ML_LCOUPLE

ML_LCOUPLE = [logical]
Default: ML_LCOUPLE = .FALSE.

Description: This tag specifies whether thermodynamic integration is activated in order to calculate the chemical potentials within the machine learning force field method.

In thermodynamic integration a coupling parameter $\lambda$ is introduced to the Hamiltonian to smoothly switch between a "non-interacting" reference state and a "fully-interacting" state. The change of the free energy along this path is written as

${\displaystyle \Delta \mu = \int\limits_{0}^{1} \langle \frac{dH(\lambda)}{d\lambda} \rangle_{\lambda} d\lambda. }$

Using machine learning force fields the Hamiltonian can be written as

${\displaystyle H (\lambda) = \sum\limits_{i=1}^{N_{a}} \frac{|\mathbf{p}_{i}|^2}{2m_{i}} + \sum\limits_{i \notin M} U_{i}(\lambda) + \lambda \sum\limits_{i \in M} U_{i}(\lambda) + \sum\limits_{i}^{N_{a}} U_{i,\mathbf{atom}}. }$

where $N_{a}$ denotes the number of atoms and $U_{i,\mathbf{atom}}$ is an atomic reference energy for a single non interacting atom. The first term in the equation describes the potential energy and the second and third term describe the potential energy of an atom $i$ . The index $M$ denotes the atoms whose interaction is controlled by a coupling parameter. The interactions of the atoms are controlled by scaling the contributions to the atom density via the coupling parameter

${\displaystyle \rho (\mathbf{r},\lambda) = \sum\limits_{j\notin M} f_{\mathrm{cut}} \left( \left| \mathbf{r}_{j} - \mathbf{r}_{i} \right| \right) g \left[ \mathbf{r} - \left( \mathbf{r}_{j} - \mathbf{r}_{i} \right) \right] + \lambda \sum\limits_{j\in M} f_{\mathrm{cut}} \left( \left| \mathbf{r}_{j} - \mathbf{r}_{i} \right| \right) g \left[ \mathbf{r} - \left( \mathbf{r}_{j} - \mathbf{r}_{i} \right) \right]. }$

Further details on the implementation can be found in reference ^[1].

For thermodynamic integration the following parameters have to be set:

ML_MODE = run.
ML_LCOUPLE = .TRUE..
The number of atoms for which a coupling parameter is introduced ( $i \in M$ ): ML_NATOM_COUPLED.
The list of atom indices that for that the coupling parameter is applied in the interaction: ML_ICOUPLE.
The strength of the coupling parameter $\lambda$ between 0 and 1: ML_RCOUPLE.

The derivative of the hamiltonian with respect to the coupling constant $dH/d\lambda$ is written out at every MD step to the ML_LOGFILE. A sample output should look like this:

# DCOUPLE ################################
# DCOUPLE This line shows the derivative of the Hamiltonian with respect to coupling constant (dH/dlambda).
# DCOUPLE 
# DCOUPLE nstep .......... MD time step or input structure counter
# DCOUPLE der_H_lambda ... dH/dlambda
# DCOUPLE ################################
# DCOUPLE           nstep     der_H_lambda
# DCOUPLE               2                3
# DCOUPLE ################################
DCOUPLE                 1  -1.08332135E+01
DCOUPLE                 2  -1.08132321E+01
DCOUPLE                 3  -1.07631992E+01
DCOUPLE                 4  -1.06786675E+01
DCOUPLE                 5  -1.05493088E+01
DCOUPLE                 6  -1.03561161E+01
DCOUPLE                 7  -1.00762443E+01
DCOUPLE                 8  -9.69961878E+00
DCOUPLE                 9  -9.25531640E+00
DCOUPLE                10  -8.82525354E+00
...

References

↑ R. Jinnouchi, F. Karsai, and G. Kresse, Phys. Rev. B 101, 060201 (2020).

References

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