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# 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 ${\displaystyle \lambda }$ 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 ${\displaystyle N_{a}}$ denotes the number of atoms and ${\displaystyle U_{i,\mathbf {atom} }}$ 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 ${\displaystyle i}$. The index ${\displaystyle M}$ denotes the atoms whose interaction is controlled by a coupling parameter. The interaction 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_ISTART=2.
• ML_LCOUPLE=.TRUE..
• The number of atoms for which a coupling parameter is introduced (${\displaystyle i\notin M}$): 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 ${\displaystyle lambda}$ between 0 and 1: ML_RCOUPLE.

The derivative of the hamiltonian with respect to the coupling constant ${\displaystyle dH/d\lambda }$ 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
...