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<H3>fix qbmsst command
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</H3>
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<P><B>Syntax:</B>
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</P>
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<PRE>fix ID group-ID qbmsst dir shockvel keyword value ...
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</PRE>
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<UL><LI>ID, group-ID are documented in <A HREF = "fix.html">fix</A> command
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<LI>qbmsst = style name of this fix
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<LI>dir = <I>x</I> or <I>y</I> or <I>z</I>
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<LI>shockvel = shock velocity (strictly positive, velocity units)
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<LI>zero or more keyword/value pairs may be appended
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<LI>keyword = <I>q</I> or <I>mu</I> or <I>p0</I> or <I>v0</I> or <I>e0</I> or <I>tscale</I> or <I>damp</I> or <I>seed</I>or <I>f_max</I> or <I>N_f</I> or <I>eta</I> or <I>beta</I> or <I>T_init</I>
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<PRE> <I>q</I> value = cell mass-like parameter (mass^2/distance^4 units)
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<I>mu</I> value = artificial viscosity (mass/distance/time units)
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<I>p0</I> value = initial pressure in the shock equations (pressure units)
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<I>v0</I> value = initial simulation cell volume in the shock equations (distance^3 units)
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<I>e0</I> value = initial total energy (energy units)
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<I>tscale</I> value = reduction in initial temperature (unitless fraction between 0.0 and 1.0)
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<I>damp</I> value = damping parameter (time units) inverse of friction <i>γ</i>
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<I>seed</I> value = random number seed (positive integer)
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<I>f_max</I> value = upper cutoff frequency of the vibration spectrum (1/time units)
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<I>N_f</I> value = number of frequency bins (positive integer)
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<I>eta</I> value = coupling constant between the shock system and the quantum thermal bath (positive unitless)
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<I>beta</I> value = the quantum temperature is updated every beta time steps (positive integer)
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<I>T_init</I> value = quantum temperature for the initial state (temperature units)
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</PRE>
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</UL>
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<P><B>Examples:</B>
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</P>
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<PRE>fix 1 all qbmsst z 0.122 q 25 mu 0.9 tscale 0.01 damp 200 seed 35082 f_max 0.3 N_f 100 eta 1 beta 400 T_init 110 (liquid methane modeled with the REAX force field, real units)
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fix 2 all qbmsst z 72 q 40 tscale 0.05 damp 1 seed 47508 f_max 120.0 N_f 100 eta 1.0 beta 500 T_init 300 (quartz modeled with the BKS force field, metal units)
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</PRE>
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<P>Two example input scripts are given, including shocked alpha quartz
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<div itemprop="articleBody">
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<div class="section" id="fix-qbmsst-command">
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<span id="index-0"></span><h1>fix qbmsst command<a class="headerlink" href="#fix-qbmsst-command" title="Permalink to this headline">¶</a></h1>
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<div class="section" id="syntax">
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<h2>Syntax<a class="headerlink" href="#syntax" title="Permalink to this headline">¶</a></h2>
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<div class="highlight-python"><div class="highlight"><pre>fix ID group-ID qbmsst dir shockvel keyword value ...
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</pre></div>
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</div>
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<ul class="simple">
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<li>ID, group-ID are documented in <a class="reference internal" href="fix.html"><em>fix</em></a> command</li>
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<li>qbmsst = style name of this fix</li>
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<li>dir = <em>x</em> or <em>y</em> or <em>z</em></li>
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<li>shockvel = shock velocity (strictly positive, velocity units)</li>
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<li>zero or more keyword/value pairs may be appended</li>
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<li>keyword = <em>q</em> or <em>mu</em> or <em>p0</em> or <em>v0</em> or <em>e0</em> or <em>tscale</em> or <em>damp</em> or <em>seed*or *f_max</em> or <em>N_f</em> or <em>eta</em> or <em>beta</em> or <em>T_init</em></li>
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</ul>
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<pre class="literal-block">
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<em>q</em> value = cell mass-like parameter (mass^2/distance^4 units)
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<em>mu</em> value = artificial viscosity (mass/distance/time units)
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<em>p0</em> value = initial pressure in the shock equations (pressure units)
|
||||
<em>v0</em> value = initial simulation cell volume in the shock equations (distance^3 units)
|
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<em>e0</em> value = initial total energy (energy units)
|
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<em>tscale</em> value = reduction in initial temperature (unitless fraction between 0.0 and 1.0)
|
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<em>damp</em> value = damping parameter (time units) inverse of friction <i>&gamma;</i>
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<em>seed</em> value = random number seed (positive integer)
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<em>f_max</em> value = upper cutoff frequency of the vibration spectrum (1/time units)
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<em>N_f</em> value = number of frequency bins (positive integer)
|
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<em>eta</em> value = coupling constant between the shock system and the quantum thermal bath (positive unitless)
|
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<em>beta</em> value = the quantum temperature is updated every beta time steps (positive integer)
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<em>T_init</em> value = quantum temperature for the initial state (temperature units)
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</pre>
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</div>
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<div class="section" id="examples">
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<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2>
|
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<div class="highlight-python"><div class="highlight"><pre>fix 1 all qbmsst z 0.122 q 25 mu 0.9 tscale 0.01 damp 200 seed 35082 f_max 0.3 N_f 100 eta 1 beta 400 T_init 110 (liquid methane modeled with the REAX force field, real units)
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fix 2 all qbmsst z 72 q 40 tscale 0.05 damp 1 seed 47508 f_max 120.0 N_f 100 eta 1.0 beta 500 T_init 300 (quartz modeled with the BKS force field, metal units)
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</pre></div>
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</div>
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<p>Two example input scripts are given, including shocked alpha quartz
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and shocked liquid methane. The input script first equilibrate an
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initial state with the quantum thermal bath at the target temperature
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and then apply the qbmsst to simulate shock compression with quantum
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nuclear correction. The following two figures plot related quantities
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for shocked alpha quartz.
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</P>
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<CENTER><IMG SRC = "JPG/qbmsst_init.jpg">
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</CENTER>
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<P>Figure 1. Classical temperature <i>T</i><sup>cl</sup> = ∑
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<i>m<sub>i</sub>v<sub>i</sub><sup>2</sup>/3Nk</i><sub>B</sub> vs. time
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for shocked alpha quartz.</p>
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<img alt="_images/qbmsst_init.jpg" class="align-center" src="_images/qbmsst_init.jpg" />
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<p>Figure 1. Classical temperature <i>T</i><sup>cl</sup> = &sum;
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<i>m<sub>i</sub>v<sub>i</sub><sup>2</sup>/3Nk</i><sub>B</sub> vs. time
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for coupling the alpha quartz initial state with the quantum thermal
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bath at target quantum temperature <i>T</i><sup>qm</sup> = 300 K. The
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bath at target quantum temperature <i>T</i><sup>qm</sup> = 300 K. The
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NpH ensemble is used for time integration while QTB provides the
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colored random force. <i>T</i><sup>cl</sup> converges at the timescale
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of <I>damp</I> which is set to be 1 ps.
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</P>
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<CENTER><IMG SRC = "JPG/qbmsst_shock.jpg">
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</CENTER>
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<P>Figure 2. Quantum temperature and pressure vs. time for simulating
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colored random force. <i>T</i><sup>cl</sup> converges at the timescale
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of <em>damp</em> which is set to be 1 ps.</p>
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<img alt="_images/qbmsst_shock.jpg" class="align-center" src="_images/qbmsst_shock.jpg" />
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<p>Figure 2. Quantum temperature and pressure vs. time for simulating
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shocked alpha quartz with the QBMSST. The shock propagates along the z
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direction. Restart of the QBMSST command is demonstrated in the
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example input script. Thermodynamic quantities stay continuous before
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and after the restart.
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</P>
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<P><B>Description:</B>
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</P>
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<P>This command performs the Quantum-Bath coupled Multi-Scale Shock
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Technique (QBMSST) integration. See <A HREF = "#Qi">(Qi)</A> for a detailed
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and after the restart.</p>
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</div>
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<div class="section" id="description">
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<h2>Description<a class="headerlink" href="#description" title="Permalink to this headline">¶</a></h2>
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<p>This command performs the Quantum-Bath coupled Multi-Scale Shock
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Technique (QBMSST) integration. See <a class="reference internal" href="#qi"><span>(Qi)</span></a> for a detailed
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description of this method. The QBMSST provides description of the
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thermodynamics and kinetics of shock processes while incorporating
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quantum nuclear effects. The <I>shockvel</I> setting determines the steady
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shock velocity that will be simulated along direction <I>dir</I>.
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</P>
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<P>Quantum nuclear effects <A HREF = "fix_qtb.html">(fix qtb)</A> can be crucial
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quantum nuclear effects. The <em>shockvel</em> setting determines the steady
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shock velocity that will be simulated along direction <em>dir</em>.</p>
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<p>Quantum nuclear effects <a class="reference internal" href="fix_qtb.html"><em>(fix qtb)</em></a> can be crucial
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especially when the temperature of the initial state is below the
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classical limit or there is a great change in the zero point energies
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between the initial and final states. Theoretical post processing
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quantum corrections of shock compressed water and methane have been
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reported as much as 30% of the temperatures <A HREF = "#Goldman">(Goldman)</A>. A
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reported as much as 30% of the temperatures <a class="reference internal" href="#goldman"><span>(Goldman)</span></a>. A
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self-consistent method that couples the shock to a quantum thermal
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bath described by a colored noise Langevin thermostat has been
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developed by Qi et al <A HREF = "#Qi">(Qi)</A> and applied to shocked methane. The
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developed by Qi et al <a class="reference internal" href="#qi"><span>(Qi)</span></a> and applied to shocked methane. The
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onset of chemistry is reported to be at a pressure on the shock
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Hugoniot that is 40% lower than observed with classical molecular
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dynamics.
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</P>
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<P>It is highly recommended that the system be already in an equilibrium
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state with a quantum thermal bath at temperature of <I>T_init</I>. The fix
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command <A HREF = "fix_qtb.html">fix qtb</A> at constant temperature <I>T_init</I> could
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dynamics.</p>
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<p>It is highly recommended that the system be already in an equilibrium
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state with a quantum thermal bath at temperature of <em>T_init</em>. The fix
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command <a class="reference internal" href="fix_qtb.html"><em>fix qtb</em></a> at constant temperature <em>T_init</em> could
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be used before applying this command to introduce self-consistent
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quantum nuclear effects into the initial state.
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</P>
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<P>The parameters <I>q</I>, <I>mu</I>, <I>e0</I>, <I>p0</I>, <I>v0</I> and <I>tscale</I> are described
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in the command <A HREF = "fix_msst.html">fix msst</A>. The values of <I>e0</I>, <I>p0</I>, or
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<I>v0</I> will be calculated on the first step if not specified. The
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parameter of <I>damp</I>, <I>f_max</I>, and <I>N_f</I> are described in the command
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<A HREF = "fix_qtb.html">fix qtb</A>.
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</P>
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<P>The fix qbmsst command couples the shock system to a quantum thermal
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quantum nuclear effects into the initial state.</p>
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<p>The parameters <em>q</em>, <em>mu</em>, <em>e0</em>, <em>p0</em>, <em>v0</em> and <em>tscale</em> are described
|
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in the command <a class="reference internal" href="fix_msst.html"><em>fix msst</em></a>. The values of <em>e0</em>, <em>p0</em>, or
|
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<em>v0</em> will be calculated on the first step if not specified. The
|
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parameter of <em>damp</em>, <em>f_max</em>, and <em>N_f</em> are described in the command
|
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<a class="reference internal" href="fix_qtb.html"><em>fix qtb</em></a>.</p>
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<p>The fix qbmsst command couples the shock system to a quantum thermal
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bath with a rate that is proportional to the change of the total
|
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energy of the shock system, <i>etot</i> - <i>etot</i><sub>0</sub>.
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Here <i>etot</i> consists of both the system energy and a thermal
|
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term, see <A HREF = "#Qi">(Qi)</A>, and <i>etot</i><sub>0</sub> = <I>e0</I> is the
|
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initial total energy.
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</P>
|
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<P>The <I>eta</I> (<i>η</i>) parameter is a unitless coupling constant
|
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between the shock system and the quantum thermal bath. A small <I>eta</I>
|
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energy of the shock system, <i>etot</i> - <i>etot</i><sub>0</sub>.
|
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Here <i>etot</i> consists of both the system energy and a thermal
|
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term, see <a class="reference internal" href="#qi"><span>(Qi)</span></a>, and <i>etot</i><sub>0</sub> = <em>e0</em> is the
|
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initial total energy.</p>
|
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<p>The <em>eta</em> (<i>&eta;</i>) parameter is a unitless coupling constant
|
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between the shock system and the quantum thermal bath. A small <em>eta</em>
|
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value cannot adjust the quantum temperature fast enough during the
|
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temperature ramping period of shock compression while large <I>eta</I>
|
||||
leads to big temperature oscillation. A value of <I>eta</I> between 0.3 and
|
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temperature ramping period of shock compression while large <em>eta</em>
|
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leads to big temperature oscillation. A value of <em>eta</em> between 0.3 and
|
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1 is usually appropriate for simulating most systems under shock
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compression. We observe that different values of <I>eta</I> lead to almost
|
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the same final thermodynamic state behind the shock, as expected.
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</P>
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<P>The quantum temperature is updated every <I>beta</I> (<i>β</i>) steps
|
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with an integration time interval <I>beta</I> times longer than the
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simulation time step. In that case, <i>etot</i> is taken as its
|
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average over the past <I>beta</I> steps. The temperature of the quantum
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thermal bath <i>T</i><sup>qm</sup> changes dynamically according to
|
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the following equation where Δ<i>t</i> is the MD time step and
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<i>γ</i> is the friction constant which is equal to the inverse
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of the <I>damp</I> parameter.
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</P>
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compression. We observe that different values of <em>eta</em> lead to almost
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the same final thermodynamic state behind the shock, as expected.</p>
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<p>The quantum temperature is updated every <em>beta</em> (<i>&beta;</i>) steps
|
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with an integration time interval <em>beta</em> times longer than the
|
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simulation time step. In that case, <i>etot</i> is taken as its
|
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average over the past <em>beta</em> steps. The temperature of the quantum
|
||||
thermal bath <i>T</i><sup>qm</sup> changes dynamically according to
|
||||
the following equation where &Delta;<i>t</i> is the MD time step and
|
||||
<i>&gamma;</i> is the friction constant which is equal to the inverse
|
||||
of the <em>damp</em> parameter.</p>
|
||||
<center><font size="4"> <i>dT</i><sup>qm</sup>/<i>dt =
|
||||
γη</i>∑<i><sup>β</sup><sub>l =
|
||||
1</sub></i>[<i>etot</i>(<i>t-l</i>Δ<i>t</i>)-<i>etot</i><sub>0</sub>]/<i>3βNk</i><sub>B</sub>
|
||||
</font></center>
|
||||
|
||||
<P>The parameter <I>T_init</I> is the initial temperature of the quantum
|
||||
thermal bath and the system before shock loading.
|
||||
</P>
|
||||
<P>For all pressure styles, the simulation box stays orthorhombic in
|
||||
</font></center><p>The parameter <em>T_init</em> is the initial temperature of the quantum
|
||||
thermal bath and the system before shock loading.</p>
|
||||
<p>For all pressure styles, the simulation box stays orthorhombic in
|
||||
shape. Parrinello-Rahman boundary conditions (tilted box) are
|
||||
supported by LAMMPS, but are not implemented for QBMSST.
|
||||
</P>
|
||||
<HR>
|
||||
|
||||
<P><B>Restart, fix_modify, output, run start/stop, minimize info:</B>
|
||||
</P>
|
||||
<P>Because the state of the random number generator is not written to
|
||||
<A HREF = "restart.html">binary restart files</A>, this fix cannot be restarted
|
||||
"exactly" in an uninterrupted fashion. However, in a statistical
|
||||
supported by LAMMPS, but are not implemented for QBMSST.</p>
|
||||
</div>
|
||||
<hr class="docutils" />
|
||||
<div class="section" id="restart-fix-modify-output-run-start-stop-minimize-info">
|
||||
<h2>Restart, fix_modify, output, run start/stop, minimize info<a class="headerlink" href="#restart-fix-modify-output-run-start-stop-minimize-info" title="Permalink to this headline">¶</a></h2>
|
||||
<p>Because the state of the random number generator is not written to
|
||||
<a class="reference internal" href="restart.html"><em>binary restart files</em></a>, this fix cannot be restarted
|
||||
“exactly” in an uninterrupted fashion. However, in a statistical
|
||||
sense, a restarted simulation should produce similar behaviors of the
|
||||
system as if it is not interrupted. To achieve such a restart, one
|
||||
should write explicitly the same value for <I>q</I>, <I>mu</I>, <I>damp</I>, <I>f_max</I>,
|
||||
<I>N_f</I>, <I>eta</I>, and <I>beta</I> and set <I>tscale</I> = 0 if the system is
|
||||
compressed during the first run.
|
||||
</P>
|
||||
<P>The progress of the QBMSST can be monitored by printing the global
|
||||
should write explicitly the same value for <em>q</em>, <em>mu</em>, <em>damp</em>, <em>f_max</em>,
|
||||
<em>N_f</em>, <em>eta</em>, and <em>beta</em> and set <em>tscale</em> = 0 if the system is
|
||||
compressed during the first run.</p>
|
||||
<p>The progress of the QBMSST can be monitored by printing the global
|
||||
scalar and global vector quantities computed by the fix. The global
|
||||
vector contains five values in this order:
|
||||
</P>
|
||||
<P>[<I>dhugoniot</I>, <I>drayleigh</I>, <I>lagrangian_speed</I>, <I>lagrangian_position</I>,
|
||||
<I>quantum_temperature</I>]
|
||||
</P>
|
||||
<OL><LI><I>dhugoniot</I> is the departure from the Hugoniot (temperature units).
|
||||
<LI><I>drayleigh</I> is the departure from the Rayleigh line (pressure units).
|
||||
<LI><I>lagrangian_speed</I> is the laboratory-frame Lagrangian speed (particle velocity) of the computational cell (velocity units).
|
||||
<LI><I>lagrangian_position</I> is the computational cell position in the reference frame moving at the shock speed. This is the distance of the computational cell behind the shock front.
|
||||
<LI><I>quantum_temperature</I> is the temperature of the quantum thermal bath <i>T</i><sup>qm</sup>.
|
||||
</OL>
|
||||
<P>To print these quantities to the log file with descriptive column
|
||||
vector contains five values in this order:</p>
|
||||
<p>[<em>dhugoniot</em>, <em>drayleigh</em>, <em>lagrangian_speed</em>, <em>lagrangian_position</em>,
|
||||
<em>quantum_temperature</em>]</p>
|
||||
<ol class="arabic simple">
|
||||
<li><em>dhugoniot</em> is the departure from the Hugoniot (temperature units).</li>
|
||||
<li><em>drayleigh</em> is the departure from the Rayleigh line (pressure units).</li>
|
||||
<li><em>lagrangian_speed</em> is the laboratory-frame Lagrangian speed (particle velocity) of the computational cell (velocity units).</li>
|
||||
<li><em>lagrangian_position</em> is the computational cell position in the reference frame moving at the shock speed. This is the distance of the computational cell behind the shock front.</li>
|
||||
<li><em>quantum_temperature</em> is the temperature of the quantum thermal bath <i>T</i><sup>qm</sup>.</li>
|
||||
</ol>
|
||||
<p>To print these quantities to the log file with descriptive column
|
||||
headers, the following LAMMPS commands are suggested. Here the
|
||||
<A HREF = "fix_modify.html">fix_modify</A> energy command is also enabled to allow
|
||||
the thermo keyword <I>etotal</I> to print the quantity <i>etot</i>. See
|
||||
also the <A HREF = "thermo_style.html">thermo_style</A> command.
|
||||
</P>
|
||||
<PRE>fix fix_id all msst z
|
||||
fix_modify fix_id energy yes
|
||||
variable dhug equal f_fix_id[1]
|
||||
variable dray equal f_fix_id[2]
|
||||
variable lgr_vel equal f_fix_id[3]
|
||||
variable lgr_pos equal f_fix_id[4]
|
||||
variable T_qm equal f_fix_id[5]
|
||||
thermo_style custom step temp ke pe lz pzz etotal v_dhug v_dray v_lgr_vel v_lgr_pos v_T_qm f_fix_id
|
||||
</PRE>
|
||||
<P>The global scalar under the entry f_fix_id is the quantity of thermo
|
||||
energy as an extra part of <i>etot</i>. This global scalar and the
|
||||
vector of 5 quantities can be accessed by various <A HREF = "Section_howto.html#howto_15">output
|
||||
commands</A>. It is worth noting that the
|
||||
temp keyword under the <A HREF = "thermo_style.html">thermo_style</A> command print
|
||||
the instantaneous classical temperature <i>T</i><sup>cl</sup> as
|
||||
described in the command <A HREF = "fix_qtb.html">fix qtb</A>.
|
||||
</P>
|
||||
<HR>
|
||||
|
||||
<P><B>Restrictions:</B>
|
||||
</P>
|
||||
<P>This fix style is part of the USER-QTB package. It is only enabled if
|
||||
LAMMPS was built with that package. See the <A HREF = "Section_start.html#start_3">Making
|
||||
LAMMPS</A> section for more info.
|
||||
</P>
|
||||
<P>All cell dimensions must be periodic. This fix can not be used with a
|
||||
<a class="reference internal" href="fix_modify.html"><em>fix_modify</em></a> energy command is also enabled to allow
|
||||
the thermo keyword <em>etotal</em> to print the quantity <i>etot</i>. See
|
||||
also the <a class="reference internal" href="thermo_style.html"><em>thermo_style</em></a> command.</p>
|
||||
<div class="highlight-python"><div class="highlight"><pre>fix fix_id all msst z
|
||||
fix_modify fix_id energy yes
|
||||
variable dhug equal f_fix_id[1]
|
||||
variable dray equal f_fix_id[2]
|
||||
variable lgr_vel equal f_fix_id[3]
|
||||
variable lgr_pos equal f_fix_id[4]
|
||||
variable T_qm equal f_fix_id[5]
|
||||
thermo_style custom step temp ke pe lz pzz etotal v_dhug v_dray v_lgr_vel v_lgr_pos v_T_qm f_fix_id
|
||||
</pre></div>
|
||||
</div>
|
||||
<p>The global scalar under the entry f_fix_id is the quantity of thermo
|
||||
energy as an extra part of <i>etot</i>. This global scalar and the
|
||||
vector of 5 quantities can be accessed by various <a class="reference internal" href="Section_howto.html#howto-15"><span>output commands</span></a>. It is worth noting that the
|
||||
temp keyword under the <a class="reference internal" href="thermo_style.html"><em>thermo_style</em></a> command print
|
||||
the instantaneous classical temperature <i>T</i><sup>cl</sup> as
|
||||
described in the command <a class="reference internal" href="fix_qtb.html"><em>fix qtb</em></a>.</p>
|
||||
</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 fix style is part of the USER-QTB package. It is only enabled if
|
||||
LAMMPS was built with that package. See the <a class="reference internal" href="Section_start.html#start-3"><span>Making LAMMPS</span></a> section for more info.</p>
|
||||
<p>All cell dimensions must be periodic. This fix can not be used with a
|
||||
triclinic cell. The QBMSST fix has been tested only for the group-ID
|
||||
all.
|
||||
</P>
|
||||
<HR>
|
||||
|
||||
<P><B>Related commands:</B>
|
||||
</P>
|
||||
<P><A HREF = "fix_qtb.html">fix qtb</A>, <A HREF = "fix_msst.html">fix msst</A>
|
||||
</P>
|
||||
<HR>
|
||||
|
||||
<P><B>Default:</B>
|
||||
</P>
|
||||
<P>The keyword defaults are q = 10, mu = 0, tscale = 0.01, damp = 1, seed
|
||||
all.</p>
|
||||
</div>
|
||||
<hr class="docutils" />
|
||||
<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="fix_qtb.html"><em>fix qtb</em></a>, <a class="reference internal" href="fix_msst.html"><em>fix msst</em></a></p>
|
||||
</div>
|
||||
<hr class="docutils" />
|
||||
<div class="section" id="default">
|
||||
<h2>Default<a class="headerlink" href="#default" title="Permalink to this headline">¶</a></h2>
|
||||
<p>The keyword defaults are q = 10, mu = 0, tscale = 0.01, damp = 1, seed
|
||||
= 880302, f_max = 200.0, N_f = 100, eta = 1.0, beta = 100, and
|
||||
T_init=300.0. e0, p0, and v0 are calculated on the first step.
|
||||
</P>
|
||||
<HR>
|
||||
T_init=300.0. e0, p0, and v0 are calculated on the first step.</p>
|
||||
<hr class="docutils" />
|
||||
<p id="goldman"><strong>(Goldman)</strong> Goldman, Reed and Fried, J. Chem. Phys. 131, 204103 (2009)</p>
|
||||
<p id="qi"><strong>(Qi)</strong> Qi and Reed, J. Phys. Chem. A 116, 10451 (2012).</p>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
<A NAME = "Goldman"></A>
|
||||
|
||||
<P><B>(Goldman)</B> Goldman, Reed and Fried, J. Chem. Phys. 131, 204103 (2009)
|
||||
</P>
|
||||
<A NAME = "Qi"></A>
|
||||
</div>
|
||||
</div>
|
||||
<footer>
|
||||
|
||||
|
||||
<P><B>(Qi)</B> Qi and Reed, J. Phys. Chem. A 116, 10451 (2012).
|
||||
</P>
|
||||
</HTML>
|
||||
<hr/>
|
||||
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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Reference in New Issue
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