# 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*fixedpoint*or*update**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*cauchystat*cauchystat values = alpha continue alpha = strength of Cauchy stress control parameter continue =*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
```

## 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 sub-domain.
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 doc 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, fix npt/cauchy can be used with compute commands that calculate a temperature after removing a “bias” from the atom velocities. E.g. removing the center-of-mass velocity from a group of atoms or only calculating temperature on the x-component of velocity or only calculating temperature for atoms in a geometric region. 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 commands to determine which ones include a bias. In this case, the thermostat works in the following manner: the current temperature is calculated taking the bias into account, 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):

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 fix_modify *energy* option is supported by this
fix to add the energy change induced by Nose/Hoover thermostatting
and barostatting to the system’s potential energy as part of
thermodynamic output.

This fix computes a global scalar and a global vector of quantities, which can be accessed by various output commands. The scalar value calculated by this fix is “extensive”; the vector values are “intensive”.

The scalar is the cumulative energy change due to the fix.

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 USER-MISC package. It is only enabled if LAMMPS was built with that package. See the Build package doc 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*.

Without the *cauchystat* keyword, the barostat algorithm
controls the Second-Piola Kirchhoff stress, which is a stress measure
referred to the unmodified (initial) simulation box. If the box
deforms substantially during the equilibration, the difference between
the set values and the final true (Cauchy) stresses can be
considerable.

The *cauchystat* 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, the
cauchystat flag is used and the simulation box equilibrates to the
correct shape for the desired stresses. In the second, the *fix*
statement is identical except that the *cauchystat* keyword is removed
(along with related *alpha* and *continue* values). This restores the
original Parrinello-Rahman algorithm, but now with the correct simulation
box shape from the first fix.

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.

## 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).