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# Slow-growth approach

The free-energy profile along a geometric parameter $\xi$ can be scanned by an approximate slow-growth approach. In this method, the value of $\xi$ is linearly changed from the value characteristic for the initial state (1) to that for the final state (2) with a velocity of transformation ${\dot {\xi }}$ . The resulting work needed to perform a transformation $1\rightarrow 2$ can be computed as:

$w_{{1\rightarrow 2}}^{{irrev}}=\int _{{{\xi (1)}}}^{{{\xi (2)}}}\left({\frac {\partial {V(q)}}{\partial \xi }}\right)\cdot {\dot {\xi }}\,dt.$ In the limit of infinitesimally small ${\dot {\xi }}$ , the work $w_{{1\rightarrow 2}}^{{irrev}}$ corresponds to the free-energy difference between the the final and initial state. In the general case, $w_{{1\rightarrow 2}}^{{irrev}}\$ is the irreversible work related to the free energy via Jarzynski's identity:

$exp^{{-{\frac {\Delta A_{{1\rightarrow 2}}}{k_{B}\,T}}}}={\bigg \langle }exp^{{-{\frac {w_{{1\rightarrow 2}}^{{irrev}}}{k_{B}\,T}}}}{\bigg \rangle }.$ Note that calculation of the free-energy via this equation requires averaging of the term ${{\rm {exp}}}\left\{-{\frac {w_{{1\rightarrow 2}}^{{irrev}}}{k_{B}\,T}}\right\}$ over many realizations of the $1\rightarrow 2$ transformation. Detailed description of the simulation protocol that employs Jarzynski's identity can be found in reference .

## Anderson thermostat

• For a slow-growth simulation, one has to perform a calcualtion very similar to Constrained molecular dynamics but additionally the transformation velocity-related INCREM-tag for each geometric parameter with STATUS=0 has to be specified. For a slow-growth approach 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.
1. Specify the transformation velocity-related INCREM-tag for each geometric parameter with STATUS=0.

## Nose-Hoover thermostat

• For a slow-growth approach run with Nose-Hoover thermostat, one has to:
1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
2. Set MDALGO=2, and choose an appropriate setting for SMASS
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.
1. 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