\(\renewcommand{\AA}{\text{Å}}\)

# fix npt/cauchy command

## Syntax

    fix ID group-ID style_name keyword value ...

- ID, group-ID are documented in fix command

- style_name = npt/cauchy

- one or more keyword/value pairs may be appended

- keyword = temp or iso or aniso or tri or x or y or z or xy or yz or xz or couple or tchain or pchain or mtk or tloop or ploop or nreset or drag or dilate or scalexy or scaleyz or scalexz or flip or alpha or continue or fixedpoint

      temp values = Tstart Tstop Tdamp
        Tstart,Tstop = external temperature at start/end of run
        Tdamp = temperature damping parameter (time units)
      iso or aniso or tri values = Pstart Pstop Pdamp
        Pstart,Pstop = scalar external pressure at start/end of run (pressure units)
        Pdamp = pressure damping parameter (time units)
      x or y or z or xy or yz or xz values = Pstart Pstop Pdamp
        Pstart,Pstop = external stress tensor component at start/end of run (pressure units)
        Pdamp = stress damping parameter (time units)
      couple = none or xyz or xy or yz or xz
      tchain value = N
        N = length of thermostat chain (1 = single thermostat)
      pchain values = N
        N length of thermostat chain on barostat (0 = no thermostat)
      mtk value = yes or no = add in MTK adjustment term or not
      tloop value = M
        M = number of sub-cycles to perform on thermostat
      ploop value = M
        M = number of sub-cycles to perform on barostat thermostat
      nreset value = reset reference cell every this many timesteps
      drag value = Df
        Df = drag factor added to barostat/thermostat (0.0 = no drag)
      dilate value = dilate-group-ID
        dilate-group-ID = only dilate atoms in this group due to barostat volume changes
      scalexy value = yes or no = scale xy with ly
      scaleyz value = yes or no = scale yz with lz
      scalexz value = yes or no = scale xz with lz
      flip value = yes or no = allow or disallow box flips when it becomes highly skewed
      alpha value = strength of Cauchy stress control parameter
      continue value = yes or no = whether of not to continue from a previous run
      fixedpoint values = x y z
        x,y,z = perform barostat dilation/contraction around this point (distance units)

## Examples

    fix 1 water npt/cauchy temp 300.0 300.0 100.0 iso 0.0 0.0 1000.0 alpha 0.001

## Description

This command performs time integration on Nose-Hoover style non-Hamiltonian equations of motion which are designed to generate positions and velocities sampled from the isothermal-isobaric (npt) ensembles. This updates the position and velocity for atoms in the group each timestep and the box dimensions.

The thermostatting and barostatting is achieved by adding some dynamic variables which are coupled to the particle velocities (thermostatting) and simulation domain dimensions (barostatting). In addition to basic thermostatting and barostatting, this fix can also create a chain of thermostats coupled to the particle thermostat, and another chain of thermostats coupled to the barostat variables. The barostat can be coupled to the overall box volume, or to individual dimensions, including the xy, xz and yz tilt dimensions. The external pressure of the barostat can be specified as either a scalar pressure (isobaric ensemble) or as components of a symmetric stress tensor (constant stress ensemble). When used correctly, the time-averaged temperature and stress tensor of the particles will match the target values specified by Tstart/Tstop and Pstart/Pstop.

The equations of motion used are those of Shinoda et al in (Shinoda), which combine the hydrostatic equations of Martyna, Tobias and Klein in (Martyna) with the strain energy proposed by Parrinello and Rahman in (Parrinello). The time integration schemes closely follow the time-reversible measure-preserving Verlet and rRESPA integrators derived by Tuckerman et al in (Tuckerman).

The thermostat parameters are specified using the temp keyword. Other thermostat-related keywords are tchain, tloop and drag, which are discussed below.

The thermostat is applied to only the translational degrees of freedom for the particles. The translational degrees of freedom can also have a bias velocity removed before thermostatting takes place; see the description below. The desired temperature at each timestep is a ramped value during the run from Tstart to Tstop. The Tdamp parameter is specified in time units and determines how rapidly the temperature is relaxed. For example, a value of 10.0 means to relax the temperature in a timespan of (roughly) 10 time units (e.g. \(\tau\) or fs or ps - see the units command). The atoms in the fix group are the only ones whose velocities and positions are updated by the velocity/position update portion of the integration.

Note

A Nose-Hoover thermostat will not work well for arbitrary values of Tdamp. If Tdamp is too small, the temperature can fluctuate wildly; if it is too large, the temperature will take a very long time to equilibrate. A good choice for many models is a Tdamp of around 100 timesteps. Note that this is NOT the same as 100 time units for most units settings.

The barostat parameters are specified using one or more of the iso, aniso, tri, x, y, z, xy, xz, yz, and couple keywords. These keywords give you the ability to specify all 6 components of an external stress tensor, and to couple various of these components together so that the dimensions they represent are varied together during a constant-pressure simulation.

Other barostat-related keywords are pchain, mtk, ploop, nreset, drag, and dilate, which are discussed below.

Orthogonal simulation boxes have 3 adjustable dimensions (x,y,z). Triclinic (non-orthogonal) simulation boxes have 6 adjustable dimensions (x,y,z,xy,xz,yz). The create_box, read data, and read_restart commands specify whether the simulation box is orthogonal or non-orthogonal (triclinic) and explain the meaning of the xy,xz,yz tilt factors.

The target pressures for each of the 6 components of the stress tensor can be specified independently via the x, y, z, xy, xz, yz keywords, which correspond to the 6 simulation box dimensions. For each component, the external pressure or tensor component at each timestep is a ramped value during the run from Pstart to Pstop. If a target pressure is specified for a component, then the corresponding box dimension will change during a simulation. For example, if the y keyword is used, the y-box length will change. If the xy keyword is used, the xy tilt factor will change. A box dimension will not change if that component is not specified, although you have the option to change that dimension via the fix deform command.

Note that in order to use the xy, xz, or yz keywords, the simulation box must be triclinic, even if its initial tilt factors are 0.0.

For all barostat keywords, the Pdamp parameter operates like the Tdamp parameter, determining the time scale on which pressure is relaxed. For example, a value of 10.0 means to relax the pressure in a timespan of (roughly) 10 time units (e.g. \(\tau\) or fs or ps - see the units command).

Note

A Nose-Hoover barostat will not work well for arbitrary values of Pdamp. If Pdamp is too small, the pressure and volume can fluctuate wildly; if it is too large, the pressure will take a very long time to equilibrate. A good choice for many models is a Pdamp of around 1000 timesteps. However, note that Pdamp is specified in time units, and that timesteps are NOT the same as time units for most units settings.

Regardless of what atoms are in the fix group (the only atoms which are time integrated), a global pressure or stress tensor is computed for all atoms. Similarly, when the size of the simulation box is changed, all atoms are re-scaled to new positions, unless the keyword dilate is specified with a dilate-group-ID for a group that represents a subset of the atoms. This can be useful, for example, to leave the coordinates of atoms in a solid substrate unchanged and controlling the pressure of a surrounding fluid. This option should be used with care, since it can be unphysical to dilate some atoms and not others, because it can introduce large, instantaneous displacements between a pair of atoms (one dilated, one not) that are far from the dilation origin. Also note that for atoms not in the fix group, a separate time integration fix like fix nve or fix nvt can be used on them, independent of whether they are dilated or not.

The couple keyword allows two or three of the diagonal components of the pressure tensor to be “coupled” together. The value specified with the keyword determines which are coupled. For example, xz means the Pxx and Pzz components of the stress tensor are coupled. Xyz means all 3 diagonal components are coupled. Coupling means two things: the instantaneous stress will be computed as an average of the corresponding diagonal components, and the coupled box dimensions will be changed together in lockstep, meaning coupled dimensions will be dilated or contracted by the same percentage every timestep. The Pstart, Pstop, Pdamp parameters for any coupled dimensions must be identical. Couple xyz can be used for a 2d simulation; the z dimension is simply ignored.

The iso, aniso, and tri keywords are simply shortcuts that are equivalent to specifying several other keywords together.

The keyword iso means couple all 3 diagonal components together when pressure is computed (hydrostatic pressure), and dilate/contract the dimensions together. Using “iso Pstart Pstop Pdamp” is the same as specifying these 4 keywords:

    x Pstart Pstop Pdamp
    y Pstart Pstop Pdamp
    z Pstart Pstop Pdamp
    couple xyz

The keyword aniso means x, y, and z dimensions are controlled independently using the Pxx, Pyy, and Pzz components of the stress tensor as the driving forces, and the specified scalar external pressure. Using “aniso Pstart Pstop Pdamp” is the same as specifying these 4 keywords:

    x Pstart Pstop Pdamp
    y Pstart Pstop Pdamp
    z Pstart Pstop Pdamp
    couple none

The keyword tri means x, y, z, xy, xz, and yz dimensions are controlled independently using their individual stress components as the driving forces, and the specified scalar pressure as the external normal stress. Using “tri Pstart Pstop Pdamp” is the same as specifying these 7 keywords:

    x Pstart Pstop Pdamp
    y Pstart Pstop Pdamp
    z Pstart Pstop Pdamp
    xy 0.0 0.0 Pdamp
    yz 0.0 0.0 Pdamp
    xz 0.0 0.0 Pdamp
    couple none

In some cases (e.g. for solids) the pressure (volume) and/or temperature of the system can oscillate undesirably when a Nose/Hoover barostat and thermostat is applied. The optional drag keyword will damp these oscillations, although it alters the Nose/Hoover equations. A value of 0.0 (no drag) leaves the Nose/Hoover formalism unchanged. A non-zero value adds a drag term; the larger the value specified, the greater the damping effect. Performing a short run and monitoring the pressure and temperature is the best way to determine if the drag term is working. Typically a value between 0.2 to 2.0 is sufficient to damp oscillations after a few periods. Note that use of the drag keyword will interfere with energy conservation and will also change the distribution of positions and velocities so that they do not correspond to the nominal NVT, NPT, or NPH ensembles.

An alternative way to control initial oscillations is to use chain thermostats. The keyword tchain determines the number of thermostats in the particle thermostat. A value of 1 corresponds to the original Nose-Hoover thermostat. The keyword pchain specifies the number of thermostats in the chain thermostatting the barostat degrees of freedom. A value of 0 corresponds to no thermostatting of the barostat variables.

The mtk keyword controls whether or not the correction terms due to Martyna, Tuckerman, and Klein are included in the equations of motion (Martyna). Specifying no reproduces the original Hoover barostat, whose volume probability distribution function differs from the true NPT and NPH ensembles by a factor of 1/V. Hence using yes is more correct, but in many cases the difference is negligible.

The keyword tloop can be used to improve the accuracy of integration scheme at little extra cost. The initial and final updates of the thermostat variables are broken up into tloop sub-steps, each of length dt/tloop. This corresponds to using a first-order Suzuki-Yoshida scheme (Tuckerman). The keyword ploop does the same thing for the barostat thermostat.

The keyword nreset controls how often the reference dimensions used to define the strain energy are reset. If this keyword is not used, or is given a value of zero, then the reference dimensions are set to those of the initial simulation domain and are never changed. If the simulation domain changes significantly during the simulation, then the final average pressure tensor will differ significantly from the specified values of the external stress tensor. A value of nstep means that every nstep timesteps, the reference dimensions are set to those of the current simulation domain.

The scaleyz, scalexz, and scalexy keywords control whether or not the corresponding tilt factors are scaled with the associated box dimensions when barostatting triclinic periodic cells. The default values yes will turn on scaling, which corresponds to adjusting the linear dimensions of the cell while preserving its shape. Choosing no ensures that the tilt factors are not scaled with the box dimensions. See below for restrictions and default values in different situations. In older versions of LAMMPS, scaling of tilt factors was not performed. The old behavior can be recovered by setting all three scale keywords to no.

The flip keyword allows the tilt factors for a triclinic box to exceed half the distance of the parallel box length, as discussed below. If the flip value is set to yes, the bound is enforced by flipping the box when it is exceeded. If the flip value is set to no, the tilt will continue to change without flipping. Note that if applied stress induces large deformations (e.g. in a liquid), this means the box shape can tilt dramatically and LAMMPS will run less efficiently, due to the large volume of communication needed to acquire ghost atoms around a processor’s irregular-shaped subdomain. For extreme values of tilt, LAMMPS may also lose atoms and generate an error.

The fixedpoint keyword specifies the fixed point for barostat volume changes. By default, it is the center of the box. Whatever point is chosen will not move during the simulation. For example, if the lower periodic boundaries pass through (0,0,0), and this point is provided to fixedpoint, then the lower periodic boundaries will remain at (0,0,0), while the upper periodic boundaries will move twice as far. In all cases, the particle trajectories are unaffected by the chosen value, except for a time-dependent constant translation of positions.

Note

Using a barostat coupled to tilt dimensions xy, xz, yz can sometimes result in arbitrarily large values of the tilt dimensions, i.e. a dramatically deformed simulation box. LAMMPS allows the tilt factors to grow a small amount beyond the normal limit of half the box length (0.6 times the box length), and then performs a box “flip” to an equivalent periodic cell. See the discussion of the flip keyword above, to allow this bound to be exceeded, if desired.

The flip operation is described in more detail in the page for fix deform. Both the barostat dynamics and the atom trajectories are unaffected by this operation. However, if a tilt factor is incremented by a large amount (1.5 times the box length) on a single timestep, LAMMPS can not accommodate this event and will terminate the simulation with an error. This error typically indicates that there is something badly wrong with how the simulation was constructed, such as specifying values of Pstart that are too far from the current stress value, or specifying a timestep that is too large. Triclinic barostatting should be used with care. This also is true for other barostat styles, although they tend to be more forgiving of insults. In particular, it is important to recognize that equilibrium liquids can not support a shear stress and that equilibrium solids can not support shear stresses that exceed the yield stress.

One exception to this rule is if the first dimension in the tilt factor (x for xy) is non-periodic. In that case, the limits on the tilt factor are not enforced, since flipping the box in that dimension does not change the atom positions due to non-periodicity. In this mode, if you tilt the system to extreme angles, the simulation will simply become inefficient due to the highly skewed simulation box.

Note

Unlike the fix temp/berendsen command which performs thermostatting but NO time integration, this fix performs thermostatting/barostatting AND time integration. Thus you should not use any other time integration fix, such as fix nve on atoms to which this fix is applied. Likewise, fix npt/cauchy should not normally be used on atoms that also have their temperature controlled by another fix - e.g. by fix langevin or fix temp/rescale commands.

See the Howto thermostat and Howto barostat doc pages for a discussion of different ways to compute temperature and perform thermostatting and barostatting.

This fix compute a temperature and pressure each timestep. To do this, the fix creates its own computes of style “temp” and “pressure”, as if one of these sets of commands had been issued:

    compute fix-ID_temp all temp
    compute fix-ID_press all pressure fix-ID_temp

The group for both the new temperature and pressure compute is “all” since pressure is computed for the entire system. See the compute temp and compute pressure commands for details. Note that the IDs of the new computes are the fix-ID + underscore + “temp” or fix_ID + underscore + “press”.

Note that these are NOT the computes used by thermodynamic output (see the thermo_style command) with ID = thermo_temp and thermo_press. This means you can change the attributes of these fix’s temperature or pressure via the compute_modify command. Or you can print this temperature or pressure during thermodynamic output via the thermo_style custom command using the appropriate compute-ID. It also means that changing attributes of thermo_temp or thermo_press will have no effect on this fix.

Like other fixes that perform thermostatting, this fix can be used with compute commands that remove a “bias” from the atom velocities. E.g. to apply the thermostat only to atoms within a spatial region, or to remove the center-of-mass velocity from a group of atoms, or to remove the x-component of velocity from the calculation.

This is not done by default, but only if the fix_modify command is used to assign a temperature compute to this fix that includes such a bias term. See the doc pages for individual compute temp commands to determine which ones include a bias. In this case, the thermostat works in the following manner: bias is removed from each atom, thermostatting is performed on the remaining thermal degrees of freedom, and the bias is added back in.

This fix can be used with either the verlet or respa integrators. When using this fix with respa, LAMMPS uses an integrator constructed according to the following factorization of the Liouville propagator (for two rRESPA levels):

\[\begin{split}\exp \left(\mathrm{i} L \Delta t \right) = & \hat{E} \exp \left(\mathrm{i} L_\mathrm{T\textrm{-}baro} \frac{\Delta t}{2} \right) \exp \left(\mathrm{i} L_\mathrm{T\textrm{-}part} \frac{\Delta t}{2} \right) \exp \left(\mathrm{i} L_{\epsilon , 2} \frac{\Delta t}{2} \right) \exp \left(\mathrm{i} L_{2}^{(2)} \frac{\Delta t}{2} \right) \\ &\times \left[ \exp \left(\mathrm{i} L_{2}^{(1)} \frac{\Delta t}{2n} \right) \exp \left(\mathrm{i} L_{\epsilon , 1} \frac{\Delta t}{2n} \right) \exp \left(\mathrm{i} L_1 \frac{\Delta t}{n} \right) \exp \left(\mathrm{i} L_{\epsilon , 1} \frac{\Delta t}{2n} \right) \exp \left(\mathrm{i} L_{2}^{(1)} \frac{\Delta t}{2n} \right) \right]^n \\ &\times \exp \left(\mathrm{i} L_{2}^{(2)} \frac{\Delta t}{2} \right) \exp \left(\mathrm{i} L_{\epsilon , 2} \frac{\Delta t}{2} \right) \exp \left(\mathrm{i} L_\mathrm{T\textrm{-}part} \frac{\Delta t}{2} \right) \exp \left(\mathrm{i} L_\mathrm{T\textrm{-}baro} \frac{\Delta t}{2} \right) \\ &+ \mathcal{O} \left(\Delta t^3 \right)\end{split}\]

This factorization differs somewhat from that of Tuckerman et al, in that the barostat is only updated at the outermost rRESPA level, whereas Tuckerman’s factorization requires splitting the pressure into pieces corresponding to the forces computed at each rRESPA level. In theory, the latter method will exhibit better numerical stability. In practice, because Pdamp is normally chosen to be a large multiple of the outermost rRESPA timestep, the barostat dynamics are not the limiting factor for numerical stability. Both factorizations are time-reversible and can be shown to preserve the phase space measure of the underlying non-Hamiltonian equations of motion.

Note

Under NPT dynamics, for a system with zero initial total linear momentum, the total momentum fluctuates close to zero. It may occasionally undergo brief excursions to non-negligible values, before returning close to zero. Over long simulations, this has the effect of causing the center-of-mass to undergo a slow random walk. This can be mitigated by resetting the momentum at infrequent intervals using the fix momentum command.

## Restart, fix_modify, output, run start/stop, minimize info

This fix writes the state of all the thermostat and barostat variables to binary restart files. See the read_restart command for info on how to re-specify a fix in an input script that reads a restart file, so that the operation of the fix continues in an uninterrupted fashion.

The fix_modify temp and press options are supported by this fix. You can use them to assign a compute you have defined to this fix which will be used in its thermostatting or barostatting procedure, as described above. If you do this, note that the kinetic energy derived from the compute temperature should be consistent with the virial term computed using all atoms for the pressure. LAMMPS will warn you if you choose to compute temperature on a subset of atoms.

Note

If both the temp and press keywords are used in a single thermo_modify command (or in two separate commands), then the order in which the keywords are specified is important. Note that a pressure compute defines its own temperature compute as an argument when it is specified. The temp keyword will override this (for the pressure compute being used by fix npt), but only if the temp keyword comes after the press keyword. If the temp keyword comes before the press keyword, then the new pressure compute specified by the press keyword will be unaffected by the temp setting.

The cumulative energy change in the system imposed by this fix, due to thermostatting and/or barostatting, is included in the thermodynamic output keywords ecouple and econserve. See the thermo_style page for details.

This fix computes a global scalar which can be accessed by various output commands. The scalar is the same cumulative energy change due to this fix described in the previous paragraph. The scalar value calculated by this fix is “extensive”.

This fix also computes a global vector of quantities, which can be accessed by various output commands. Rhe vector values are “intensive”.

The vector stores internal Nose/Hoover thermostat and barostat variables. The number and meaning of the vector values depends on which fix is used and the settings for keywords tchain and pchain, which specify the number of Nose/Hoover chains for the thermostat and barostat. If no thermostatting is done, then tchain is 0. If no barostatting is done, then pchain is 0. In the following list, “ndof” is 0, 1, 3, or 6, and is the number of degrees of freedom in the barostat. Its value is 0 if no barostat is used, else its value is 6 if any off-diagonal stress tensor component is barostatted, else its value is 1 if couple xyz is used or couple xy for a 2d simulation, otherwise its value is 3.

The order of values in the global vector and their meaning is as follows. The notation means there are tchain values for eta, followed by tchain for eta_dot, followed by ndof for omega, etc:

- eta[tchain] = particle thermostat displacements (unitless)

- eta_dot[tchain] = particle thermostat velocities (1/time units)

- omega[ndof] = barostat displacements (unitless)

- omega_dot[ndof] = barostat velocities (1/time units)

- etap[pchain] = barostat thermostat displacements (unitless)

- etap_dot[pchain] = barostat thermostat velocities (1/time units)

- PE_eta[tchain] = potential energy of each particle thermostat displacement (energy units)

- KE_eta_dot[tchain] = kinetic energy of each particle thermostat velocity (energy units)

- PE_omega[ndof] = potential energy of each barostat displacement (energy units)

- KE_omega_dot[ndof] = kinetic energy of each barostat velocity (energy units)

- PE_etap[pchain] = potential energy of each barostat thermostat displacement (energy units)

- KE_etap_dot[pchain] = kinetic energy of each barostat thermostat velocity (energy units)

- PE_strain[1] = scalar strain energy (energy units)

This fix can ramp its external temperature and pressure over multiple runs, using the start and stop keywords of the run command. See the run command for details of how to do this.

This fix is not invoked during energy minimization.

## Restrictions

This fix is part of the EXTRA-FIX package. It is only enabled if LAMMPS was built with that package. See the Build package page for more info.

X, y, z cannot be barostatted if the associated dimension is not periodic. Xy, xz, and yz can only be barostatted if the simulation domain is triclinic and the second dimension in the keyword (y dimension in xy) is periodic. Z, xz, and yz, cannot be barostatted for 2D simulations. The create_box, read data, and read_restart commands specify whether the simulation box is orthogonal or non-orthogonal (triclinic) and explain the meaning of the xy,xz,yz tilt factors.

For the temp keyword, the final Tstop cannot be 0.0 since it would make the external T = 0.0 at some timestep during the simulation which is not allowed in the Nose/Hoover formulation.

The scaleyz yes and scalexz yes keyword/value pairs can not be used for 2D simulations. scaleyz yes, scalexz yes, and scalexy yes options can only be used if the second dimension in the keyword is periodic, and if the tilt factor is not coupled to the barostat via keywords tri, yz, xz, and xy.

The alpha keyword modifies the barostat as per Miller et al. (Miller)_”#nc-Miller” so that the Cauchy stress is controlled. alpha is the non-dimensional parameter, typically set to 0.001 or 0.01 that determines how aggressively the algorithm drives the system towards the set Cauchy stresses. Larger values of alpha will modify the system more quickly, but can lead to instabilities. Smaller values will lead to longer convergence time. Since alpha also influences how much the stress fluctuations deviate from the equilibrium fluctuations, it should be set as small as possible.

A continue value of yes indicates that the fix is subsequent to a previous run with the npt/cauchy fix, and the intention is to continue from the converged stress state at the end of the previous run. This may be required, for example, when implementing a multi-step loading/unloading sequence over several fixes.

Setting alpha to zero is not permitted. To “turn off” the cauchystat control and thus restore the equilibrium stress fluctuations, two subsequent fixes should be used. In the first, fix npt/cauchy is used and the simulation box equilibrates to the correct shape for the desired stresses. In the second, fix npt is used instead which uses the original Parrinello-Rahman algorithm, but now with the corrected simulation box shape from using fix npt/cauchy.

This fix can be used with dynamic groups as defined by the group command. Likewise it can be used with groups to which atoms are added or deleted over time, e.g. a deposition simulation. However, the conservation properties of the thermostat and barostat are defined for systems with a static set of atoms. You may observe odd behavior if the atoms in a group vary dramatically over time or the atom count becomes very small.

## Related commands

fix nve, fix_modify, run_style

## Default

The keyword defaults are tchain = 3, pchain = 3, mtk = yes, tloop = ploop = 1, nreset = 0, drag = 0.0, dilate = all, couple = none, cauchystat = no, scaleyz = scalexz = scalexy = yes if periodic in second dimension and not coupled to barostat, otherwise no.

(Martyna) Martyna, Tobias and Klein, J Chem Phys, 101, 4177 (1994).

(Parrinello) Parrinello and Rahman, J Appl Phys, 52, 7182 (1981).

(Tuckerman) Tuckerman, Alejandre, Lopez-Rendon, Jochim, and Martyna, J Phys A: Math Gen, 39, 5629 (2006).

(Shinoda) Shinoda, Shiga, and Mikami, Phys Rev B, 69, 134103 (2004).

(Miller) Miller, Tadmor, Gibson, Bernstein and Pavia, J Chem Phys, 144, 184107 (2016).