Collective changes in atom positions often have drastic influences on the physical properties of solids. An example is the lifting of spatial inversion symmetry caused by atomic displacement, from which a lot of intriguing states of matter emerge, including ferroelectricity, Weyl semimetals, spin-valley locked state, and so on. Predicting physical property requires knowledge of the underlying atomic structure, or at least of the space group symmetry, provided by experiments or computations. Finding the most stable structure of a given crystal is, however, difficult due to the lack of a priori knowledge on what distortion can reduce total energy.

In this talk, I will first introduce the process of stable structure exploration based on phonon calculations, through which one can obtain the most stable low-symmetry structure only assuming the "parent" structure. As an example, cases of 2D layered perovskite materials will be presented, where previous experimental attempts missed the distortion leading to a hybrid improper ferroelectric state. The second part will discuss the quantum-mechanical origin of atomic displacements based on the pseudo (or second-order) Jahn-Teller effect. This theory has been successfully applied to describe the electronic origin of zone-center displacements and molecular deformations. We have recently revealed that the pseudo Jahn-Teller effect can also uncover the driving mechanism of zone-boundary distortion, which offers a general approach to understanding the origin of structural phase transitions in crystals.