Constrained molecular dynamics

Constrained molecular dynamics is performed using the SHAKE algorithm.. In this algorithm, the Lagrangian for the system Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathcal {L}} is extended as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathcal {L}}^{*}({\mathbf {q,{\dot {q}}}})={\mathcal {L}}({\mathbf {q,{\dot {q}}}})+\sum _{{i=1}}^{{r}}\lambda _{i}\sigma _{i}(q),

where the summation is over r geometric constraints, ${\mathcal {L}}^{*}$ is the Lagrangian for the extended system, and λ_i is a Lagrange multiplier associated with a geometric constraint σ_i:

$\sigma _{i}(q)=\xi _{i}({q})-\xi _{i}\;$

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

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

1. Perform a standard MD step (leap-frog algorithm):

   $v_{i}^{t+{\Delta }t/2}=v_{i}^{t-{\Delta }t/2}+{\frac {a_{i}^{t}}{m_{i}}}{\Delta }t$

   $q_{i}^{t+{\Delta }t}=q_{i}^{t}+v_{i}^{t+{\Delta }t/2}{\Delta }t$

2. Use the new positions q(t+Δt) to compute Lagrange multipliers for all constraints:

   ${\lambda }_{k}={\frac {1}{{\Delta }t^{2}}}{\frac {\sigma _{k}(q^{t+{\Delta }t})}{\sum _{i=1}^{N}m_{i}^{-1}\bigtriangledown _{i}{\sigma }_{k}(q^{t})\bigtriangledown _{i}{\sigma }_{k}(q^{t+{\Delta }t})}}$

3. Update the velocities and positions by adding a contribution due to restoring forces (proportional to λ_k):

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): v_{i}^{{t+{\Delta }t/2}}=v_{i}^{{t-{\Delta }t/2}}+\left(a_{i}^{{t}}-\sum _{k}{\frac {{\lambda }_{k}}{m_{i}}}\bigtriangledown _{i}{\sigma }_{k}(q^{{t}})\right){\Delta }t

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): q_{i}^{{t+{\Delta }t}}=q_{i}^{{t}}+v_{i}^{{t+{\Delta }t/2}}{\Delta }t

4. repeat steps 2-4 until either |σ_i(q)| are smaller than a predefined tolerance (determined by SHAKETOL), or the number of iterations exceeds SHAKEMAXITER.

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