2014-11-25 05:13:20 +08:00
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2014-11-27 00:10:26 +08:00
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<CENTER><A HREF = "http://lammps.sandia.gov">LAMMPS WWW Site</A> - <A HREF = "Manual.html">LAMMPS Documentation</A> - <A HREF = "Section_commands.html#comm">LAMMPS Commands</A>
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<HR>
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<H3>fix pimd 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 pimd 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>pimd = style name of this fix command
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<LI>zero or more keyword/value pairs may be appended
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<LI>keyword = <I>method</I> or <I>fmass</I> or <I>sp</I> or <I>temp</I> or <I>nhc</I>
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<PRE> <I>method</I> value = <I>pimd</I> or <I>nmpimd</I> or <I>cmd</I>
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<I>fmass</I> value = scaling factor on mass
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<I>sp</I> value = scaling factor on Planck constant
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<I>temp</I> value = temperature (temperarate units)
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<I>nhc</I> value = Nc = number of chains in Nose-Hoover thermostat
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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 pimd method nmpimd fmass 1.0 sp 2.0 temp 300.0 nhc 4
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</PRE>
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<P><B>Description:</B>
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</P>
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<P>This command performs quantum molecular dynamics simulations based on
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the Feynman path integral to include effects of tunneling and
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zero-point motion. In this formalism, the isomorphism of a quantum
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partition function for the original system to a classical partition
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function for a ring-polymer system is exploited, to efficiently sample
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configurations from the canonical ensemble <A HREF = "#Feynman">(Feynman)</A>.
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The classical partition function and its components are given
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by the following equations:
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</P>
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<CENTER><IMG SRC = "Eqs/fix_pimd.jpg">
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</CENTER>
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<P>The interested user is referred to any of the numerous references on
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this methodology, but briefly, each quantum particle in a path
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integral simulation is represented by a ring-polymer of P quasi-beads,
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labeled from 1 to P. During the simulation, each quasi-bead interacts
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with beads on the other ring-polymers with the same imaginary time
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index (the second term in the effective potential above). The
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quasi-beads also interact with the two neighboring quasi-beads through
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the spring potential in imaginary-time space (first term in effective
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potential). To sample the canonical ensemble, a Nose-Hoover massive
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chain thermostat is applied <A HREF = "#Tuckerman">(Tuckerman)</A>. With the
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massive chain algorithm, a chain of NH thermostats is coupled to each
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degree of freedom for each quasi-bead. The keyword <I>temp</I> sets the
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target temperature for the system and the keyword <I>nhc</I> sets the
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number <I>Nc</I> of thermostats in each chain. For example, for a
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simulation of N particles with P beads in each ring-polymer, the total
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number of NH thermostats would be 3 x N x P x Nc.
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</P>
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<P>IMPORTANT NOTE: This fix implements a complete velocity-verlet
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integrator combined with NH massive chain thermostat, so no
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other time integration fix should be used.
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</P>
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<P>The <I>method</I> keyword determines what style of PIMD is performed. A
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value of <I>pimd</I> is standard PIMD. A value of <I>nmpimd</I> is for
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normal-mode PIMD. A value of <I>cmd</I> is for centroid molecular dynamics
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(CMD). The difference between the styles is as follows.
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</P>
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<P>In standard PIMD, the value used for a bead's fictitious mass is
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arbitrary. A common choice is to use Mi = m/P, which results in the
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mass of the entire ring-polymer being equal to the real quantum
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particle. But it can be difficult to efficiently integrate the
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equations of motion for the stiff harmonic interactions in the ring
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polymers.
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</P>
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<P>A useful way to resolve this issue is to integrate the equations of
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motion in a normal mode representation, using Normal Mode
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Path-Integral Molecular Dynamics (NMPIMD) <A HREF = "#Cao1">(Cao1)</A>. In NMPIMD,
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the NH chains are attached to each normal mode of the ring-polymer and
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the fictitious mass of each mode is chosen as Mk = the eigenvalue of
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the Kth normal mode for k > 0. The k = 0 mode, referred to as the
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zero-frequency mode or centroid, corresponds to overall translation of
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the ring-polymer and is assigned the mass of the real particle.
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</P>
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<P>Motion of the centroid can be effectively uncoupled from the other
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normal modes by scaling the fictitious masses to achieve a partial
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adiabatic separation. This is called a Centroid Molecular Dynamics
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(CMD) approximation <A HREF = "#Cao2">(Cao2)</A>. The time-evolution (and resulting
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dynamics) of the quantum particles can be used to obtain centroid time
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correlation functions, which can be further used to obtain the true
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quantum correlation function for the original system. The CMD method
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also uses normal modes to evolve the system, except only the k > 0
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modes are thermostatted, not the centroid degrees of freedom.
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</P>
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<P>The keyword <I>fmass</I> sets a further scaling factor for the fictitious
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masses of beads, which can be used for the Partial Adiabatic CMD
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<A HREF = "#Hone">(Hone)</A>, or to be set as P, which results in the fictitious
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masses to be equal to the real particle masses.
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</P>
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<P>The keyword <I>sp</I> is a scaling factor on Planck's constant, which can
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be useful for debugging or other purposes. The default value of 1.0
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is appropriate for most situations.
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</P>
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<P>The PIMD algorithm in LAMMPS is implemented as a hyper-parallel scheme
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as described in <A HREF = "#Calhoun">(Calhoun)</A>. In LAMMPS this is done by using
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<A HREF = "Section_howto.html#howto_5">multi-replica feature</A> in LAMMPS, where
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each quasi-particle system is stored and simulated on a separate
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partition of processors. The following diagram illustrates this
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approach. The original system with 2 ring polymers is shown in red.
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Since each ring has 4 quasi-beads (imaginary time slices), there are 4
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replicas of the system, each running on one of the 4 partitions of
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processors. Each replica (shown in green) owns one quasi-bead in each
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ring.
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</P>
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<CENTER><IMG SRC = "JPG/pimd.jpg">
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</CENTER>
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<P>To run a PIMD simulation with M quasi-beads in each ring polymer using
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N MPI tasks for each partition's domain-decomposition, you would use P
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= MxN processors (cores) and run the simulation as follows:
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</P>
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<PRE>mpirun -np P lmp_mpi -partition MxN -in script
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</PRE>
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<P>Note that in the LAMMPS input script for a multi-partition simulation,
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it is often very useful to define a <A HREF = "variable.html">uloop-style
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variable</A> such as
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</P>
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<PRE>variable ibead uloop M pad
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</PRE>
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<P>where M is the number of quasi-beads (partitions) used in the
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calculation. The uloop variable can then be used to manage I/O
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related tasks for each of the partitions, e.g.
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</P>
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<PRE>dump dcd all dcd 10 system_${ibead}.dcd
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restart 1000 system_${ibead}.restart1 system_${ibead}.restart2
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read_restart system_${ibead}.restart2
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</PRE>
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<P><B>Restrictions:</B>
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</P>
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<P>This fix is part of the USER-MISC package. It is only enabled if
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LAMMPS was built with that package. See the <A HREF = "Section_start.html#start_3">Making
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LAMMPS</A> section for more info.
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</P>
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<P>A PIMD simulation can be initialized with a single data file read via
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the <A HREF = "read_data.html">read_data</A> command. However, this means all
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quasi-beads in a ring polymer will have identical positions and
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velocities, resulting in identical trajectories for all quasi-beads.
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To avoid this, users can simply initialize velocities with different
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random number seeds assigned to each partition, as defined by the
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uloop variable, e.g.
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</P>
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<PRE>velocity all create 300.0 1234${ibead} rot yes dist gaussian
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</PRE>
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<P><B>Default:</B>
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</P>
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<P>The keyword defaults are method = pimd, fmass = 1.0, sp = 1.0, temp = 300.0,
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and nhc = 2.
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</P>
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<HR>
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<A NAME = "Feynman"></A>
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<P><B>(Feynman)</B> R. Feynman and A. Hibbs, Chapter 7, Quantum Mechanics and
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Path Integrals, McGraw-Hill, New York (1965).
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</P>
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<A NAME = "Tuckerman"></A>
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<P><B>(Tuckerman)</B> M. Tuckerman and B. Berne, J Chem Phys, 99, 2796 (1993).
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</P>
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<A NAME = "Cao1"></A>
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<P><B>(Cao1)</B> J. Cao and B. Berne, J Chem Phys, 99, 2902 (1993).
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</P>
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<A NAME = "Cao2"></A>
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<P><B>(Cao2)</B> J. Cao and G. Voth, J Chem Phys, 100, 5093 (1994).
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</P>
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<A NAME = "Hone"></A>
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<P><B>(Hone)</B> T. Hone, P. Rossky, G. Voth, J Chem Phys, 124,
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154103 (2006).
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</P>
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<A NAME = "Calhoun"></A>
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<P><B>(Calhoun)</B> A. Calhoun, M. Pavese, G. Voth, Chem Phys Letters, 262,
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415 (1996).
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</P>
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</HTML>
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