Liquid Neon

To initially test that the D&C algorithm returns an acceptable approximation of the density matrix, a simple test case was used. The D&C algorithm is most appropriate for weakly interacting systems. Long range Coulombic interactions are not yet included in the CRYSTAL implementation and periodic covalently bonded systems cause inconsistencies in the subsystems' electronic structure, see Section 6. A 10 Å$^3$ box of neon atoms created using a short classical molecular dynamics simulation¹ was used as a test case, see Figure 2.

Figure 2: The system used to test that the density matrix partitioning and reconstruction functioned correctly. 10 Å$^3$ of neon atoms in positions determined by a short classical molecular dynamics computation.

The system was partitioned into subsystems using one core-atom subsystems, and a set of partition matrices as defined in Equation (4). The convergence of the system energy as a function of halo radius is reported in Table 1. It can be seen that as the size of the halo is increased the energy converges on the result of the traditional Hamiltonian diagonalisation used in standard CRYSTAL calculations. This is unsurprising considering the weak interatomic interactions in this system, but indicates that the divide and conquer implementation correctly partitions the system and reconstructs the density matrix based on the subsystem computations.

Table 1: Convergence on the DFT energy for the D&C algorithm as the radius of the halo increases. $E_{\mathtt {D\&C}}$ is the energy computed using the D&C algorithm, $E_{\mathtt {trad}}$ is the energy computed using Hamiltonian Matrix diagonalisation -- the standard CRYSTAL algorithm.