  The calculations of thermal conductivity requires to know anharmonic
properties of the crystal. For this purpose a non-perturbative anharmonic
theory is applied, which do not make use of the potential energy expansion over
atomic displacements, but instead, runs ab initio calculations of
Hellmann-Feynman forces for atomic patterns of atoms with specific
displacements to rebuild the anharmonic phonon frequencies, and group
velocities. see [K.Parlinski, Phys.Rev. B 98, 054305 (2018),] The Green-Kubo
equation for the thermal conductivity needs to know the above quantities and
the phonon relaxation times, which are related to the 4th-order phonon
correlation function expressed in terms of phonon anihilation and creation Bose
operators. In currect formulation of anharmonic theory the relaxation times can
be derived as analitical expression. The Green-Kubo formulae was succesfully
applied to find thermal conductivity of Si and conductivities, related to the
phonon and elastic waves, respectivily, were computed.
