Slow-growth approach

In general, constrained molecular dynamics generates biased statistical averages. It can be shown that the correct average for a quantity [math]\displaystyle{ a(\xi) }[/math] can be obtained using the formula:

[math]\displaystyle{ a(\xi)=\frac{\langle |\mathbf{Z}|^{-1/2} a(\xi^*) \rangle_{\xi^*}}{\langle |\mathbf{Z}|^{-1/2}\rangle_{\xi^*}}, }[/math]

where [math]\displaystyle{ \langle ... \rangle_{\xi^*} }[/math] stands for the statistical average of the quantity enclosed in angular parentheses computed for a constrained ensemble and [math]\displaystyle{ Z }[/math] is a mass metric tensor defined as:

[math]\displaystyle{ 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:

[math]\displaystyle{ \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]\displaystyle{ \lambda_{\xi_k} }[/math] is the Lagrange multiplier associated with the parameter [math]\displaystyle{ {\xi_k} }[/math] used in the SHAKE algorithm.

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

[math]\displaystyle{ {\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.

• 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.

For a slow-growth simulation, one has to additionally:

5. Specify the transformation velocity-related INCREM-tag for each geometric parameter with STATUS=0

VASP can handle multiple (even redundant) constraints. Note, however, that a too large number of constraints can cause problems with the stability of the SHAKE algorithm. In problematic cases, it is recommended to use a looser convergence criterion (see SHAKETOL) and to allow a larger number of iterations (see SHAKEMAXITER) in the SHAKE algorithm. Hard constraints may also be used in metadynamics simulations (see MDALGO=11 | 21). Information about the constraints is written onto the REPORT-file: check the lines following the string: Const_coord