\(\renewcommand{\AA}{\text{Å}}\)

# fix pimd/langevin command

# fix pimd/nvt command

## Syntax

```
fix ID group-ID style keyword value ...
```

ID, group-ID are documented in fix command

style =

*pimd/langevin*or*pimd/nvt*= style name of this fix commandzero or more keyword/value pairs may be appended

keywords for style

*pimd/nvt**keywords*=*method*or*fmass*or*sp*or*temp*or*nhc**method*value =*pimd*or*nmpimd*or*cmd**fmass*value = scaling factor on mass*sp*value = scaling factor on Planck constant*temp*value = temperature (temperature units)*nhc*value = Nc = number of chains in Nose-Hoover thermostatkeywords for style

*pimd/langevin**keywords*=*method*or*integrator*or*ensemble*or*fmmode*or*fmass*or*scale*or*temp*or*thermostat*or*tau*or*iso*or*aniso*or*barostat*or*taup*or*fixcom*or*lj**method*value =*nmpimd*(default) or*pimd**integrator*value =*obabo*or*baoab**fmmode*value =*physical*or*normal**fmass*value = scaling factor on mass*temp*value = temperature (temperature unit) temperature = target temperature of the thermostat*thermostat*values = style seed style value =*PILE_L*seed = random number generator seed*tau*value = thermostat damping parameter (time unit)*scale*value = scaling factor of the damping times of non-centroid modes of PILE_L thermostat*iso*or*aniso*values = pressure (pressure unit) pressure = scalar external pressure of the barostat*barostat*value =*BZP*or*MTTK**taup*value = barostat damping parameter (time unit)*fixcom*value =*yes*or*no**lj*values = epsilon sigma mass planck mvv2e epsilon = energy scale for reduced units (energy units) sigma = length scale for reduced units (length units) mass = mass scale for reduced units (mass units) planck = Planck's constant for other unit style mvv2e = mass * velocity^2 to energy conversion factor for other unit style

## Examples

```
fix 1 all pimd/nvt method nmpimd fmass 1.0 sp 2.0 temp 300.0 nhc 4
fix 1 all pimd/langevin ensemble npt integrator obabo temp 113.15 thermostat PILE_L 1234 tau 1.0 iso 1.0 barostat BZP taup 1.0
```

## Description

Changed in version 28Mar2023.

Fix pimd was renamed to fix *pimd/nvt* and fix *pimd/langevin* was added.

These fix commands perform quantum molecular dynamics simulations based on the Feynman path-integral to include effects of tunneling and zero-point motion. In this formalism, the isomorphism of a quantum partition function for the original system to a classical partition function for a ring-polymer system is exploited, to efficiently sample configurations from the canonical ensemble (Feynman).

The classical partition function and its components are given by the following equations:

\(M_i\) is the fictitious mass of the \(i\)-th mode, and m is the actual mass of the atoms.

The interested user is referred to any of the numerous references on this methodology, but briefly, each quantum particle in a path integral simulation is represented by a ring-polymer of P quasi-beads, labeled from 1 to P. During the simulation, each quasi-bead interacts with beads on the other ring-polymers with the same imaginary time index (the second term in the effective potential above). The quasi-beads also interact with the two neighboring quasi-beads through the spring potential in imaginary-time space (first term in effective potential). To sample the canonical ensemble, any thermostat can be applied.

Fix *pimd/nvt* applies a Nose-Hoover massive chain thermostat
(Tuckerman). With the massive chain
algorithm, a chain of NH thermostats is coupled to each degree of
freedom for each quasi-bead. The keyword *temp* sets the target
temperature for the system and the keyword *nhc* sets the number *Nc* of
thermostats in each chain. For example, for a simulation of N particles
with P beads in each ring-polymer, the total number of NH thermostats
would be 3 x N x P x Nc.

Fix *pimd/langevin* implements a Langevin thermostat in the normal mode
representation, and also provides a barostat to sample the NPH/NPT ensembles.

Note

Both these *fix* styles implement a complete velocity-verlet integrator
combined with a thermostat, so no other time integration fix should be used.

The *method* keyword determines what style of PIMD is performed. A
value of *pimd* is standard PIMD. A value of *nmpimd* is for
normal-mode PIMD. A value of *cmd* is for centroid molecular dynamics
(CMD). The difference between the styles is as follows.

In standard PIMD, the value used for a bead’s fictitious mass is arbitrary. A common choice is to use \(M_i = m/P\), which results in the mass of the entire ring-polymer being equal to the real quantum particle. But it can be difficult to efficiently integrate the equations of motion for the stiff harmonic interactions in the ring polymers.

A useful way to resolve this issue is to integrate the equations of motion in a normal mode representation, using Normal Mode Path-Integral Molecular Dynamics (NMPIMD) (Cao1). In NMPIMD, the NH chains are attached to each normal mode of the ring-polymer and the fictitious mass of each mode is chosen as Mk = the eigenvalue of the Kth normal mode for k > 0. The k = 0 mode, referred to as the zero-frequency mode or centroid, corresponds to overall translation of the ring-polymer and is assigned the mass of the real particle.

Note

Motion of the centroid can be effectively uncoupled from the other normal modes by scaling the fictitious masses to achieve a partial adiabatic separation. This is called a Centroid Molecular Dynamics (CMD) approximation (Cao2). The time-evolution (and resulting dynamics) of the quantum particles can be used to obtain centroid time correlation functions, which can be further used to obtain the true quantum correlation function for the original system. The CMD method also uses normal modes to evolve the system, except only the k > 0 modes are thermostatted, not the centroid degrees of freedom.

Added in version 21Nov2023: Mode *pimd* added to fix pimd/langevin.

Fix pimd/langevin supports the *method* values *nmpimd* and *pimd*. The default value is *nmpimd*.
If *method* is *nmpimd*, the normal mode representation is used to integrate the equations of motion.
The exact solution of harmonic oscillator is used to propagate the free ring polymer part of the Hamiltonian.
If *method* is *pimd*, the Cartesian representation is used to integrate the equations of motion.
The harmonic force is added to the total force of the system, and the numerical integrator is used to propagate the Hamiltonian.

The keyword *integrator* specifies the Trotter splitting method used by *fix pimd/langevin*.
See (Liu) for a discussion on the OBABO and BAOAB splitting schemes. Typically
either of the two should work fine.

The keyword *fmass* sets a further scaling factor for the fictitious
masses of beads, which can be used for the Partial Adiabatic CMD
(Hone), or to be set as P, which results in the fictitious
masses to be equal to the real particle masses.

The keyword *fmmode* of *fix pimd/langevin* determines the mode of fictitious
mass preconditioning. There are two options: *physical* and *normal*. If *fmmode* is
*physical*, then the physical mass of the particles are used (and then multiplied by
*fmass*). If *fmmode* is *normal*, then the physical mass is first multiplied by the
eigenvalue of each normal mode, and then multiplied by *fmass*. More precisely, the
fictitious mass of *fix pimd/langevin* is determined by two factors: *fmmode* and *fmass*.
If *fmmode* is *physical*, then the fictitious mass is

If *fmmode* is *normal*, then the fictitious mass is

where \(\lambda_i\) is the eigenvalue of the \(i\)-th normal mode.

Note

Fictitious mass is only used in the momentum of the equation of motion (\(\mathbf{p}_i=M_i\mathbf{v}_i\)), and not used in the spring elastic energy (\(\sum_{i=1}^P \frac{1}{2}m\omega_P^2(q_i - q_{i+1})^2\), \(m\) is always the actual mass of the particles).

The keyword *sp* is a scaling factor on Planck’s constant, which can
be useful for debugging or other purposes. The default value of 1.0
is appropriate for most situations.

The keyword *ensemble* for fix style *pimd/langevin* determines which ensemble is it
going to sample. The value can be *nve* (microcanonical), *nvt* (canonical), *nph* (isoenthalpic),
and *npt* (isothermal-isobaric).

The keyword *temp* specifies temperature parameter for fix styles *pimd/nvt* and *pimd/langevin*. It should read
a positive floating-point number.

Note

For pimd simulations, a temperature values should be specified even for nve ensemble. Temperature will make a difference for nve pimd, since the spring elastic frequency between the beads will be affected by the temperature.

The keyword *thermostat* reads *style* and *seed* of thermostat for fix style *pimd/langevin*. *style* can only
be *PILE_L* (path integral Langevin equation local thermostat, as described in Ceriotti), and *seed* should a positive integer number, which serves as the seed of the pseudo random number generator.

Note

The fix style *pimd/langevin* uses the stochastic PILE_L thermostat to control temperature. This thermostat works on the normal modes
of the ring polymer. The *tau* parameter controls the centroid mode, and the *scale* parameter controls the non-centroid modes.

The keyword *tau* specifies the thermostat damping time parameter for fix style *pimd/langevin*. It is in time unit. It only works on the centroid mode.

The keyword *scale* specifies a scaling parameter for the damping times of the non-centroid modes for fix style *pimd/langevin*. The default
damping time of the non-centroid mode \(i\) is \(\frac{P}{\beta\hbar}\sqrt{\lambda_i\times\mathrm{fmass}}\) (*fmmode* is *physical*) or \(\frac{P}{\beta\hbar}\sqrt{\mathrm{fmass}}\) (*fmmode* is *normal*). The damping times of all non-centroid modes are the default values divided by *scale*.

The barostat parameters for fix style *pimd/langevin* with *npt* or *nph* ensemble is specified using one of *iso* and *aniso*
keywords. A *pressure* value should be given with pressure unit. The keyword *iso* means couple all 3 diagonal components together when pressure is computed (hydrostatic pressure), and dilate/contract the dimensions together. 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.

The keyword *barostat* reads *style* of barostat for fix style *pimd/langevin*. *style* can
be *BZP* (Bussi-Zykova-Parrinello, as described in Bussi) or *MTTK* (Martyna-Tuckerman-Tobias-Klein, as described in Martyna1 and Martyna2).

The keyword *taup* specifies the barostat damping time parameter for fix style *pimd/langevin*. It is in time unit.

The keyword *fixcom* specifies whether the center-of-mass of the extended ring-polymer system is fixed during the pimd simulation.
Once *fixcom* is set to be *yes*, the center-of-mass velocity will be distracted from the centroid-mode velocities in each step.

The keyword *lj* should be used if lj units is used for *fix pimd/langevin*. Typically one may want to use
reduced units to run the simulation, and then convert the results into some physical units (for example, metal units). In this case, the 5 quantities in the physical mass units are needed: epsilon (energy scale), sigma (length scale), mass, Planck’s constant, mvv2e (mass * velocity^2 to energy conversion factor). Planck’s constant and mvv2e can be found in src/update.cpp. If there is no need to convert reduced units to physical units, you can omit the keyword *lj* and these five values will be set to 1.

The PIMD algorithm in LAMMPS is implemented as a hyper-parallel scheme as described in Calhoun. In LAMMPS this is done by using multi-replica feature in LAMMPS, where each quasi-particle system is stored and simulated on a separate partition of processors. The following diagram illustrates this approach. The original system with 2 ring polymers is shown in red. Since each ring has 4 quasi-beads (imaginary time slices), there are 4 replicas of the system, each running on one of the 4 partitions of processors. Each replica (shown in green) owns one quasi-bead in each ring.

To run a PIMD simulation with M quasi-beads in each ring polymer using N MPI tasks for each partition’s domain-decomposition, you would use P = MxN processors (cores) and run the simulation as follows:

```
mpirun -np P lmp_mpi -partition MxN -in script
```

Note that in the LAMMPS input script for a multi-partition simulation, it is often very useful to define a uloop-style variable such as

```
variable ibead uloop M pad
```

where M is the number of quasi-beads (partitions) used in the calculation. The uloop variable can then be used to manage I/O related tasks for each of the partitions, e.g.

```
dump dcd all dcd 10 system_${ibead}.dcd
dump 1 all custom 100 ${ibead}.xyz id type x y z vx vy vz ix iy iz fx fy fz
restart 1000 system_${ibead}.restart1 system_${ibead}.restart2
read_restart system_${ibead}.restart2
```

Note

Fix *pimd/langevin* dumps the Cartesian coordinates, but dumps the velocities and
forces in the normal mode representation. If the Cartesian velocities and forces are
needed, it is easy to perform the transformation when doing post-processing.

It is recommended to dump the image flags (*ix iy iz*) for fix *pimd/langevin*. It
will be useful if you want to calculate some estimators during post-processing.

Major differences of *fix pimd/nvt* and *fix pimd/langevin* are:

Fix pimd/nvtincludes Cartesian pimd, normal mode pimd, and centroid md.Fix pimd/langevinonly intends to support normal mode pimd, as it is commonly enough for thermodynamic sampling.

Fix pimd/nvtuses Nose-Hoover chain thermostat.Fix pimd/langevinuses Langevin thermostat.

Fix pimd/langevinprovides barostat, so the npt ensemble can be sampled.Fix pimd/nvtonly support nvt ensemble.

Fix pimd/langevinprovides several quantum estimators in output.

Fix pimd/langevinallows multiple processes for each bead. Forfix pimd/nvt, there is a large chance that multi-process tasks for each bead may fail.The dump of

fix pimd/nvtare all Cartesian.Fix pimd/langevindumps normal-mode velocities and forces, and Cartesian coordinates.

Initially, the inter-replica communication and normal mode transformation parts of *fix pimd/langevin* are written based on
those of *fix pimd/nvt*, but are significantly revised.

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

Fix *pimd/nvt* writes the state of the Nose/Hoover thermostat over all
quasi-beads 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.

Fix *pimd/langevin* writes the state of the barostat overall beads to
binary restart files. Since it uses a stochastic thermostat,
the state of the thermostat is not written. However, the state of the system
can be restored by reading the restart file, except that it will re-initialize
the random number generator.

None of the fix_modify options are relevant to fix pimd/nvt.

Fix *pimd/nvt* computes a global 3-vector, which can be accessed by
various output commands. The three quantities in
the global vector are:

the total spring energy of the quasi-beads,

the current temperature of the classical system of ring polymers,

the current value of the scalar virial estimator for the kinetic energy of the quantum system (Herman).

The vector values calculated by fix *pimd/nvt* are “extensive”, except for the
temperature, which is “intensive”.

Fix *pimd/langevin* computes a global vector of quantities, which
can be accessed by various output commands. Note that
it outputs multiple log files, and different log files contain information
about different beads or modes (see detailed explanations below). If *ensemble*
is *nve* or *nvt*, the vector has 10 values:

kinetic energy of the normal mode

spring elastic energy of the normal mode

potential energy of the bead

total energy of all beads (conserved if

ensembleisnve)primitive kinetic energy estimator

virial energy estimator

centroid-virial energy estimator

primitive pressure estimator

thermodynamic pressure estimator

centroid-virial pressure estimator

The first 3 are different for different log files, and the others are the same for different log files.

If *ensemble* is *nph* or *npt*, the vector stores internal variables of the barostat. If *iso* is used,
the vector has 15 values:

kinetic energy of the normal mode

spring elastic energy of the normal mode

potential energy of the bead

total energy of all beads (conserved if

ensembleisnve)primitive kinetic energy estimator

virial energy estimator

centroid-virial energy estimator

primitive pressure estimator

thermodynamic pressure estimator

centroid-virial pressure estimator

barostat velocity

barostat kinetic energy

barostat potential energy

barostat cell Jacobian

enthalpy of the extended system (sum of 4, 12, 13, and 14; conserved if

ensembleisnph)

If *aniso* or *x* or *y* or *z* is used for the barostat, the vector has 17 values:

kinetic energy of the normal mode

spring elastic energy of the normal mode

potential energy of the bead

total energy of all beads (conserved if

ensembleisnve)primitive kinetic energy estimator

virial energy estimator

centroid-virial energy estimator

primitive pressure estimator

thermodynamic pressure estimator

centroid-virial pressure estimator

x component of barostat velocity

y component of barostat velocity

z component of barostat velocity

barostat kinetic energy

barostat potential energy

barostat cell Jacobian

enthalpy of the extended system (sum of 4, 14, 15, and 16; conserved if

ensembleisnph)

No parameter of fix *pimd/nvt* or *pimd/langevin* can be used with the *start/stop* keywords
of the run command. Fix *pimd/nvt* or *pimd/langevin* is not invoked during
energy minimization.

## Restrictions

These fixes are part of the REPLICA package. They are only enabled if LAMMPS was built with that package. See the Build package page for more info.

Fix *pimd/nvt* cannot be used with lj units.
Fix *pimd/langevin* can be used with lj units. See the above part for how to use it.

A PIMD simulation can be initialized with a single data file read via the read_data command. However, this means all quasi-beads in a ring polymer will have identical positions and velocities, resulting in identical trajectories for all quasi-beads. To avoid this, users can simply initialize velocities with different random number seeds assigned to each partition, as defined by the uloop variable, e.g.

```
velocity all create 300.0 1234${ibead} rot yes dist gaussian
```

## Default

The keyword defaults for fix *pimd/nvt* are method = pimd, fmass = 1.0, sp
= 1.0, temp = 300.0, and nhc = 2.

**(Feynman)** R. Feynman and A. Hibbs, Chapter 7, Quantum Mechanics and
Path Integrals, McGraw-Hill, New York (1965).

**(Tuckerman)** M. Tuckerman and B. Berne, J Chem Phys, 99, 2796 (1993).

**(Cao1)** J. Cao and B. Berne, J Chem Phys, 99, 2902 (1993).

**(Cao2)** J. Cao and G. Voth, J Chem Phys, 100, 5093 (1994).

**(Hone)** T. Hone, P. Rossky, G. Voth, J Chem Phys, 124,
154103 (2006).

**(Calhoun)** A. Calhoun, M. Pavese, G. Voth, Chem Phys Letters, 262,
415 (1996).

**(Herman)** M. F. Herman, E. J. Bruskin, B. J. Berne, J Chem Phys, 76, 5150 (1982).

**(Bussi)** G. Bussi, T. Zykova-Timan, M. Parrinello, J Chem Phys, 130, 074101 (2009).

**(Ceriotti)** M. Ceriotti, M. Parrinello, T. Markland, D. Manolopoulos, J. Chem. Phys. 133, 124104 (2010).

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**(Martyna2)** G. Martyna, A. Hughes, M. Tuckerman, J. Chem. Phys. 110, 3275 (1999).

**(Liu)** J. Liu, D. Li, X. Liu, J. Chem. Phys. 145, 024103 (2016).