Charge Densities and Triply-Periodic Minimal Surfaces in Crystalline Materials

The relationship between surfaces of constant charge density and triply-periodic minimal surfaces (TPMS) has been the subject of considerable speculation over many years. Zero-potential surfaces generated by an electrostatic field from a distribution of point charges corresponding to a crystal provide an approximate description of the TPMS for the structure of that crystal. We have recently provided a first-principles alternative to such phenomenological comparisons based on the Vienna {\it ab initio} simulation package (VASP). We showed that the surfaces of zero charge density calculated for the crystal structure of a material converges to the TPMS of the corresponding crystalline material. The exchange-correlation potentials are chosen for the particular material based on the benchmarking of various approximations for these potentials carried out by others. Here, we report an extension of our previous work by giving additional examples of our theory that shows the zero electron density of an equilibrium structure corresponds to a TPMS for a variety of materials and crystalline structures. We study the ground states of elemental materials that differ electronically and structurally, Na, Cu, Al, Zr, and a compound, NiTi, as well as different phases of the elemental solids that are observed by varying the different thermodynamic conditions.