Probing the limitations of isotropic pair potentials to produce ground-state structural extremes via inverse statistical mechanics

Inverse statistical-mechanical methods have recently been employed to design optimized short-range radial (isotropic) pair potentials that robustly produce novel targeted classical ground-state many-particle configurations. The target structures considered in those studies were low-coordinated crystals with a high degree of symmetry. In this paper, we further test the fundamental limitations of radial pair potentials by targeting crystal structures with appreciably less symmetry, including those in which the particles have different local structural environments. These challenging target configurations demanded that we modify previous inverse optimization techniques. In particular, we first find local minima of a candidate enthalpy surface and determine the enthalpy difference $\Delta H$ between such inherent structures and the target structure. Then we determine the lowest positive eigenvalue ${\lambda }_0$ of the Hessian matrix of the enthalpy surface at the target configuration. Finally, we maximize ${\lambda }_0\Delta H$ so that the target structure is both locally stable and globally stable with respect to the inherent structures. Using this modified optimization technique, we have designed short-range radial pair potentials that stabilize the two-dimensional kagome crystal, the rectangular kagome crystal, and rectangular lattices, as well as the three-dimensional structure of the CaF${}_2$ crystal inhabited by a single-particle species. We verify our results by cooling liquid configurations to absolute zero temperature via simulated annealing and ensuring that such states have stable phonon spectra. Except for the rectangular kagome structure, all of the target structures can be stabilized with monotonic repulsive potentials. Our work demonstrates that single-component systems with short-range radial pair potentials can counterintuitively self-assemble into crystal ground states with low symmetry and different local structural environments. Finally, we present general principles that offer guidance in determining whether certain target structures can be achieved as ground states by radial pair potentials.

Figure 1

A schematic plot of the enthalpy surface (equivalent of potential energy surface at constant pressure). If we simply define $\Delta h$ as the enthalpy difference between the target and the lowest competitor and maximize it, we will encounter the “close-competitor problem.” If the competitor list contains structurally close competitors, $\Delta h$ will be controlled by a structurally close competitor, causing an abnormal lifting of the enthalpy of structurally close competitors.

Figure 2

A schematic plot of the enthalpy surface, illustrating our definition of $\Delta h$. (a) If the target structure is not a local minimum of the enthalpy surface, the inherent structure of the target will not be identical to the target and will have a lower enthalpy. $\Delta h$ becomes negative. (b) and (c) If the target structure is a local minimum of the enthalpy surface, the inherent structure of the target will be identical to the target. $\Delta h$ becomes the enthalpy difference between the target and a different inherent structure. Thus, $\Delta h$ might be positive. (b) However, after maximizing $\Delta h$, the curvature near the target structure might be very small, leading to an undesirable phonon spectrum. (c) By maximizing ${\lambda }_0\Delta h/(1+r_c^d)$, we sacrifice some $\Delta h$ to increase the curvature near the target structure while favoring short-range potentials. Note that we usually cannot find all inherent structures in the complex, multidimensional enthalpy surface. If we miss an inherent structure that has a lower enthalpy than our target, that inherent structure will be discovered in the later verification step by simulated annealing.

Figure 3

Result of a 108-particle simulated annealing for the potential given by Eq. (9). This is a perfect kagome crystal.

Figure 4

(left panel) The kagome potential $u_2\left(r\right)$ vs distance corresponding to Eq. (9). (right) The phonon frequency squared ${\omega }^2$ vs the wave vector of the kagome crystal.

Figure 5

(left) Lower-order potential $u_2\left(r\right)$ vs distance for the rectangular lattice with aspect ratio $b/a=2$, corresponding to Eq. (10). (right) The phonon frequency squared ${\omega }^2$ vs the wave vector of the target.

Figure 6

Result of a 108-particle simulated annealing for the potential given by Eq. (10). The particles show a tendency to self-assemble into the rectangular lattice with aspect ratio $b/a=2$, but many defects exist in the resulting configuration.

Figure 7

(left) Higher-order potential $u_2\left(r\right)$ vs distance for a rectangular lattice with aspect ratio $b/a=2$, corresponding to Eq. (11). (right) The phonon frequency squared ${\omega }^2$ vs the wave vector of the target.

Figure 8

Result of a 108-particle simulated annealing for the potential given by Eq. (11). This is a perfect rectangular lattice with aspect ratio $b/a=2$.

Figure 9

(left panel) The potential $u_2\left(r\right)$ vs distance for a rectangular lattice with aspect ratio $b/a=\pi$, corresponding to Eq. (12). (right) The phonon frequency squared ${\omega }^2$ vs the wave vector of the target.

Figure 10

Result of a 24-particle simulated annealing for the potential given by Eq. (12). This is a perfect rectangular lattice with aspect ratio $b/a=\pi$.

Figure 11

The rectangular kagome crystal structure. The particle indicated by an arrow has three nearest neighbors on the left and one nearest neighbor on the right; thus, it is very hard to be stabilized.

Figure 12

(left) The rectangular kagome potential $u_2\left(r\right)$ vs distance corresponding to Eq. (13). (right) The phonon frequency squared ${\omega }^2$ vs the wave vector of the rectangular kagome crystal.

Figure 13

Result of a 24-particle simulated annealing for the potential given by Eq. (13). This is a perfect rectangular kagome crystal.

Figure 14

The conventional unit cell of a CaF${}_2$ crystal. Blue (dark gray) spheres are Ca${}^{2+}$ ions, yellow (light gray) spheres are F${}^{-}$ ions. Particle radii are drawn proportionally to their crystal ionic radii [34] $r$(Ca${}^{2+}$) $=$ 126 pm, $r$(F${}^{-}$) $=$ 117 pm.

Figure 15

(left) The CaF${}_2$ potential $u_2\left(r\right)$ vs distance corresponding to Eq. (14). (right) The phonon frequency squared ${\omega }^2$ vs the wave vector of the CaF${}_2$ crystal inhabited by a single-particle species.

Figure 16

Result of a 12 000-particle MD based simulated annealing for the potential given by Eq. (14). Yellow (light gray) particles are fixed into the CaF${}_2$ structure during the simulation. Green (dark gray) particles self-assemble into the same structure.