Adaptive Finite Element Method for Solving the Exact Kohn-Sham Equation of Density Functional Theory

Results of the application of an adaptive piecewise linear finite element (FE) based solution using the FETK library of M. Holst to a density functional theory (DFT) approximation to the electronic structure of atoms and molecules are reported. The severe problem associated with the rapid variation of the electronic wave functions in the near singular regions of the atomic centers is treated by implementing completely unstructured simplex meshes that resolve these features around atomic nuclei. This concentrates the computational work in the regions in which the shortest length scales are necessary and provides for low resolution in regions for which there is no electron density. The accuracy of the solutions significantly improved when adaptive mesh refinement was applied, and it was found that the essential difficulties of the Kohn-Sham eigenvalues equation were the result of the singular behavior of the atomic potentials. Even though the matrix representations of the discrete Hamiltonian operator in the adaptive finite element basis are always sparse with a linear complexity in the number of discretization points, the overall memory and computational requirements for the solver implemented were found to be quite high. The number of mesh vertices per atom as a function of the atomic number Z and the required accuracy e (in atomic units) was estimated to be v(epsilon, Z) approximate to 122.37(Z(2.2346)/epsilon(1.1173)), and the number of floating point operations per minimization step for a system of N-A atoms was found to be O(N(A)(3)v(epsilon, Z)) (e.g., with Z = 26, epsilon = 0.0015 au, and N-A = 100, the memory requirement and computational cost would be similar to 0.2 terabytes and similar to 25 petaflops). It was found that the high cost of the method could be reduced somewhat by using a geometric-based refinement strategy to fix the error near the singularities.