# Thermodynamic integration

A detailed description of thermodynamic integration is given in reference ^{[1]}.

The free energy of a fully interacting system can be written as the sum of the free energy a non-interacting reference system and the difference in the free energy of the fully interacting system and the non-interacting system

- .

Using thermodynamic integration the free energy difference between the two systems is written as

.

Here and describe the potential energies of a fully-interacting and a non-interacting reference system, respectively. The coupling strength of the systems is controlled via the coupling parameter . It is neccessary that the connection of the two systems via the coupling constant is reversible. The notation denotes an ensemble average of a system driven by the following classical Hamiltonian

- .

## Thermodynamic integration with harmonic reference

The Helmholtz free energy () of a fully interacting system (1) can be expressed in terms of that of system harmonic in Cartesian coordinates (0,) as follows

where is anharmonic free energy. The latter term can be determined by means of thermodynamic integration^{[2]} (TI)

with being the potential energy of system , is a coupling constant and is the NVT ensemble average of the system driven by the Hamiltonian

Free energy of harmonic reference system within the quasi-classical theory writes

with the electronic free energy for the configuration corresponding to the potential energy minimum with the atomic position vector , the number of vibrational degrees of freedom , and the angular frequency of vibrational mode obtained using the Hesse matrix . Finally, the harmonic potential energy is expressed as

Thus, a conventional TI calculation consists of the following steps:

- determine and in structural relaxation
- compute in vibrational analysis
- use the data obtained in the point 2 to determine that defines the harmonic forcefield
- perform NVT MD simulations for several values of and determine
- integrate over the grid and compute

Unfortunately, there are several problems linked with such a straightforward approach. First, the systems with rotational and/or translational degrees of freedom cannot be treated in a straightforward manner because is not invariant under rotations and translations. Conventional TI is thus unsuitable for simulations of gas phase molecules or adsorbate-substrate systems. and this problem also imposes restrictions on the choice of thermostat used in NVT simulation (Langevin thermostat, for instance, does not conserve position of the center of mass and is therefore unsuitable for the use in conventional TI). Furthermore, if the Hesse matrix of the harmonic system has one or more eigenvalues that nearly vanish, the simulations with 0 is likely to generate unphysical configurations causing serious convergence issues. These problems have been addressed in series of works by Amsler et al.^{[3]}^{[4]}

First, the method was formulated in terms of rotationally and translationally invariant internal coordinates , whereby the free energy of interacting system is repartitioned as follows:

where is the free energy change due to transformation from the system harmonic in into the system harmonic in and is that for the transformation of the latter into a fully interacting system. The force field for the system harmonic in is defined as:

where and the Hesse matrix defined for a potential energy minimum is related to via with being the Wilson matrix. Note that the calculation of the term is inexpensive as it corresponds to a force field to force field transformation. Furthermore, this term vanishes in the case of phase volume conserving coordinates, such as interatomic distances.

## References

- ↑ F. Dorner, Z. Sukurma, C. Dellago, and G. Kresse, Phys. Rev. Lett.
**121**, 195701 (2018). - ↑ J. G. Kirkwood,
*J. Chem. Phys*.**3,**300–313 (1935) - ↑ J. Amsler, P. N. Plessow, F. Studt, T. Bucko, J. Chem. Theory Comput. 17, 1155-1169 (2021)
- ↑ J. Amsler, P. N. Plessow, F. Studt, T. Bučko, J. Chem. Theory Comput. 19, 2455-2468 (2023)