Slow-growth approach

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

[math]\displaystyle{ w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}} \left ( \frac{\partial {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt. }[/math]

In the limit of infinitesimally small [math]\displaystyle{ \dot{\xi} }[/math], the work [math]\displaystyle{ w^{irrev}_{1 \rightarrow 2} }[/math] corresponds to the free-energy difference between the the final and initial state. In the general case, [math]\displaystyle{ w^{irrev}_{1 \rightarrow 2}$ }[/math]is the irreversible work related to the free energy via Jarzynski's identity:

[math]\displaystyle{ {\rm exp}\left\{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T} \right \}= \bigg \langle {\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \} \bigg\rangle. }[/math]

Note that calculation of the free-energy via this equation requires averaging of the term [math]\displaystyle{ {\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \} }[/math] over many realizations of the [math]\displaystyle{ 1 \rightarrow 2 }[/math] 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.

5. 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.

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

References

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