Normal modes of vibrations can be extracted from atomistic simulations by projecting the real space trajectories onto the reciprocal space. Normal modes coordinates are not unique and their primary function is to transform the coupled Hamiltonian to a set of independent harmonic oscillators. Several descriptors of normal modes exist but not all of them are appropriate for analyzing thermal transport. In most normal mode analyses (NMA), complex normal mode coordinates are employed, which combine the modal contributions of waves moving in opposite directions. The wave-vector q that is represented in the complex normal modes does not uniquely represent a wave traveling in the +q or –q direction, instead it denotes an average of both directions. Thus the popular complex normal modes have a theoretical inability to resolve a real heat current along a specific direction-dependent wave-vector q.
In this work, we employ a set of real asymmetric normal mode coordinates, which can distinguish lattice waves moving in opposite directions – a virtue that immediately endows the ability to discriminate a heat current in a certain direction. These normal mode coordinates have a real amplitude A(q,p) that is not equal to A(−q,p). We then derive an expression for heat current that is real, and which can be expressed as a difference between the squares of the amplitudes in +q and –q directions. Finally, we drive a correction term for phonon lifetime that arises from the correlation between modes moving along opposite directions.