
| 4553 |
< |
-r& {\tt -{}-rcorr} & compute mean square displacement \\ |
| 4554 |
< |
-v& {\tt -{}-vcorr} & compute velocity correlation function \\ |
| 4555 |
< |
-d& {\tt -{}-dcorr} & compute dipole correlation function |
| 4553 |
> |
-s& {\tt -{}-selecorr} & selection correlation function \\ |
| 4554 |
> |
-r& {\tt -{}-rcorr} & compute mean squared displacement \\ |
| 4555 |
> |
-v& {\tt -{}-vcorr} & velocity autocorrelation function \\ |
| 4556 |
> |
-d& {\tt -{}-dcorr} & dipole correlation function \\ |
| 4557 |
> |
-l& {\tt -{}-lcorr} & Lengendre correlation function \\ |
| 4558 |
> |
& {\tt -{}-lcorrZ} & Lengendre correlation function binned by $z$ \\ |
| 4559 |
> |
& {\tt -{}-cohZ} & Lengendre correlation function for OH bond vectors binned by $z$\\ |
| 4560 |
> |
-M& {\tt -{}-sdcorr} & System dipole correlation function\\ |
| 4561 |
> |
& {\tt -{}-r\_rcorr} & Radial mean squared displacement\\ |
| 4562 |
> |
& {\tt -{}-thetacorr} & Angular mean squared displacement\\ |
| 4563 |
> |
& {\tt -{}-drcorr} & Directional mean squared displacement for particles with unit vectors\\ |
| 4564 |
> |
& {\tt -{}-helfandEcorr} & Helfand moment for thermal conductvity\\ |
| 4565 |
> |
-p& {\tt -{}-momentum} & Helfand momentum for viscosity\\ |
| 4566 |
> |
& {\tt -{}-stresscorr} & Stress tensor correlation function |
| 4567 |
|
\end{longtable} |
| 4568 |
|
|
| 4569 |
|
\chapter{\label{section:PreparingInput} Preparing Input Configurations} |
| 4630 |
|
to {\tt atom2md}, but they use a specific input format and do not |
| 4631 |
|
expect the the input specifier on the command line. |
| 4632 |
|
|
| 4633 |
+ |
|
| 4634 |
|
\section{\label{section:SimpleBuilder}SimpleBuilder} |
| 4635 |
|
|
| 4636 |
|
{\tt SimpleBuilder} creates simple lattice structures. It requires an |
| 4657 |
|
& {\tt -{}-nz=INT} & number of unit cells in z |
| 4658 |
|
\end{longtable} |
| 4659 |
|
|
| 4660 |
+ |
\section{\label{section:icosahedralBuilder}icosahedralBuilder} |
| 4661 |
+ |
|
| 4662 |
+ |
{\tt icosahedralBuilder} creates single-component geometric solids |
| 4663 |
+ |
that can be useful in simulating nanostructures. Like the other |
| 4664 |
+ |
builders, it requires an initial, but skeletal {\sc OpenMD} file to |
| 4665 |
+ |
specify the component that is to be placed on the lattice. The total |
| 4666 |
+ |
number of placed molecules will be shown at the top of the |
| 4667 |
+ |
configuration file that is generated, and that number may not match |
| 4668 |
+ |
the original meta-data file, so a new meta-data file is also generated |
| 4669 |
+ |
which matches the lattice structure. |
| 4670 |
+ |
|
| 4671 |
+ |
The options available for icosahedralBuilder are as follows: |
| 4672 |
+ |
\begin{longtable}[c]{|EFG|} |
| 4673 |
+ |
\caption{icosahedralBuilder Command-line Options} |
| 4674 |
+ |
\\ \hline |
| 4675 |
+ |
{\bf option} & {\bf verbose option} & {\bf behavior} \\ \hline |
| 4676 |
+ |
\endhead |
| 4677 |
+ |
\hline |
| 4678 |
+ |
\endfoot |
| 4679 |
+ |
-h& {\tt -{}-help} & Print help and exit\\ |
| 4680 |
+ |
-V& {\tt -{}-version} & Print version and exit\\ |
| 4681 |
+ |
-o& {\tt -{}-output=STRING} & Output file name\\ |
| 4682 |
+ |
-n& {\tt -{}-shells=INT} & Nanoparticle shells\\ |
| 4683 |
+ |
-d& {\tt -{}-latticeConstant=DOUBLE} & Lattice spacing in Angstroms for cubic lattice.\\ |
| 4684 |
+ |
-c& {\tt -{}-columnAtoms=INT} & Number of atoms along central |
| 4685 |
+ |
column (Decahedron only)\\ |
| 4686 |
+ |
-t& {\tt -{}-twinAtoms=INT} & Number of atoms along twin |
| 4687 |
+ |
boundary (Decahedron only) \\ |
| 4688 |
+ |
-p& {\tt -{}-truncatedPlanes=INT} & Number of truncated planes (Curling-stone Decahedron only)\\ |
| 4689 |
+ |
\hline |
| 4690 |
+ |
\multicolumn{3}{|l|}{One option from the following group of options is required:} \\ |
| 4691 |
+ |
\hline |
| 4692 |
+ |
& {\tt -{}-ico} & Create an Icosahedral cluster \\ |
| 4693 |
+ |
& {\tt -{}-deca} & Create a regualar Decahedral cluster\\ |
| 4694 |
+ |
& {\tt -{}-ino} & Create an Ino Decahedral cluster\\ |
| 4695 |
+ |
& {\tt -{}-marks} & Create a Marks Decahedral cluster\\ |
| 4696 |
+ |
& {\tt -{}-stone} & Create a Curling-stone Decahedral cluster |
| 4697 |
+ |
\end{longtable} |
| 4698 |
+ |
|
| 4699 |
+ |
|
| 4700 |
|
\section{\label{section:Hydro}Hydro} |
| 4701 |
|
{\tt Hydro} generates resistance tensor ({\tt .diff}) files which are |
| 4702 |
|
required when using the Langevin integrator using complex rigid |
| 4730 |
|
\end{longtable} |
| 4731 |
|
|
| 4732 |
|
|
| 4733 |
+ |
|
| 4734 |
+ |
|
| 4735 |
+ |
|
| 4736 |
|
\chapter{\label{section:parallelization} Parallel Simulation Implementation} |
| 4737 |
|
|
| 4738 |
|
Although processor power is continually improving, it is still |
| 4816 |
|
DMR-0079647. |
| 4817 |
|
|
| 4818 |
|
|
| 4819 |
< |
\bibliographystyle{jcc} |
| 4819 |
> |
\bibliographystyle{aip} |
| 4820 |
|
\bibliography{openmdDoc} |
| 4821 |
|
|
| 4822 |
|
\end{document} |
