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1 < \chapter{\label{chap:intro}INTRODUCTION AND BACKGROUND}
1 > \chapter{\label{chap:intro}INTRODUCTION AND BACKGROUND}
2  
3 + The following dissertation presents the primary aspects of the
4 + research I have performed and been involved with over the last several
5 + years. Rather than presenting the topics in a chronological fashion,
6 + they were arranged to form a series where the later topics apply and
7 + extend the findings of the former topics. This layout does lead to
8 + occasional situations where knowledge gleaned from earlier chapters
9 + (particularly chapter \ref{chap:electrostatics}) is not applied
10 + outright in the later work; however, I feel that this organization is
11 + more instructive and provides a more cohesive progression of research
12 + efforts.
13 +
14 + This chapter gives a general overview of molecular simulations, with
15 + particular emphasis on considerations that need to be made in order to
16 + apply the technique properly. This leads quite naturally into chapter
17 + \ref{chap:electrostatics}, where we investigate correction techniques
18 + for proper handling of long-ranged electrostatic interactions. In
19 + particular we develop and evaluate some new simple pairwise
20 + methods. These techniques make an appearance in the later chapters, as
21 + they are applied to specific systems of interest, showing how it they
22 + can improve the quality of various molecular simulations.
23 +
24 + In chapter \ref{chap:water}, we focus on simple water models,
25 + specifically the single-point soft sticky dipole (SSD) family of water
26 + models. These single-point models are more efficient than the common
27 + multi-point partial charge models and, in many cases, better capture
28 + the dynamic properties of water. We discuss improvements to these
29 + models in regards to long-range electrostatic corrections and show
30 + that these models can work well with the techniques discussed in
31 + chapter \ref{chap:electrostatics}. By investigating and improving
32 + simple water models, we are extending the limits of computational
33 + efficiency for systems that depend heavily on water calculations.
34 +
35 + In chapter \ref{chap:ice}, we study a unique polymorph of ice that we
36 + discovered while performing water simulations with the fast simple
37 + water models discussed in the previous chapter. This form of ice,
38 + which we called ``imaginary ice'' (Ice-$i$), has a low-density
39 + structure which is different from any known polymorph from either
40 + experiment or other simulations. In this study, we perform a free
41 + energy analysis and see that this structure is in fact the
42 + thermodynamically preferred form of ice for both the single-point and
43 + commonly used multi-point water models under the chosen simulation
44 + conditions. We also consider electrostatic corrections, again
45 + including the techniques discussed in chapter
46 + \ref{chap:electrostatics}, to see how they influence the free energy
47 + results. This work, to some degree, addresses the appropriateness of
48 + using these simplistic water models outside of the conditions for
49 + which they were developed.
50 +
51 + Finally, in chapter \ref{chap:conclusion}, we summarize the work
52 + presented in the previous chapters and connect ideas together with
53 + some final comments. The supporting information follows in the
54 + appendix, and it gives a more detailed look at systems discussed in
55 + chapter \ref{chap:electrostatics}.
56 +
57 + \section{On Molecular Simulation}
58 +
59   In order to advance our understanding of natural chemical and physical
60   processes, researchers develop explanations for events observed
61   experimentally. These hypotheses, supported by a body of corroborating
# Line 35 | Line 91 | molecular dynamics near exclusively, so we will presen
91   molecular dynamics can be used to investigate dynamical
92   quantities. The research presented in the following chapters utilized
93   molecular dynamics near exclusively, so we will present a general
94 < introduction to molecular dynamics and not Monte Carlo. There are
95 < several resources available for those desiring a more in-depth
96 < presentation of either of these
41 < techniques.\cite{Allen87,Frenkel02,Leach01}
94 > introduction to molecular dynamics. There are several resources
95 > available for those desiring a more in-depth presentation of either of
96 > these techniques.\cite{Allen87,Frenkel02,Leach01}
97  
98   \section{\label{sec:MolecularDynamics}Molecular Dynamics}
99  
# Line 48 | Line 103 | given configuration of particles at time $t_1$, basica
103   configuration to evolve in a manner that mimics real molecular
104   systems. To do this, we start with clarifying what we know about a
105   given configuration of particles at time $t_1$, basically that each
106 < particle in the configuration has a position ($q$) and velocity
107 < ($\dot{q}$). We now want to know what the configuration will be at
106 > particle in the configuration has a position ($\mathbf{q}$) and velocity
107 > ($\dot{\mathbf{q}}$). We now want to know what the configuration will be at
108   time $t_2$. To find out, we need the classical equations of
109   motion, and one useful formulation of them is the Lagrangian form.
110  
111 < The Lagrangian ($L$) is a function of the positions and velocites that
111 > The Lagrangian ($L$) is a function of the positions and velocities that
112   takes the form,
113   \begin{equation}
114   L = K - V,
# Line 65 | Line 120 | over time is zero,\cite{Tolman38}
120   motion, to say that the variation of the integral of the Lagrangian
121   over time is zero,\cite{Tolman38}
122   \begin{equation}
123 < \delta\int_{t_1}^{t_2}L(q,\dot{q})dt = 0.
123 > \delta\int_{t_1}^{t_2}L(\mathbf{q},\dot{\mathbf{q}})dt = 0.
124   \end{equation}
125 < This can be written as a summation of integrals to give
125 > The variation can be transferred to the variables that make up the
126 > Lagrangian,
127   \begin{equation}
128 < \int_{t_1}^{t_2}\sum_{i=1}^{3N}\left(\frac{\partial L}{\partial q_i}\delta q_i
129 <        + \frac{\partial L}{\partial \dot{q}_i}\delta \dot{q}_i\right)dt = 0.
128 > \int_{t_1}^{t_2}\sum_{i=1}^{3N}\left(
129 >        \frac{\partial L}{\partial \mathbf{q}_i}\delta \mathbf{q}_i
130 >        + \frac{\partial L}{\partial \dot{\mathbf{q}}_i}\delta
131 >                \dot{\mathbf{q}}_i\right)dt = 0.
132   \end{equation}
133 < Using fact that $\dot q$ is the derivative with respect to time of $q$
134 < and integrating the second partial derivative in the parenthesis by
135 < parts, this equation simplifies to
133 > Using the fact that $\dot{\mathbf{q}}$ is the derivative of
134 > $\mathbf{q}$ with respect to time and integrating the second partial
135 > derivative in the parenthesis by parts, this equation simplifies to
136   \begin{equation}
137   \int_{t_1}^{t_2}\sum_{i=1}^{3N}\left(
138 <        \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i}
139 <        - \frac{\partial L}{\partial q_i}\right)\delta {q}_i dt = 0,
138 >        \frac{d}{dt}\frac{\partial L}{\partial \dot{\mathbf{q}}_i}
139 >        - \frac{\partial L}{\partial \mathbf{q}_i}\right)
140 >                \delta {\mathbf{q}}_i dt = 0,
141   \end{equation}
142 < and the above equation is only valid for
142 > and since each variable is independent, we can separate the
143 > contribution from each of the variables:
144   \begin{equation}
145 < \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i}
146 <        - \frac{\partial L}{\partial q_i} = 0\quad\quad(i=1,2,\dots,3N).
145 > \frac{d}{dt}\frac{\partial L}{\partial \dot{\mathbf{q}}_i}
146 >        - \frac{\partial L}{\partial \mathbf{q}_i} = 0
147 >                \quad\quad(i=1,2,\dots,3N).
148   \label{eq:formulation}
149   \end{equation}
150   To obtain the classical equations of motion for the particles, we can
# Line 126 | Line 187 | differential equation. The velocities are necessary fo
187   interesting to note that equation (\ref{eq:verlet}) does not include
188   velocities, and this makes sense since they are not present in the
189   differential equation. The velocities are necessary for the
190 < calculation of the kinetic energy, and they can be calculated at each
191 < time step with the following equation:
190 > calculation of the kinetic energy and can be calculated at each time
191 > step with the equation:
192   \begin{equation}
193   \mathbf{v}_i(t) = \frac{\mathbf{r}_i(t+\delta t)-\mathbf{r}_i(t-\delta t)}
194                         {2\delta t}.
# Line 141 | Line 202 | symplectic, meaning that it preserves area in phase-sp
202   higher order information that is discarded after integrating
203   steps. Another interesting property of this algorithm is that it is
204   symplectic, meaning that it preserves area in phase-space. Symplectic
205 < algorithms system stays in the region of phase-space dictated by the
206 < ensemble and enjoy greater long-time energy
207 < conservations.\cite{Frenkel02}
205 > algorithms keep the system evolving in the region of phase-space
206 > dictated by the ensemble and enjoy a greater degree of energy
207 > conservation.\cite{Frenkel02}
208  
209   While the error in the positions calculated using the Verlet algorithm
210   is small ($\mathcal{O}(\delta t^4)$), the error in the velocities is
211 < quite a bit larger ($\mathcal{O}(\delta t^2)$).\cite{Allen87} Swope
212 < {\it et al.}  developed a corrected for of this algorithm, the
211 > substantially larger ($\mathcal{O}(\delta t^2)$).\cite{Allen87} Swope
212 > {\it et al.}  developed a corrected version of this algorithm, the
213   `velocity Verlet' algorithm, which improves the error of the velocity
214   calculation and thus the energy conservation.\cite{Swope82} This
215   algorithm involves a full step of the positions,
# Line 161 | Line 222 | and a half step of the velocities,
222   \mathbf{v}\left(t+\frac{1}{2}\delta t\right) = \mathbf{v}(t)
223                                          + \frac{1}{2}\delta t\mathbf{a}(t).
224   \end{equation}
225 < After calculating new forces, the velocities can be updated to a full
226 < step,
225 > After forces are calculated at the new positions, the velocities can
226 > be updated to a full step,
227   \begin{equation}
228   \mathbf{v}(t+\delta t) = \mathbf{v}\left(t+\frac{1}{2}\delta t\right)
229                                  + \frac{1}{2}\delta t\mathbf{a}(t+\delta t).
# Line 230 | Line 291 | $\mathsf{A}$ can be expressed as arithmetic operations
291   representation of the orientation of a rigid
292   body.\cite{Evans77,Evans77b,Allen87} Thus, the elements of
293   $\mathsf{A}$ can be expressed as arithmetic operations involving the
294 < four quaternions ($q_w, q_x, q_y,$ and $q_z$),
294 > four quaternions ($q_0, q_1, q_2,$ and $q_3$),
295   \begin{equation}
296   \mathsf{A} = \left( \begin{array}{l@{\quad}l@{\quad}l}
297   q_0^2+q_1^2-q_2^2-q_3^2 & 2(q_1q_2+q_0q_3) & 2(q_1q_3-q_0q_2) \\
# Line 241 | Line 302 | performance enhancements over Euler angles, particular
302   Integration of the equations of motion involves a series of arithmetic
303   operations involving the quaternions and angular momenta and leads to
304   performance enhancements over Euler angles, particularly for very
305 < small systems.\cite{Evans77} This integration methods works well for
305 > small systems.\cite{Evans77} This integration method works well for
306   propagating orientational motion in the canonical ensemble ($NVT$);
307   however, energy conservation concerns arise when using the simple
308   quaternion technique under the microcanonical ($NVE$) ensemble.  An
309 < earlier implementation of {\sc oopse} utilized quaternions for
309 > earlier implementation of our simulation code utilized quaternions for
310   propagation of rotational motion; however, a detailed investigation
311   showed that they resulted in a steady drift in the total energy,
312   something that has been observed by Kol {\it et al.} (also see
# Line 334 | Line 395 | propagator.\cite{Trotter59} This has three effects:
395   {\it symplectic}),
396   \item the integrator is time-{\it reversible}, and
397   \item the error for a single time step is of order
398 < $\mathcal{O}\left(\delta t^4\right)$ for time steps of length $\delta t$.
398 > $\mathcal{O}\left(\delta t^3\right)$ for time steps of length $\delta t$.
399   \end{enumerate}
400  
401   After the initial half-step ({\tt moveA}), the forces and torques are
# Line 421 | Line 482 | integrator, while the quaternion scheme will require ~
482   0.001~kcal~mol$^{-1}$ per particle is desired, a nanosecond of
483   simulation time will require ~19 hours of CPU time with the {\sc dlm}
484   integrator, while the quaternion scheme will require ~154 hours of CPU
485 < time. This demonstrates the computational advantage of the integration
486 < scheme utilized in {\sc oopse}.
485 > time. This demonstrates the computational advantage of the {\sc dlm}
486 > integration scheme.
487  
488   \section{Accumulating Interactions}
489  
# Line 432 | Line 493 | interactions increases by $\mathcal{O}(N(N-1)/2)$ if y
493   multipole) on each particle from their surroundings. This can quickly
494   become a cumbersome task for large systems since the number of pair
495   interactions increases by $\mathcal{O}(N(N-1)/2)$ if you accumulate
496 < interactions between all particles in the system, note the utilization
497 < of Newton's third law to reduce the interaction number from
498 < $\mathcal{O}(N^2)$. The case of periodic boundary conditions further
499 < complicates matters by turning the finite system into an infinitely
500 < repeating one. Fortunately, we can reduce the scale of this problem by
501 < using spherical cutoff methods.
496 > interactions between all particles in the system. (Note the
497 > utilization of Newton's third law to reduce the interaction number
498 > from $\mathcal{O}(N^2)$.) The case of periodic boundary conditions
499 > further complicates matters by turning the finite system into an
500 > infinitely repeating one. Fortunately, we can reduce the scale of this
501 > problem by using spherical cutoff methods.
502  
503   \begin{figure}
504   \centering
# Line 450 | Line 511 | interactions between all particles in the simulation,
511   \end{figure}
512   With spherical cutoffs, rather than accumulating the full set of
513   interactions between all particles in the simulation, we only
514 < explicitly consider interactions between local particles out to a
515 < specified cutoff radius distance, $R_\textrm{c}$, (see figure
514 > explicitly consider interactions between particles separated by less
515 > than a specified cutoff radius distance, $R_\textrm{c}$, (see figure
516   \ref{fig:sphereCut}). This reduces the scaling of the interaction to
517   $\mathcal{O}(N\cdot\textrm{c})$, where `c' is a value that depends on
518   the size of $R_\textrm{c}$ (c $\approx R_\textrm{c}^3$). Determination
# Line 460 | Line 521 | list radius $R_\textrm{l}$, which is larger than $R_\t
521   the this expense, we can use neighbor lists.\cite{Verlet67,Thompson83}
522   With neighbor lists, we have a second list of particles built from a
523   list radius $R_\textrm{l}$, which is larger than $R_\textrm{c}$. Once
524 < any of the particles within $R_\textrm{l}$ move a distance of
524 > any particle within $R_\textrm{l}$ moves half the distance of
525   $R_\textrm{l}-R_\textrm{c}$ (the ``skin'' thickness), we rebuild the
526   list with the full $N(N-1)/2$ loop.\cite{Verlet67} With an appropriate
527 < skin thickness, these updates are only performed every $\sim$20
528 < time steps, significantly reducing the time spent on pair-list
529 < bookkeeping operations.\cite{Allen87} If these neighbor lists are
530 < utilized, it is important that these list updates occur
531 < regularly. Incorrect application of this technique leads to
532 < non-physical dynamics, such as the ``flying block of ice'' behavior
533 < for which improper neighbor list handling was identified a one of the
534 < possible causes.\cite{Harvey98,Sagui99}
527 > skin thickness, these updates are only performed every $\sim$20 time
528 > steps, significantly reducing the time spent on pair-list bookkeeping
529 > operations.\cite{Allen87} If these neighbor lists are utilized, it is
530 > important that these list updates occur regularly. Incorrect
531 > application of this technique leads to non-physical dynamics, such as
532 > the ``flying block of ice'' behavior for which improper neighbor list
533 > handling was identified a one of the possible
534 > causes.\cite{Harvey98,Sagui99}
535  
536   \subsection{Correcting Cutoff Discontinuities}
537   \begin{figure}
# Line 485 | Line 546 | region in order to smooth out the discontinuity.}
546   region in order to smooth out the discontinuity.}
547   \label{fig:ljCutoff}
548   \end{figure}
549 < As particle move in and out of $R_\textrm{c}$, there will be sudden
550 < discontinuous jumps in the potential (and forces) due to their
551 < appearance and disappearance. In order to prevent heating and poor
552 < energy conservation in the simulations, this discontinuity needs to be
553 < smoothed out. There are several ways to modify the function so that it
554 < crosses $R_\textrm{c}$ in a continuous fashion, and the easiest
555 < methods is shifting the potential. To shift the potential, we simply
556 < subtract out the value we calculate at $R_\textrm{c}$ from the whole
557 < potential. For the shifted form of the Lennard-Jones potential is
549 > As the distance between a pair of particles fluctuates around
550 > $R_\textrm{c}$, there will be sudden discontinuous jumps in the
551 > potential (and forces) due to their inclusion and exclusion from the
552 > interaction loop. In order to prevent heating and poor energy
553 > conservation in the simulations, this discontinuity needs to be
554 > smoothed out. There are several ways to modify the potential function
555 > to eliminate these discontinuties, and the easiest methods is shifting
556 > the potential. To shift the potential, we simply subtract out the
557 > value we calculate at $R_\textrm{c}$ from the whole potential. The
558 > shifted form of the Lennard-Jones potential is
559   \begin{equation}
560   V_\textrm{SLJ} = \left\{\begin{array}{l@{\quad\quad}l}
561          V_\textrm{LJ}(r_{ij}) - V_\textrm{LJ}(R_\textrm{c}) & r_{ij} < R_\textrm{c}, \\
# Line 502 | Line 564 | where
564   \end{equation}
565   where
566   \begin{equation}
567 < V_\textrm{LJ} = 4\epsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12} -
568 <                                \left(\frac{\sigma}{r_{ij}}\right)^6\right].
567 > V_\textrm{LJ}(r_{ij}) =
568 >        4\epsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12} -
569 >        \left(\frac{\sigma}{r_{ij}}\right)^6\right].
570   \end{equation}
571   In figure \ref{fig:ljCutoff}, the shifted form of the potential
572   reaches zero at the cutoff radius at the expense of the correct
573 < magnitude below $R_\textrm{c}$. This correction method also does
573 > magnitude inside $R_\textrm{c}$. This correction method also does
574   nothing to correct the discontinuity in the forces. We can account for
575   this force discontinuity by shifting in the by applying the shifting
576 < in the forces rather than just the potential via
576 > in the forces as well as in the potential via
577   \begin{equation}
578   V_\textrm{SFLJ} = \left\{\begin{array}{l@{\quad\quad}l}
579          V_\textrm{LJ}({r_{ij}}) - V_\textrm{LJ}(R_\textrm{c}) -
# Line 542 | Line 605 | switching transition.
605   that the higher the order of the polynomial, the more abrupt the
606   switching transition.
607  
608 < \subsection{Long-Range Interaction Corrections}
608 > \subsection{\label{sec:LJCorrections}Long-Range Interaction Corrections}
609  
610 < While a good approximation, accumulating interaction only from the
611 < nearby particles can lead to some issues, because the further away
612 < surrounding particles do still have a small effect. For instance,
613 < while the strength of the Lennard-Jones interaction is quite weak at
614 < $R_\textrm{c}$ in figure \ref{fig:ljCutoff}, we are discarding all of
615 < the attractive interaction that extends out to extremely long
610 > While a good approximation, accumulating interactions only from nearby
611 > particles can lead to some issues, because particles at distances
612 > greater than $R_\textrm{c}$ do still have a small effect. For
613 > instance, while the strength of the Lennard-Jones interaction is quite
614 > weak at $R_\textrm{c}$ in figure \ref{fig:ljCutoff}, we are discarding
615 > all of the attractive interactions that extend out to extremely long
616   distances. Thus, the potential is a little too high and the pressure
617   on the central particle from the surroundings is a little too low. For
618 < homogeneous Lennard-Jones systems, we can correct for this neglect by
618 > homogeneous Lennard-Jones systems, we can correct for this effect by
619   assuming a uniform density and integrating the missing part,
620   \begin{equation}
621   V_\textrm{full}(r_{ij}) \approx V_\textrm{c}
622                  + 2\pi N\rho\int^\infty_{R_\textrm{c}}r^2V_\textrm{LJ}(r)dr,
623   \end{equation}
624   where $V_\textrm{c}$ is the truncated Lennard-Jones
625 < potential.\cite{Allen87} Like the potential, other Lennard-Jones
626 < properties can be corrected by integration over the relevant
627 < function. Note that with heterogeneous systems, this correction begins
628 < to break down because the density is no longer uniform.
625 > potential.\cite{Allen87} Like the potential, other properties can be
626 > corrected by integration over the relevant function. Note that with
627 > heterogeneous systems, this correction breaks down because the density
628 > is no longer uniform.
629  
630   Correcting long-range electrostatic interactions is a topic of great
631   importance in the field of molecular simulations. There have been
# Line 574 | Line 637 | In typical molecular dynamics simulations there are no
637   \subsection{Periodic Boundary Conditions}
638  
639   In typical molecular dynamics simulations there are no restrictions
640 < placed on the motion of particles outside of what the interparticle
640 > placed on the motion of particles outside of what the inter-particle
641   interactions dictate. This means that if a particle collides with
642   other particles, it is free to move away from the site of the
643   collision. If we consider the entire system as a collection of
# Line 607 | Line 670 | unrestricted fashion while calculating interactions us
670   region in space. To avoid this ``soft'' confinement, it is common
671   practice to allow the particle coordinates to expand in an
672   unrestricted fashion while calculating interactions using a wrapped
673 < set of ``minimum image'' coordinates. These minimum image coordinates
674 < need not be stored because they are easily calculated on the fly when
675 < determining particle distances.
673 > set of ``minimum image'' coordinates. These coordinates need not be
674 > stored because they are easily calculated while determining particle
675 > distances.
676  
677   \section{Calculating Properties}
678  
679 < In order to use simulations to model experimental process and evaluate
680 < theories, we need to be able to extract properties from the
679 > In order to use simulations to model experimental processes and
680 > evaluate theories, we need to be able to extract properties from the
681   results. In experiments, we can measure thermodynamic properties such
682   as the pressure and free energy. In computer simulations, we can
683   calculate properties from the motion and configuration of particles in
# Line 623 | Line 686 | isobaric-isothermal ($NPT$), and microcanonical ($NVE$
686  
687   The work presented in the later chapters use the canonical ($NVT$),
688   isobaric-isothermal ($NPT$), and microcanonical ($NVE$) statistical
689 < mechanics ensembles. The different ensembles lend themselves to more
689 > mechanical ensembles. The different ensembles lend themselves to more
690   effectively calculating specific properties. For instance, if we
691   concerned ourselves with the calculation of dynamic properties, which
692   are dependent upon the motion of the particles, it is better to choose
693 < an ensemble that does not add system motions to keep properties like
694 < the temperature or pressure constant. In this case, the $NVE$ ensemble
695 < would be the most appropriate choice. In chapter \ref{chap:ice}, we
696 < discuss calculating free energies, which are non-mechanical
697 < thermodynamic properties, and these calculations also depend on the
698 < chosen ensemble.\cite{Allen87} The Helmholtz free energy ($A$) depends
699 < on the $NVT$ partition function ($Q_{NVT}$),
693 > an ensemble that does not add artificial motions to keep properties
694 > like the temperature or pressure constant. In this case, the $NVE$
695 > ensemble would be the most appropriate choice. In chapter
696 > \ref{chap:ice}, we discuss calculating free energies, which are
697 > non-mechanical thermodynamic properties, and these calculations also
698 > depend on the chosen ensemble.\cite{Allen87} The Helmholtz free energy
699 > ($A$) depends on the $NVT$ partition function ($Q_{NVT}$),
700   \begin{equation}
701   A = -k_\textrm{B}T\ln Q_{NVT},
702   \end{equation}
# Line 652 | Line 715 | calculated from an average over the densities for the
715   chosen property like the density in the $NPT$ ensemble, where the
716   volume is allowed to fluctuate. The density for the simulation is
717   calculated from an average over the densities for the individual
718 < configurations. Being that the configurations from the integration
719 < process are related to one another by the time evolution of the
720 < interactions, this average is technically a time average. In
721 < calculating thermodynamic properties, we would actually prefer an
722 < ensemble average that is representative of all accessible states of
723 < the system. We can calculate thermodynamic properties from the time
724 < average by taking advantage of the ergodic hypothesis, which states
725 < that over a long period of time, the time average and the ensemble
726 < average are the same.
718 > configurations. Since the configurations from the integration process
719 > are related to one another by the time evolution of the interactions,
720 > this average is technically a time average. In calculating
721 > thermodynamic properties, we would actually prefer an ensemble average
722 > that is representative of all accessible states of the system. We can
723 > calculate thermodynamic properties from the time average by taking
724 > advantage of the ergodic hypothesis, which states that for a
725 > sufficiently chaotic system, and over a long enough period of time,
726 > the time and ensemble averages are the same.
727  
728 < In addition to the average, the fluctuations of that particular
729 < property can be determined via the standard deviation. Fluctuations
730 < are useful for measuring various thermodynamic properties in computer
728 > In addition to the average, the fluctuations of a particular property
729 > can be determined via the standard deviation. Not only are
730 > fluctuations useful for determining the spread of values around the
731 > average and the error in the calculation of the value, they are also
732 > useful for measuring various thermodynamic properties in computer
733   simulations. In section \ref{sec:t5peThermo}, we use fluctuations in
734 < propeties like the enthalpy and volume to calculate various
734 > properties like the enthalpy and volume to calculate other
735   thermodynamic properties, such as the constant pressure heat capacity
736   and the isothermal compressibility.
737  
738 < \section{Application of the Techniques}
738 > \section{OOPSE}
739  
740   In the following chapters, the above techniques are all utilized in
741   the study of molecular systems. There are a number of excellent
# Line 678 | Line 743 | Because of our interest in rigid body dynamics, point
743   incorporate many of these
744   methods.\cite{Brooks83,MacKerell98,Pearlman95,Berendsen95,Lindahl01,Smith96,Ponder87}
745   Because of our interest in rigid body dynamics, point multipoles, and
746 < systems where the orientational degrees cannot be handled using the
747 < common constraint procedures (like {\sc shake}), the majority of the
748 < following work was performed using {\sc oopse}, the object-oriented
749 < parallel simulation engine.\cite{Meineke05} The {\sc oopse} package
750 < started out as a collection of separate programs written within our
751 < group, and has developed into one of the few parallel molecular
752 < dynamics packages capable of accurately integrating rigid bodies and
753 < point multipoles.
746 > systems where the orientational degrees of freedom cannot be handled
747 > using the common constraint procedures (like {\sc shake}), the
748 > majority of the following work was performed using {\sc oopse}, the
749 > object-oriented parallel simulation engine.\cite{Meineke05} The {\sc
750 > oopse} package started out as a collection of separate programs
751 > written within our group, and has developed into one of the few
752 > parallel molecular dynamics packages capable of accurately integrating
753 > rigid bodies and point multipoles. This simulation package is
754 > open-source software, available to anyone interested in performing
755 > molecular dynamics simulations. More information about {\sc oopse} can
756 > be found in reference \cite{Meineke05} or at the {\tt
757 > http://oopse.org} website.
758  
690 – In chapter \ref{chap:electrostatics}, we investigate correction
691 – techniques for proper handling of long-ranged electrostatic
692 – interactions. In particular we develop and evaluate some new pairwise
693 – methods which we have incorporated into {\sc oopse}. These techniques
694 – make an appearance in the later chapters, as they are applied to
695 – specific systems of interest, showing how it they can improve the
696 – quality of various molecular simulations.
759  
698 – In chapter \ref{chap:water}, we focus on simple water models,
699 – specifically the single-point soft sticky dipole (SSD) family of water
700 – models. These single-point models are more efficient than the common
701 – multi-point partial charge models and, in many cases, better capture
702 – the dynamic properties of water. We discuss improvements to these
703 – models in regards to long-range electrostatic corrections and show
704 – that these models can work well with the techniques discussed in
705 – chapter \ref{chap:electrostatics}. By investigating and improving
706 – simple water models, we are extending the limits of computational
707 – efficiency for systems that heavily depend on water calculations.
708 –
709 – In chapter \ref{chap:ice}, we study a unique polymorph of ice that we
710 – discovered while performing water simulations with the fast simple
711 – water models discussed in the previous chapter. This form of ice,
712 – which we called ``imaginary ice'' (Ice-$i$), has a low-density
713 – structure which is different from any known polymorph from either
714 – experiment or other simulations. In this study, we perform a free
715 – energy analysis and see that this structure is in fact the
716 – thermodynamically preferred form of ice for both the single-point and
717 – commonly used multi-point water models under the chosen simulation
718 – conditions. We also consider electrostatic corrections, again
719 – including the techniques discussed in chapter
720 – \ref{chap:electrostatics}, to see how they influence the free energy
721 – results. This work, to some degree, addresses the appropriateness of
722 – using these simplistic water models outside of the conditions for
723 – which they were developed.
724 –
725 – In the final chapter we summarize the work presented previously. We
726 – also close with some final comments before the supporting information
727 – presented in the appendices.

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