I understand that a harmonic picture of the potential in a material isn't enough to study lattice dynamics thoroughly. The quasi-harmonic approximation is a good workaround and helps incorporate thermal expansion effects in the lattice.

I've read and heard at times that a so-and-so material is anharmonic. The implication being that neither of harmonic or quasi-harmonic approximations suffices and a non-harmonic expansion of the potential is required.

How can we decide which material needs what, or specifically that a particular material is anharmonic?

A pseudo-related question but one that I thought would do better as a separate thread: Is there an upper limit of temperature after which the quasi-harmonic approximation (QHA) fails?.


This is a difficult question without a straight-forward answer. In general you have to perform a test to decide whether the harmonic approximation is sufficient or whether you need to include higher order anharmonic terms in the potential expansion. Due to the computational cost of including anharmonic terms, very often systems are assumed to be harmonic without further checks, which can be problematic.

Having said this, here are a few thoughts one can also use as a starting point. At equilibrium, a material sits at a local minimum of the potential energy surface. The harmonic approximation is then based on the assumption that the atomic nuclei/ions do not move very far away from this minimum, and that a second order expansion of the energy around the minimum is sufficient to describe atomic vibrations. Therefore, the harmonic approximation will break down when the atoms move significantly far away from equilibrium. A few examples include:

  1. High temperature. At sufficiently high temperature, solids melt, and all materials will behave in an anharmonic manner sufficiently close to melting. But what temperature is that? It is strongly material dependent. A starting estimate could be to use the Lindemann criterion which, roughly speaking, states that the melting temperature of a material corresponds to the atomic vibrational amplitudes approaching 15-30% of the interatomic distance. Therefore, if your atoms are vibrating anywhere near these amplitudes, it is likely anharmonic terms are important.
  2. Light elements. The vibrational amplitudes of an element are larger the smaller the mass. This means that anharmonic terms tend to be larger for the lighter elements, and indeed in some like hydrogen (lightest of all elements) they can dominate even at zero temperature (quantum fluctuations are anharmonic).
  3. Structural phase transitions. Even if your system is well below the melting point or not made of light elements, structural phase transitions can be dominated by anharmonic vibrational terms. The most well-known example of this are perhaps the perovskite family, which typically exhibits a series of temperature-induced structural phase transitions from high symmetry high temperature cubic to lower temperature lower symmetry tetragonal, orthorhombic, etc. The higher symmetry structures correspond to saddle points of the potential energy surface (rather than minima), and the structures are stabilized to those points through anharmonic vibrations. A purely harmonic description would lead to the presence of imaginary modes that would drive you towards the lowest energy structure, and fail to describe the stability of the higher symmetry higher temperature structures.
  • $\begingroup$ Thank you as always, ProfM. With regard to the $3^{rd}$ point, prior experimental evidence of such phase transitions and a subsequent failure of the harmonic approximation (HA) to account for them could be a pointer to the need of the anharmonic expansion. Given there is no prior experimental evidence and the HA gives imaginary modes, is there any way to decide if an anharmonic treatment could reveal more useful information? Or is there only one way out - try it and see. $\endgroup$ Nov 23 '20 at 11:08
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    $\begingroup$ @HitanshuSachania, if there are imaginary phonons then one of two things can happen. The first is that the structure you have is not the experimentally relevant one, and following the distortion associated with the imaginary phonon will take you to a lower energy structure that will not have any more imaginary phonons (you may have to iterate a few times until you find the proper local minimum). After you find a structure that has no imaginary phonons, then the harmonic approximation should be a good starting point. $\endgroup$
    – ProfM
    Nov 23 '20 at 18:40
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    $\begingroup$ The second thing that can happen is that your structure really sits at a saddle point of the potential energy landscape, in which case it is stabilised by anharmonic vibrations and there is no way of treating the system at the harmonic level. I would say that the only definite answer would be to try, but you can also look at the energy difference between the saddle point structure and local minima and compare it to a typical thermal energy to have an estimate of whether anharmonic vibrations driven by thermal energy could potentially stabilize the structure. $\endgroup$
    – ProfM
    Nov 23 '20 at 18:42
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    $\begingroup$ Excellent answer by Prof! As a small note about working implementation - Is it possible to capture this anharmonicity using either the Frozen Phonon or DFPT approach? $\endgroup$
    – Xivi76
    Nov 24 '20 at 5:07
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    $\begingroup$ @Xivi76 anharmonic calculations are typically implemented in a finite difference setting using a variety of methods (systematic superposition of frozen phonons, stochastic superposition of frozen phonons, or MD-like superposition of frozen phonons). DFPT is typically limited to harmonic phonons. $\endgroup$
    – ProfM
    Nov 24 '20 at 8:24

A "pure harmonic system" does not allow opportunity for evolution. It is the equivalent of a fixed point. In initial consideration, its stability seems appealing, as it appears to be a goal (or “the goal”) of an imperfect system. However, it only embodies that moniker once, and change is the only real constant. Pure harmonics are fragile, brittle, and only stable conditionally. What's really appealing is resonance.

  • $\begingroup$ This does not seem to address the question. You seem to using a more philosophical definition of harmonic rather than the physical meaning meant here. Please consider editing the answer or it is likely to be removed. $\endgroup$
    – Tyberius
    Nov 23 '20 at 18:14
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    $\begingroup$ I actually think this answer even being less physical meaning does capture the general problem here. The harmonic approximation is mainly incorrect when a change is occurring (phase transition etc). By intention or chance, this is a decent supplementary answer to ProfM's answer for someone who doesn't want to get into the details. $\endgroup$ Nov 23 '20 at 18:58

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