Homec4science
Diffusion lammps (lammm-devel) Browse

lammps/doc/neb.html
60334f9b7221lammm-devel

Last Change
Raw File

neb.html
View Options

<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>neb command &mdash; LAMMPS 15 May 2015 version documentation</title>
<link rel="stylesheet" href="_static/css/theme.css" type="text/css" />
<link rel="stylesheet" href="_static/sphinxcontrib-images/LightBox2/lightbox2/css/lightbox.css" type="text/css" />
<link rel="top" title="LAMMPS 15 May 2015 version documentation" href="index.html"/>
<script src="_static/js/modernizr.min.js"></script>
</head>
<body class="wy-body-for-nav" role="document">
<div class="wy-grid-for-nav">
<nav data-toggle="wy-nav-shift" class="wy-nav-side">
<div class="wy-side-nav-search">
<a href="Manual.html" class="icon icon-home"> LAMMPS
</a>
<div role="search">
<form id="rtd-search-form" class="wy-form" action="search.html" method="get">
<input type="text" name="q" placeholder="Search docs" />
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
</div>
<div class="wy-menu wy-menu-vertical" data-spy="affix" role="navigation" aria-label="main navigation">
<ul>
<li class="toctree-l1"><a class="reference internal" href="Section_intro.html">1. Introduction</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_start.html">2. Getting Started</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_commands.html">3. Commands</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_packages.html">4. Packages</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_accelerate.html">5. Accelerating LAMMPS performance</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_howto.html">6. How-to discussions</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_example.html">7. Example problems</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_perf.html">8. Performance &amp; scalability</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_tools.html">9. Additional tools</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_modify.html">10. Modifying &amp; extending LAMMPS</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_python.html">11. Python interface to LAMMPS</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_errors.html">12. Errors</a></li>
<li class="toctree-l1"><a class="reference internal" href="Section_history.html">13. Future and history</a></li>
</ul>
</div>
&nbsp;
</nav>
<section data-toggle="wy-nav-shift" class="wy-nav-content-wrap">
<nav class="wy-nav-top" role="navigation" aria-label="top navigation">
<i data-toggle="wy-nav-top" class="fa fa-bars"></i>
<a href="Manual.html">LAMMPS</a>
</nav>
<div class="wy-nav-content">
<div class="rst-content">
<div role="navigation" aria-label="breadcrumbs navigation">
<ul class="wy-breadcrumbs">
<li><a href="Manual.html">Docs</a> &raquo;</li>
<li>neb command</li>
<li class="wy-breadcrumbs-aside">
<a href="http://lammps.sandia.gov">Homepage</a>
<a href="Section_commands.html#comm">Commands</a>
</li>
</ul>
<hr/>
</div>
<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
<div itemprop="articleBody">
<div class="section" id="neb-command">
<span id="index-0"></span><h1>neb command<a class="headerlink" href="#neb-command" title="Permalink to this headline"></a></h1>
<div class="section" id="syntax">
<h2>Syntax<a class="headerlink" href="#syntax" title="Permalink to this headline"></a></h2>
<div class="highlight-python"><div class="highlight"><pre>neb etol ftol N1 N2 Nevery file-style arg
</pre></div>
</div>
<ul class="simple">
<li>etol = stopping tolerance for energy (energy units)</li>
<li>ftol = stopping tolerance for force (force units)</li>
<li>N1 = max # of iterations (timesteps) to run initial NEB</li>
<li>N2 = max # of iterations (timesteps) to run barrier-climbing NEB</li>
<li>Nevery = print replica energies and reaction coordinates every this many timesteps</li>
<li>file-style= <em>final</em> or <em>each</em> or <em>none</em></li>
</ul>
<pre class="literal-block">
<em>final</em> arg = filename
filename = file with initial coords for final replica
coords for intermediate replicas are linearly interpolated between first and last replica
<em>each</em> arg = filename
filename = unique filename for each replica (except first) with its initial coords
<em>none</em> arg = no argument
all replicas assumed to already have their initial coords
</pre>
</div>
<div class="section" id="examples">
<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline"></a></h2>
<div class="highlight-python"><div class="highlight"><pre>neb 0.1 0.0 1000 500 50 final coords.final
neb 0.0 0.001 1000 500 50 each coords.initial.$i
neb 0.0 0.001 1000 500 50 none
</pre></div>
</div>
</div>
<div class="section" id="description">
<h2>Description<a class="headerlink" href="#description" title="Permalink to this headline"></a></h2>
<p>Perform a nudged elastic band (NEB) calculation using multiple
replicas of a system. Two or more replicas must be used; the first
and last are the end points of the transition path.</p>
<p>NEB is a method for finding both the atomic configurations and height
of the energy barrier associated with a transition state, e.g. for an
atom to perform a diffusive hop from one energy basin to another in a
coordinated fashion with its neighbors. The implementation in LAMMPS
follows the discussion in these 3 papers: <a class="reference internal" href="#henkelman1"><span>(Henkelman1)</span></a>,
<a class="reference internal" href="#henkelman2"><span>(Henkelman2)</span></a>, and <a class="reference internal" href="#nakano"><span>(Nakano)</span></a>.</p>
<p>Each replica runs on a partition of one or more processors. Processor
partitions are defined at run-time using the -partition command-line
switch; see <a class="reference internal" href="Section_start.html#start-7"><span>Section_start 7</span></a> of the
manual. Note that if you have MPI installed, you can run a
multi-replica simulation with more replicas (partitions) than you have
physical processors, e.g you can run a 10-replica simulation on just
one or two processors. You will simply not get the performance
speed-up you would see with one or more physical processors per
replica. See <a class="reference internal" href="Section_howto.html#howto-5"><span>this section</span></a> of the manual
for further discussion.</p>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">The current NEB implementation in LAMMPS only allows
there to be one processor per replica.</p>
</div>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">As explained below, a NEB calculation perfoms a damped
dynamics minimization across all the replicas. The mimimizer uses
whatever timestep you have defined in your input script, via the
<a class="reference internal" href="timestep.html"><em>timestep</em></a> command. Often NEB will converge more
quickly if you use a timestep about 10x larger than you would normally
use for dynamics simulations.</p>
</div>
<p>When a NEB calculation is performed, it is assumed that each replica
is running the same system, though LAMMPS does not check for this.
I.e. the simulation domain, the number of atoms, the interaction
potentials, and the starting configuration when the neb command is
issued should be the same for every replica.</p>
<p>In a NEB calculation each atom in a replica is connected to the same
atom in adjacent replicas by springs, which induce inter-replica
forces. These forces are imposed by the <a class="reference internal" href="fix_neb.html"><em>fix neb</em></a>
command, which must be used in conjunction with the neb command. The
group used to define the fix neb command defines the NEB atoms which
are the only ones that inter-replica springs are applied to. If the
group does not include all atoms, then non-NEB atoms have no
inter-replica springs and the forces they feel and their motion is
computed in the usual way due only to other atoms within their
replica. Conceptually, the non-NEB atoms provide a background force
field for the NEB atoms. They can be allowed to move during the NEB
minimiation procedure (which will typically induce different
coordinates for non-NEB atoms in different replicas), or held fixed
using other LAMMPS commands such as <a class="reference external" href="fix_setforce">fix setforce</a>. Note
that the <a class="reference internal" href="partition.html"><em>partition</em></a> command can be used to invoke a
command on a subset of the replicas, e.g. if you wish to hold NEB or
non-NEB atoms fixed in only the end-point replicas.</p>
<p>The initial atomic configuration for each of the replicas can be
specified in different manners via the <em>file-style</em> setting, as
discussed below. Only atoms whose initial coordinates should differ
from the current configuration need be specified.</p>
<p>Conceptually, the initial configuration for the first replica should
be a state with all the atoms (NEB and non-NEB) having coordinates on
one side of the energy barrier. A perfect energy minimum is not
required, since atoms in the first replica experience no spring forces
from the 2nd replica. Thus the damped dynamics minimizaiton will
drive the first replica to an energy minimum if it is not already
there. However, you will typically get better convergence if the
initial state is already at a minimum. For example, for a system with
a free surface, the surface should be fully relaxed before attempting
a NEB calculation.</p>
<p>Likewise, the initial configuration of the final replica should be a
state with all the atoms (NEB and non-NEB) on the other side of the
energy barrier. Again, a perfect energy minimum is not required,
since the atoms in the last replica also experience no spring forces
from the next-to-last replica, and thus the damped dynamics
minimization will drive it to an energy minimum.</p>
<p>As explained below, the initial configurations of intermediate
replicas can be atomic coordinates interpolated in a linear fashion
between the first and last replicas. This is often adequate state for
simple transitions. For more complex transitions, it may lead to slow
convergence or even bad results if the minimum energy path (MEP, see
below) of states over the barrier cannot be correctly converged to
from such an initial configuration. In this case, you will want to
generate initial states for the intermediate replicas that are
geometrically closer to the MEP and read them in.</p>
<hr class="docutils" />
<p>For a <em>file-style</em> setting of <em>final</em>, a filename is specified which
contains atomic coordinates for zero or more atoms, in the format
described below. For each atom that appears in the file, the new
coordinates are assigned to that atom in the final replica. Each
intermediate replica also assigns a new position to that atom in an
interpolated manner. This is done by using the current position of
the atom as the starting point and the read-in position as the final
point. The distance between them is calculated, and the new position
is assigned to be a fraction of the distance. E.g. if there are 10
replicas, the 2nd replica will assign a position that is 10% of the
distance along a line between the starting and final point, and the
9th replica will assign a position that is 90% of the distance along
the line. Note that this procedure to produce consistent coordinates
across all the replicas, the current coordinates need to be the same
in all replicas. LAMMPS does not check for this, but invalid initial
configurations will likely result if it is not the case.</p>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">The &#8220;distance&#8221; between the starting and final point is
calculated in a minimum-image sense for a periodic simulation box.
This means that if the two positions are on opposite sides of a box
(periodic in that dimension), the distance between them will be small,
because the periodic image of one of the atoms is close to the other.
Similarly, even if the assigned position resulting from the
interpolation is outside the periodic box, the atom will be wrapped
back into the box when the NEB calculation begins.</p>
</div>
<p>For a <em>file-style</em> setting of <em>each</em>, a filename is specified which is
assumed to be unique to each replica. This can be done by
using a variable in the filename, e.g.</p>
<div class="highlight-python"><div class="highlight"><pre>variable i equal part
neb 0.0 0.001 1000 500 50 each coords.initial.$i
</pre></div>
</div>
<p>which in this case will substitute the partition ID (0 to N-1) for the
variable I, which is also effectively the replica ID. See the
<a class="reference internal" href="variable.html"><em>variable</em></a> command for other options, such as using
world-, universe-, or uloop-style variables.</p>
<p>Each replica (except the first replica) will read its file, formatted
as described below, and for any atom that appears in the file, assign
the specified coordinates to its atom. The various files do not need
to contain the same set of atoms.</p>
<p>For a <em>file-style</em> setting of <em>none</em>, no filename is specified. Each
replica is assumed to already be in its initial configuration at the
time the neb command is issued. This allows each replica to define
its own configuration by reading a replica-specific data or restart or
dump file, via the <a class="reference internal" href="read_data.html"><em>read_data</em></a>,
<a class="reference internal" href="read_restart.html"><em>read_restart</em></a>, or <a class="reference internal" href="read_dump.html"><em>read_dump</em></a>
commands. The replica-specific names of these files can be specified
as in the discussion above for the <em>each</em> file-style. Also see the
section below for how a NEB calculation can produce restart files, so
that a long calculation can be restarted if needed.</p>
<div class="admonition warning">
<p class="first admonition-title">Warning</p>
<p class="last">None of the <em>file-style</em> settings change the initial
configuration of any atom in the first replica. The first replica
must thus be in the correct initial configuration at the time the neb
command is issued.</p>
</div>
<hr class="docutils" />
<p>A NEB calculation proceeds in two stages, each of which is a
minimization procedure, performed via damped dynamics. To enable
this, you must first define a damped dynamics
<a class="reference internal" href="min_style.html"><em>min_style</em></a>, such as <em>quickmin</em> or <em>fire</em>. The <em>cg</em>,
<em>sd</em>, and <em>hftn</em> styles cannot be used, since they perform iterative
line searches in their inner loop, which cannot be easily synchronized
across multiple replicas.</p>
<p>The minimizer tolerances for energy and force are set by <em>etol</em> and
<em>ftol</em>, the same as for the <a class="reference internal" href="minimize.html"><em>minimize</em></a> command.</p>
<p>A non-zero <em>etol</em> means that the NEB calculation will terminate if the
energy criterion is met by every replica. The energies being compared
to <em>etol</em> do not include any contribution from the inter-replica
forces, since these are non-conservative. A non-zero <em>ftol</em> means
that the NEB calculation will terminate if the force criterion is met
by every replica. The forces being compared to <em>ftol</em> include the
inter-replica forces between an atom and its images in adjacent
replicas.</p>
<p>The maximum number of iterations in each stage is set by <em>N1</em> and
<em>N2</em>. These are effectively timestep counts since each iteration of
damped dynamics is like a single timestep in a dynamics
<a class="reference internal" href="run.html"><em>run</em></a>. During both stages, the potential energy of each
replica and its normalized distance along the reaction path (reaction
coordinate RD) will be printed to the screen and log file every
<em>Nevery</em> timesteps. The RD is 0 and 1 for the first and last replica.
For intermediate replicas, it is the cumulative distance (normalized
by the total cumulative distance) between adjacent replicas, where
&#8220;distance&#8221; is defined as the length of the 3N-vector of differences in
atomic coordinates, where N is the number of NEB atoms involved in the
transition. These outputs allow you to monitor NEB&#8217;s progress in
finding a good energy barrier. <em>N1</em> and <em>N2</em> must both be multiples
of <em>Nevery</em>.</p>
<p>In the first stage of NEB, the set of replicas should converge toward
the minimum energy path (MEP) of conformational states that transition
over the barrier. The MEP for a barrier is defined as a sequence of
3N-dimensional states that cross the barrier at its saddle point, each
of which has a potential energy gradient parallel to the MEP itself.
The replica states will also be roughly equally spaced along the MEP
due to the inter-replica spring force added by the <a class="reference internal" href="fix_neb.html"><em>fix neb</em></a> command.</p>
<p>In the second stage of NEB, the replica with the highest energy
is selected and the inter-replica forces on it are converted to a
force that drives its atom coordinates to the top or saddle point of
the barrier, via the barrier-climbing calculation described in
<span class="xref std std-ref">(Henkelman2)</span>. As before, the other replicas rearrange
themselves along the MEP so as to be roughly equally spaced.</p>
<p>When both stages are complete, if the NEB calculation was successful,
one of the replicas should be an atomic configuration at the top or
saddle point of the barrier, the potential energies for the set of
replicas should represent the energy profile of the barrier along the
MEP, and the configurations of the replicas should be a sequence of
configurations along the MEP.</p>
<hr class="docutils" />
<p>A few other settings in your input script are required or advised to
perform a NEB calculation. See the IMPORTANT NOTE about the choice of
timestep at the beginning of this doc page.</p>
<p>An atom map must be defined which it is not by default for <a class="reference internal" href="atom_style.html"><em>atom_style atomic</em></a> problems. The <a class="reference internal" href="atom_modify.html"><em>atom_modify map</em></a> command can be used to do this.</p>
<p>The &#8220;atom_modify sort 0 0.0&#8221; command should be used to turn off atom
sorting.</p>
<div class="admonition note">
<p class="first admonition-title">Note</p>
<p class="last">This sorting restriction will be removed in a future version of
NEB in LAMMPS.</p>
</div>
<p>The minimizers in LAMMPS operate on all atoms in your system, even
non-NEB atoms, as defined above. To prevent non-NEB atoms from moving
during the minimization, you should use the <a class="reference internal" href="fix_setforce.html"><em>fix setforce</em></a> command to set the force on each of those
atoms to 0.0. This is not required, and may not even be desired in
some cases, but if those atoms move too far (e.g. because the initial
state of your system was not well-minimized), it can cause problems
for the NEB procedure.</p>
<p>The damped dynamics <a class="reference internal" href="min_style.html"><em>minimizers</em></a>, such as <em>quickmin</em>
and <em>fire</em>), adjust the position and velocity of the atoms via an
Euler integration step. Thus you must define an appropriate
<a class="reference internal" href="timestep.html"><em>timestep</em></a> to use with NEB. As mentioned above, NEB
will often converge more quickly if you use a timestep about 10x
larger than you would normally use for dynamics simulations.</p>
<hr class="docutils" />
<p>Each file read by the neb command containing atomic coordinates used
to initialize one or more replicas must be formatted as follows.</p>
<p>The file can be ASCII text or a gzipped text file (detected by a .gz
suffix). The file can contain initial blank lines or comment lines
starting with &#8220;#&#8221; which are ignored. The first non-blank, non-comment
line should list N = the number of lines to follow. The N successive
lines contain the following information:</p>
<div class="highlight-python"><div class="highlight"><pre>ID1 x1 y1 z1
ID2 x2 y2 z2
...
IDN xN yN zN
</pre></div>
</div>
<p>The fields are the the atom ID, followed by the x,y,z coordinates.
The lines can be listed in any order. Additional trailing information
on the line is OK, such as a comment.</p>
<p>Note that for a typical NEB calculation you do not need to specify
initial coordinates for very many atoms to produce differing starting
and final replicas whose intermediate replicas will converge to the
energy barrier. Typically only new coordinates for atoms
geometrically near the barrier need be specified.</p>
<p>Also note there is no requirement that the atoms in the file
correspond to the NEB atoms in the group defined by the <a class="reference internal" href="fix_neb.html"><em>fix neb</em></a> command. Not every NEB atom need be in the file,
and non-NEB atoms can be listed in the file.</p>
<hr class="docutils" />
<p>Four kinds of output can be generated during a NEB calculation: energy
barrier statistics, thermodynamic output by each replica, dump files,
and restart files.</p>
<p>When running with multiple partitions (each of which is a replica in
this case), the print-out to the screen and master log.lammps file
contains a line of output, printed once every <em>Nevery</em> timesteps. It
contains the timestep, the maximum force per replica, the maximum
force per atom (in any replica), potential gradients in the initial,</p>
<blockquote>
<div>final, and climbing replicas,</div></blockquote>
<p>the forward and backward energy barriers,
the total reaction coordinate (RDT), and
the normalized reaction coordinate and potential energy of each replica.</p>
<p>The &#8220;maximum force per replica&#8221; is
the two-norm of the 3N-length force vector for the atoms in each
replica, maximized across replicas, which is what the <em>ftol</em> setting
is checking against. In this case, N is all the atoms in each
replica. The &#8220;maximum force per atom&#8221; is the maximum force component
of any atom in any replica. The potential gradients are the two-norm
of the 3N-length force vector solely due to the interaction potential i.e.
without adding in inter-replica forces. Note that inter-replica forces
are zero in the initial and final replicas, and only affect
the direction in the climbing replica. For this reason, the &#8220;maximum
force per replica&#8221; is often equal to the potential gradient in the
climbing replica. In the first stage of NEB, there is no climbing
replica, and so the potential gradient in the highest energy replica
is reported, since this replica will become the climbing replica
in the second stage of NEB.</p>
<p>The &#8220;reaction coordinate&#8221; (RD) for each
replica is the two-norm of the 3N-length vector of distances between
its atoms and the preceding replica&#8217;s atoms, added to the RD of the
preceding replica. The RD of the first replica RD1 = 0.0;
the RD of the final replica RDN = RDT, the total reaction coordinate.
The normalized RDs are divided by RDT,
so that they form a monotonically increasing sequence
from zero to one. When computing RD, N only includes the atoms
being operated on by the fix neb command.</p>
<p>The forward (reverse) energy barrier is the potential energy of the highest
replica minus the energy of the first (last) replica.</p>
<p>When running on multiple partitions, LAMMPS produces additional log
files for each partition, e.g. log.lammps.0, log.lammps.1, etc. For a
NEB calculation, these contain the thermodynamic output for each
replica.</p>
<p>If <a class="reference internal" href="dump.html"><em>dump</em></a> commands in the input script define a filename
that includes a <em>universe</em> or <em>uloop</em> style <a class="reference internal" href="variable.html"><em>variable</em></a>,
then one dump file (per dump command) will be created for each
replica. At the end of the NEB calculation, the final snapshot in
each file will contain the sequence of snapshots that transition the
system over the energy barrier. Earlier snapshots will show the
convergence of the replicas to the MEP.</p>
<p>Likewise, <a class="reference internal" href="restart.html"><em>restart</em></a> filenames can be specified with a
<em>universe</em> or <em>uloop</em> style <a class="reference internal" href="variable.html"><em>variable</em></a>, to generate
restart files for each replica. These may be useful if the NEB
calculation fails to converge properly to the MEP, and you wish to
restart the calculation from an intermediate point with altered
parameters.</p>
<p>There are 2 Python scripts provided in the tools/python directory,
neb_combine.py and neb_final.py, which are useful in analyzing output
from a NEB calculation. Assume a NEB simulation with M replicas, and
the NEB atoms labelled with a specific atom type.</p>
<p>The neb_combine.py script extracts atom coords for the NEB atoms from
all M dump files and creates a single dump file where each snapshot
contains the NEB atoms from all the replicas and one copy of non-NEB
atoms from the first replica (presumed to be identical in other
replicas). This can be visualized/animated to see how the NEB atoms
relax as the NEB calculation proceeds.</p>
<p>The neb_final.py script extracts the final snapshot from each of the M
dump files to create a single dump file with M snapshots. This can be
visualized to watch the system make its transition over the energy
barrier.</p>
<p>To illustrate, here are images from the final snapshot produced by the
neb_combine.py script run on the dump files produced by the two
example input scripts in examples/neb. Click on them to see a larger
image.</p>
<a data-lightbox="group-0eb1e253-2977-45e3-863d-e69fe78196e9"
href="_images/hop1.jpg"
class=""
title=""
data-title=""
><img src="_images/hop1.jpg"
class=""
width="25%"
height="auto"
alt=""/>
</a><a data-lightbox="group-fb58d79f-e00a-4e78-b97a-b1bff65370eb"
href="_images/hop2.jpg"
class=""
title=""
data-title=""
><img src="_images/hop2.jpg"
class=""
width="25%"
height="auto"
alt=""/>
</a></div>
<hr class="docutils" />
<div class="section" id="restrictions">
<h2>Restrictions<a class="headerlink" href="#restrictions" title="Permalink to this headline"></a></h2>
<p>This command can only be used if LAMMPS was built with the REPLICA
package. See the <a class="reference internal" href="Section_start.html#start-3"><span>Making LAMMPS</span></a> section
for more info on packages.</p>
</div>
<div class="section" id="related-commands">
<h2>Related commands<a class="headerlink" href="#related-commands" title="Permalink to this headline"></a></h2>
<p><a class="reference internal" href="prd.html"><em>prd</em></a>, <a class="reference internal" href="temper.html"><em>temper</em></a>, <a class="reference internal" href="fix_langevin.html"><em>fix langevin</em></a>, <a class="reference internal" href="fix_viscous.html"><em>fix viscous</em></a></p>
<p><strong>Default:</strong> none</p>
<hr class="docutils" />
<p id="henkelman1"><strong>(Henkelman1)</strong> Henkelman and Jonsson, J Chem Phys, 113, 9978-9985 (2000).</p>
<p id="henkelman2"><strong>(Henkelman2)</strong> Henkelman, Uberuaga, Jonsson, J Chem Phys, 113,
9901-9904 (2000).</p>
<p id="nakano"><strong>(Nakano)</strong> Nakano, Comp Phys Comm, 178, 280-289 (2008).</p>
</div>
</div>
</div>
</div>
<footer>
<hr/>
<div role="contentinfo">
<p>
&copy; Copyright .
</p>
</div>
Built with <a href="http://sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/snide/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>.
</footer>
</div>
</div>
</section>
</div>
<script type="text/javascript">
var DOCUMENTATION_OPTIONS = {
URL_ROOT:'./',
VERSION:'15 May 2015 version',
COLLAPSE_INDEX:false,
FILE_SUFFIX:'.html',
HAS_SOURCE: true
};
</script>
<script type="text/javascript" src="_static/jquery.js"></script>
<script type="text/javascript" src="_static/underscore.js"></script>
<script type="text/javascript" src="_static/doctools.js"></script>
<script type="text/javascript" src="_static/sphinxcontrib-images/LightBox2/lightbox2/js/jquery-1.11.0.min.js"></script>
<script type="text/javascript" src="_static/sphinxcontrib-images/LightBox2/lightbox2/js/lightbox.min.js"></script>
<script type="text/javascript" src="_static/sphinxcontrib-images/LightBox2/lightbox2-customize/jquery-noconflict.js"></script>
<script type="text/javascript" src="_static/js/theme.js"></script>
<script type="text/javascript">
jQuery(function () {
SphinxRtdTheme.StickyNav.enable();
});
</script>
</body>
</html>
c4science · Help