| 42 |
|
|
| 43 |
|
\section{\label{sec:MolecularDynamics}Molecular Dynamics} |
| 44 |
|
|
| 45 |
< |
\section{\label{sec:MovingParticles}Propagating Particle Motion} |
| 45 |
> |
As stated above, in molecular dynamics we focus on evolving |
| 46 |
> |
configurations of molecules over time. In order to use this as a tool |
| 47 |
> |
for understanding experiments and testing theories, we want the |
| 48 |
> |
configuration to evolve in a manner that mimics real molecular |
| 49 |
> |
systems. To do this, we start with clarifying what we know about a |
| 50 |
> |
given configuration of particles at time $t_1$, basically that each |
| 51 |
> |
particle in the configuration has a position ($q$) and velocity |
| 52 |
> |
($\dot{q}$). We now want to know what the configuration will be at |
| 53 |
> |
time $t_2$. To find out, we need the classical equations of |
| 54 |
> |
motion, and one useful formulation of them is the Lagrangian form. |
| 55 |
> |
|
| 56 |
> |
The Lagrangian ($L$) is a function of the positions and velocites that |
| 57 |
> |
takes the form, |
| 58 |
> |
\begin{equation} |
| 59 |
> |
L = K - V, |
| 60 |
> |
\label{eq:lagrangian} |
| 61 |
> |
\end{equation} |
| 62 |
> |
where $K$ is the kinetic energy and $V$ is the potential energy. We |
| 63 |
> |
can use Hamilton's principle, which states that the integral of the |
| 64 |
> |
Lagrangian over time has a stationary value for the correct path of |
| 65 |
> |
motion, to say that the variation of the integral of the Lagrangian |
| 66 |
> |
over time is zero,\cite{Tolman38} |
| 67 |
> |
\begin{equation} |
| 68 |
> |
\delta\int_{t_1}^{t_2}L(q,\dot{q})dt = 0. |
| 69 |
> |
\end{equation} |
| 70 |
> |
This can be written as a summation of integrals to give |
| 71 |
> |
\begin{equation} |
| 72 |
> |
\int_{t_1}^{t_2}\sum_{i=1}^{3N}\left(\frac{\partial L}{\partial q_i}\delta q_i |
| 73 |
> |
+ \frac{\partial L}{\partial \dot{q}_i}\delta \dot{q}_i\right)dt = 0. |
| 74 |
> |
\end{equation} |
| 75 |
> |
Using fact that $\dot q$ is the derivative with respect to time of $q$ |
| 76 |
> |
and integrating the second partial derivative in the parenthesis by |
| 77 |
> |
parts, this equation simplifies to |
| 78 |
> |
\begin{equation} |
| 79 |
> |
\int_{t_1}^{t_2}\sum_{i=1}^{3N}\left( |
| 80 |
> |
\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} |
| 81 |
> |
- \frac{\partial L}{\partial q_i}\right)\delta {q}_i dt = 0, |
| 82 |
> |
\end{equation} |
| 83 |
> |
and the above equation is only valid for |
| 84 |
> |
\begin{equation} |
| 85 |
> |
\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} |
| 86 |
> |
- \frac{\partial L}{\partial q_i} = 0\quad\quad(i=1,2,\dots,3N). |
| 87 |
> |
\label{eq:formulation} |
| 88 |
> |
\end{equation} |
| 89 |
> |
To obtain the classical equations of motion for the particles, we can |
| 90 |
> |
substitute equation (\ref{eq:lagrangian}) into the above equation with |
| 91 |
> |
$m\dot{\mathbf{r}}^2/2$ for the kinetic energy, giving |
| 92 |
> |
\begin{equation} |
| 93 |
> |
\frac{d}{dt}(m\dot{\mathbf{r}})+\frac{dV}{d\mathbf{r}}=0, |
| 94 |
> |
\end{equation} |
| 95 |
> |
or more recognizably, |
| 96 |
> |
\begin{equation} |
| 97 |
> |
\mathbf{f} = m\mathbf{a}, |
| 98 |
> |
\end{equation} |
| 99 |
> |
where $\mathbf{f} = -dV/d\mathbf{r}$ and $\mathbf{a} = |
| 100 |
> |
d^2\mathbf{r}/dt^2$. This Lagrangian formulation shown in equation |
| 101 |
> |
(\ref{eq:formulation}) is generalized, and it can be used to determine |
| 102 |
> |
equations of motion in forms outside of the typical Cartesian case |
| 103 |
> |
shown here. |
| 104 |
> |
|
| 105 |
> |
\subsection{\label{sec:Verlet}Verlet Integration} |
| 106 |
> |
|
| 107 |
> |
In order to perform molecular dynamics, we need an algorithm that |
| 108 |
> |
integrates the equations of motion described above. Ideal algorithms |
| 109 |
> |
are both simple in implementation and highly accurate. There have been |
| 110 |
> |
a large number of algorithms developed for this purpose; however, for |
| 111 |
> |
reasons discussed below, we are going to focus on the Verlet class of |
| 112 |
> |
integrators.\cite{Gear66,Beeman76,Berendsen86,Allen87,Verlet67,Swope82} |
| 113 |
> |
|
| 114 |
> |
In Verlet's original study of computer ``experiments'', he directly |
| 115 |
> |
integrated the Newtonian second order differential equation of motion, |
| 116 |
> |
\begin{equation} |
| 117 |
> |
m\frac{d^2\mathbf{r}_i}{dt^2} = \sum_{j\ne i}\mathbf{f}(r_{ij}), |
| 118 |
> |
\end{equation} |
| 119 |
> |
with the following simple algorithm: |
| 120 |
> |
\begin{equation} |
| 121 |
> |
\mathbf{r}_i(t+\delta t) = -\mathbf{r}_i(t-\delta t) + 2\mathbf{r}_i(t) |
| 122 |
> |
+ \sum_{j\ne i}\mathbf{f}(r_{ij}(t))\delta t^2, |
| 123 |
> |
\label{eq:verlet} |
| 124 |
> |
\end{equation} |
| 125 |
> |
where $\delta t$ is the time step of integration.\cite{Verlet67} It is |
| 126 |
> |
interesting to note that equation (\ref{eq:verlet}) does not include |
| 127 |
> |
velocities, and this makes sense since they are not present in the |
| 128 |
> |
differential equation. The velocities are necessary for the |
| 129 |
> |
calculation of the kinetic energy, and they can be calculated at each |
| 130 |
> |
time step with the following equation: |
| 131 |
> |
\begin{equation} |
| 132 |
> |
\mathbf{v}_i(t) = \frac{\mathbf{r}_i(t+\delta t)-\mathbf{r}_i(t-\delta t)} |
| 133 |
> |
{2\delta t}. |
| 134 |
> |
\end{equation} |
| 135 |
> |
|
| 136 |
> |
Like the equation of motion it solves, the Verlet algorithm has the |
| 137 |
> |
beneficial property of being time-reversible, meaning that you can |
| 138 |
> |
integrate the configuration forward and then backward and end up at |
| 139 |
> |
the original configuration. Some other methods for integration, like |
| 140 |
> |
predictor-corrector methods, lack this property in that they require |
| 141 |
> |
higher order information that is discarded after integrating |
| 142 |
> |
steps. Another interesting property of this algorithm is that it is |
| 143 |
> |
symplectic, meaning that it preserves area in phase-space. Symplectic |
| 144 |
> |
algorithms system stays in the region of phase-space dictated by the |
| 145 |
> |
ensemble and enjoy greater long-time energy |
| 146 |
> |
conservations.\cite{Frenkel02} |
| 147 |
> |
|
| 148 |
> |
While the error in the positions calculated using the Verlet algorithm |
| 149 |
> |
is small ($\mathcal{O}(\delta t^4)$), the error in the velocities is |
| 150 |
> |
quite a bit larger ($\mathcal{O}(\delta t^2)$).\cite{Allen87} Swope |
| 151 |
> |
{\it et al.} developed a corrected for of this algorithm, the |
| 152 |
> |
`velocity Verlet' algorithm, which improves the error of the velocity |
| 153 |
> |
calculation and thus the energy conservation.\cite{Swope82} This |
| 154 |
> |
algorithm involves a full step of the positions, |
| 155 |
> |
\begin{equation} |
| 156 |
> |
\mathbf{r}(t+\delta t) = \mathbf{r}(t) + \delta t\mathbf{v}(t) |
| 157 |
> |
+ \frac{1}{2}\delta t^2\mathbf{a}(t), |
| 158 |
> |
\end{equation} |
| 159 |
> |
and a half step of the velocities, |
| 160 |
> |
\begin{equation} |
| 161 |
> |
\mathbf{v}\left(t+\frac{1}{2}\delta t\right) = \mathbf{v}(t) |
| 162 |
> |
+ \frac{1}{2}\delta t\mathbf{a}(t). |
| 163 |
> |
\end{equation} |
| 164 |
> |
After calculating new forces, the velocities can be updated to a full |
| 165 |
> |
step, |
| 166 |
> |
\begin{equation} |
| 167 |
> |
\mathbf{v}(t+\delta t) = \mathbf{v}\left(t+\frac{1}{2}\delta t\right) |
| 168 |
> |
+ \frac{1}{2}\delta t\mathbf{a}(t+\delta t). |
| 169 |
> |
\end{equation} |
| 170 |
> |
By integrating in this manner, the error in the velocities reduces to |
| 171 |
> |
$\mathcal{O}(\delta t^3)$. It should be noted that the error in the |
| 172 |
> |
positions increases to $\mathcal{O}(\delta t^3)$, but the resulting |
| 173 |
> |
improvement in the energies coupled with the maintained simplicity, |
| 174 |
> |
time-reversibility, and symplectic nature make it an improvement over |
| 175 |
> |
the original form. |
| 176 |
|
|
| 177 |
|
\subsection{\label{sec:IntroIntegrate}Rigid Body Motion} |
| 178 |
|
|
| 179 |
|
Rigid bodies are non-spherical particles or collections of particles |
| 180 |
|
(e.g. $\mbox{C}_{60}$) that have a constant internal potential and |
| 181 |
|
move collectively.\cite{Goldstein01} Discounting iterative constraint |
| 182 |
< |
procedures like {\sc shake} and {\sc rattle}, they are not included in |
| 183 |
< |
most simulation packages because of the algorithmic complexity |
| 184 |
< |
involved in propagating orientational degrees of |
| 185 |
< |
freedom.\cite{Ryckaert77,Andersen83,Krautler01} Integrators which |
| 186 |
< |
propagate orientational motion with an acceptable level of energy |
| 187 |
< |
conservation for molecular dynamics are relatively new inventions. |
| 182 |
> |
procedures like {\sc shake} and {\sc rattle} for approximating rigid |
| 183 |
> |
bodies, they are not included in most simulation packages because of |
| 184 |
> |
the algorithmic complexity involved in propagating orientational |
| 185 |
> |
degrees of freedom.\cite{Ryckaert77,Andersen83,Krautler01} Integrators |
| 186 |
> |
which propagate orientational motion with an acceptable level of |
| 187 |
> |
energy conservation for molecular dynamics are relatively new |
| 188 |
> |
inventions. |
| 189 |
|
|
| 190 |
|
Moving a rigid body involves determination of both the force and |
| 191 |
|
torque applied by the surroundings, which directly affect the |
| 424 |
|
time. This demonstrates the computational advantage of the integration |
| 425 |
|
scheme utilized in {\sc oopse}. |
| 426 |
|
|
| 296 |
– |
\subsection{Periodic Boundary Conditions} |
| 297 |
– |
|
| 298 |
– |
\begin{figure} |
| 299 |
– |
\centering |
| 300 |
– |
\includegraphics[width=4.5in]{./figures/periodicImage.pdf} |
| 301 |
– |
\caption{With periodic boundary conditions imposed, when particles |
| 302 |
– |
move out of one side the simulation box, they wrap back in the |
| 303 |
– |
opposite side. In this manner, a finite system of particles behaves as |
| 304 |
– |
an infinite system.} |
| 305 |
– |
\label{fig:periodicImage} |
| 306 |
– |
\end{figure} |
| 307 |
– |
|
| 427 |
|
\section{Accumulating Interactions} |
| 428 |
|
|
| 429 |
|
In the force calculation between {\tt moveA} and {\tt moveB} mentioned |
| 571 |
|
this problem, as well as useful corrective techniques, is presented in |
| 572 |
|
chapter \ref{chap:electrostatics}. |
| 573 |
|
|
| 574 |
< |
\section{Measuring Properties} |
| 574 |
> |
\subsection{Periodic Boundary Conditions} |
| 575 |
|
|
| 576 |
+ |
In typical molecular dynamics simulations there are no restrictions |
| 577 |
+ |
placed on the motion of particles outside of what the interparticle |
| 578 |
+ |
interactions dictate. This means that if a particle collides with |
| 579 |
+ |
other particles, it is free to move away from the site of the |
| 580 |
+ |
collision. If we consider the entire system as a collection of |
| 581 |
+ |
particles, they are not confined by walls of the ``simulation box'' |
| 582 |
+ |
and can freely move away from the other particles. With no boundary |
| 583 |
+ |
considerations, particles moving outside of the simulation box |
| 584 |
+ |
enter a vacuum. This is correct behavior for cluster simulations in a |
| 585 |
+ |
vacuum; however, if we want to simulate bulk or spatially infinite |
| 586 |
+ |
systems, we need to use periodic boundary conditions. |
| 587 |
+ |
|
| 588 |
+ |
\begin{figure} |
| 589 |
+ |
\centering |
| 590 |
+ |
\includegraphics[width=4.5in]{./figures/periodicImage.pdf} |
| 591 |
+ |
\caption{With periodic boundary conditions imposed, when particles |
| 592 |
+ |
move out of one side the simulation box, they wrap back in the |
| 593 |
+ |
opposite side. In this manner, a finite system of particles behaves as |
| 594 |
+ |
an infinite system.} |
| 595 |
+ |
\label{fig:periodicImage} |
| 596 |
+ |
\end{figure} |
| 597 |
+ |
In periodic boundary conditions, as a particle moves outside one wall |
| 598 |
+ |
of the simulation box, the coordinates are remapped such that the |
| 599 |
+ |
particle enters the opposing side of the box. This process is easy to |
| 600 |
+ |
visualize in two dimensions as shown in figure \ref{fig:periodicImage} |
| 601 |
+ |
and can occur in three dimensions, though it is not as easy to |
| 602 |
+ |
visualize. Remapping the actual coordinates of the particles can be |
| 603 |
+ |
problematic in that the we are restricting the distance a particle can |
| 604 |
+ |
move from it's point of origin to a diagonal of the simulation |
| 605 |
+ |
box. Thus, even though we are not confining the system with hard |
| 606 |
+ |
walls, we are confining the particle coordinates to a particular |
| 607 |
+ |
region in space. To avoid this ``soft'' confinement, it is common |
| 608 |
+ |
practice to allow the particle coordinates to expand in an |
| 609 |
+ |
unrestricted fashion while calculating interactions using a wrapped |
| 610 |
+ |
set of ``minimum image'' coordinates. These minimum image coordinates |
| 611 |
+ |
need not be stored because they are easily calculated on the fly when |
| 612 |
+ |
determining particle distances. |
| 613 |
+ |
|
| 614 |
+ |
\section{Calculating Properties} |
| 615 |
+ |
|
| 616 |
+ |
In order to use simulations to model experimental process and evaluate |
| 617 |
+ |
theories, we need to be able to extract properties from the |
| 618 |
+ |
results. In experiments, we can measure thermodynamic properties such |
| 619 |
+ |
as the pressure and free energy. In computer simulations, we can |
| 620 |
+ |
calculate properties from the motion and configuration of particles in |
| 621 |
+ |
the system and make connections between these properties and the |
| 622 |
+ |
experimental thermodynamic properties through statistical mechanics. |
| 623 |
+ |
|
| 624 |
+ |
The work presented in the later chapters use the canonical ($NVT$), |
| 625 |
+ |
isobaric-isothermal ($NPT$), and microcanonical ($NVE$) statistical |
| 626 |
+ |
mechanics ensembles. The different ensembles lend themselves to more |
| 627 |
+ |
effectively calculating specific properties. For instance, if we |
| 628 |
+ |
concerned ourselves with the calculation of dynamic properties, which |
| 629 |
+ |
are dependent upon the motion of the particles, it is better to choose |
| 630 |
+ |
an ensemble that does not add system motions to keep properties like |
| 631 |
+ |
the temperature or pressure constant. In this case, the $NVE$ ensemble |
| 632 |
+ |
would be the most appropriate choice. In chapter \ref{chap:ice}, we |
| 633 |
+ |
discuss calculating free energies, which are non-mechanical |
| 634 |
+ |
thermodynamic properties, and these calculations also depend on the |
| 635 |
+ |
chosen ensemble.\cite{Allen87} The Helmholtz free energy ($A$) depends |
| 636 |
+ |
on the $NVT$ partition function ($Q_{NVT}$), |
| 637 |
+ |
\begin{equation} |
| 638 |
+ |
A = -k_\textrm{B}T\ln Q_{NVT}, |
| 639 |
+ |
\end{equation} |
| 640 |
+ |
while the Gibbs free energy ($G$) depends on the $NPT$ partition |
| 641 |
+ |
function ($Q_{NPT}$), |
| 642 |
+ |
\begin{equation} |
| 643 |
+ |
G = -k_\textrm{B}T\ln Q_{NPT}. |
| 644 |
+ |
\end{equation} |
| 645 |
+ |
It is also useful to note that the conserved quantities of the $NVT$ |
| 646 |
+ |
and $NPT$ ensembles are related to the Helmholtz and Gibbs free |
| 647 |
+ |
energies respectively.\cite{Melchionna93} |
| 648 |
+ |
|
| 649 |
+ |
Integrating the equations of motion is a simple method to obtain a |
| 650 |
+ |
sequence of configurations that sample the chosen ensemble. For each |
| 651 |
+ |
of these configurations, we can calculate an instantaneous value for a |
| 652 |
+ |
chosen property like the density in the $NPT$ ensemble, where the |
| 653 |
+ |
volume is allowed to fluctuate. The density for the simulation is |
| 654 |
+ |
calculated from an average over the densities for the individual |
| 655 |
+ |
configurations. Being that the configurations from the integration |
| 656 |
+ |
process are related to one another by the time evolution of the |
| 657 |
+ |
interactions, this average is technically a time average. In |
| 658 |
+ |
calculating thermodynamic properties, we would actually prefer an |
| 659 |
+ |
ensemble average that is representative of all accessible states of |
| 660 |
+ |
the system. We can calculate thermodynamic properties from the time |
| 661 |
+ |
average by taking advantage of the ergodic hypothesis, which states |
| 662 |
+ |
that over a long period of time, the time average and the ensemble |
| 663 |
+ |
average are the same. |
| 664 |
+ |
|
| 665 |
+ |
In addition to the average, the fluctuations of that particular |
| 666 |
+ |
property can be determined via the standard deviation. Fluctuations |
| 667 |
+ |
are useful for measuring various thermodynamic properties in computer |
| 668 |
+ |
simulations. In section \ref{sec:t5peThermo}, we use fluctuations in |
| 669 |
+ |
propeties like the enthalpy and volume to calculate various |
| 670 |
+ |
thermodynamic properties, such as the constant pressure heat capacity |
| 671 |
+ |
and the isothermal compressibility. |
| 672 |
+ |
|
| 673 |
|
\section{Application of the Techniques} |
| 674 |
|
|
| 675 |
|
In the following chapters, the above techniques are all utilized in |
