Methodology and meaning of computing heat flux via atomic stress in systems with constraint dynamics

Reliably obtaining thermal properties of complex systems, which often involves computing heat flux to obtain thermal conductivity via either Fourier's law or the Green-Kubo relation, is an important task in modern molecular dynamics simulations. In our previous work [Surblys et al., Phys. Rev. E 99, 051301(R) (2019)], we have demonstrated that atomic stress could be used to efficiently compute heat flux for molecules with angle, dihedral, or improper many-body interactions, provided a newly derived "centroid"form was used. This was later successfully implemented in the LAMMPS simulation package. On the other hand, small rigid molecules, like water and partial constraints in semi-flexible molecules, are often implemented via constraint force algorithms. There has been a lack of clarification if the constraint forces that maintain geometric constraints and can also be considered as many-body forces contribute to the overall heat flux and how to compute them correctly and efficiently. To address this, we investigate how to apply the centroid atomic stress form to reliably compute heat flux for systems with constraint or rigid body dynamics. We successfully apply the centroid atomic stress form to flexible, semi-flexible, and rigid water models; decompose the computed thermal conductivity into separate components; and demonstrate that the contribution from constraint forces to the overall heat flux and thermal conductivity is small but non-negligible. We also show that while the centroid formulation produces correct heat flux values, the original "group"formulation produces incorrect and sometimes unphysical results. Finally, we provide insight into the meaning of constraint force contribution.