Extended-system Adaptive Biasing Force (eABF)

Extended-system ABF (eABF) is a variant of ABF () where the bias is not applied directly to the collective variable, but to an extended coordinate (``fictitious variable'') $\lambda$ that evolves dynamically according to Newtonian or Langevin dynamics. Such an extended coordinate is enabled for a given colvar using the extendedLagrangian and associated keywords (). The theory of eABF and the present implementation are documented in detail in reference [66].

Defining an ABF bias on a colvar wherein the extendedLagrangian option is active will perform eABF automatically; there is no dedicated option.

The extended variable $\lambda$ is coupled to the colvar $z=\xi(q)$ by the harmonic potential $(k/2) (z - \lambda)^2$ . Under eABF dynamics, the adaptive bias on $\lambda$ is the running estimate of the average spring force:

$$F^{\mathrm{bias}}(\lambda^{*}) = \left\langle k(\lambda{} - z) \right\rangle_{\lambda^{*}}$$ (13.27)

where the angle brackets indicate a canonical average conditioned by $\lambda=\lambda^*$ . At long simulation times, eABF produces a flat histogram of the extended variable $\lambda$ , and a flattened histogram of $\xi$ , whose exact shape depends on the strength of the coupling as defined by extendedFluctuation in the colvar. Coupling should be somewhat loose for faster exploration and convergence, but strong enough that the bias does help overcome barriers along the colvar $\xi$ .[66] Distribution of the colvar may be assessed by plotting its histogram, which is written to the outputName.zcount file in every eABF simulation. Note that a histogram bias () applied to an extended-Lagrangian colvar will access the extended degree of freedom $\lambda$ , not the original colvar $\xi$ ; however, the joint histogram may be explicitly requested by listing the name of the colvar twice in a row within the colvars parameter of the histogram block.

The eABF PMF is that of the coordinate $\lambda$ , it is not exactly the free energy profile of $\xi$ . That quantity can be calculated based on the CZAR estimator.

CZAR estimator of the free energy

The corrected z-averaged restraint (CZAR) estimator is described in detail in reference [66]. It is computed automatically in eABF simulations, regardless of the number of colvars involved. Note that ABF may also be applied on a combination of extended and non-extended colvars; in that case, CZAR still provides an unbiased estimate of the free energy gradient.

CZAR estimates the free energy gradient as:

$$A'(z) = - \frac{1}{\beta} \frac{d\ln \tilde \rho (z)}{dz} + k (\langle\lambda\rangle_z - z).$$ (13.28)

where $z=\xi(q)$ is the colvar, $\lambda$ is the extended variable harmonically coupled to $z$ with a force constant $k$ , and $\tilde\rho (z)$ is the observed distribution (histogram) of $z$ , affected by the eABF bias.

Parameters for the CZAR estimator are:

• CZARestimator $\langle\,$ Calculate CZAR estimator of the free energy?$\,\rangle$
  Context: abf
  Acceptable values: boolean
  Default value: yes
  Description: This option is only available when ABF is performed on extended-Lagrangian colvars. When enabled, it triggers calculation of the free energy following the CZAR estimator.

• writeCZARwindowFile $\langle\,$ Write internal data from CZAR to a separate file?$\,\rangle$
  Context: abf
  Acceptable values: boolean
  Default value: no
  Description: When this option is enabled, eABF simulations will write a file containing the $z$ -averaged restraint force under the name outputName.zgrad. The same information is always included in the colvars state file, which is sufficient for restarting an eABF simulation. These separate file is only useful when joining adjacent windows from a stratified eABF simulation, either to continue the simulation in a broader window or to compute a CZAR estimate of the PMF over the full range of the coordinate(s). Important warning. Unbiased free-energy estimators from eABF dynamics rely on some form of sampling histogram. When running stratified (windowed) calculations this histogram becomes discontinuous, and as a result the free energy gradient estimated by CZAR is inaccurate at the window boundary, resulting in visible "blips" in the PMF. As a workaround, we recommend manually replacing the two free energy gradient values at the boundary, either with the ABF values from .grad files (accurate in the limit of tight coupling), or with values interpolated for the neighboring values of the CZAR gradient.

Similar to ABF, the CZAR estimator produces two output files in multicolumn text format ():

• outputName.czar.grad: current estimate of the free energy gradient (grid), in multicolumn;
• outputName.czar.pmf: only for one-dimensional calculations, integrated free energy profile or PMF.

The sampling histogram associated with the CZAR estimator is the $z$ -histogram, which is written in the file outputName.zcount.