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1  
2   \section{\label{sec:empericalEnergy}The Emperical Energy Functions}
3  
4 < \subsection{\label{sec:atomsMolecules}Atoms and Molecules}
4 > \subsection{\label{sec:atomsMolecules}Atoms, Molecules and Rigid Bodies}
5  
6   The basic unit of an {\sc oopse} simulation is the atom. The parameters
7   describing the atom are generalized to make the atom as flexible a
# Line 9 | Line 9 | are not currently suporrted by {\sc oopse}.
9   element, or be used for collections of atoms such as a methyl
10   group. The atoms are also capable of having a directional component
11   associated with them, often in the form of a dipole. Charges on atoms
12 < are not currently suporrted by {\sc oopse}.
12 > are not currently suported by {\sc oopse}.
13  
14 + \begin{lstlisting}[caption={[Specifier for molecules and atoms] An example specifing the simple Ar molecule},label=sch:AtmMole]
15 + molecule{
16 +  name = "Ar";
17 +  nAtoms = 1;
18 +  atom[0]{
19 +     type="Ar";
20 +     position( 0.0, 0.0, 0.0 );
21 +  }
22 + }
23 + \end{lstlisting}
24 +
25   The second most basic building block of a simulation is the
26 < molecule. The molecule is a way for {\sc oopse} to keep track of the atoms
27 < in a simulation in logical manner. This particular unit will store the
28 < identities of all the atoms associated with itself and is responsible
29 < for the evaluation of its own bonded interaction (i.e.~bonds, bends,
30 < and torsions).
26 > molecule. The molecule is a way for {\sc oopse} to keep track of the
27 > atoms in a simulation in logical manner. This particular unit will
28 > store the identities of all the atoms associated with itself and is
29 > responsible for the evaluation of its own bonded interaction
30 > (i.e.~bonds, bends, and torsions).
31  
32 + As stated previously, one of the features that sets {\sc oopse} apart
33 + from most of the current molecular simulation packages is the ability
34 + to handle rigid body dynamics. Rigid bodies are non-spherical
35 + particles or collections of particles that have a constant internal
36 + potential and move collectively.\cite{Goldstein01} They are not
37 + included in most simulation packages because of the requirement to
38 + propagate the orientational degrees of freedom. Until recently,
39 + integrators which propagate orientational motion have been lacking.
40 +
41 + Moving a rigid body involves determination of both the force and
42 + torque applied by the surroundings, which directly affect the
43 + translational and rotational motion in turn. In order to accumulate
44 + the total force on a rigid body, the external forces and torques must
45 + first be calculated for all the internal particles. The total force on
46 + the rigid body is simply the sum of these external forces.
47 + Accumulation of the total torque on the rigid body is more complex
48 + than the force in that it is the torque applied on the center of mass
49 + that dictates rotational motion. The torque on rigid body {\it i} is
50 + \begin{equation}
51 + \boldsymbol{\tau}_i=
52 +        \sum_{a}(\mathbf{r}_{ia}-\mathbf{r}_i)\times \mathbf{f}_{ia}
53 +        + \boldsymbol{\tau}_{ia},
54 + \label{eq:torqueAccumulate}
55 + \end{equation}
56 + where $\boldsymbol{\tau}_i$ and $\mathbf{r}_i$ are the torque on and
57 + position of the center of mass respectively, while $\mathbf{f}_{ia}$,
58 + $\mathbf{r}_{ia}$, and $\boldsymbol{\tau}_{ia}$ are the force on,
59 + position of, and torque on the component particles of the rigid body.
60 +
61 + The summation of the total torque is done in the body fixed axis of
62 + the rigid body. In order to move between the space fixed and body
63 + fixed coordinate axes, parameters describing the orientation must be
64 + maintained for each rigid body. At a minimum, the rotation matrix
65 + (\textbf{A}) can be described by the three Euler angles ($\phi,
66 + \theta,$ and $\psi$), where the elements of \textbf{A} are composed of
67 + trigonometric operations involving $\phi, \theta,$ and
68 + $\psi$.\cite{Goldstein01} In order to avoid numerical instabilities
69 + inherent in using the Euler angles, the four parameter ``quaternion''
70 + scheme is often used. The elements of \textbf{A} can be expressed as
71 + arithmetic operations involving the four quaternions ($q_0, q_1, q_2,$
72 + and $q_3$).\cite{allen87:csl} Use of quaternions also leads to
73 + performance enhancements, particularly for very small
74 + systems.\cite{Evans77}
75 +
76 + {\sc oopse} utilizes a relatively new scheme that propagates the
77 + entire nine parameter rotation matrix internally. Further discussion
78 + on this choice can be found in Sec.~\ref{sec:integrate}.
79 +
80 + \subsection{\label{sec:LJPot}The Lennard Jones Potential}
81 +
82 + The most basic force field implemented in OOPSE is the Lennard-Jones
83 + potential. The Lennard-Jones potential. Which mimics the Van der Waals
84 + interaction at long distances, and uses an emperical repulsion at
85 + short distances. The Lennard-Jones potential is given by:
86 + \begin{equation}
87 + V_{\text{LJ}}(r_{ij}) =
88 +        4\epsilon_{ij} \biggl[
89 +        \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12}
90 +        - \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6}
91 +        \biggr]
92 + \label{eq:lennardJonesPot}
93 + \end{equation}
94 + Where $r_{ij}$ is the distance between particle $i$ and $j$,
95 + $\sigma_{ij}$ scales the length of the interaction, and
96 + $\epsilon_{ij}$ scales the well depth of the potential.
97 +
98 + Because this potential is calculated between all pairs, the force
99 + evaluation can become computationally expensive for large systems. To
100 + keep the pair evaluation to a manegable number, OOPSE employs a
101 + cut-off radius.\cite{allen87:csl} The cutoff radius is set to be
102 + $2.5\sigma_{ii}$, where $\sigma_{ii}$ is the largest Lennard-Jones length
103 + parameter in the system. Truncating the calculation at
104 + $r_{\text{cut}}$ introduces a discontinuity into the potential
105 + energy. To offset this discontinuity, the energy value at
106 + $r_{\text{cut}}$ is subtracted from the entire potential. This causes
107 + the potential to go to zero at the cut-off radius.
108 +
109 + Interactions between dissimilar particles requires the generation of
110 + cross term parameters for $\sigma$ and $\epsilon$. These are
111 + calculated through the Lorentz-Berthelot mixing
112 + rules:\cite{allen87:csl}
113 + \begin{equation}
114 + \sigma_{ij} = \frac{1}{2}[\sigma_{ii} + \sigma_{jj}]
115 + \label{eq:sigmaMix}
116 + \end{equation}
117 + and
118 + \begin{equation}
119 + \epsilon_{ij} = \sqrt{\epsilon_{ii} \epsilon_{jj}}
120 + \label{eq:epsilonMix}
121 + \end{equation}
122 +
123 +
124   \subsection{\label{sec:DUFF}Dipolar Unified-Atom Force Field}
125  
126 < The \underline{D}ipolar \underline{U}nified-Atom
127 < \underline{F}orce \underline{F}ield ({\sc duff}) was developed to
25 < simulate lipid bilayers. We needed a model capable of forming
126 > The Dipolar Unified-atom Force Field ({\sc duff}) was developed to
127 > simulate lipid bilayers. The systems require a model capable of forming
128   bilayers, while still being sufficiently computationally efficient to
129   allow simulations of large systems ($\approx$100's of phospholipids,
130   $\approx$1000's of waters) for long times ($\approx$10's of
131   nanoseconds).
132  
133 < With this goal in mind, we have eliminated all point charges. Charge
134 < distributions were replaced with dipoles, and charge-neutral
135 < distributions were reduced to Lennard-Jones interaction sites. This
133 > With this goal in mind, {\sc duff} has no point charges. Charge
134 > neutral distributions were replaced with dipoles, while most atoms and
135 > groups of atoms were reduced to Lennard-Jones interaction sites. This
136   simplification cuts the length scale of long range interactions from
137   $\frac{1}{r}$ to $\frac{1}{r^3}$, allowing us to avoid the
138 < computationally expensive Ewald-Sum. Instead, we can use
139 < neighbor-lists and cutoff radii for the dipolar interactions.
138 > computationally expensive Ewald sum. Instead, we can use
139 > neighbor-lists, reaction field, and cutoff radii for the dipolar
140 > interactions.
141  
142 < As an example, lipid head groups in {\sc duff} are represented as point
143 < dipole interaction sites.PC and PE Lipid head groups are typically
144 < zwitterionic in nature, with charges separated by as much as
145 < 6~$\mbox{\AA}$. By placing a dipole of 20.6~Debye at the head group
146 < center of mass, our model mimics the head group of PC.\cite{Cevc87}
147 < Additionally, a Lennard-Jones site is located at the
148 < pseudoatom's center of mass. The model is illustrated by the dark grey
149 < atom in Fig.~\ref{fig:lipidModel}.
142 > As an example, lipid head-groups in {\sc duff} are represented as
143 > point dipole interaction sites. By placing a dipole of 20.6~Debye at
144 > the head group center of mass, our model mimics the head group of
145 > phosphatidylcholine.\cite{Cevc87} Additionally, a Lennard-Jones site
146 > is located at the pseudoatom's center of mass. The model is
147 > illustrated by the dark grey atom in Fig.~\ref{fig:lipidModel}. The water model we use to complement the dipoles of the lipids is out
148 > repaarameterization of the soft sticky dipole (SSD) model of Ichiye
149 > \emph{et al.}\cite{liu96:new_model}
150  
151   \begin{figure}
152 < \epsfxsize=6in
153 < \epsfbox{lipidModel.epsi}
152 > \epsfxsize=\linewidth
153 > \epsfbox{lipidModel.eps}
154   \caption{A representation of the lipid model. $\phi$ is the torsion angle, $\theta$ %
155 < is the bend angle, $\mu$ is the dipole moment of the head group, and n is the chain length.}
155 > is the bend angle, $\mu$ is the dipole moment of the head group, and n
156 > is the chain length.}
157   \label{fig:lipidModel}
158   \end{figure}
159  
56 – The water model we use to complement the dipoles of the lipids is
57 – the soft sticky dipole (SSD) model of Ichiye \emph{et
58 – al.}\cite{liu96:new_model} This model is discussed in greater detail
59 – in Sec.~\ref{sec:SSD}. In all cases we reduce water to a single
60 – Lennard-Jones interaction site. The site also contains a dipole to
61 – mimic the partial charges on the hydrogens and the oxygen. However,
62 – what makes the SSD model unique is the inclusion of a tetrahedral
63 – short range potential to recover the hydrogen bonding of water, an
64 – important factor when modeling bilayers, as it has been shown that
65 – hydrogen bond network formation is a leading contribution to the
66 – entropic driving force towards lipid bilayer formation.\cite{Cevc87}
67 –
68 –
160   Turning to the tails of the phospholipids, we have used a set of
161   scalable parameters to model the alkyl groups with Lennard-Jones
162   sites. For this, we have used the TraPPE force field of Siepmann
# Line 91 | Line 182 | V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Interna
182   The total energy of function in {\sc duff} is given by the following:
183   \begin{equation}
184   V_{\text{Total}} = \sum^{N}_{I=1} V^{I}_{\text{Internal}}
185 <        + \sum^{N}_{I=1} \sum^{N}_{J=1} V^{IJ}_{\text{Cross}}
185 >        + \sum^{N}_{I=1} \sum_{J>I} V^{IJ}_{\text{Cross}}
186   \label{eq:totalPotential}
187   \end{equation}
188   Where $V^{I}_{\text{Internal}}$ is the internal potential of a molecule:
189   \begin{equation}
190   V^{I}_{\text{Internal}} =
191          \sum_{\theta_{ijk} \in I} V_{\text{bend}}(\theta_{ijk})
192 <        + \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\theta_{ijkl})
192 >        + \sum_{\phi_{ijkl} \in I} V_{\text{torsion}}(\phi_{ijkl})
193          + \sum_{i \in I} \sum_{(j>i+4) \in I}
194          \biggl[ V_{\text{LJ}}(r_{ij}) +  V_{\text{dipole}}
195          (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
# Line 106 | Line 197 | within in the molecule. $V_{\text{torsion}}$ is the to
197   \label{eq:internalPotential}
198   \end{equation}
199   Here $V_{\text{bend}}$ is the bend potential for all 1, 3 bonded pairs
200 < within in the molecule. $V_{\text{torsion}}$ is the torsion The
201 < pairwise portions of the internal potential are excluded for pairs
202 < that ar closer than three bonds, i.e.~atom pairs farther away than a
203 < torsion are included in the pair-wise loop.
200 > within the molecule, and $V_{\text{torsion}}$ is the torsion potential
201 > for all 1, 4 bonded pairs. The pairwise portions of the internal
202 > potential are excluded for pairs that are closer than three bonds,
203 > i.e.~atom pairs farther away than a torsion are included in the
204 > pair-wise loop.
205  
114 – The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is
115 – as follows:
116 – \begin{equation}
117 – V^{IJ}_{\text{Cross}} =
118 –        \sum_{i \in I} \sum_{j \in J}
119 –        \biggl[ V_{\text{LJ}}(r_{ij}) +  V_{\text{dipole}}
120 –        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
121 –        + V_{\text{sticky}}
122 –        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
123 –        \biggr]
124 – \label{eq:crossPotentail}
125 – \end{equation}
126 – Where $V_{\text{LJ}}$ is the Lennard Jones potential,
127 – $V_{\text{dipole}}$ is the dipole dipole potential, and
128 – $V_{\text{sticky}}$ is the sticky potential defined by the SSD model.
206  
207   The bend potential of a molecule is represented by the following function:
208   \begin{equation}
# Line 140 | Line 217 | V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi]
217   The torsion potential and parameters are also taken from TraPPE. It is
218   of the form:
219   \begin{equation}
220 < V_{\text{torsion}}(\phi_{ijkl}) = c_1[1 + \cos \phi]
220 > V_{\text{torsion}}(\phi) = c_1[1 + \cos \phi]
221          + c_2[1 + \cos(2\phi)]
222          + c_3[1 + \cos(3\phi)]
223   \label{eq:origTorsionPot}
224   \end{equation}
225   Here $\phi_{ijkl}$ is the angle defined by four bonded neighbors $i$,
226 < $j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}).  However,
227 < for computaional efficency, the torsion potentail has been recast
228 < after the method of CHARMM\cite{charmm1983} whereby the angle series
229 < is converted to a power series of the form:
226 > $j$, $k$, and $l$ (again, see Fig.~\ref{fig:lipidModel}). For
227 > computaional efficency, the torsion potential has been recast after
228 > the method of CHARMM\cite{charmm1983} whereby the angle series is
229 > converted to a power series of the form:
230   \begin{equation}
231   V_{\text{torsion}}(\phi_{ijkl}) =  
232          k_3 \cos^3 \phi + k_2 \cos^2 \phi + k_1 \cos \phi + k_0
# Line 165 | Line 242 | The Lennard-Jones potential is given by:
242   By recasting the equation to a power series, repeated trigonometric
243   evaluations are avoided during the calculation of the potential.
244  
245 < The Lennard-Jones potential is given by:
245 >
246 > The cross portion of the total potential, $V^{IJ}_{\text{Cross}}$, is
247 > as follows:
248   \begin{equation}
249 < V_{\text{LJ}}(r_{ij}) =
250 <        4\epsilon_{ij} \biggl[
251 <        \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{12}
252 <        - \biggl(\frac{\sigma_{ij}}{r_{ij}}\biggr)^{6}
249 > V^{IJ}_{\text{Cross}} =
250 >        \sum_{i \in I} \sum_{j \in J}
251 >        \biggl[ V_{\text{LJ}}(r_{ij}) +  V_{\text{dipole}}
252 >        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
253 >        + V_{\text{sticky}}
254 >        (\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},\boldsymbol{\Omega}_{j})
255          \biggr]
256 < \label{eq:lennardJonesPot}
256 > \label{eq:crossPotentail}
257   \end{equation}
258 < Where $r_ij$ is the distance between atoms $i$ and $j$, $\sigma_{ij}$
259 < scales the length of the interaction, and $\epsilon_{ij}$ scales the
260 < energy of the potential.
258 > Where $V_{\text{LJ}}$ is the Lennard Jones potential,
259 > $V_{\text{dipole}}$ is the dipole dipole potential, and
260 > $V_{\text{sticky}}$ is the sticky potential defined by the SSD
261 > model. Note that not all atom types include all interactions.
262  
263   The dipole-dipole potential has the following form:
264   \begin{equation}
265   V_{\text{dipole}}(\mathbf{r}_{ij},\boldsymbol{\Omega}_{i},
266 <        \boldsymbol{\Omega}_{j}) = \frac{1}{4\pi\epsilon_{0}} \biggl[
267 <        \frac{\boldsymbol{\mu}_{i} \cdot \boldsymbol{\mu}_{j}}{r^{3}_{ij}}
266 >        \boldsymbol{\Omega}_{j}) = \frac{|\mu_i||\mu_j|}{4\pi\epsilon_{0}r_{ij}^{3}} \biggl[
267 >        \boldsymbol{\hat{u}}_{i} \cdot \boldsymbol{\hat{u}}_{j}
268          -
269 <        \frac{3(\boldsymbol{\mu}_i \cdot \mathbf{r}_{ij}) %
270 <                (\boldsymbol{\mu}_j \cdot \mathbf{r}_{ij}) }
271 <                {r^{5}_{ij}} \biggr]
269 >        \frac{3(\boldsymbol{\hat{u}}_i \cdot \mathbf{r}_{ij}) %
270 >                (\boldsymbol{\hat{u}}_j \cdot \mathbf{r}_{ij}) }
271 >                {r^{2}_{ij}} \biggr]
272   \label{eq:dipolePot}
273   \end{equation}
274   Here $\mathbf{r}_{ij}$ is the vector starting at atom $i$ pointing
275   towards $j$, and $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$
276 < are the Euler angles of atom $i$ and $j$
277 < respectively. $\boldsymbol{\mu}_i$ is the dipole vector of atom
278 < $i$ it takes its orientation from $\boldsymbol{\Omega}_i$.
276 > are the Euler angles of atom $i$ and $j$ respectively. $|\mu_i|$ is
277 > the magnitude of the dipole moment of atom $i$ and
278 > $\boldsymbol{\hat{u}}_i$ is the standard unit orientation vector of
279 > $\boldsymbol{\Omega}_i$.
280  
281  
282 < \subsection{\label{sec:SSD}Water Model: SSD and Derivatives}
282 > \subsection{\label{sec:SSD}The {\sc DUFF} Water Models: SSD/E and SSD/RF}
283  
284 < In the interest of computational efficiency, the native solvent used
285 < in {\sc oopse} is the Soft Sticky Dipole (SSD) water model. SSD was
286 < developed by Ichiye \emph{et al.} as a modified form of the
287 < hard-sphere water model proposed by Bratko, Blum, and
284 > In the interest of computational efficiency, the default solvent used
285 > by {\sc oopse} is the extended Soft Sticky Dipole (SSD/E) water
286 > model.\cite{Gezelter04} The original SSD was developed by Ichiye
287 > \emph{et al.}\cite{Ichiye96} as a modified form of the hard-sphere
288 > water model proposed by Bratko, Blum, and
289   Luzar.\cite{Bratko85,Bratko95} It consists of a single point dipole
290   with a Lennard-Jones core and a sticky potential that directs the
291   particles to assume the proper hydrogen bond orientation in the first
292   solvation shell. Thus, the interaction between two SSD water molecules
293   \emph{i} and \emph{j} is given by the potential
294   \begin{equation}
295 < u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp}
296 < (\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
297 < u_{ij}^{sp}
298 < (\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
295 > V_{ij} =
296 >        V_{ij}^{LJ} (r_{ij})\ + V_{ij}^{dp}
297 >        (\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ +
298 >        V_{ij}^{sp}
299 >        (\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j),
300 > \label{eq:ssdPot}
301   \end{equation}
302   where the $\mathbf{r}_{ij}$ is the position vector between molecules
303 < \emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and
303 > \emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and
304   $\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the
305 < orientations of the respective molecules. The Lennard-Jones, dipole,
306 < and sticky parts of the potential are giving by the following
307 < equations,
305 > orientations of the respective molecules. The Lennard-Jones and dipole
306 > parts of the potential are given by equations \ref{eq:lennardJonesPot}
307 > and \ref{eq:dipolePot} respectively. The sticky part is described by
308 > the following,
309   \begin{equation}
310 < u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right],
310 > u_{ij}^{sp}(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=
311 >        \frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},
312 >        \boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) +
313 >        s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},
314 >        \boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
315 > \label{eq:stickyPot}
316   \end{equation}
317 + where $\nu_0$ is a strength parameter for the sticky potential, and
318 + $s$ and $s^\prime$ are cubic switching functions which turn off the
319 + sticky interaction beyond the first solvation shell. The $w$ function
320 + can be thought of as an attractive potential with tetrahedral
321 + geometry:
322   \begin{equation}
323 < 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}\ ,
323 > w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=
324 >        \sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
325 > \label{eq:stickyW}
326   \end{equation}
327 + while the $w^\prime$ function counters the normal aligned and
328 + anti-aligned structures favored by point dipoles:
329   \begin{equation}
330 < \begin{split}
331 < u_{ij}^{sp}
332 < (\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)
232 < &=
233 < \frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\
234 < & \quad \ +
235 < s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ ,
236 < \end{split}
330 > w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=
331 >        (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0,
332 > \label{eq:stickyWprime}
333   \end{equation}
334 < where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole
335 < unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D,
336 < $\nu_0$ scales the strength of the overall sticky potential, $s$ and
337 < $s^\prime$ are cubic switching functions. The $w$ and $w^\prime$
338 < functions take the following forms,
334 > It should be noted that $w$ is proportional to the sum of the $Y_3^2$
335 > and $Y_3^{-2}$ spherical harmonics (a linear combination which
336 > enhances the tetrahedral geometry for hydrogen bonded structures),
337 > while $w^\prime$ is a purely empirical function.  A more detailed
338 > description of the functional parts and variables in this potential
339 > can be found in the original SSD
340 > articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03}
341 >
342 > Since SSD is a single-point {\it dipolar} model, the force
343 > calculations are simplified significantly relative to the standard
344 > {\it charged} multi-point models. In the original Monte Carlo
345 > simulations using this model, Ichiye {\it et al.} reported that using
346 > SSD decreased computer time by a factor of 6-7 compared to other
347 > models.\cite{Ichiye96} What is most impressive is that these savings
348 > did not come at the expense of accurate depiction of the liquid state
349 > properties.  Indeed, SSD maintains reasonable agreement with the Soper
350 > diffraction data for the structural features of liquid
351 > water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties
352 > exhibited by SSD agree with experiment better than those of more
353 > computationally expensive models (like TIP3P and
354 > SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction
355 > of solvent properties makes SSD a very attractive model for the
356 > simulation of large scale biochemical simulations.
357 >
358 > Recent constant pressure simulations revealed issues in the original
359 > SSD model that led to lower than expected densities at all target
360 > pressures.\cite{Ichiye03,Gezelter04} The default model in {\sc oopse}
361 > is therefore SSD/E, a density corrected derivative of SSD that
362 > exhibits improved liquid structure and transport behavior. If the use
363 > of a reaction field long-range interaction correction is desired, it
364 > is recommended that the parameters be modified to those of the SSD/RF
365 > model. Solvent parameters can be easily modified in an accompanying
366 > {\sc BASS} file as illustrated in the scheme below. A table of the
367 > parameter values and the drawbacks and benefits of the different
368 > density corrected SSD models can be found in reference
369 > \ref{Gezelter04}.
370 >
371 > !!!Place a {\sc BASS} scheme showing SSD parameters around here!!!
372 >
373 > \subsection{\label{sec:eam}Embedded Atom Method}
374 >
375 > Several other molecular dynamics packages\cite{dynamo86} exist which have the
376 > capacity to simulate metallic systems, including some that have
377 > parallel computational abilities\cite{plimpton93}. Potentials that
378 > describe bonding transition metal
379 > systems\cite{Finnis84,Ercolessi88,Chen90,Qi99,Ercolessi02} have a
380 > attractive interaction which models  ``Embedding''
381 > a positively charged metal ion in the electron density due to the
382 > free valance ``sea'' of electrons created by the surrounding atoms in
383 > the system. A mostly repulsive pairwise part of the potential
384 > describes the interaction of the positively charged metal core ions
385 > with one another. A particular potential description called the
386 > Embedded Atom Method\cite{Daw84,FBD86,johnson89,Lu97}({\sc eam}) that has
387 > particularly wide adoption has been selected for inclusion in OOPSE. A
388 > good review of {\sc eam} and other metallic potential formulations was done
389 > by Voter.\cite{voter}
390 >
391 > The {\sc eam} potential has the form:
392 > \begin{eqnarray}
393 > V & = & \sum_{i} F_{i}\left[\rho_{i}\right] + \sum_{i} \sum_{j \neq i}
394 > \phi_{ij}({\bf r}_{ij})  \\
395 > \rho_{i}  & = & \sum_{j \neq i} f_{j}({\bf r}_{ij})
396 > \end{eqnarray}S
397 >
398 > where $F_{i} $ is the embedding function that equates the energy required to embed a
399 > positively-charged core ion $i$ into a linear superposition of
400 > sperically averaged atomic electron densities given by
401 > $\rho_{i}$.  $\phi_{ij}$ is a primarily repulsive pairwise interaction
402 > between atoms $i$ and $j$. In the original formulation of
403 > {\sc eam} cite{Daw84}, $\phi_{ij}$ was an entirely repulsive term, however
404 > in later refinements to EAM have shown that non-uniqueness between $F$
405 > and $\phi$ allow for more general forms for $\phi$.\cite{Daw89}
406 > There is a cutoff distance, $r_{cut}$, which limits the
407 > summations in the {\sc eam} equation to the few dozen atoms
408 > surrounding atom $i$ for both the density $\rho$ and pairwise $\phi$
409 > 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}.
410 >
411 >
412 > \subsection{\label{Sec:pbc}Periodic Boundary Conditions}
413 >
414 > \newcommand{\roundme}{\operatorname{round}}
415 >
416 > \textit{Periodic boundary conditions} are widely used to simulate truly
417 > macroscopic systems with a relatively small number of particles. The
418 > simulation box is replicated throughout space to form an infinite
419 > lattice.  During the simulation, when a particle moves in the primary
420 > cell, its image in other boxes move in exactly the same direction with
421 > exactly the same orientation.Thus, as a particle leaves the primary
422 > cell, one of its images will enter through the opposite face.If the
423 > simulation box is large enough to avoid "feeling" the symmetries of
424 > the periodic lattice, surface effects can be ignored. Cubic,
425 > orthorhombic and parallelepiped are the available periodic cells In
426 > OOPSE. We use a matrix to describe the property of the simulation
427 > box. Therefore, both the size and shape of the simulation box can be
428 > changed during the simulation. The transformation from box space
429 > vector $\mathbf{s}$ to its corresponding real space vector
430 > $\mathbf{r}$ is defined by
431   \begin{equation}
432 < w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij},
432 > \mathbf{r}=\underline{\underline{H}}\cdot\mathbf{s}%
433   \end{equation}
434 +
435 +
436 + where $H=(h_{x},h_{y},h_{z})$ is a transformation matrix made up of
437 + the three box axis vectors. $h_{x},h_{y}$ and $h_{z}$ represent the
438 + three sides of the simulation box respectively.
439 +
440 + To find the minimum image, we convert the real vector to its
441 + corresponding vector in box space first, \bigskip%
442   \begin{equation}
443 < 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,
443 > \mathbf{s}=\underline{\underline{H}}^{-1}\cdot\mathbf{r}%
444   \end{equation}
445 < where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive
446 < term that promotes hydrogen bonding orientations within the first
447 < solvation shell, and $w^\prime$ is a dipolar repulsion term that
448 < repels unrealistic dipolar arrangements within the first solvation
449 < shell. A more detailed description of the functional parts and
254 < variables in this potential can be found in other
255 < articles.\cite{liu96:new_model,chandra99:ssd_md}
445 > And then, each element of $\mathbf{s}$ is wrapped to lie between -0.5 to 0.5,
446 > \begin{equation}
447 > s_{i}^{\prime}=s_{i}-\roundme(s_{i})
448 > \end{equation}
449 > where
450  
451 < Since SSD is a one-site point dipole model, the force calculations are
258 < simplified significantly from a computational standpoint, in that the
259 < number of long-range interactions is dramatically reduced. In the
260 < original Monte Carlo simulations using this model, Ichiye \emph{et
261 < al.} reported a calculation speed up of up to an order of magnitude
262 < over other comparable models while maintaining the structural behavior
263 < of water.\cite{liu96:new_model} In the original molecular dynamics studies of
264 < SSD, it was shown that it actually improves upon the prediction of
265 < water's dynamical properties over TIP3P and SPC/E.\cite{chandra99:ssd_md}
451 > %
452  
453 < Recent constant pressure simulations revealed issues in the original
454 < SSD model that led to lower than expected densities at all target
455 < pressures.\cite{Ichiye03,Gezelter04} Reparameterizations of the
456 < original SSD have resulted in improved density behavior, as well as
457 < alterations in the water structure and transport behavior. {\sc oopse} is
458 < easily modified to impliment these new potential parameter sets for
459 < the derivative water models: SSD1, SSD/E, and SSD/RF. All of the
460 < variable parameters are listed in the accompanying BASS file, and
461 < these parameters simply need to be changed to the updated values.
453 > \begin{equation}
454 > \roundme(x)=\left\{
455 > \begin{array}{cc}
456 > \lfloor{x+0.5}\rfloor & \text{if \ }x\geqslant0\\
457 > \lceil{x-0.5}\rceil & \text{otherwise}%
458 > \end{array}
459 > \right.
460 > \end{equation}
461 > For example, $\roundme(3.6)=4$, $\roundme(3.1)=3$, $\roundme(-3.6)=-4$,
462 > $\roundme(-3.1)=-3$.
463  
464 + Finally, we obtain the minimum image coordinates by transforming back
465 + to real space,%
466  
467 < \subsection{\label{sec:eam}Embedded Atom Model}
467 > \begin{equation}
468 > \mathbf{r}^{\prime}=\underline{\underline{H}}^{-1}\cdot\mathbf{s}^{\prime}%
469 > \end{equation}
470  
280 – here there be Monsters

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