Slow-growth approach

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

$a(\xi )={\frac {\langle |\mathbf {Z} |^{-1/2}a(\xi ^{*})\rangle _{\xi ^{*}}}{\langle |\mathbf {Z} |^{-1/2}\rangle _{\xi ^{*}}}},$

where $\langle ...\rangle _{\xi ^{*}}$ stands for the statistical average of the quantity enclosed in angular parentheses computed for a constrained ensemble and $Z$ is a mass metric tensor defined as:

$Z_{\alpha ,\beta }={\sum }_{i=1}^{3N}m_{i}^{-1}\nabla _{i}\xi _{\alpha }\cdot \nabla _{i}\xi _{\beta },\,\alpha =1,...,r,\,\beta =1,...,r,$

It can be shown that the free energy gradient can be computed using the equation:

${\Bigl (}{\frac {\partial A}{\partial \xi _{k}}}{\Bigr )}_{\xi ^{*}}={\frac {1}{\langle |Z|^{-1/2}\rangle _{\xi ^{*}}}}\langle |Z|^{-1/2}[\lambda _{k}+{\frac {k_{B}T}{2|Z|}}\sum _{j=1}^{r}(Z^{-1})_{kj}\sum _{i=1}^{3N}m_{i}^{-1}\nabla _{i}\xi _{j}\cdot \nabla _{i}|Z|]\rangle _{\xi ^{*}},$

where $\lambda _{\xi _{k}}$ is the Lagrange multiplier associated with the parameter ${\xi _{k}}$ 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:

${\Delta }A_{1\rightarrow 2}=\int _{\xi (1)}^{\xi (2)}{\Bigl (}{\frac {\partial {A}}{\partial \xi }}{\Bigr )}_{\xi ^{*}}\cdot d{\xi }.$

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