| 154 |
|
system is said to be using conducting (or ``tin-foil'') boundary |
| 155 |
|
conditions, $\epsilon_{\rm S} = \infty$. Figure |
| 156 |
|
\ref{fig:ewaldTime} shows how the Ewald sum has been applied over |
| 157 |
< |
time. Initially, due to the small sizes of the systems that could be |
| 158 |
< |
feasibly simulated, the entire simulation box was replicated to |
| 159 |
< |
convergence. In more modern simulations, the simulation boxes have |
| 160 |
< |
grown large enough that a real-space cutoff could potentially give |
| 161 |
< |
convergent behavior. Indeed, it has often been observed that the |
| 162 |
< |
reciprocal-space portion of the Ewald sum can be small and rapidly |
| 163 |
< |
convergent compared to the real-space portion with the choice of small |
| 164 |
< |
$\alpha$.\cite{Karasawa89,Kolafa92} |
| 157 |
> |
time. Initially, due to the small system sizes that could be |
| 158 |
> |
simulated feasibly, the entire simulation box was replicated to |
| 159 |
> |
convergence. In more modern simulations, the systems have grown large |
| 160 |
> |
enough that a real-space cutoff could potentially give convergent |
| 161 |
> |
behavior. Indeed, it has been observed that with the choice of a |
| 162 |
> |
small $\alpha$, the reciprocal-space portion of the Ewald sum can be |
| 163 |
> |
rapidly convergent and small relative to the real-space |
| 164 |
> |
portion.\cite{Karasawa89,Kolafa92} |
| 165 |
|
|
| 166 |
|
\begin{figure} |
| 167 |
|
\centering |
| 168 |
< |
\includegraphics[width = \linewidth]{./ewaldProgression.pdf} |
| 169 |
< |
\caption{How the application of the Ewald summation has changed with |
| 170 |
< |
the increase in computer power. Initially, only small numbers of |
| 171 |
< |
particles could be studied, and the Ewald sum acted to replicate the |
| 172 |
< |
unit cell charge distribution out to convergence. Now, much larger |
| 173 |
< |
systems of charges are investigated with fixed distance cutoffs. The |
| 174 |
< |
calculated structure factor is used to sum out to great distance, and |
| 175 |
< |
a surrounding dielectric term is included.} |
| 168 |
> |
\includegraphics[width = \linewidth]{./ewaldProgression2.pdf} |
| 169 |
> |
\caption{The change in the application of the Ewald sum with |
| 170 |
> |
increasing computational power. Initially, only small systems could |
| 171 |
> |
be studied, and the Ewald sum replicated the simulation box to |
| 172 |
> |
convergence. Now, much larger systems of charges are investigated |
| 173 |
> |
with fixed-distance cutoffs.} |
| 174 |
|
\label{fig:ewaldTime} |
| 175 |
|
\end{figure} |
| 176 |
|
|
| 680 |
|
%(Fig. \ref{fig:argonSlice}). |
| 681 |
|
|
| 682 |
|
These procedures guaranteed us a set of representative configurations |
| 683 |
< |
from chemically-relevant systems sampled from an appropriate |
| 684 |
< |
ensemble. Force field parameters for the ions and Argon were taken |
| 683 |
> |
from chemically-relevant systems sampled from appropriate |
| 684 |
> |
ensembles. Force field parameters for the ions and Argon were taken |
| 685 |
|
from the force field utilized by {\sc oopse}.\cite{Meineke05} |
| 686 |
|
|
| 687 |
|
%\begin{figure} |
| 708 |
|
were utilized for the reaction field simulations. Additionally, we |
| 709 |
|
investigated the use of these cutoffs with the SP, SF, and pure |
| 710 |
|
cutoff. The SPME electrostatics were performed using the TINKER |
| 711 |
< |
implementation of SPME,\cite{Ponder87} while all other method |
| 712 |
< |
calculations were performed using the OOPSE molecular mechanics |
| 711 |
> |
implementation of SPME,\cite{Ponder87} while all other calculations |
| 712 |
> |
were performed using the {\sc oopse} molecular mechanics |
| 713 |
|
package.\cite{Meineke05} All other portions of the energy calculation |
| 714 |
|
(i.e. Lennard-Jones interactions) were handled in exactly the same |
| 715 |
|
manner across all systems and configurations. |
| 721 |
|
Typical molecular mechanics packages set this to a value dependent on |
| 722 |
|
the cutoff radius and a tolerance (typically less than $1 \times |
| 723 |
|
10^{-4}$ kcal/mol). Smaller tolerances are typically associated with |
| 724 |
< |
increased accuracy at the expense of increased time spent calculating |
| 725 |
< |
the reciprocal-space portion of the |
| 726 |
< |
summation.\cite{Perram88,Essmann95} The default TINKER tolerance of $1 |
| 727 |
< |
\times 10^{-8}$ kcal/mol was used in all SPME calculations, resulting |
| 728 |
< |
in Ewald Coefficients of 0.4200, 0.3119, and 0.2476 \AA$^{-1}$ for |
| 729 |
< |
cutoff radii of 9, 12, and 15 \AA\ respectively. |
| 724 |
> |
increasing accuracy at the expense of computational time spent on the |
| 725 |
> |
reciprocal-space portion of the summation.\cite{Perram88,Essmann95} |
| 726 |
> |
The default TINKER tolerance of $1 \times 10^{-8}$ kcal/mol was used |
| 727 |
> |
in all SPME calculations, resulting in Ewald coefficients of 0.4200, |
| 728 |
> |
0.3119, and 0.2476 \AA$^{-1}$ for cutoff radii of 9, 12, and 15 \AA\ |
| 729 |
> |
respectively. |
| 730 |
|
|
| 731 |
|
\section{Results and Discussion} |
| 732 |
|
|
| 768 |
|
readers can consult the accompanying supporting information for a |
| 769 |
|
comparison where all groups are neutral. |
| 770 |
|
|
| 771 |
< |
For the {\sc sp} method, inclusion of potential damping improves the |
| 772 |
< |
agreement with Ewald, and using an $\alpha$ of 0.2 \AA $^{-1}$ shows |
| 773 |
< |
an excellent correlation and quality of fit with the SPME results, |
| 774 |
< |
particularly with a cutoff radius greater than 12 |
| 771 |
> |
For the {\sc sp} method, inclusion of electrostatic damping improves |
| 772 |
> |
the agreement with Ewald, and using an $\alpha$ of 0.2 \AA $^{-1}$ |
| 773 |
> |
shows an excellent correlation and quality of fit with the SPME |
| 774 |
> |
results, particularly with a cutoff radius greater than 12 |
| 775 |
|
\AA . Use of a larger damping parameter is more helpful for the |
| 776 |
|
shortest cutoff shown, but it has a detrimental effect on simulations |
| 777 |
|
with larger cutoffs. |
| 778 |
|
|
| 779 |
< |
In the {\sc sf} sets, increasing damping results in progressively |
| 780 |
< |
worse correlation with Ewald. Overall, the undamped case is the best |
| 779 |
> |
In the {\sc sf} sets, increasing damping results in progressively {\it |
| 780 |
> |
worse} correlation with Ewald. Overall, the undamped case is the best |
| 781 |
|
performing set, as the correlation and quality of fits are |
| 782 |
|
consistently superior regardless of the cutoff distance. The undamped |
| 783 |
|
case is also less computationally demanding (because no evaluation of |
| 792 |
|
|
| 793 |
|
Evaluation of pairwise methods for use in Molecular Dynamics |
| 794 |
|
simulations requires consideration of effects on the forces and |
| 795 |
< |
torques. Investigation of the force and torque vector magnitudes |
| 796 |
< |
provides a measure of the strength of these values relative to SPME. |
| 797 |
< |
Figures \ref{fig:frcMag} and \ref{fig:trqMag} respectively show the |
| 798 |
< |
force and torque vector magnitude regression results for the |
| 801 |
< |
accumulated analysis over all the system types. |
| 795 |
> |
torques. Figures \ref{fig:frcMag} and \ref{fig:trqMag} show the |
| 796 |
> |
regression results for the force and torque vector magnitudes, |
| 797 |
> |
respectively. The data in these figures was generated from an |
| 798 |
> |
accumulation of the statistics from all of the system types. |
| 799 |
|
|
| 800 |
|
\begin{figure} |
| 801 |
|
\centering |
| 810 |
|
\label{fig:frcMag} |
| 811 |
|
\end{figure} |
| 812 |
|
|
| 813 |
+ |
Again, it is striking how well the Shifted Potential and Shifted Force |
| 814 |
+ |
methods are doing at reproducing the SPME forces. The undamped and |
| 815 |
+ |
weakly-damped {\sc sf} method gives the best agreement with Ewald. |
| 816 |
+ |
This is perhaps expected because this method explicitly incorporates a |
| 817 |
+ |
smooth transition in the forces at the cutoff radius as well as the |
| 818 |
+ |
neutralizing image charges. |
| 819 |
+ |
|
| 820 |
|
Figure \ref{fig:frcMag}, for the most part, parallels the results seen |
| 821 |
|
in the previous $\Delta E$ section. The unmodified cutoff results are |
| 822 |
|
poor, but using group based cutoffs and a switching function provides |
| 823 |
< |
a improvement much more significant than what was seen with $\Delta |
| 824 |
< |
E$. Looking at the {\sc sp} sets, the slope and $R^2$ |
| 825 |
< |
improve with the use of damping to an optimal result of 0.2 \AA |
| 826 |
< |
$^{-1}$ for the 12 and 15 \AA\ cutoffs. Further increases in damping, |
| 823 |
> |
an improvement much more significant than what was seen with $\Delta |
| 824 |
> |
E$. |
| 825 |
> |
|
| 826 |
> |
With moderate damping and a large enough cutoff radius, the {\sc sp} |
| 827 |
> |
method is generating usable forces. Further increases in damping, |
| 828 |
|
while beneficial for simulations with a cutoff radius of 9 \AA\ , is |
| 829 |
< |
detrimental to simulations with larger cutoff radii. The undamped |
| 830 |
< |
{\sc sf} method gives forces in line with those obtained using |
| 831 |
< |
SPME, and use of a damping function results in minor improvement. The |
| 827 |
< |
reaction field results are surprisingly good, considering the poor |
| 829 |
> |
detrimental to simulations with larger cutoff radii. |
| 830 |
> |
|
| 831 |
> |
The reaction field results are surprisingly good, considering the poor |
| 832 |
|
quality of the fits for the $\Delta E$ results. There is still a |
| 833 |
< |
considerable degree of scatter in the data, but it correlates well in |
| 834 |
< |
general. To be fair, we again note that the reaction field |
| 835 |
< |
calculations do not encompass NaCl crystal and melt systems, so these |
| 833 |
> |
considerable degree of scatter in the data, but the forces correlate |
| 834 |
> |
well with the Ewald forces in general. We note that the reaction |
| 835 |
> |
field calculations do not include the pure NaCl systems, so these |
| 836 |
|
results are partly biased towards conditions in which the method |
| 837 |
|
performs more favorably. |
| 838 |
|
|
| 849 |
|
\label{fig:trqMag} |
| 850 |
|
\end{figure} |
| 851 |
|
|
| 852 |
< |
To evaluate the torque vector magnitudes, the data set from which |
| 853 |
< |
values are drawn is limited to rigid molecules in the systems |
| 854 |
< |
(i.e. water molecules). In spite of this smaller sampling pool, the |
| 851 |
< |
torque vector magnitude results in figure \ref{fig:trqMag} are still |
| 852 |
< |
similar to those seen for the forces; however, they more clearly show |
| 853 |
< |
the improved behavior that comes with increasing the cutoff radius. |
| 854 |
< |
Moderate damping is beneficial to the {\sc sp} and helpful |
| 855 |
< |
yet possibly unnecessary with the {\sc sf} method, and they also |
| 856 |
< |
show that over-damping adversely effects all cutoff radii rather than |
| 857 |
< |
showing an improvement for systems with short cutoffs. The reaction |
| 858 |
< |
field method performs well when calculating the torques, better than |
| 859 |
< |
the Shifted Force method over this limited data set. |
| 852 |
> |
Molecular torques were only available from the systems which contained |
| 853 |
> |
rigid molecules (i.e. the systems containing water). The data in |
| 854 |
> |
fig. \ref{fig:trqMag} is taken from this smaller sampling pool. |
| 855 |
|
|
| 856 |
+ |
Torques appear to be much more sensitive to charges at a longer |
| 857 |
+ |
distance. The striking feature in comparing the new electrostatic |
| 858 |
+ |
methods with SPME is how much the agreement improves with increasing |
| 859 |
+ |
cutoff radius. Again, the weakly damped and undamped {\sc sf} method |
| 860 |
+ |
appears to be reproducing the SPME torques most accurately. |
| 861 |
+ |
|
| 862 |
+ |
Water molecules are dipolar, and the reaction field method reproduces |
| 863 |
+ |
the effect of the surrounding polarized medium on each of the |
| 864 |
+ |
molecular bodies. Therefore it is not surprising that reaction field |
| 865 |
+ |
performs best of all of the methods on molecular torques. |
| 866 |
+ |
|
| 867 |
|
\subsection{Directionality of the Force and Torque Vectors} |
| 868 |
|
|
| 869 |
< |
Having force and torque vectors with magnitudes that are well |
| 870 |
< |
correlated to SPME is good, but if they are not pointing in the proper |
| 871 |
< |
direction the results will be incorrect. These vector directions were |
| 872 |
< |
investigated through measurement of the angle formed between them and |
| 873 |
< |
those from SPME. The results (Fig. \ref{fig:frcTrqAng}) are compared |
| 874 |
< |
through the variance ($\sigma^2$) of the Gaussian fits of the angle |
| 875 |
< |
error distributions of the combined set over all system types. |
| 869 |
> |
It is clearly important that a new electrostatic method can reproduce |
| 870 |
> |
the magnitudes of the force and torque vectors obtained via the Ewald |
| 871 |
> |
sum. However, the {\it directionality} of these vectors will also be |
| 872 |
> |
vital in calculating dynamical quantities accurately. Force and |
| 873 |
> |
torque directionalities were investigated by measuring the angles |
| 874 |
> |
formed between these vectors and the same vectors calculated using |
| 875 |
> |
SPME. The results (Fig. \ref{fig:frcTrqAng}) are compared through the |
| 876 |
> |
variance ($\sigma^2$) of the Gaussian fits of the angle error |
| 877 |
> |
distributions of the combined set over all system types. |
| 878 |
|
|
| 879 |
|
\begin{figure} |
| 880 |
|
\centering |
| 881 |
|
\includegraphics[width=5.5in]{./frcTrqAngplot.pdf} |
| 882 |
< |
\caption{Statistical analysis of the quality of the Gaussian fit of |
| 883 |
< |
the force and torque vector angular distributions for a given |
| 884 |
< |
electrostatic method compared with the reference Ewald sum. Results |
| 885 |
< |
with a variance ($\sigma^2$) equal to zero (dashed line) indicate |
| 886 |
< |
force and torque directions indistinguishable from those obtained |
| 887 |
< |
using SPME. Different values of the cutoff radius are indicated with |
| 888 |
< |
different symbols (9\AA\ = circles, 12\AA\ = squares, and 15\AA\ = |
| 889 |
< |
inverted triangles).} |
| 882 |
> |
\caption{Statistical analysis of the width of the angular distribution |
| 883 |
> |
that the force and torque vectors from a given electrostatic method |
| 884 |
> |
make with their counterparts obtained using the reference Ewald sum. |
| 885 |
> |
Results with a variance ($\sigma^2$) equal to zero (dashed line) |
| 886 |
> |
indicate force and torque directions indistinguishable from those |
| 887 |
> |
obtained using SPME. Different values of the cutoff radius are |
| 888 |
> |
indicated with different symbols (9\AA\ = circles, 12\AA\ = squares, |
| 889 |
> |
and 15\AA\ = inverted triangles).} |
| 890 |
|
\label{fig:frcTrqAng} |
| 891 |
|
\end{figure} |
| 892 |
|
|
| 893 |
|
Both the force and torque $\sigma^2$ results from the analysis of the |
| 894 |
|
total accumulated system data are tabulated in figure |
| 895 |
< |
\ref{fig:frcTrqAng}. All of the sets, aside from the over-damped case |
| 896 |
< |
show the improvement afforded by choosing a longer simulation cutoff. |
| 897 |
< |
Increasing the cutoff from 9 to 12 \AA\ typically results in a halving |
| 898 |
< |
of the distribution widths, with a similar improvement going from 12 |
| 899 |
< |
to 15 \AA . The undamped {\sc sf}, Group Based Cutoff, and |
| 892 |
< |
Reaction Field methods all do equivalently well at capturing the |
| 893 |
< |
direction of both the force and torque vectors. Using damping |
| 894 |
< |
improves the angular behavior significantly for the {\sc sp} |
| 895 |
< |
and moderately for the {\sc sf} methods. Increasing the damping |
| 896 |
< |
too far is destructive for both methods, particularly to the torque |
| 897 |
< |
vectors. Again it is important to recognize that the force vectors |
| 898 |
< |
cover all particles in the systems, while torque vectors are only |
| 899 |
< |
available for neutral molecular groups. Damping appears to have a |
| 900 |
< |
more beneficial effect on non-neutral bodies, and this observation is |
| 901 |
< |
investigated further in the accompanying supporting information. |
| 895 |
> |
\ref{fig:frcTrqAng}. Here it is clear that the Shifted Potential ({\sc |
| 896 |
> |
sp}) method would be essentially unusable for molecular dynamics until |
| 897 |
> |
the damping function is added. The Shifted Force ({\sc sf}) method, |
| 898 |
> |
however, is generating force and torque vectors which are within a few |
| 899 |
> |
degrees of the Ewald results even with weak (or no) damping. |
| 900 |
|
|
| 901 |
+ |
All of the sets (aside from the over-damped case) show the improvement |
| 902 |
+ |
afforded by choosing a larger cutoff radius. Increasing the cutoff |
| 903 |
+ |
from 9 to 12 \AA\ typically results in a halving of the width of the |
| 904 |
+ |
distribution, with a similar improvement going from 12 to 15 |
| 905 |
+ |
\AA . |
| 906 |
+ |
|
| 907 |
+ |
The undamped {\sc sf}, group-based cutoff, and reaction field methods |
| 908 |
+ |
all do equivalently well at capturing the direction of both the force |
| 909 |
+ |
and torque vectors. Using damping improves the angular behavior |
| 910 |
+ |
significantly for the {\sc sp} and moderately for the {\sc sf} |
| 911 |
+ |
methods. Overdamping is detrimental to both methods. Again it is |
| 912 |
+ |
important to recognize that the force vectors cover all particles in |
| 913 |
+ |
the systems, while torque vectors are only available for neutral |
| 914 |
+ |
molecular groups. Damping appears to have a more beneficial effect on |
| 915 |
+ |
charged bodies, and this observation is investigated further in the |
| 916 |
+ |
accompanying supporting information. |
| 917 |
+ |
|
| 918 |
+ |
Although not discussed previously, group based cutoffs can be applied |
| 919 |
+ |
to both the {\sc sp} and {\sc sf} methods. Use of a switching |
| 920 |
+ |
function corrects for the discontinuities that arise when atoms of the |
| 921 |
+ |
two groups exit the cutoff radius before the group centers leave each |
| 922 |
+ |
other's cutoff. Though there are no significant benefits or drawbacks |
| 923 |
+ |
observed in $\Delta E$ and vector magnitude results when doing this, |
| 924 |
+ |
there is a measurable improvement in the vector angle results. Table |
| 925 |
+ |
\ref{tab:groupAngle} shows the angular variance values obtained using |
| 926 |
+ |
group based cutoffs and a switching function alongside the results |
| 927 |
+ |
seen in figure \ref{fig:frcTrqAng}. The {\sc sp} shows much narrower |
| 928 |
+ |
angular distributions for both the force and torque vectors when using |
| 929 |
+ |
an $\alpha$ of 0.2 \AA$^{-1}$ or less, while {\sc sf} shows |
| 930 |
+ |
improvements in the undamped and lightly damped cases. Thus, by |
| 931 |
+ |
calculating the electrostatic interactions in terms of molecular pairs |
| 932 |
+ |
rather than atomic pairs, the direction of the force and torque |
| 933 |
+ |
vectors can be determined more accurately. |
| 934 |
+ |
|
| 935 |
|
\begin{table}[htbp] |
| 936 |
|
\centering |
| 937 |
|
\caption{Variance ($\sigma^2$) of the force (top set) and torque |
| 966 |
|
\label{tab:groupAngle} |
| 967 |
|
\end{table} |
| 968 |
|
|
| 937 |
– |
Although not discussed previously, group based cutoffs can be applied |
| 938 |
– |
to both the {\sc sp} and {\sc sf} methods. Use off a |
| 939 |
– |
switching function corrects for the discontinuities that arise when |
| 940 |
– |
atoms of a group exit the cutoff before the group's center of mass. |
| 941 |
– |
Though there are no significant benefit or drawbacks observed in |
| 942 |
– |
$\Delta E$ and vector magnitude results when doing this, there is a |
| 943 |
– |
measurable improvement in the vector angle results. Table |
| 944 |
– |
\ref{tab:groupAngle} shows the angular variance values obtained using |
| 945 |
– |
group based cutoffs and a switching function alongside the standard |
| 946 |
– |
results seen in figure \ref{fig:frcTrqAng} for comparison purposes. |
| 947 |
– |
The {\sc sp} shows much narrower angular distributions for |
| 948 |
– |
both the force and torque vectors when using an $\alpha$ of 0.2 |
| 949 |
– |
\AA$^{-1}$ or less, while {\sc sf} shows improvements in the |
| 950 |
– |
undamped and lightly damped cases. Thus, by calculating the |
| 951 |
– |
electrostatic interactions in terms of molecular pairs rather than |
| 952 |
– |
atomic pairs, the direction of the force and torque vectors are |
| 953 |
– |
determined more accurately. |
| 954 |
– |
|
| 969 |
|
One additional trend to recognize in table \ref{tab:groupAngle} is |
| 970 |
< |
that the $\sigma^2$ values for both {\sc sp} and |
| 971 |
< |
{\sc sf} converge as $\alpha$ increases, something that is easier |
| 972 |
< |
to see when using group based cutoffs. Looking back on figures |
| 973 |
< |
\ref{fig:delE}, \ref{fig:frcMag}, and \ref{fig:trqMag}, show this |
| 974 |
< |
behavior clearly at large $\alpha$ and cutoff values. The reason for |
| 975 |
< |
this is that the complimentary error function inserted into the |
| 976 |
< |
potential weakens the electrostatic interaction as $\alpha$ increases. |
| 977 |
< |
Thus, at larger values of $\alpha$, both the summation method types |
| 978 |
< |
progress toward non-interacting functions, so care is required in |
| 979 |
< |
choosing large damping functions lest one generate an undesirable loss |
| 980 |
< |
in the pair interaction. Kast \textit{et al.} developed a method for |
| 981 |
< |
choosing appropriate $\alpha$ values for these types of electrostatic |
| 982 |
< |
summation methods by fitting to $g(r)$ data, and their methods |
| 983 |
< |
indicate optimal values of 0.34, 0.25, and 0.16 \AA$^{-1}$ for cutoff |
| 984 |
< |
values of 9, 12, and 15 \AA\ respectively.\cite{Kast03} These appear |
| 985 |
< |
to be reasonable choices to obtain proper MC behavior |
| 986 |
< |
(Fig. \ref{fig:delE}); however, based on these findings, choices this |
| 987 |
< |
high would introduce error in the molecular torques, particularly for |
| 988 |
< |
the shorter cutoffs. Based on the above findings, empirical damping |
| 975 |
< |
up to 0.2 \AA$^{-1}$ proves to be beneficial, but damping is arguably |
| 976 |
< |
unnecessary when using the {\sc sf} method. |
| 970 |
> |
that the $\sigma^2$ values for both {\sc sp} and {\sc sf} converge as |
| 971 |
> |
$\alpha$ increases, something that is easier to see when using group |
| 972 |
> |
based cutoffs. The reason for this is that the complimentary error |
| 973 |
> |
function inserted into the potential weakens the electrostatic |
| 974 |
> |
interaction as $\alpha$ increases. Thus, at larger values of |
| 975 |
> |
$\alpha$, both summation methods progress toward non-interacting |
| 976 |
> |
functions, so care is required in choosing large damping functions |
| 977 |
> |
lest one generate an undesirable loss in the pair interaction. Kast |
| 978 |
> |
\textit{et al.} developed a method for choosing appropriate $\alpha$ |
| 979 |
> |
values for these types of electrostatic summation methods by fitting |
| 980 |
> |
to $g(r)$ data, and their methods indicate optimal values of 0.34, |
| 981 |
> |
0.25, and 0.16 \AA$^{-1}$ for cutoff values of 9, 12, and 15 \AA\ |
| 982 |
> |
respectively.\cite{Kast03} These appear to be reasonable choices to |
| 983 |
> |
obtain proper MC behavior (Fig. \ref{fig:delE}); however, based on |
| 984 |
> |
these findings, choices this high would introduce error in the |
| 985 |
> |
molecular torques, particularly for the shorter cutoffs. Based on the |
| 986 |
> |
above findings, empirical damping up to 0.2 \AA$^{-1}$ proves to be |
| 987 |
> |
beneficial, but damping may be unnecessary when using the {\sc sf} |
| 988 |
> |
method. |
| 989 |
|
|
| 990 |
|
\subsection{Short-Time Dynamics: Velocity Autocorrelation Functions of NaCl Crystals} |
| 991 |
|
|
| 992 |
< |
In the previous studies using a {\sc sf} variant of the damped |
| 993 |
< |
Wolf coulomb potential, the structure and dynamics of water were |
| 994 |
< |
investigated rather extensively.\cite{Zahn02,Kast03} Their results |
| 995 |
< |
indicated that the damped {\sc sf} method results in properties |
| 996 |
< |
very similar to those obtained when using the Ewald summation. |
| 997 |
< |
Considering the statistical results shown above, the good performance |
| 998 |
< |
of this method is not that surprising. Rather than consider the same |
| 999 |
< |
systems and simply recapitulate their results, we decided to look at |
| 1000 |
< |
the solid state dynamical behavior obtained using the best performing |
| 1001 |
< |
summation methods from the above results. |
| 992 |
> |
Zahn {\it et al.} investigated the structure and dynamics of water |
| 993 |
> |
using eqs. (\ref{eq:ZahnPot}) and |
| 994 |
> |
(\ref{eq:WolfForces}).\cite{Zahn02,Kast03} Their results indicated |
| 995 |
> |
that a method similar (but not identical with) the damped {\sc sf} |
| 996 |
> |
method resulted in properties very similar to those obtained when |
| 997 |
> |
using the Ewald summation. The properties they studied (pair |
| 998 |
> |
distribution functions, diffusion constants, and velocity and |
| 999 |
> |
orientational correlation functions) may not be particularly sensitive |
| 1000 |
> |
to the long-range and collective behavior that governs the |
| 1001 |
> |
low-frequency behavior in crystalline systems. |
| 1002 |
|
|
| 1003 |
+ |
We are using two separate measures to probe the effects of these |
| 1004 |
+ |
alternative electrostatic methods on the dynamics in crystalline |
| 1005 |
+ |
materials. For short- and intermediate-time dynamics, we are |
| 1006 |
+ |
computing the velocity autocorrelation function, and for long-time |
| 1007 |
+ |
and large length-scale collective motions, we are looking at the |
| 1008 |
+ |
low-frequency portion of the power spectrum. |
| 1009 |
+ |
|
| 1010 |
|
\begin{figure} |
| 1011 |
|
\centering |
| 1012 |
|
\includegraphics[width = \linewidth]{./vCorrPlot.pdf} |
| 1013 |
|
\caption{Velocity auto-correlation functions of NaCl crystals at |
| 1014 |
< |
1000 K while using SPME, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and |
| 1015 |
< |
{\sc sp} ($\alpha$ = 0.2). The inset is a magnification of the first |
| 1016 |
< |
trough. The times to first collision are nearly identical, but the |
| 1017 |
< |
differences can be seen in the peaks and troughs, where the undamped |
| 1018 |
< |
to weakly damped methods are stiffer than the moderately damped and |
| 1019 |
< |
SPME methods.} |
| 1014 |
> |
1000 K using SPME, {\sc sf} ($\alpha$ = 0.0, 0.1, \& 0.2), and {\sc |
| 1015 |
> |
sp} ($\alpha$ = 0.2). The inset is a magnification of the area around |
| 1016 |
> |
the first minimum. The times to first collision are nearly identical, |
| 1017 |
> |
but differences can be seen in the peaks and troughs, where the |
| 1018 |
> |
undamped and weakly damped methods are stiffer than the moderately |
| 1019 |
> |
damped and SPME methods.} |
| 1020 |
|
\label{fig:vCorrPlot} |
| 1021 |
|
\end{figure} |
| 1022 |
|
|
