Extended Lagrangian.

The following options enable extended-system dynamics, where a colvar is coupled to an additional degree of freedom (fictitious particle) by a harmonic spring. All biasing and confining forces are then applied to the extended degree of freedom, and the actual, geometric colvar (function of Cartesian coordinates) only feels the force from the harmonic spring.

• extendedLagrangian $\langle\,$ Add extended degree of freedom$\,\rangle$
  Context: colvar
  Acceptable values: boolean
  Default value: off
  Description: Adds a fictitious particle to be coupled to the colvar by a harmonic spring. The fictitious mass and the force constant of the coupling potential are derived from the parameters extendedTimeConstant and extendedFluctuation, described below. Biasing forces on the colvar are applied to this fictitious particle, rather than to the atoms directly. This implements the extended Lagrangian formalism used in some metadynamics simulations [33].

• extendedFluctuation $\langle\,$ Standard deviation between the colvar and the fictitious particle (colvar unit)$\,\rangle$
  Context: colvar
  Acceptable values: positive decimal
  Description: Defines the spring stiffness for the extendedLagrangian mode, by setting the typical deviation between the colvar and the extended degree of freedom due to thermal fluctuation. The spring force constant is calculated internally as $k_B T / \sigma^2$ , where $\sigma$ is the value of extendedFluctuation.

• extendedTimeConstant $\langle\,$ Oscillation period of the fictitious particle (fs)$\,\rangle$
  Context: colvar
  Acceptable values: positive decimal
  Default value: 200
  Description: Defines the inertial mass of the fictitious particle, by setting the oscillation period of the harmonic oscillator formed by the fictitious particle and the spring. The period should be much larger than the MD time step to ensure accurate integration of the extended particle's equation of motion. The fictitious mass is calculated internally as $k_B T (\tau/2 \pi \sigma)^2$ , where $\tau$ is the period and $\sigma$ is the typical fluctuation (see above).

• extendedTemp $\langle\,$ Temperature for the extended degree of freedom (K)$\,\rangle$
  Context: colvar
  Acceptable values: positive decimal
  Default value: thermostat temperature
  Description: Temperature used for calculating the coupling force constant of the extended coordinate (see extendedFluctuation) and, if needed, as a target temperature for extended Langevin dynamics (see extendedLangevinDamping). This should normally be left at its default value.

• extendedLangevinDamping $\langle\,$ Damping factor for extended Langevin dynamics (ps$^{-1}$ )$\,\rangle$
  Context: colvar
  Acceptable values: positive decimal
  Default value: 1.0
  Description: If this is non-zero, the extended degree of freedom undergoes Langevin dynamics at temperature extendedTemp. The friction force is minus extendedLangevinDamping times the velocity. This is useful because the extended dynamics coordinate may heat up in the transient non-equilibrium regime of ABF. Use moderate damping values, to limit viscous friction (potentially slowing down diffusive sampling) and stochastic noise (increasing the variance of statistical measurements). In doubt, use the default value.