Expanded Ensemble mode

Expanded ensemble (EE) mode can be used for calculations of solvation free energies by gradual removal or insertion of molecules in the system. The theory and computational details behind the expanded ensemble method are described in papers (J.Chem.Phys.96, 1776 (1992); Molec.Simulations , 18, 43 (1996); J.Chem.Phys.,108,227 (1998)). Implementation of the EE approach here differs from the cited above articles, in a way that interaction parameters of the choosen solute molecule with the rest of the system are scaled with the coupling parameter $\alpha$, $0 \le \alpha \le 1$ according the folowing rules:

$$\sigma = const$$ (12)

$$\epsilon \to \alpha^{n1} \epsilon$$

$$q \to \alpha^{n2} q$$

The coupling parameter $\alpha$ assumes a number of fixed values between 0 and 1, where ``1'' corresponds to the solute molecule properly inserted in the solvent, while value ``0'' correspons to the fully eleminated solute (which then can be considered as an a gas phase). The program computes probability distribution over subensembles with different values of $\alpha$, from which the free energy deffirence can be obtained. Additional bias for transition probabilities between subensembles is given by the so-called balancing factors, which need to be tuned in order to provide reasonable homogenious distribution of probabilities over subensembles.

Note 1: Special care should be taken in MDEE simulations of molecules having zero Lennard-Jones potential for hydrogen atoms (like SPC or TIP3P water). Simulation may become unstable when repulsive potential between oxygens becoms weeker while electrostatic attraction between hydrogen and oxygen still present. It is advisable to prescribe some small Lennard-Jones potential to such hydrogens.

Note 2: It is always recommended to use option ``Cut_forces'' with threshold level 0.05 - 0.1 in the case of EE simulation to avoid ``collisions'' with ``almost deleted'' particles

The expanded ensemble is specified by the following keywords:

• MDEE <num_ensembles> <mode>

  Run expanded ensemble simulations with <num_ensembles> subensembles, Parameter <mode> can be either ``manual'' (manual setup of EE parameters) or ``auto'' (automatic setup of EE parameters using Wang-Landau algorithm)

  In the case of manual choice of the expanded ensemble parameters this option should be followed by exactly num_ensembles lines, with two number on each: the value of coupling parameter $\alpha$ and balancing factor, corresponding to this subensemble. Values of balancing factors are given per inserted molecule in units of ``$kT$''.

  In the case of automatic setup of the expanded ensemble parameters, the set of coupling parameters $\alpha$ is choosen uniformly in the range [0:1] (that is, $\alpha_m = (M-m)/(M-1)$ where $M$ is the total number of subensembles <num_ensembles>. The balancing factors are set up automatically, starting from zero values, using the Wang-Landau algorithm (Wang F., Landau D. P., Phys. Rev. Lett. 86, P. 2050-2053 (2001)) in expanded ensemble simulations. After each attempt of visiting a subensemble, an increment is added to the corresponding balancing factor. After completing a series (sweep) (the number of MD steps in a serie is determined by parameter ``Serie_average''), the value of the increment is decreased by a factor (default value 0.5). The tuning of balancing factors is performed before the final averaging begins, which is defined by parameter Average_from. After processing Average_from series, the balancing factors become fixed, and final accumulation of subensemble probabilities and other averages is carried out.

  Notes:

  1. In case of automatic tuning of balancing factors, it is resonable to reserve at least 10 series for their optimization (parameter Average_from). The balancing factors are adjusted in the end of each series. The number of MD steps per series (parameter Serie_average) should be also large enough to ensure proper optimization (as a rule of thumb it can be 100*(number of atoms in the inserted molecule)).After the simulation has run the number of series given by parameter Average_from, the balancing factors become constant, and the final averaging can begin. It is reasonable to run 10-20 series more for the final averaging.

  2. While evaluating results, ensure that the system has walked at least several times between the extreeme subensembles (the first and the last). Look at the ``table of transition'' in the end of the output.

  3. Pay attention at the acceptance ratios for transitions between subensembles, they can be found in the last two columns of the section ``Distribution over subensembles'' in the output. The acceptance ratios should not be too low. If they are two low (a few percent or less), the number of subensembles should be increased.

• EE_start-in <start_in>

  Start EE simulation in subensemble number <start_in>. Default: in the last subensemble (that is, with completely deleted EE molecule(s))

• EE_typ <i_typ>

  Specifies that the molecules of type i_typ are those which are inserted/deleted in the EE procedure.

• EE_freq <steps> Try EE transition after steps MD steps.

  Default: 10

• EE_new

  Change balancing factors after restart (manual mode only). Otherwize old values written in the restart file are used

• EE_scaling <n1> <n2>

  Powers of scaling of Lennerd-Jones and electrostatic interactions, see (12) Default: n1=4, n2=2

• WL_parameters <increment> <factor>

  Sets up initial increment in the Wang-Landau optimization of balancing factors, and scaling factor by which the increment is scaled after each series.

  Default values: increment = 0.1, factor = 0.5.

• Probability_threshold <P_low> Assing value P_low to computed subensembles probabilities which are lower than this number

  Default: 0.001