Final tweaks to HMA documentation

HMA should not be used if atoms are expected to diffuse.  It is also

restricted to simulations in the NVT ensemble.  While this compute may be

used with any potential in LAMMPS, it may not provide efficiency improvements

for potentials that do not go to 0 smoothly at the truncation distance;

used with any potential in LAMMPS, it will provide inaccurate results

for potentials that do not go to 0 at the truncation distance;

"pair_lj_smooth_linear"_pair_lj_smooth_linear.html and Ewald summation should

work fine, while "pair_lj"_pair_lj.html will perform poorly unless 

the cutoff is very large.  Furthermore, computation of the heat capacity with

the potential is shifted (via "pair_modify"_pair_modify.html shift) or the cutoff is large.  Furthermore, computation of the heat capacity with

this compute is restricted to those that implement the single_hessian method

in Pair.  Implementing single_hessian in additional pair styles is simple.

Please contact Andrew Schultz (ajs42 at buffalo.edu) and David Kofke (kofke at

A detailed description of this method can be found in ("Moustafa"_#hma-Moustafa). The potential energy is computed by the formula:

\begin\{equation\}

\left< U\right>_\{HMA\} = \frac\{d(N-1)\}\{2\beta\}  + \left< U + \frac\{1\}\{2\} F\bullet\Delta r \right>

\left< U\right>_\{HMA\} = \frac\{d\}\{2\} (N-1) k_B T  + \left< U + \frac\{1\}\{2\} F\bullet\Delta r \right>

\end\{equation\}

where \(N\) is the number of atoms in the system, \(\beta\) is the reciprocal of the thermodynamic temperature, \(d\) is the

where \(N\) is the number of atoms in the system, \(k_B\) is Boltzmann's

constant, \(T\) is the temperature, \(d\) is the

dimensionality of the system (2 or 3 for 2d/3d), \(F\bullet\Delta r\) is the sum of dot products of the 

atomic force vectors and displacement (from lattice sites) vectors, and \(U\) is the sum of 

pair, bond, angle, dihedral, improper, kspace (long-range), and fix energies. 

\end\{equation\}

where \(\rho\) is the number density of the system, \(\Delta \hat P\) is the

difference between the harmonic and lattice pressure, and \(P_\{vir\}\) is

difference between the harmonic and lattice pressure, \(P_\{vir\}\) is

the virial pressure computed as the sum of pair, bond, angle, dihedral,

improper, kspace (long-range), and fix contributions to the force on each

atom.  Although the method will work for any value of \(\Delta \hat P\)

atom, and \(k_B=1/k_B T\).  Although the method will work for any value of \(\Delta \hat P\)

specified (use pressure "units"_units.html), the precision of the resultant

pressure is sensitive to \(\Delta \hat P\); the precision tends to be

best when \(\Delta \hat P\) is the actual the difference between the lattice

pressure and harmonic pressure.

\begin\{equation\}

\left<C_V \right>_\{HMA\} = \frac\{d k_B (N-1)\}\{2\} + \beta \left( \left<

U_\{HMA\}^2 \right> - \left<U_\{HMA\}\right>^2 \right)/T + \frac\{1\}\{4 T\}

\left<C_V \right>_\{HMA\} = \frac\{d\}\{2\} (N-1) k_B + \frac\{1\}\{k_B T^2\} \left( \left<

U_\{HMA\}^2 \right> - \left<U_\{HMA\}\right>^2 \right) + \frac\{1\}\{4 T\}

\left< F\bullet\Delta r + \Delta r \bullet \Phi \bullet \Delta r \right>

\end\{equation\}

thermo_modify format float '%22.15e' :pre

The {anharmonic} keyword will instruct the compute to return anharmonic

properties rather than the full properties (lattice, harmonic and anharmonic).

properties rather than the full properties, which include lattice, harmonic

and anharmonic contributions.

When using this keyword, the compute must be first active (it must be included

via a "thermo_style custom"_thermo_style.html command) while the atoms are

still at their lattice sites (before equilibration).

[Output info:]

This compute calculates a global vector that includes the n properties

requested as arguments to the command (the potential energy, pressure or heat

requested as arguments to the command (the potential energy, pressure and/or heat

capacity).  The elements of the vector can be accessed by indices 1-n by any

command that uses global vector values as input.  See the "Howto

output"_Howto_output.html doc page for an overview of LAMMPS output options.

[Restrictions:]

This compute is part of the USER-MISC package.  It is only

enabled if LAMMPS was built with that package.  See the "Build

This compute is part of the USER-MISC package.  It is enabled only

if LAMMPS was built with that package.  See the "Build

package"_Build_package.html doc page for more info.

Usage restricted to canonical (NVT) ensemble simulation only.