Action: LOGMFD

Module	logmfd
Description	Usage
Used to perform LogMFD, LogPD, and TAMD/d-AFED.

Details and examples

Used to perform LogMFD, LogPD, and TAMD/d-AFED.

LogMFD

Consider a physical system of $N_q$ particles, for which the Hamiltonian is given as

$${H_{\rm MD}}\left( {\bf{\Gamma}} \right) = \sum\limits_{j = 1}^{{N_q}} {\frac{{{\bf{p}}_j^2}}{{2{m_j}}}} + \Phi \left( {\bf{q}} \right)$$

where ${\bf q}_j$, ${\bf p}_j$ ($\bf{\Gamma}={\bf q},{\bf p}$), and $m_j$ are the position, momentum, and mass of particle $j$ respectively, and $\Phi$ is the potential energy function for ${\bf q}$. The free energy $F({\bf X})$ as a function of a set of $N$ collective variables (CVs) is given as

$$\begin{aligned} F\left( {{\bf X}} \right) &= - {k_B}T\log \int {\exp \left[ { - \beta {H_{\rm MD}}} \right]\prod\limits_{i = 1}^N {\delta \left( {{s_i}\left( {{\bf q}} \right) - {X_i}} \right)} d{\bf{\Gamma }}} \\ &\simeq - {k_B}T\log \int {\exp \left[ { - \beta \left\{ {{H_{\rm MD}} + \sum\limits_i^N {\frac{{{K_i}}}{2}{{\left( {{s_i}\left( {{\bf q}} \right) - {X_i}} \right)}^2}} } \right\}} \right]} d{\bf{\Gamma }} \end{aligned}$$

where $\bf{s}$ are the CVs, $k_B$ is Boltzmann constant, $\beta=1/k_BT$, and $K_i$ is the spring constant which is large enough to invoke

$$\delta \left( x \right) = \lim_{k \to \infty } \sqrt {\beta k/2\pi} \exp \left( -\beta kx^2/2 \right)$$

In mean-force dynamics, ${\bf X}$ are treated as fictitious dynamical variables, which are associated with the following Hamiltonian,

$${H_{\rm log}} = \sum\limits_{i = 1}^N {\frac{{P_{{X_i}}^2}}{{2{M_i}}} + \Psi \left( {{\bf X}} \right)}$$

where ${P_{{X_i}}}$ and $M_i$ are the momentum and mass of $X_i$, respectively, and $\Psi$ is the potential function for ${\bf X}$. We assume that $\Psi$ is a functional of $F({\bf X})$; $Ψ[F({\bf X})]$. The simplest form of $\Psi$ is $\Psi = F({\bf X})$, which corresponds to the TAMD/d-AFED methods that are discussed in the first two papers cited below (or the extended Lagrangian dynamics in the limit of the adiabatic decoupling between $\bf{q}$ and ${\bf X}$). In the case of LogMFD, the following form of $\Psi$ is introduced and discussed in the third paper cited below.

$${\Psi _{\rm log}}\left( {{\bf X}} \right) = \gamma \log \left[ {\alpha F\left( {{\bf X}} \right) + 1} \right]$$

where $\alpha$ (ALPHA) and $\gamma$ (GAMMA) are positive parameters. The logarithmic form of $\Psi_{\rm log}$ ensures the dynamics of ${\bf X}$ on a much smoother energy surface [i.e., $\Psi_{\rm log}({\bf X})$] than $F({\bf X})$, thus enhancing the sampling in the ${\bf X}$-space. The parameters $\alpha$ and $\gamma$ determine the degree of flatness of $\Psi_{\rm log}$, but adjusting only $\alpha$ is normally sufficient to have a relatively flat surface (with keeping the relation $\gamma=1/\alpha$).

The equation of motion for $X_i$ in LogMFD (no thermostat) is

$${M_i}{\ddot X_i} = - \left( {\frac{{\alpha \gamma }}{{\alpha F + 1}}} \right)\frac{{\partial F}}{{\partial {X_i}}}$$

where $-\partial F/\partial X_i$ is evaluated as a canonical average under the condition that ${\bf X}$ is fixed;

$$\begin{aligned} - \frac{{\partial F}}{{\partial {X_i}}} &\simeq \frac{1}{Z}\int {{K_i}\left( {{s_i}\left( {{\bf q}} \right) - {X_i}} \right)\exp \left[ { - \beta \left\{ {{H_{\rm MD}} + \sum\limits_i^N {\frac{{{K_i}}}{2}{{\left( {{s_i}\left( {{\bf q}} \right) - {X_i}} \right)}^2}} } \right\}} \right]} d{\bf{\Gamma }}\\ &\equiv {\left\langle {{K_i}\left( {{s_i}\left( {{\bf q}} \right) - {X_i}} \right)} \right\rangle _{{\bf X}}} \end{aligned}$$

where

$$Z = \int {\exp \left[ { - \beta \left\{ {{H_{\rm MD}} + \sum\limits_i^N {\frac{{{K_i}}}{2}{{\left( {{s_i}\left( {{\bf q}} \right) - {X_i}} \right)}^2}} } \right\}} \right]} d{\bf{\Gamma }}$$

The mean-force (MF) is practically evaluated by performing a shot-time canonical MD run of $N_m$ steps each time ${\bf X}$ is updated according to the equation of motion for ${\bf X}$.

If the canonical average for the MF is effectively converged, the dynamical variables ${\bf q}$ and ${\bf X}$ are decoupled and they evolve adiabatically, which can be exploited for the on-the-fly evaluation of $F({\bf X})$. I.e., $H_{\rm log}$ should be a constant of motion in this case, thus $F({\bf X})$ can be evaluated each time ${\bf X}$ is updated as

$$F\left( {{{\bf X}}\left( t \right)} \right) = \frac{1}{\alpha} \left[ \exp \frac{1}{\gamma} \left\{ \left( H_{\rm log} - \sum_i \frac{P_{X_i}^2}{2M_i} \right) \right\} - 1 \right]$$

This means that $F({\bf X})$ can be constructed without post processing (on-the-fly free energy reconstruction). Note that the on-the-fly free energy reconstruction is also possible in TAMD/d-AFED if the Hamiltonian-like conserved quantity is available (e.g., the Nose-Hoover type dynamics).

LogPD

The accuracy in the MF is critical to the on-the-fly free energy reconstruction. To improve the evaluation of the MF, parallel-dynamics (PD) is incorporated into LogMFD, leading to logarithmic parallel-dynamics (LogPD) that is discussed in the fourth paper in the reference section.

In PD, the MF is evaluated by a non-equilibrium path-ensemble based on the Crooks-Jarzynski non-equilibrium work relation. To this end, multiple replicas of the MD system which run in parallel are introduced. The CVs [${\bf s}({\bf q})$] in each replica is restrained to the same value of ${\bf X}(t)$. A canonical MD run with $N_m$ steps is performed in each replica, then the MF on $X_i$ is evaluated using the MD trajectories from all replicas. The MF is practically calculated as

$$- \frac{{\partial F}}{{\partial {X_i}}} = \sum\limits_{k = 1}^{{N_r}} {{W_k}} \sum\limits_{n = 1}^{{N_m}} {\frac{1}{{{N_m}}}{K_i}\left[ {{s_i}\left( {{{\bf q}}_n^k} \right) - {X_i}} \right]}$$

where ${\bf q}^k_n$ indicates the ${\bf q}$-configuration at time step $n$ in the $N_m$-step shot-time MD run in the $k$th replica among $N_r$ replicas. $W_k$ is given as

$${W_k} = \frac{{{e^{ - \beta {w_k}\left( t \right)}}}}{{\sum\limits_{k=1}^{{N_r}} {{e^{ - \beta {w_k}\left( t \right)}}} }}$$

where

$${w_k}\left( t \right) = \int\limits_{{X_0}}^{X\left( t \right)} {\sum\limits_{i=1}^N {\frac{{\partial H_{\rm MD}^k}}{{\partial {X_i}}}d{X_i}} }$$

$$H_{\rm MD}^k\left( {{\bf{\Gamma }},{{\bf X}}} \right) = {H_{\rm MD}}\left( {{{\bf{\Gamma }}^k}} \right) + \sum\limits_{i = 1}^N {\frac{{{K_i}}}{2}{{\left( {s_i^k - {X_i}} \right)}^2}}$$

and $s^k_i$ is the $i$th CV in the $k$th replica.

$W_k$ comes from the Crooks-Jarzynski non-equilibrium work relation by which we can evaluate an equilibrium ensemble average from a set of non-equilibrium trajectories. Note that, to avoid possible numerical errors in the exponential function, the following form of $W_k$ is instead used in PLUMED,

$${W_k}\left( t \right) = \frac{{\exp \left[ { - \beta \left\{ {{w_k}\left( t \right) - {w_{\min }}\left( t \right)} \right\}} \right]}}{{\sum\nolimits_k {\exp \left[ { - \beta \left\{ {{w_k}\left( t \right) - {w_{\min }}\left( t \right)} \right\}} \right]} }}$$

where

$${w_{\min }}\left( t \right) = {\rm Min}_k\left[ {{w_k}\left( t \right)} \right]$$

With the MF evaluated using the PD approach, free energy profiles can be reconstructed more efficiently (requiring less elapsed computing time) in LogPD than with a single MD system in LogMFD. In the case that there exist more than one stable state separated by high energy barriers in the hidden subspace orthogonal to the CV-subspace, LogPD is particularly of use to incorporate all the contributions from such hidden states with appropriate weights (in the limit $N_r\to\infty$ ).

Note that LogPD calculations should always be initiated with an equilibrium ${\bf q}$-configuration in each replica, because the Crooks-Jarzynski non-equilibrium work relation is invoked. Also note that LogPD is currently available only with Gromacs, while LogMFD can be performed with LAMMPS, Gromacs, Quantum Espresso, NAMD, and so on.

Using LogMFD/PD with a thermostat

Introducing a thermostat on ${\bf X}$ is often recommended in LogMFD/PD to maintain the adiabatic decoupling between ${\bf q}$ and ${\bf X}$. In the framework of the LogMFD approach, the Nose-Hoover type thermostat and the Gaussian isokinetic (velocity scaling) thermostat can be used to control the kinetic energy of ${\bf X}$.

Nose-Hoover LogMFD/PD

The equation of motion for $X_i$ coupled to a Nose-Hoover thermostat variable $\eta$ (single heat bath) is

$${M_i}{\ddot X_i} = - \left( {\frac{{\alpha \gamma }}{{\alpha F + 1}}} \right)\frac{{\partial F}}{{\partial {X_i}}} - {M_i}{\dot X_i}\dot \eta$$

The equation of motion for $\eta$ is

$$Q\ddot \eta = \sum\limits_{i = 1}^N {\frac{{P_{{X_i}}^2}}{{{M_i}}} - N{k_B}T}$$

where $N$ is the number of the CVs. Since the following pseudo-Hamiltonian is a constant of motion in Nose-Hoover LogMFD/PD,

$$H_{\rm log}^{\rm NH} = \sum\limits_{i = 1}^N {\frac{{P_{{X_i}}^2}}{{2{M_i}}} + \gamma \log \left[ {\alpha F\left( {{\bf X}} \right) + 1} \right] + \frac{1}{2}Q{{\dot \eta }^2} + N{k_B}T\eta}$$

$F({\bf X}(t))$ is obtained at each MFD step as

$$F\left( {{{\bf X}}\left( t \right)} \right) = \frac{1}{\alpha }\left[ {\exp \left\{ {{{ \frac{1}{\gamma} \left( {H_{\rm log}^{\rm NH} - \sum_i {\frac{{P_{{X_i}}^2}}{{2{M_i}}}} - \frac{1}{2}Q{{\dot \eta }^2} - N{k_B}T\eta} \right)} }} \right\} - 1} \right]$$

Velocity scaling LogMFD/PD

The velocity scaling algorithm (which is equivalent to the Gaussian isokinetic dynamics in the limit $\Delta t\to 0$) can also be employed to control the velocity of ${\bf X}$, $\bf{V}_x$.

The following algorithm is introduced to perform the isokinetic LogMFD calculations that are discussed in the fifth paper cited belo;

$$\begin{aligned} {V_{{X_i}}}\left( {{t_{n + 1}}} \right) &= V_{X_i}^\prime \left( {{t_n}} \right) + \Delta t \left[ { - \left( {\frac{{\alpha \gamma }}{{\alpha F\left( {{t_n}} \right) + 1}}} \right) \frac{{\partial F\left( {{t_n}} \right)}}{{\partial {X_i}}}} \right]\\ S\left( {{t_{n + 1}}} \right) &= \sqrt {\frac{{N{k_B}T}}{{\sum\limits_i {{M_i}V_{{X_i}}^2\left( {{t_{n + 1}}} \right)} }}} \\ {V_{{X_i}}}^\prime \left( {{t_{n + 1}}} \right) &= S\left( {{t_{n + 1}}} \right){V_{{X_i}}}\left( {{t_{n + 1}}} \right)\\ {X_i}\left( {{t_{n + 1}}} \right) &= {X_i}\left( {{t_n}} \right) + \Delta t V_{X_i}^\prime \left( {{t_{n + 1}}} \right)\\ {\Psi_{\rm log}}\left( {{t_{n + 1}}} \right) &= N{k_B}T\log S\left( {{t_{n + 1}}} \right) + {\Psi_{\rm log}}\left( {{t_n}} \right)\\ F\left( {{t_{n + 1}}} \right) &= \frac{1}{\alpha} \left[ \exp \left\{ \Psi_{\rm log} \left( t_{n+1} \right) / \gamma \right\} - 1 \right] \end{aligned}$$

Note that $V_{X_i}^\prime\left( {{t_0}} \right)$ is assumed to be initially given, which meets the following relation,

$$\sum\limits_{i = 1}^N M_i V_{X_i}^{\prime 2} \left( t_0 \right) = N{k_B}{T}$$

It should be stressed that, in the same way as in the NVE and Nose-Hoover LogMFD/PD, $F({\bf X}(t))$ can be evaluated at each MFD step (on-the-fly free energy reconstruction) in Velocity Scaling LogMFD/PD.

Examples

Example-LoGMFD Example of LogMFD

The following input file is called plumed.dat and tells plumed to restrain collective variables to the fictitious dynamical variables in LogMFD/PD.

Click on the labels of the actions for more information on what each action computes

UNITSThis command sets the internal units for the code. More details TIMEthe units of time=fs LENGTHthe units of lengths=1.0 ENERGYthe units of energy=kcal/mol MASSthe units of masses=1.0 CHARGEthe units of charges=1.0
The UNITS action with label  calculates somethingphiThe TORSION action with label phi calculates the following quantities:
 Quantity   
 Type   
 Description  
phi
scalar
the TORSION involving these atoms: TORSIONCalculate one or multiple torsional angles. More details ATOMSthe four atoms involved in the torsional angle=5,7,9,15
psiThe TORSION action with label psi calculates the following quantities:
 Quantity   
 Type   
 Description  
psi
scalar
the TORSION involving these atoms: TORSIONCalculate one or multiple torsional angles. More details ATOMSthe four atoms involved in the torsional angle=7,9,15,17

# LogMFD
LOGMFDUsed to perform LogMFD, LogPD, and TAMD/d-AFED. More details ...
LABELa label for the action so that its output can be referenced in the input to other actions=logmfdThe LOGMFD action with label logmfd calculates the following quantities:
 Quantity   
 Type   
 Description  
logmfd.bias
scalar
the instantaneous value of the bias potential
logmfd.phi_fict
scalar
For example, the fictitious collective variable for LogMFD is specified as ARG=dist12 and LABEL=logmfd in LOGMFD section in Plumed input file, the associated fictitious dynamical variable can be specified as PRINT ARG=dist12,logmfd.dist12_fict FILE=COLVAR This particular component measures this quantity for the input CV named phi
logmfd.phi_vfict
scalar
For example, the fictitious collective variable for LogMFD is specified as ARG=dist12 and LABEL=logmfd in LOGMFD section in Plumed input file, the velocity of the associated fictitious dynamical variable can be specified as PRINT ARG=dist12,logmfd.dist12_vfict FILE=COLVAR This particular component measures this quantity for the input CV named phi
logmfd.psi_fict
scalar
For example, the fictitious collective variable for LogMFD is specified as ARG=dist12 and LABEL=logmfd in LOGMFD section in Plumed input file, the associated fictitious dynamical variable can be specified as PRINT ARG=dist12,logmfd.dist12_fict FILE=COLVAR This particular component measures this quantity for the input CV named psi
logmfd.psi_vfict
scalar
For example, the fictitious collective variable for LogMFD is specified as ARG=dist12 and LABEL=logmfd in LOGMFD section in Plumed input file, the velocity of the associated fictitious dynamical variable can be specified as PRINT ARG=dist12,logmfd.dist12_vfict FILE=COLVAR This particular component measures this quantity for the input CV named psi
ARGthe labels of the scalars on which the bias will act=phi,psi
KAPPASpring constant of the harmonic restraining potential=1000.0,1000.0
DELTA_TTime step for the fictitious dynamical variables (DELTA_T=1 often works)=1.0
INTERVALPeriod of MD steps (N_m) to update fictitious dynamical variables=200
TEMPTarget temperature for the fictitious dynamical variables in LogMFD/PD=300.0
FLOGThe initial free energy value in the LogMFD/PD run=2.0
MFICTMasses of each fictitious dynamical variable=5000000,5000000
VFICTThe initial velocities of the fictitious dynamical variables=3.5e-4,3.5e-4
ALPHAAlpha parameter for LogMFD=4.0
THERMOSTATType of thermostat for the fictitious dynamical variables=NVE
FICT_MAXMaximum values reachable for the fictitious dynamical variables=3.15,3.15
FICT_MINMinimum values reachable for the fictitious dynamical variables=-3.15,-3.15
... LOGMFD

To submit this simulation with Gromacs, use the following command line to execute a LogMFD run with Gromacs-MD. Here topol.tpr is the input file which contains atomic coordinates and Gromacs parameters.

gmx_mpi mdrun -s topol.tpr -plumed

This command will output files named logmfd.out and replica.out.

The output file logmfd.out records free energy and all fictitious dynamical variables at every MFD step.

# LogMFD
# CVs : phi psi
# Mass for CV particles : 5000000.000000 5000000.000000
# Mass for thermostat   :   11923.224809
# 1:iter_mfd, 2:Flog, 3:2*Ekin/gkb[K], 4:eta, 5:Veta,
# 6:phi_fict(t), 7:phi_vfict(t), 8:phi_force(t),
# 9:psi_fict(t), 10:psi_vfict(t), 11:psi_force(t),
       0       2.000000     308.221983       0.000000       0.000000      -2.442363       0.000350       5.522717       2.426650       0.000350       7.443177
       1       1.995466     308.475775       0.000000       0.000000      -2.442013       0.000350      -4.406246       2.427000       0.000350      11.531345
       2       1.992970     308.615664       0.000000       0.000000      -2.441663       0.000350      -3.346513       2.427350       0.000350      15.763196
       3       1.988619     308.859946       0.000000       0.000000      -2.441313       0.000350       6.463092       2.427701       0.000351       6.975422
...

The output file replica.out records all collective variables at every MFD step.

 Replica No. 0 of 1.
# 1:iter_mfd, 2:work, 3:weight,
# 4:phi(q)
# 5:psi(q)
       0    0.000000e+00     1.000000e+00       -2.436841       2.434093
       1   -4.539972e-03     1.000000e+00       -2.446420       2.438531
       2   -7.038516e-03     1.000000e+00       -2.445010       2.443114
       3   -1.139672e-02     1.000000e+00       -2.434850       2.434677
...

Example of LogPD

Use the following command line to execute a LogPD run using two MD replicas (note that only Gromacs is currently available for LogPD). Here 0/topol.tpr and 1/topol.tpr are the input files, each containing the atomic coordinates for the corresponding replica and Gromacs parameters. All the directories, 0/ and 1/, should contain the same plumed.dat.

mpirun -np 2 gmx_mpi mdrun -s topol -plumed -multidir 0 1

This command will output files named logmfd.out in 0/, and replica.out.0 and replica.out.1 in 0/ and 1/, respectively.

The output file logmfd.out records free energy and all fictitious dynamical variables at every MFD step.

# LogPD, replica parallel of LogMFD
# number of replica : 2
# CVs : phi psi
# Mass for CV particles : 5000000.000000 5000000.000000
# Mass for thermostat   :   11923.224809
# 1:iter_mfd, 2:Flog, 3:2*Ekin/gkb[K], 4:eta, 5:Veta,
# 6:phi_fict(t), 7:phi_vfict(t), 8:phi_force(t),
# 9:psi_fict(t), 10:psi_vfict(t), 11:psi_force(t),
       0       2.000000     308.221983       0.000000       0.000000      -2.417903       0.000350       4.930899       2.451475       0.000350      -3.122024
       1       1.999367     308.257404       0.000000       0.000000      -2.417552       0.000350       0.851133       2.451825       0.000350      -1.552718
       2       1.999612     308.243683       0.000000       0.000000      -2.417202       0.000350      -1.588175       2.452175       0.000350       1.601274
       3       1.999608     308.243922       0.000000       0.000000      -2.416852       0.000350       4.267745       2.452525       0.000350      -4.565621
...

The output file replica.out.0 records all collective variables and the Jarzynski weight of replica No.0 at every MFD step.

# Replica No. 0 of 2.
# 1:iter_mfd, 2:work, 3:weight,
# 4:phi(q)
# 5:psi(q)
       0    0.000000e+00     5.000000e-01       -2.412607       2.446191
       1   -4.649101e-06     4.994723e-01       -2.421403       2.451318
       2    1.520985e-03     4.983996e-01       -2.420269       2.455049
       3    1.588855e-03     4.983392e-01       -2.411081       2.447394
...

The output file replica.out.1 records all collective variables and the Jarzynski weight of replica No.1 at every MFD step.

# Replica No. 1 of 2.
# 1:iter_mfd, 2:work, 3:weight,
# 4:phi(q)
# 5:psi(q)
       0    0.000000e+00     5.000000e-01       -2.413336       2.450516
       1   -1.263077e-03     5.005277e-01       -2.412009       2.449229
       2   -2.295444e-03     5.016004e-01       -2.417322       2.452512
       3   -2.371507e-03     5.016608e-01       -2.414078       2.448521
...

Input

The arguments that serve as the input for this action are specified using one or more of the keywords in the following table.

Keyword	Type	Description
ARG	scalar	the labels of the scalars on which the bias will act

Output components

This action calculates the values in the following table. These values can be referenced elsewhere in the input by using this Action's label followed by a dot and the name of the value required from the list below.

Name	Type	Description
bias	scalar	the instantaneous value of the bias potential
_fict	scalar	For example, the fictitious collective variable for LogMFD is specified as ARG=dist12 and LABEL=logmfd in LOGMFD section in Plumed input file, the associated fictitious dynamical variable can be specified as PRINT ARG=dist12,logmfd
_vfict	scalar	For example, the fictitious collective variable for LogMFD is specified as ARG=dist12 and LABEL=logmfd in LOGMFD section in Plumed input file, the velocity of the associated fictitious dynamical variable can be specified as PRINT ARG=dist12,logmfd

Full list of keywords

The following table describes the keywords and options that can be used with this action

Keyword	Type	Default	Description
ARG	input	none	the labels of the scalars on which the bias will act
INTERVAL	compulsory	none	Period of MD steps (N_m) to update fictitious dynamical variables
DELTA_T	compulsory	none	Time step for the fictitious dynamical variables (DELTA_T=1 often works)
THERMOSTAT	compulsory	none	Type of thermostat for the fictitious dynamical variables
KAPPA	compulsory	none	Spring constant of the harmonic restraining potential
FICT_MAX	compulsory	none	Maximum values reachable for the fictitious dynamical variables
FICT_MIN	compulsory	none	Minimum values reachable for the fictitious dynamical variables
FLOG	compulsory	none	The initial free energy value in the LogMFD/PD run
NUMERICAL_DERIVATIVESThis keyword do not have examples	optional	false	calculate the derivatives for these quantities numerically
TEMP	optional	not used	Target temperature for the fictitious dynamical variables in LogMFD/PD
TAMDThis keyword do not have examples	optional	not used	When TAMD=1, TAMD/d-AFED calculations can be performed instead of LogMFD
ALPHA	optional	not used	Alpha parameter for LogMFD
GAMMAThis keyword do not have examples	optional	not used	Gamma parameter for LogMFD
FICTThis keyword do not have examples	optional	not used	The initial values of the fictitious dynamical variables
VFICT	optional	not used	The initial velocities of the fictitious dynamical variables
MFICT	optional	not used	Masses of each fictitious dynamical variable
XETAThis keyword do not have examples	optional	not used	The initial eta variable of the Nose-Hoover thermostat for the fictitious dynamical variables
VETAThis keyword do not have examples	optional	not used	The initial velocity of eta variable
METAThis keyword do not have examples	optional	not used	Mass of eta variable
WORKThis keyword do not have examples	optional	not used	The initial value of the work done by fictitious dynamical variables in each replica
TEMPPDThis keyword do not have examples	optional	not used	Temperature of the Boltzmann factor in the Jarzynski weight in LogPD (Gromacs only)

References

More information about how this action can be used is available in the following articles:

• J. B. Abrams, M. E. Tuckerman, Efficient and Direct Generation of Multidimensional Free Energy Surfaces via Adiabatic Dynamics without Coordinate Transformations. The Journal of Physical Chemistry B. 112, 15742–15757 (2008)
• L. Maragliano, E. Vanden-Eijnden, A temperature accelerated method for sampling free energy and determining reaction pathways in rare events simulations. Chemical Physics Letters. 426, 168–175 (2006)
• T. Morishita, S. G. Itoh, H. Okumura, M. Mikami, Free-energy calculation via mean-force dynamics using a logarithmic energy landscape. Physical Review E. 85 (2012)
• T. Morishita, Y. Yonezawa, A. M. Ito, Free Energy Reconstruction from Logarithmic Mean-Force Dynamics Using Multiple Nonequilibrium Trajectories. Journal of Chemical Theory and Computation. 13, 3106–3119 (2017)
• T. Morishita, T. Nakamura, W. Shinoda, A. M. Ito, Isokinetic approach in logarithmic mean-force dynamics for on-the-fly free energy reconstruction. Chemical Physics Letters. 706, 633–640 (2018)