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Constrained molecular dynamics

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In general, constrained molecular dynamics generates biased statistical averages. It can be shown that the correct average for a quantity can be obtained using the formula:

where stands for the statistical average of the quantity enclosed in angular parentheses computed for a constrained ensemble and is a mass metric tensor defined as:

It can be shown that the free energy gradient can be computed using the equation:[1][2][3][4]

where is the Lagrange multiplier associated with the parameter used in the SHAKE algorithm.[5]

The free-energy difference between states (1) and (2) can be computed by integrating the free-energy gradients over a connecting path:

Note that as the free-energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant.

Constrained molecular dynamics is performed using the SHAKE algorithm.[5]. In this algorithm, the Lagrangian for the system is extended as follows:

where the summation is over r geometric constraints, is the Lagrangian for the extended system, and λi is a Lagrange multiplier associated with a geometric constraint σi:

with ξi(q) being a geometric parameter and ξi is the value of ξi(q) fixed during the simulation.

In the SHAKE algorithm, the Lagrange multipliers λi are determined in the iterative procedure:

  1. Perform a standard MD step (leap-frog algorithm):
  2. Use the new positions q(tt) to compute Lagrange multipliers for all constraints:
  3. Update the velocities and positions by adding a contribution due to restoring forces (proportional to λk):
  4. repeat steps 2-4 until either |σi(q)| are smaller than a predefined tolerance (determined by SHAKETOL), or the number of iterations exceeds SHAKEMAXITER.

Anderson thermostat

  • For a constrained molecular dynamics run with Andersen thermostat, one has to:
  1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW.
  2. Set MDALGO=1, and choose an appropriate setting for ANDERSEN_PROB.
  3. Define geometric constraints in the ICONST-file, and set the STATUS parameter for the constrained coordinates to 0.
  4. When the free-energy gradient is to be computed, set LBLUEOUT=.TRUE.