Constrained molecular dynamics
Constrained molecular dynamics is performed using the SHAKE algorithm.[1]. 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:
where the summation is over r geometric constraints, 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 the Lagrangian for the extended system, and λi is a Lagrange multiplier associated with a geometric constraint σi:
- 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/":): \sigma _{i}(q)=\xi _{i}({q})-\xi _{i}\;
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:
- Perform a standard MD step (leap-frog algorithm):
- 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}}+{\frac {a_{i}^{{t}}}{m_{i}}}{\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
- Use the new positions q(t+Δt) to compute Lagrange multipliers for all constraints:
- 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/":): {\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}})}}
- Update the velocities and positions by adding a contribution due to restoring forces (proportional to λk):
- 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
- repeat steps 2-4 until either |σi(q)| are smaller than a predefined tolerance (determined by SHAKETOL), or the number of iterations exceeds SHAKEMAXITER.
- ↑ Cite error: Invalid
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