in the physical sciences is that it remains in general impossible to predict the structure of even the simplest crystalline solids from a knowledge of their chemical composition.” J. Maddox writing in Nature (1988) M. C. Escher, “Horseman” (1946) J. Maddox, Nature 335, 201 (1988) CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 2
can be made to show that the number of potential-energy minima is likely to rise exponentially with system size Stillinger, Phys. Rev. E 59 (1), 48 (1999) Dellago and Bolhuis in Advances in Polymer Science 221, pp. 167 (2009) CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 4
modes of a crystal are (orthogonal) combinations of the atomic degrees of freedom with an associated energy, and which can describe the natural thermal motion of a crystal The presence of imaginary modes in the harmonic phonon spectrum identifies the structure as a maximum on the potential-energy surface The atomic displacements associated with the imaginary modes tell us how to distort the structure to lower the energy (i.e. to go “downhill”) If we know of (or can guess) a high-energy structure (usually high-temperature, often also high-symmetry), we can “follow” the imaginary modes to map out the PES and locate the connected minima If we start from a high-temperature structure and progressively “freeze in” the imaginary modes, we are likely to be mimicking how the crystal explores its PES during cooling CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 17
problem: the number of minima on the PES likely grows exponentially with system size The presence or absence of imaginary harmonic modes in the phonon spectrum can be used to identify a structure as a minimum or maximum on the PES In cases where a high-temperature phase with imaginary modes is known, or can be obtained e.g. as an “educated guess”, iteratively “following” the imaginary modes allows the PES to be mapped out to locate connected minima Where applicable (it isn’t always...!), this approach may have some advantages: o The 3𝑛𝑎 degrees of freedom are effectively reduced a smaller number of imaginary modes o Off-Γ imaginary modes indicate when a distortion requires an expansion of the cell volume o Possibly mimics how the system explores the PES during cooling CCG Autumn Meeting 2023, 15th Nov 2023 | Slide 31