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Figure 4
Structure stabilization in the limiting case T → 0 K, with classical versus quantum mechanical treatment. The same symbols are used as in Fig. 2[link]: a is the rotational angle of the BX3 octahedron, V0(a) and V0′(a) are maps of the PES before and after structural relaxation, and red corresponds to ionic wave functions φ(a) and φ′(a). Structural optimizations of V0(a) → V0′(a) using e.g. ab initio calculations (green arrows, `relaxation') lead to a reduction in the minimum of the potential energy ΔV (classical treatment). Generally, the symmetry of the structure is reduced by relaxation. (a) Quantum mechanical treatment: in the limiting case T → 0 K, an additional part of the total energy is due to the kinetic energy – the zero-point energy EZP. This scenario could include the transition from Immm to P21/n in cryolite at the critical temperature TC = 885 K (Anthony et al., 2005BB8), but also could include the cubic to Pmmm transition in PbTiO3-perovskite (Cole & Espenschied, 1937BB23) between 600 and 800 K (Zhu et al., 2011BB95), and some of the transitions in CaTiO3. Only if EZP before relaxation is comparable to EZP′ after relaxation, is it sufficient to compare the minima of V0(a) and V0′(a) to determine the stability. (b) If, on the other hand, the reduction of the energy ΔV is rather small during the optimization and/or the resulting potential is narrow, the situation arises that ΔV < 0 but [V_{0,\min}+E_{\rm ZP}\lt V^{\prime}_{0,\min}+E^{\prime}_{\rm ZP}] (quantum mechanical treatment), and therefore the high symmetry phase is stabilized even at low temperatures. This kind of stabilization is possible for CeAlO3, TlCuF3, SrFeO3, CsAuCl3, CeGaO3 CeAlO3, TlCuF3, SrFeO3, CsAuCl3 or CeGaO3 at low temperatures, as they remain in the P4/mmm phase.

IUCrJ
Volume 10| Part 3| May 2023| Pages 309-320
ISSN: 2052-2525
https://doi.org/10.1107/S2052252523002208