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\section{\label{sec:empericalEnergy}The Emperical Energy Functions} |
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\section{\label{sec:empiricalEnergy}The Empirical Energy Functions} |
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|
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\subsection{\label{sec:atomsMolecules}Atoms, Molecules and Rigid Bodies} |
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|
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The basic unit of an {\sc oopse} simulation is the atom. The parameters |
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describing the atom are generalized to make the atom as flexible a |
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representation as possible. They may represent specific atoms of an |
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element, or be used for collections of atoms such as a methyl |
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group. The atoms are also capable of having a directional component |
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associated with them, often in the form of a dipole. Charges on atoms |
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are not currently suporrted by {\sc oopse}. |
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The basic unit of an {\sc oopse} simulation is the atom. The |
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parameters describing the atom are generalized to make the atom as |
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flexible a representation as possible. They may represent specific |
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atoms of an element, or be used for collections of atoms such as |
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methyl and carbonyl groups. The atoms are also capable of having |
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directional components associated with them (\emph{e.g.}~permanent |
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dipoles). Charges on atoms are not currently supported by {\sc oopse}. |
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|
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The second most basic building block of a simulation is the |
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molecule. The molecule is a way for {\sc oopse} to keep track of the atoms |
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in a simulation in logical manner. This particular unit will store the |
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identities of all the atoms associated with itself and is responsible |
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for the evaluation of its own bonded interaction (i.e.~bonds, bends, |
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and torsions). |
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\begin{lstlisting}[caption={[Specifier for molecules and atoms] A sample specification of the simple Ar molecule},label=sch:AtmMole] |
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molecule{ |
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name = "Ar"; |
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nAtoms = 1; |
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atom[0]{ |
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type="Ar"; |
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position( 0.0, 0.0, 0.0 ); |
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} |
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} |
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\end{lstlisting} |
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|
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As stated in the previously, one of the features that sets OOPSE apart |
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Atoms can be collected into secondary srtructures such as rigid bodies |
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or molecules. The molecule is a way for {\sc oopse} to keep track of |
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the atoms in a simulation in logical manner. Molecular units store the |
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identities of all the atoms associated with themselves, and are |
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responsible for the evaluation of their own internal interactions |
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(\emph{i.e.}~bonds, bends, and torsions). Scheme \ref{sch:AtmMole} |
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shws how one creates a molecule in the \texttt{.mdl} files. The |
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position of the atoms given in the declaration are relative to the |
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origin of the molecule, and is used when creating a system containing |
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the molecule. |
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|
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As stated previously, one of the features that sets {\sc oopse} apart |
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from most of the current molecular simulation packages is the ability |
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to handle rigid body dynamics. Rigid bodies are non-spherical |
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particles or collections of particles that have a constant internal |
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potential and move collectively.\cite{Goldstein01} They are not |
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included in many standard simulation packages because of the need to |
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consider orientational degrees of freedom and include an integrator |
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that accurately propagates these motions in time. |
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included in most simulation packages because of the requirement to |
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propagate the orientational degrees of freedom. Until recently, |
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integrators which propagate orientational motion have been lacking. |
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Moving a rigid body involves determination of both the force and |
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torque applied by the surroundings, which directly affect the |
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translation and rotation in turn. In order to accumulate the total |
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force on a rigid body, the external forces must first be calculated |
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for all the interal particles. The total force on the rigid body is |
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simply the sum of these external forces. Accumulation of the total |
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torque on the rigid body is similar to the force in that it is a sum |
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of the torque applied on each internal particle, mapped onto the |
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center of mass of the rigid body. |
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translational and rotational motion in turn. In order to accumulate |
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the total force on a rigid body, the external forces and torques must |
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first be calculated for all the internal particles. The total force on |
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the rigid body is simply the sum of these external forces. |
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Accumulation of the total torque on the rigid body is more complex |
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than the force in that it is the torque applied on the center of mass |
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that dictates rotational motion. The torque on rigid body {\it i} is |
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\begin{equation} |
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\boldsymbol{\tau}_i= |
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\sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia} |
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+ \boldsymbol{\tau}_{ia}, |
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\label{eq:torqueAccumulate} |
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\end{equation} |
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where $\boldsymbol{\tau}_i$ and $\mathbf{r}_i$ are the torque on and |
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position of the center of mass respectively, while $\mathbf{f}_{ia}$, |
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$\mathbf{r}_{ia}$, and $\boldsymbol{\tau}_{ia}$ are the force on, |
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position of, and torque on the component particles of the rigid body. |
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|
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The application of the total torque is done in the body fixed axis of |
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The summation of the total torque is done in the body fixed axis of |
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the rigid body. In order to move between the space fixed and body |
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fixed coordinate axes, parameters describing the orientation be |
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maintained for each rigid body. At a minimum, the rotation matrix can |
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be described and propagated by the three Euler |
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angles.\cite{Goldstein01} In order to avoid rotational limitations |
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when using Euler angles, the four parameter ``quaternion'' scheme can |
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be used instead.\cite{allen87:csl} Use of quaternions also leads to |
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fixed coordinate axes, parameters describing the orientation must be |
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maintained for each rigid body. At a minimum, the rotation matrix |
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(\textbf{A}) can be described by the three Euler angles ($\phi, |
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\theta,$ and $\psi$), where the elements of \textbf{A} are composed of |
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trigonometric operations involving $\phi, \theta,$ and |
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$\psi$.\cite{Goldstein01} In order to avoid numerical instabilities |
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inherent in using the Euler angles, the four parameter ``quaternion'' |
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scheme is often used. The elements of \textbf{A} can be expressed as |
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arithmetic operations involving the four quaternions ($q_0, q_1, q_2,$ |
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and $q_3$).\cite{allen87:csl} Use of quaternions also leads to |
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performance enhancements, particularly for very small |
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systems.\cite{Evans77} OOPSE utilizes a relatively new scheme that |
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propagates the entire nine parameter rotation matrix. Further |
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discussion on this choice can be found in Sec.~\ref{sec:integrate}. |
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systems.\cite{Evans77} |
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|
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\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
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{\sc oopse} utilizes a relatively new scheme that propagates the |
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entire nine parameter rotation matrix internally. Further discussion |
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on this choice can be found in Sec.~\ref{sec:integrate}. An example |
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definition of a riged body can be seen in Scheme |
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\ref{sch:rigidBody}. The positions in the atom definitions are the |
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placements of the atoms relative to the origin of the rigid body, |
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which itself has a position relative to the origin of the molecule. |
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|
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The most basic force field implemented in OOPSE is the Lennard-Jones |
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potential. The Lennard-Jones potential mimics the attractive forces |
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two charge neutral particles experience when spontaneous dipoles are |
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induced on each other. This is the predominant intermolecular force in |
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systems of such as noble gases and simple alkanes. |
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\begin{lstlisting}[caption={[Defining rigid bodies]A sample definition of a rigid body},label={sch:rigidBody}] |
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molecule{ |
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name = "TIP3P_water"; |
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nRigidBodies = 1; |
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rigidBody[0]{ |
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nAtoms = 3; |
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atom[0]{ |
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type = "O_TIP3P"; |
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position( 0.0, 0.0, -0.06556 ); |
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} |
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atom[1]{ |
| 99 |
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type = "H_TIP3P"; |
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position( 0.0, 0.75695, 0.52032 ); |
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} |
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atom[2]{ |
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type = "H_TIP3P"; |
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position( 0.0, -0.75695, 0.52032 ); |
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} |
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position( 0.0, 0.0, 0.0 ); |
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orientation( 0.0, 0.0, 1.0 ); |
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} |
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} |
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> |
\end{lstlisting} |
| 111 |
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|
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The Lennard-Jones potential is given by: |
| 112 |
> |
\subsection{\label{sec:LJPot}The Lennard Jones Potential} |
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> |
|
| 114 |
> |
The most basic force field implemented in {\sc oopse} is the |
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Lennard-Jones potential, which mimics the van der Waals interaction at |
| 116 |
> |
long distances, and uses an empirical repulsion at short |
| 117 |
> |
distances. The Lennard-Jones potential is given by: |
| 118 |
|
\begin{equation} |
| 119 |
|
V_{\text{LJ}}(r_{ij}) = |
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4\epsilon_{ij} \biggl[ |
| 123 |
|
\biggr] |
| 124 |
|
\label{eq:lennardJonesPot} |
| 125 |
|
\end{equation} |
| 126 |
< |
Where $r_ij$ is the distance between particle $i$ and $j$, $\sigma_{ij}$ |
| 127 |
< |
scales the length of the interaction, and $\epsilon_{ij}$ scales the |
| 128 |
< |
energy well depth of the potential. |
| 126 |
> |
Where $r_{ij}$ is the distance between particles $i$ and $j$, |
| 127 |
> |
$\sigma_{ij}$ scales the length of the interaction, and |
| 128 |
> |
$\epsilon_{ij}$ scales the well depth of the potential. Scheme |
| 129 |
> |
\ref{sch:LJFF} gives and example partial \texttt{.bass} file that |
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> |
shows a system of 108 Ar particles simulated with the Lennard-Jones |
| 131 |
> |
force field. |
| 132 |
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|
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< |
Because the potential is calculated between all pairs, the force |
| 133 |
> |
\begin{lstlisting}[caption={[Invocation of the Lennard-Jones force field] A sample system using the Lennard-Jones force field.},label={sch:LJFF}] |
| 134 |
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|
| 135 |
> |
/* |
| 136 |
> |
* The Ar molecule is specified |
| 137 |
> |
* external to the.bass file |
| 138 |
> |
*/ |
| 139 |
> |
|
| 140 |
> |
#include "argon.mdl" |
| 141 |
> |
|
| 142 |
> |
nComponents = 1; |
| 143 |
> |
component{ |
| 144 |
> |
type = "Ar"; |
| 145 |
> |
nMol = 108; |
| 146 |
> |
} |
| 147 |
> |
|
| 148 |
> |
/* |
| 149 |
> |
* The initial configuration is generated |
| 150 |
> |
* before the simulation is invoked. |
| 151 |
> |
*/ |
| 152 |
> |
|
| 153 |
> |
initialConfig = "./argon.init"; |
| 154 |
> |
|
| 155 |
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forceField = "LJ"; |
| 156 |
> |
\end{lstlisting} |
| 157 |
> |
|
| 158 |
> |
Because this potential is calculated between all pairs, the force |
| 159 |
|
evaluation can become computationally expensive for large systems. To |
| 160 |
< |
keep the pair evaluation to a manegable number, OOPSE employs the use |
| 161 |
< |
of a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be |
| 162 |
< |
$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest self self length |
| 163 |
< |
parameter in the system. Truncating the calculation at |
| 164 |
< |
$r_{\text{cut}}$ introduces a discontinuity into the potential |
| 160 |
> |
keep the pair evaluations to a manageable number, {\sc oopse} employs |
| 161 |
> |
a cut-off radius.\cite{allen87:csl} The cutoff radius is set to be |
| 162 |
> |
$2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest Lennard-Jones |
| 163 |
> |
length parameter present in the simulation. Truncating the calculation |
| 164 |
> |
at $r_{\text{cut}}$ introduces a discontinuity into the potential |
| 165 |
|
energy. To offset this discontinuity, the energy value at |
| 166 |
< |
$r_{\text{cut}}$ is subtracted from the entire potential. This causes |
| 167 |
< |
the equation to go to zero at the cut-off radius. |
| 166 |
> |
$r_{\text{cut}}$ is subtracted from the potential. This causes the |
| 167 |
> |
potential to go to zero smoothly at the cut-off radius. |
| 168 |
|
|
| 169 |
|
Interactions between dissimilar particles requires the generation of |
| 170 |
|
cross term parameters for $\sigma$ and $\epsilon$. These are |
| 181 |
|
\end{equation} |
| 182 |
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|
| 183 |
|
|
| 184 |
+ |
|
| 185 |
|
\subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field} |
| 186 |
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|
| 187 |
< |
The \underline{D}ipolar \underline{U}nified-Atom |
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< |
\underline{F}orce \underline{F}ield ({\sc duff}) was developed to |
| 189 |
< |
simulate lipid bilayers. We needed a model capable of forming |
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< |
bilayers, while still being sufficiently computationally efficient to |
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allow simulations of large systems ($\approx$100's of phospholipids, |
| 192 |
< |
$\approx$1000's of waters) for long times ($\approx$10's of |
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nanoseconds). |
| 187 |
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The dipolar unified-atom force field ({\sc duff}) was developed to |
| 188 |
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simulate lipid bilayers. The simulations require a model capable of |
| 189 |
> |
forming bilayers, while still being sufficiently computationally |
| 190 |
> |
efficient to allow large systems ($\approx$100's of phospholipids, |
| 191 |
> |
$\approx$1000's of waters) to be simulated for long times |
| 192 |
> |
($\approx$10's of nanoseconds). |
| 193 |
|
|
| 194 |
< |
With this goal in mind, we have eliminated all point charges. Charge |
| 195 |
< |
distributions were replaced with dipoles, and charge-neutral |
| 196 |
< |
distributions were reduced to Lennard-Jones interaction sites. This |
| 197 |
< |
simplification cuts the length scale of long range interactions from |
| 198 |
< |
$\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the |
| 199 |
< |
computationally expensive Ewald-Sum. Instead, we can use |
| 200 |
< |
neighbor-lists and cutoff radii for the dipolar interactions. |
| 194 |
> |
With this goal in mind, {\sc duff} has no point |
| 195 |
> |
charges. Charge-neutral distributions were replaced with dipoles, |
| 196 |
> |
while most atoms and groups of atoms were reduced to Lennard-Jones |
| 197 |
> |
interaction sites. This simplification cuts the length scale of long |
| 198 |
> |
range interactions from $\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us |
| 199 |
> |
to avoid the computationally expensive Ewald sum. Instead, we can use |
| 200 |
> |
neighbor-lists, reaction field, and cutoff radii for the dipolar |
| 201 |
> |
interactions. |
| 202 |
|
|
| 203 |
< |
As an example, lipid head groups in {\sc duff} are represented as point |
| 204 |
< |
dipole interaction sites.PC and PE Lipid head groups are typically |
| 205 |
< |
zwitterionic in nature, with charges separated by as much as |
| 206 |
< |
6~$\mbox{\AA}$. By placing a dipole of 20.6~Debye at the head group |
| 207 |
< |
center of mass, our model mimics the head group of PC.\cite{Cevc87} |
| 208 |
< |
Additionally, a Lennard-Jones site is located at the |
| 209 |
< |
pseudoatom's center of mass. The model is illustrated by the dark grey |
| 210 |
< |
atom in Fig.~\ref{fig:lipidModel}. |
| 203 |
> |
As an example, lipid head-groups in {\sc duff} are represented as |
| 204 |
> |
point dipole interaction sites. By placing a dipole of 20.6~Debye at |
| 205 |
> |
the head group center of mass, our model mimics the head group of |
| 206 |
> |
phosphatidylcholine.\cite{Cevc87} Additionally, a large Lennard-Jones |
| 207 |
> |
site is located at the pseudoatom's center of mass. The model is |
| 208 |
> |
illustrated by the dark grey atom in Fig.~\ref{fig:lipidModel}. The |
| 209 |
> |
water model we use to complement the dipoles of the lipids is our |
| 210 |
> |
reparameterization of the soft sticky dipole (SSD) model of Ichiye |
| 211 |
> |
\emph{et al.}\cite{liu96:new_model} |
| 212 |
|
|
| 213 |
|
\begin{figure} |
| 214 |
+ |
\epsfxsize=\linewidth |
| 215 |
|
\epsfbox{lipidModel.eps} |
| 216 |
|
\caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ % |
| 217 |
< |
is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.} |
| 217 |
> |
is the bend angle, $\mu$ is the dipole moment of the head group, and n |
| 218 |
> |
is the chain length.} |
| 219 |
|
\label{fig:lipidModel} |
| 220 |
|
\end{figure} |
| 221 |
|
|
| 222 |
< |
The water model we use to complement the dipoles of the lipids is |
| 223 |
< |
the soft sticky dipole (SSD) model of Ichiye \emph{et |
| 224 |
< |
al.}\cite{liu96:new_model} This model is discussed in greater detail |
| 137 |
< |
in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single |
| 138 |
< |
Lennard-Jones interaction site. The site also contains a dipole to |
| 139 |
< |
mimic the partial charges on the hydrogens and the oxygen. However, |
| 140 |
< |
what makes the SSD model unique is the inclusion of a tetrahedral |
| 141 |
< |
short range potential to recover the hydrogen bonding of water, an |
| 142 |
< |
important factor when modeling bilayers, as it has been shown that |
| 143 |
< |
hydrogen bond network formation is a leading contribution to the |
| 144 |
< |
entropic driving force towards lipid bilayer formation.\cite{Cevc87} |
| 145 |
< |
|
| 146 |
< |
|
| 147 |
< |
Turning to the tails of the phospholipids, we have used a set of |
| 148 |
< |
scalable parameters to model the alkyl groups with Lennard-Jones |
| 149 |
< |
sites. For this, we have used the TraPPE force field of Siepmann |
| 222 |
> |
We have used a set of scalable parameters to model the alkyl groups |
| 223 |
> |
with Lennard-Jones sites. For this, we have borrowed parameters from |
| 224 |
> |
the TraPPE force field of Siepmann |
| 225 |
|
\emph{et al}.\cite{Siepmann1998} TraPPE is a unified-atom |
| 226 |
|
representation of n-alkanes, which is parametrized against phase |
| 227 |
< |
equilibria using Gibbs Monte Carlo simulation |
| 227 |
> |
equilibria using Gibbs ensemble Monte Carlo simulation |
| 228 |
|
techniques.\cite{Siepmann1998} One of the advantages of TraPPE is that |
| 229 |
|
it generalizes the types of atoms in an alkyl chain to keep the number |
| 230 |
|
of pseudoatoms to a minimum; the parameters for an atom such as |
| 231 |
|
$\text{CH}_2$ do not change depending on what species are bonded to |
| 232 |
|
it. |
| 233 |
|
|
| 234 |
< |
TraPPE also constrains of all bonds to be of fixed length. Typically, |
| 234 |
> |
TraPPE also constrains all bonds to be of fixed length. Typically, |
| 235 |
|
bond vibrations are the fastest motions in a molecular dynamic |
| 236 |
|
simulation. Small time steps between force evaluations must be used to |
| 237 |
< |
ensure adequate sampling of the bond potential conservation of |
| 238 |
< |
energy. By constraining the bond lengths, larger time steps may be |
| 239 |
< |
used when integrating the equations of motion. |
| 237 |
> |
ensure adequate sampling of the bond potential to ensure conservation |
| 238 |
> |
of energy. By constraining the bond lengths, larger time steps may be |
| 239 |
> |
used when integrating the equations of motion. A simulation using {\sc |
| 240 |
> |
duff} is illustrated in Scheme \ref{sch:DUFF}. |
| 241 |
|
|
| 242 |
+ |
\begin{lstlisting}[caption={[Invocation of {\sc duff}]Sample \texttt{.bass} file showing a simulation utilizing {\sc duff}},label={sch:DUFF}] |
| 243 |
|
|
| 244 |
+ |
#include "water.mdl" |
| 245 |
+ |
#include "lipid.mdl" |
| 246 |
+ |
|
| 247 |
+ |
nComponents = 2; |
| 248 |
+ |
component{ |
| 249 |
+ |
type = "simpleLipid_16"; |
| 250 |
+ |
nMol = 60; |
| 251 |
+ |
} |
| 252 |
+ |
|
| 253 |
+ |
component{ |
| 254 |
+ |
type = "SSD_water"; |
| 255 |
+ |
nMol = 1936; |
| 256 |
+ |
} |
| 257 |
+ |
|
| 258 |
+ |
initialConfig = "bilayer.init"; |
| 259 |
+ |
|
| 260 |
+ |
forceField = "DUFF"; |
| 261 |
+ |
|
| 262 |
+ |
\end{lstlisting} |
| 263 |
+ |
|
| 264 |
|
\subsubsection{\label{subSec:energyFunctions}{\sc duff} Energy Functions} |
| 265 |
|
|
| 266 |
< |
The total energy of function in {\sc duff} is given by the following: |
| 266 |
> |
The total potential energy function in {\sc duff} is |
| 267 |
|
\begin{equation} |
| 268 |
< |
V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
| 269 |
< |
+ \sum^{N}_{I=1} \sum^{N}_{J=1} V^{IJ}_{\text{Cross}} |
| 268 |
> |
V = \sum^{N}_{I=1} V^{I}_{\text{Internal}} |
| 269 |
> |
+ \sum^{N}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}} |
| 270 |
|
\label{eq:totalPotential} |
| 271 |
|
\end{equation} |
| 272 |
< |
Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule: |
| 272 |
> |
Where $V^{I}_{\text{Internal}}$ is the internal potential of molecule $I$: |
| 273 |
|
\begin{equation} |
| 274 |
|
V^{I}_{\text{Internal}} = |
| 275 |
|
\sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk}) |
| 276 |
< |
+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\theta_{ijkl}) |
| 276 |
> |
+ \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{ijkl}) |
| 277 |
|
+ \sum_{i \in I} \sum_{(j>i+4) \in I} |
| 278 |
|
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 279 |
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 281 |
|
\label{eq:internalPotential} |
| 282 |
|
\end{equation} |
| 283 |
|
Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs |
| 284 |
< |
within in the molecule. $V_{\text{torsion}}$ is the torsion The |
| 285 |
< |
pairwise portions of the internal potential are excluded for pairs |
| 286 |
< |
that ar closer than three bonds, i.e.~atom pairs farther away than a |
| 287 |
< |
torsion are included in the pair-wise loop. |
| 284 |
> |
within the molecule $I$, and $V_{\text{torsion}}$ is the torsion potential |
| 285 |
> |
for all 1, 4 bonded pairs. The pairwise portions of the internal |
| 286 |
> |
potential are excluded for pairs that are closer than three bonds, |
| 287 |
> |
i.e.~atom pairs farther away than a torsion are included in the |
| 288 |
> |
pair-wise loop. |
| 289 |
|
|
| 192 |
– |
The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is |
| 193 |
– |
as follows: |
| 194 |
– |
\begin{equation} |
| 195 |
– |
V^{IJ}_{\text{Cross}} = |
| 196 |
– |
\sum_{i \in I} \sum_{j \in J} |
| 197 |
– |
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 198 |
– |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 199 |
– |
+ V_{\text{sticky}} |
| 200 |
– |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 201 |
– |
\biggr] |
| 202 |
– |
\label{eq:crossPotentail} |
| 203 |
– |
\end{equation} |
| 204 |
– |
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 205 |
– |
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 206 |
– |
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model. |
| 290 |
|
|
| 291 |
|
The bend potential of a molecule is represented by the following function: |
| 292 |
|
\begin{equation} |
| 293 |
< |
V_{\theta_{ijk}} = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
| 293 |
> |
V_{\text{bend}}(\theta_{ijk}) = k_{\theta}( \theta_{ijk} - \theta_0 )^2 \label{eq:bendPot} |
| 294 |
|
\end{equation} |
| 295 |
|
Where $\theta_{ijk}$ is the angle defined by atoms $i$, $j$, and $k$ |
| 296 |
< |
(see Fig.~\ref{fig:lipidModel}), and $\theta_0$ is the equilibrium |
| 297 |
< |
bond angle. $k_{\theta}$ is the force constant which determines the |
| 296 |
> |
(see Fig.~\ref{fig:lipidModel}), $\theta_0$ is the equilibrium |
| 297 |
> |
bond angle, and $k_{\theta}$ is the force constant which determines the |
| 298 |
|
strength of the harmonic bend. The parameters for $k_{\theta}$ and |
| 299 |
< |
$\theta_0$ are based off of those in TraPPE.\cite{Siepmann1998} |
| 299 |
> |
$\theta_0$ are borrowed from those in TraPPE.\cite{Siepmann1998} |
| 300 |
|
|
| 301 |
< |
The torsion potential and parameters are also taken from TraPPE. It is |
| 301 |
> |
The torsion potential and parameters are also borrowed from TraPPE. It is |
| 302 |
|
of the form: |
| 303 |
|
\begin{equation} |
| 304 |
< |
V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi] |
| 304 |
> |
V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi] |
| 305 |
|
+ c_2[1 + \cos(2\phi)] |
| 306 |
|
+ c_3[1 + \cos(3\phi)] |
| 307 |
|
\label{eq:origTorsionPot} |
| 308 |
|
\end{equation} |
| 309 |
< |
Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$, |
| 310 |
< |
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). However, |
| 311 |
< |
for computaional efficency, the torsion potentail has been recast |
| 312 |
< |
after the method of CHARMM\cite{charmm1983} whereby the angle series |
| 313 |
< |
is converted to a power series of the form: |
| 309 |
> |
Here $\phi$ is the angle defined by four bonded neighbors $i$, |
| 310 |
> |
$j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). For |
| 311 |
> |
computational efficiency, the torsion potential has been recast after |
| 312 |
> |
the method of CHARMM,\cite{charmm1983} in which the angle series is |
| 313 |
> |
converted to a power series of the form: |
| 314 |
|
\begin{equation} |
| 315 |
< |
V_{\text{torsion}}(\phi_{ijkl}) = |
| 315 |
> |
V_{\text{torsion}}(\phi) = |
| 316 |
|
k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0 |
| 317 |
|
\label{eq:torsionPot} |
| 318 |
|
\end{equation} |
| 323 |
|
k_2 &= 2 c_2 \\ |
| 324 |
|
k_3 &= 4c_3 |
| 325 |
|
\end{align*} |
| 326 |
< |
By recasting the equation to a power series, repeated trigonometric |
| 327 |
< |
evaluations are avoided during the calculation of the potential. |
| 326 |
> |
By recasting the potential as a power series, repeated trigonometric |
| 327 |
> |
evaluations are avoided during the calculation of the potential energy. |
| 328 |
|
|
| 329 |
|
|
| 330 |
+ |
The cross potential between molecules $I$ and $J$, $V^{IJ}_{\text{Cross}}$, is |
| 331 |
+ |
as follows: |
| 332 |
+ |
\begin{equation} |
| 333 |
+ |
V^{IJ}_{\text{Cross}} = |
| 334 |
+ |
\sum_{i \in I} \sum_{j \in J} |
| 335 |
+ |
\biggl[ V_{\text{LJ}}(r_{ij}) + V_{\text{dipole}} |
| 336 |
+ |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 337 |
+ |
+ V_{\text{sticky}} |
| 338 |
+ |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j}) |
| 339 |
+ |
\biggr] |
| 340 |
+ |
\label{eq:crossPotentail} |
| 341 |
+ |
\end{equation} |
| 342 |
+ |
Where $V_{\text{LJ}}$ is the Lennard Jones potential, |
| 343 |
+ |
$V_{\text{dipole}}$ is the dipole dipole potential, and |
| 344 |
+ |
$V_{\text{sticky}}$ is the sticky potential defined by the SSD model |
| 345 |
+ |
(Sec.~\ref{sec:SSD}). Note that not all atom types include all |
| 346 |
+ |
interactions. |
| 347 |
|
|
| 348 |
|
The dipole-dipole potential has the following form: |
| 349 |
|
\begin{equation} |
| 350 |
|
V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i}, |
| 351 |
< |
\boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[ |
| 352 |
< |
\frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}} |
| 351 |
> |
\boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[ |
| 352 |
> |
\boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j} |
| 353 |
|
- |
| 354 |
< |
\frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) % |
| 355 |
< |
(\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) } |
| 356 |
< |
{r^{5}_{ij}} \biggr] |
| 354 |
> |
\frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) % |
| 355 |
> |
(\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) } |
| 356 |
> |
{r^{2}_{ij}} \biggr] |
| 357 |
|
\label{eq:dipolePot} |
| 358 |
|
\end{equation} |
| 359 |
|
Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing |
| 360 |
|
towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ |
| 361 |
< |
are the Euler angles of atom $i$ and $j$ |
| 362 |
< |
respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom |
| 363 |
< |
$i$ it takes its orientation from $\boldsymbol{\Omega}_i$. |
| 361 |
> |
are the orientational degrees of freedom for atoms $i$ and $j$ |
| 362 |
> |
respectively. $|\mu_i|$ is the magnitude of the dipole moment of atom |
| 363 |
> |
$i$, $\boldsymbol{\hat{u}}_i$ is the standard unit orientation |
| 364 |
> |
vector of $\boldsymbol{\Omega}_i$, and $\boldsymbol{\hat{r}}_{ij}$ is |
| 365 |
> |
the unit vector pointing along $\mathbf{r}_{ij}$. |
| 366 |
|
|
| 367 |
|
|
| 368 |
< |
\subsection{\label{sec:SSD}Water Model: SSD and Derivatives} |
| 368 |
> |
\subsubsection{\label{sec:SSD}The {\sc duff} Water Models: SSD/E and SSD/RF} |
| 369 |
|
|
| 370 |
< |
In the interest of computational efficiency, the native solvent used |
| 371 |
< |
in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was |
| 372 |
< |
developed by Ichiye \emph{et al.} as a modified form of the |
| 373 |
< |
hard-sphere water model proposed by Bratko, Blum, and |
| 370 |
> |
In the interest of computational efficiency, the default solvent used |
| 371 |
> |
by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water |
| 372 |
> |
model.\cite{Gezelter04} The original SSD was developed by Ichiye |
| 373 |
> |
\emph{et al.}\cite{liu96:new_model} as a modified form of the hard-sphere |
| 374 |
> |
water model proposed by Bratko, Blum, and |
| 375 |
|
Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole |
| 376 |
|
with a Lennard-Jones core and a sticky potential that directs the |
| 377 |
|
particles to assume the proper hydrogen bond orientation in the first |
| 378 |
|
solvation shell. Thus, the interaction between two SSD water molecules |
| 379 |
|
\emph{i} and \emph{j} is given by the potential |
| 380 |
|
\begin{equation} |
| 381 |
< |
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
| 382 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
| 383 |
< |
u_{ij}^{sp} |
| 384 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
| 381 |
> |
V_{ij} = |
| 382 |
> |
V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp} |
| 383 |
> |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
| 384 |
> |
V_{ij}^{sp} |
| 385 |
> |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
| 386 |
> |
\label{eq:ssdPot} |
| 387 |
|
\end{equation} |
| 388 |
|
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
| 389 |
< |
\emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and |
| 389 |
> |
\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and |
| 390 |
|
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
| 391 |
< |
orientations of the respective molecules. The Lennard-Jones, dipole, |
| 392 |
< |
and sticky parts of the potential are giving by the following |
| 393 |
< |
equations, |
| 391 |
> |
orientations of the respective molecules. The Lennard-Jones and dipole |
| 392 |
> |
parts of the potential are given by equations \ref{eq:lennardJonesPot} |
| 393 |
> |
and \ref{eq:dipolePot} respectively. The sticky part is described by |
| 394 |
> |
the following, |
| 395 |
|
\begin{equation} |
| 396 |
< |
u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right], |
| 396 |
> |
u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)= |
| 397 |
> |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij}, |
| 398 |
> |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + |
| 399 |
> |
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij}, |
| 400 |
> |
\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
| 401 |
> |
\label{eq:stickyPot} |
| 402 |
|
\end{equation} |
| 403 |
+ |
where $\nu_0$ is a strength parameter for the sticky potential, and |
| 404 |
+ |
$s$ and $s^\prime$ are cubic switching functions which turn off the |
| 405 |
+ |
sticky interaction beyond the first solvation shell. The $w$ function |
| 406 |
+ |
can be thought of as an attractive potential with tetrahedral |
| 407 |
+ |
geometry: |
| 408 |
|
\begin{equation} |
| 409 |
< |
u_{ij}^{dp} = \frac{\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j}{r_{ij}^3}-\frac{3(\boldsymbol{\mu}_i\cdot\mathbf{r}_{ij})(\boldsymbol{\mu}_j\cdot\mathbf{r}_{ij})}{r_{ij}^5}\ , |
| 409 |
> |
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
| 410 |
> |
\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
| 411 |
> |
\label{eq:stickyW} |
| 412 |
|
\end{equation} |
| 413 |
+ |
while the $w^\prime$ function counters the normal aligned and |
| 414 |
+ |
anti-aligned structures favored by point dipoles: |
| 415 |
|
\begin{equation} |
| 416 |
< |
\begin{split} |
| 417 |
< |
u_{ij}^{sp} |
| 418 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) |
| 299 |
< |
&= |
| 300 |
< |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\ |
| 301 |
< |
& \quad \ + |
| 302 |
< |
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
| 303 |
< |
\end{split} |
| 416 |
> |
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)= |
| 417 |
> |
(\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
| 418 |
> |
\label{eq:stickyWprime} |
| 419 |
|
\end{equation} |
| 420 |
< |
where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole |
| 421 |
< |
unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D, |
| 422 |
< |
$\nu_0$ scales the strength of the overall sticky potential, $s$ and |
| 423 |
< |
$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
| 424 |
< |
functions take the following forms, |
| 425 |
< |
\begin{equation} |
| 426 |
< |
w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
| 312 |
< |
\end{equation} |
| 313 |
< |
\begin{equation} |
| 314 |
< |
w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
| 315 |
< |
\end{equation} |
| 316 |
< |
where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive |
| 317 |
< |
term that promotes hydrogen bonding orientations within the first |
| 318 |
< |
solvation shell, and $w^\prime$ is a dipolar repulsion term that |
| 319 |
< |
repels unrealistic dipolar arrangements within the first solvation |
| 320 |
< |
shell. A more detailed description of the functional parts and |
| 321 |
< |
variables in this potential can be found in other |
| 322 |
< |
articles.\cite{liu96:new_model,chandra99:ssd_md} |
| 420 |
> |
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
| 421 |
> |
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
| 422 |
> |
enhances the tetrahedral geometry for hydrogen bonded structures), |
| 423 |
> |
while $w^\prime$ is a purely empirical function. A more detailed |
| 424 |
> |
description of the functional parts and variables in this potential |
| 425 |
> |
can be found in the original SSD |
| 426 |
> |
articles.\cite{liu96:new_model,liu96:monte_carlo,chandra99:ssd_md,Ichiye03} |
| 427 |
|
|
| 428 |
< |
Since SSD is a one-site point dipole model, the force calculations are |
| 429 |
< |
simplified significantly from a computational standpoint, in that the |
| 430 |
< |
number of long-range interactions is dramatically reduced. In the |
| 431 |
< |
original Monte Carlo simulations using this model, Ichiye \emph{et |
| 432 |
< |
al.} reported a calculation speed up of up to an order of magnitude |
| 433 |
< |
over other comparable models while maintaining the structural behavior |
| 434 |
< |
of water.\cite{liu96:new_model} In the original molecular dynamics studies of |
| 435 |
< |
SSD, it was shown that it actually improves upon the prediction of |
| 436 |
< |
water's dynamical properties over TIP3P and SPC/E.\cite{chandra99:ssd_md} |
| 428 |
> |
Since SSD is a single-point {\it dipolar} model, the force |
| 429 |
> |
calculations are simplified significantly relative to the standard |
| 430 |
> |
{\it charged} multi-point models. In the original Monte Carlo |
| 431 |
> |
simulations using this model, Ichiye {\it et al.} reported that using |
| 432 |
> |
SSD decreased computer time by a factor of 6-7 compared to other |
| 433 |
> |
models.\cite{liu96:new_model} What is most impressive is that these savings |
| 434 |
> |
did not come at the expense of accurate depiction of the liquid state |
| 435 |
> |
properties. Indeed, SSD maintains reasonable agreement with the Soper |
| 436 |
> |
diffraction data for the structural features of liquid |
| 437 |
> |
water.\cite{Soper86,liu96:new_model} Additionally, the dynamical properties |
| 438 |
> |
exhibited by SSD agree with experiment better than those of more |
| 439 |
> |
computationally expensive models (like TIP3P and |
| 440 |
> |
SPC/E).\cite{chandra99:ssd_md} The combination of speed and accurate depiction |
| 441 |
> |
of solvent properties makes SSD a very attractive model for the |
| 442 |
> |
simulation of large scale biochemical simulations. |
| 443 |
|
|
| 444 |
|
Recent constant pressure simulations revealed issues in the original |
| 445 |
|
SSD model that led to lower than expected densities at all target |
| 446 |
< |
pressures.\cite{Ichiye03,Gezelter04} Reparameterizations of the |
| 447 |
< |
original SSD have resulted in improved density behavior, as well as |
| 448 |
< |
alterations in the water structure and transport behavior. {\sc oopse} is |
| 449 |
< |
easily modified to impliment these new potential parameter sets for |
| 450 |
< |
the derivative water models: SSD1, SSD/E, and SSD/RF. All of the |
| 451 |
< |
variable parameters are listed in the accompanying BASS file, and |
| 452 |
< |
these parameters simply need to be changed to the updated values. |
| 446 |
> |
pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse} |
| 447 |
> |
is therefore SSD/E, a density corrected derivative of SSD that |
| 448 |
> |
exhibits improved liquid structure and transport behavior. If the use |
| 449 |
> |
of a reaction field long-range interaction correction is desired, it |
| 450 |
> |
is recommended that the parameters be modified to those of the SSD/RF |
| 451 |
> |
model. Solvent parameters can be easily modified in an accompanying |
| 452 |
> |
{\sc BASS} file as illustrated in the scheme below. A table of the |
| 453 |
> |
parameter values and the drawbacks and benefits of the different |
| 454 |
> |
density corrected SSD models can be found in reference |
| 455 |
> |
\ref{Gezelter04}. |
| 456 |
|
|
| 457 |
+ |
\begin{lstlisting}[caption={[A simulation of {\sc ssd} water]An example file showing a simulation including {\sc ssd} water.},label={sch:ssd}] |
| 458 |
|
|
| 459 |
< |
\subsection{\label{sec:eam}Embedded Atom Model} |
| 459 |
> |
#include "water.mdl" |
| 460 |
|
|
| 461 |
< |
Several molecular dynamics codes\cite{dynamo86} exist which have the |
| 461 |
> |
nComponents = 1; |
| 462 |
> |
component{ |
| 463 |
> |
type = "SSD_water"; |
| 464 |
> |
nMol = 864; |
| 465 |
> |
} |
| 466 |
> |
|
| 467 |
> |
initialConfig = "liquidWater.init"; |
| 468 |
> |
|
| 469 |
> |
forceField = "DUFF"; |
| 470 |
> |
|
| 471 |
> |
/* |
| 472 |
> |
* The reactionField flag toggles reaction |
| 473 |
> |
* field corrections. |
| 474 |
> |
*/ |
| 475 |
> |
|
| 476 |
> |
reactionField = false; // defaults to false |
| 477 |
> |
dielectric = 80.0; // dielectric for reaction field |
| 478 |
> |
|
| 479 |
> |
/* |
| 480 |
> |
* The following two flags set the cutoff |
| 481 |
> |
* radius for the electrostatic forces |
| 482 |
> |
* as well as the skin thickness of the switching |
| 483 |
> |
* function. |
| 484 |
> |
*/ |
| 485 |
> |
|
| 486 |
> |
electrostaticCutoffRadius = 9.2; |
| 487 |
> |
electrostaticSkinThickness = 1.38; |
| 488 |
> |
|
| 489 |
> |
\end{lstlisting} |
| 490 |
> |
|
| 491 |
> |
|
| 492 |
> |
\subsection{\label{sec:eam}Embedded Atom Method} |
| 493 |
> |
|
| 494 |
> |
Several other molecular dynamics packages\cite{dynamo86} exist which have the |
| 495 |
|
capacity to simulate metallic systems, including some that have |
| 496 |
|
parallel computational abilities\cite{plimpton93}. Potentials that |
| 497 |
|
describe bonding transition metal |
| 498 |
|
systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a |
| 499 |
< |
attractive interaction which models the stabilization of ``Embedding'' |
| 500 |
< |
a positively charged metal ion in an electron density created by the |
| 499 |
> |
attractive interaction which models ``Embedding'' |
| 500 |
> |
a positively charged metal ion in the electron density due to the |
| 501 |
|
free valance ``sea'' of electrons created by the surrounding atoms in |
| 502 |
|
the system. A mostly repulsive pairwise part of the potential |
| 503 |
|
describes the interaction of the positively charged metal core ions |
| 504 |
|
with one another. A particular potential description called the |
| 505 |
< |
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}(EAM) that has |
| 506 |
< |
particularly wide adoption has been selected for inclusion in OOPSE. A |
| 507 |
< |
good review of EAM and other metallic potential formulations was done |
| 505 |
> |
Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has |
| 506 |
> |
particularly wide adoption has been selected for inclusion in {\sc oopse}. A |
| 507 |
> |
good review of {\sc eam} and other metallic potential formulations was done |
| 508 |
|
by Voter.\cite{voter} |
| 509 |
|
|
| 510 |
|
The {\sc eam} potential has the form: |
| 512 |
|
V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i} |
| 513 |
|
\phi_{ij}({\bf r}_{ij}) \\ |
| 514 |
|
\rho_{i} & = & \sum_{j \neq i} f_{j}({\bf r}_{ij}) |
| 515 |
< |
\end{eqnarray} |
| 515 |
> |
\end{eqnarray}S |
| 516 |
|
|
| 517 |
< |
where $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 518 |
< |
between atoms $i$ and $j$.In the origional formulation of |
| 519 |
< |
EAM\cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 520 |
< |
in later refinements to EAM have shown that nonuniqueness between $F$ |
| 521 |
< |
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} The |
| 522 |
< |
embedding function $F_{i}$ is the energy required to embedded an |
| 523 |
< |
positively-charged core ion $i$ into a linear supeposition of |
| 524 |
< |
sperically averaged atomic electron densities given by |
| 525 |
< |
$\rho_{i}$. There is a cutoff distance, $r_{cut}$, which limits the |
| 517 |
> |
where $F_{i} $ is the embedding function that equates the energy required to embed a |
| 518 |
> |
positively-charged core ion $i$ into a linear superposition of |
| 519 |
> |
spherically averaged atomic electron densities given by |
| 520 |
> |
$\rho_{i}$. $\phi_{ij}$ is a primarily repulsive pairwise interaction |
| 521 |
> |
between atoms $i$ and $j$. In the original formulation of |
| 522 |
> |
{\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however |
| 523 |
> |
in later refinements to EAM have shown that non-uniqueness between $F$ |
| 524 |
> |
and $\phi$ allow for more general forms for $\phi$.\cite{Daw89} |
| 525 |
> |
There is a cutoff distance, $r_{cut}$, which limits the |
| 526 |
|
summations in the {\sc eam} equation to the few dozen atoms |
| 527 |
|
surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$ |
| 528 |
< |
interactions. |
| 528 |
> |
interactions. Foiles et al. fit EAM potentials for fcc metals Cu, Ag, Au, Ni, Pd, Pt and alloys of these metals\cite{FDB86}. These potential fits are in the DYNAMO 86 format and are included with {\sc oopse}. |
| 529 |
|
|
| 530 |
+ |
|
| 531 |
|
\subsection{\label{Sec:pbc}Periodic Boundary Conditions} |
| 532 |
< |
|
| 533 |
< |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 534 |
< |
macroscopic systems with a relatively small number of particles. Simulation |
| 535 |
< |
box is replicated throughout space to form an infinite lattice. During the |
| 536 |
< |
simulation, when a particle moves in the primary cell, its periodic image |
| 537 |
< |
particles in other boxes move in exactly the same direction with exactly the |
| 538 |
< |
same orientation.Thus, as a particle leaves the primary cell, one of its |
| 539 |
< |
images will enter through the opposite face.If the simulation box is large |
| 540 |
< |
enough to avoid "feeling" the symmetric of the periodic lattice,the surface |
| 541 |
< |
effect could be ignored. Cubic and parallelepiped are the available periodic |
| 542 |
< |
cells. \bigskip In OOPSE, we use the matrix instead of the vector to describe |
| 543 |
< |
the property of the simulation box. Therefore, not only the size of the |
| 544 |
< |
simulation box could be changed during the simulation, but also the shape of |
| 545 |
< |
it. The transformation from box space vector $\overrightarrow{s}$ to its |
| 546 |
< |
corresponding real space vector $\overrightarrow{r}$ is defined by |
| 547 |
< |
\begin{equation} |
| 548 |
< |
\overrightarrow{r}=H\overrightarrow{s}% |
| 549 |
< |
\end{equation} |
| 550 |
< |
|
| 551 |
< |
|
| 552 |
< |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
| 553 |
< |
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the sides of the |
| 554 |
< |
simulation box respectively. |
| 555 |
< |
|
| 556 |
< |
To find the minimum image, we need to convert the real vector to its |
| 557 |
< |
corresponding vector in box space first, \bigskip% |
| 558 |
< |
\begin{equation} |
| 559 |
< |
\overrightarrow{s}=H^{-1}\overrightarrow{r}% |
| 560 |
< |
\end{equation} |
| 561 |
< |
And then, each element of $\overrightarrow{s}$ is casted to lie between -0.5 |
| 562 |
< |
to 0.5, |
| 563 |
< |
\begin{equation} |
| 564 |
< |
s_{i}^{\prime}=s_{i}-round(s_{i}) |
| 565 |
< |
\end{equation} |
| 566 |
< |
where% |
| 567 |
< |
|
| 568 |
< |
\begin{equation} |
| 569 |
< |
round(x)=\lfloor{x+0.5}\rfloor\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ |
| 570 |
< |
}x\geqslant0 |
| 571 |
< |
\end{equation} |
| 572 |
< |
% |
| 573 |
< |
|
| 574 |
< |
\begin{equation} |
| 575 |
< |
round(x)=\lceil{x-0.5}\rceil\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }if\text{ }x<0 |
| 576 |
< |
\end{equation} |
| 577 |
< |
|
| 578 |
< |
|
| 579 |
< |
For example, $round(3.6)=4$,$round(3.1)=3$, $round(-3.6)=-4$, $round(-3.1)=-3$. |
| 580 |
< |
|
| 581 |
< |
Finally, we could get the minimum image by transforming back to real space,% |
| 582 |
< |
|
| 583 |
< |
\begin{equation} |
| 584 |
< |
\overrightarrow{r^{\prime}}=H^{-1}\overrightarrow{s^{\prime}}% |
| 585 |
< |
\end{equation} |
| 532 |
> |
|
| 533 |
> |
\newcommand{\roundme}{\operatorname{round}} |
| 534 |
> |
|
| 535 |
> |
\textit{Periodic boundary conditions} are widely used to simulate truly |
| 536 |
> |
macroscopic systems with a relatively small number of particles. The |
| 537 |
> |
simulation box is replicated throughout space to form an infinite lattice. |
| 538 |
> |
During the simulation, when a particle moves in the primary cell, its image in |
| 539 |
> |
other boxes move in exactly the same direction with exactly the same |
| 540 |
> |
orientation.Thus, as a particle leaves the primary cell, one of its images |
| 541 |
> |
will enter through the opposite face.If the simulation box is large enough to |
| 542 |
> |
avoid \textquotedblleft feeling\textquotedblright\ the symmetries of the |
| 543 |
> |
periodic lattice, surface effects can be ignored. Cubic, orthorhombic and |
| 544 |
> |
parallelepiped are the available periodic cells In OOPSE. We use a matrix to |
| 545 |
> |
describe the property of the simulation box. Therefore, both the size and |
| 546 |
> |
shape of the simulation box can be changed during the simulation. The |
| 547 |
> |
transformation from box space vector $\mathbf{s}$ to its corresponding real |
| 548 |
> |
space vector $\mathbf{r}$ is defined by |
| 549 |
> |
\begin{equation} |
| 550 |
> |
\mathbf{r}=\underline{\mathbf{H}}\cdot\mathbf{s}% |
| 551 |
> |
\end{equation} |
| 552 |
> |
|
| 553 |
> |
|
| 554 |
> |
where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of the three |
| 555 |
> |
box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the three sides of the |
| 556 |
> |
simulation box respectively. |
| 557 |
> |
|
| 558 |
> |
To find the minimum image of a vector $\mathbf{r}$, we convert the real vector |
| 559 |
> |
to its corresponding vector in box space first, \bigskip% |
| 560 |
> |
\begin{equation} |
| 561 |
> |
\mathbf{s}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{r}% |
| 562 |
> |
\end{equation} |
| 563 |
> |
And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5, |
| 564 |
> |
\begin{equation} |
| 565 |
> |
s_{i}^{\prime}=s_{i}-\roundme(s_{i}) |
| 566 |
> |
\end{equation} |
| 567 |
> |
where |
| 568 |
> |
|
| 569 |
> |
% |
| 570 |
> |
|
| 571 |
> |
\begin{equation} |
| 572 |
> |
\roundme(x)=\left\{ |
| 573 |
> |
\begin{array}{cc}% |
| 574 |
> |
\lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\ |
| 575 |
> |
\lceil{x-0.5}\rceil & \text{otherwise}% |
| 576 |
> |
\end{array} |
| 577 |
> |
\right. |
| 578 |
> |
\end{equation} |
| 579 |
> |
|
| 580 |
> |
|
| 581 |
> |
For example, $\roundme(3.6)=4$,$\roundme(3.1)=3$, $\roundme(-3.6)=-4$, $\roundme(-3.1)=-3$. |
| 582 |
> |
|
| 583 |
> |
Finally, we obtain the minimum image coordinates $\mathbf{r}^{\prime}$ by |
| 584 |
> |
transforming back to real space,% |
| 585 |
> |
|
| 586 |
> |
\begin{equation} |
| 587 |
> |
\mathbf{r}^{\prime}=\underline{\mathbf{H}}^{-1}\cdot\mathbf{s}^{\prime}% |
| 588 |
> |
\end{equation} |
| 589 |
> |
|
