Constrained molecular dynamics: Difference between revisions

Revision as of 14:22, 18 April 2022

Constrained molecular dynamics is performed using the SHAKE algorithm.^[1]. In this algorithm, the Lagrangian for the system $\mathcal{L}$ is extended as follows:

{\displaystyle \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:

{\displaystyle \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:

Perform a standard MD step (leap-frog algorithm):
${\displaystyle v^{t+{\Delta}t/2}_i = v^{t-{\Delta}t/2}_i + \frac{a^{t}_i}{m_i} {\Delta}t }$

${\displaystyle q^{t+{\Delta}t}_i = q^{t}_i + v^{t+{\Delta}t/2}_i{\Delta}t }$
Use the new positions q(t+Δt) to compute Lagrange multipliers for all constraints:
${\displaystyle {\lambda}_k= \frac{1}{{\Delta}t^2} \frac{\sigma_k(q^{t+{\Delta}t})}{\sum_{i=1}^N m_i^{-1} \bigtriangledown_i{\sigma}_k(q^{t}) \bigtriangledown_i{\sigma}_k(q^{t+{\Delta}t})} }$
Update the velocities and positions by adding a contribution due to restoring forces (proportional to λ_k):
${\displaystyle v^{t+{\Delta}t/2}_i = v^{t-{\Delta}t/2}_i + \left( a^{t}_i-\sum_k \frac{{\lambda}_k}{m_i} \bigtriangledown_i{\sigma}_k(q^{t}) \right ) {\Delta}t }$

${\displaystyle q^{t+{\Delta}t}_i = q^{t}_i + v^{t+{\Delta}t/2}_i{\Delta}t }$
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.

Anderson thermostat

For a constrained molecular dynamics run with Andersen thermostat, one has to:

Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW.
Set MDALGO=1, and choose an appropriate setting for ANDERSEN_PROB.
Define geometric constraints in the ICONST-file, and set the STATUS parameter for the constrained coordinates to 0.
When the free-energy gradient is to be computed, set LBLUEOUT=.TRUE.

References

↑ J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).

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[Ryckaert77-1] J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).

[1]

@@ Line 1: / Line 1: @@
-In general, constrained molecular dynamics generates biased statistical averages.
-It can be shown that the correct average for a quantity <math>a(\xi)</math> can be obtained using the formula:
-:<math>
-a(\xi)=\frac{\langle |\mathbf{Z}|^{-1/2} a(\xi^*) \rangle_{\xi^*}}{\langle |\mathbf{Z}|^{-1/2}\rangle_{\xi^*}},
-</math>
-where <math>\langle ... \rangle_{\xi^*}</math> stands for the statistical average of the quantity enclosed in angular parentheses computed for a constrained ensemble and <math>Z</math> is a mass metric tensor defined as:
-:<math>
-Z_{\alpha,\beta}={\sum}_{i=1}^{3N} m_i^{-1} \nabla_i \xi_\alpha \cdot \nabla_i \xi_\beta, \, \alpha=1,...,r, \, \beta=1,...,r,
-</math>
-It can be shown that the free energy gradient can be computed using the equation:<ref name="Carter89"/><ref name="Otter00"/><ref name="Darve02"/><ref name="Fleurat05"/>
-:<math>
-\Bigl(\frac{\partial A}{\partial \xi_k}\Bigr)_{\xi^*}=\frac{1}{\langle|Z|^{-1/2}\rangle_{\xi^*}}\langle |Z|^{-1/2} [\lambda_k +\frac{k_B T}{2 |Z|} \sum_{j=1}^{r}(Z^{-1})_{kj} \sum_{i=1}^{3N} m_i^{-1}\nabla_i \xi_j \cdot \nabla_i |Z|]\rangle_{\xi^*},
-</math>
-where <math>\lambda_{\xi_k}</math> is the Lagrange multiplier associated with the parameter <math>{\xi_k}</math> used in the [[#SHAKE|SHAKE algorithm]].<ref name="Ryckaert77"/>
-The free-energy difference between states (1) and (2) can be computed by integrating the free-energy gradients over a connecting path:
-:<math>
-{\Delta}A_{1 \rightarrow 2} = \int_{{\xi(1)}}^{{\xi(2)}}\Bigl( \frac{\partial {A}} {\partial \xi} \Bigr)_{\xi^*} \cdot d{\xi}.
-</math>
-Note that as the free-energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant.
 <div id="SHAKE"></div>
 Constrained molecular dynamics is performed using the SHAKE algorithm.<ref name="Ryckaert77"/>.