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The current state of the art for large-scale quantum-mechanical simulationsis the planewave (PW) pseudopotential method, as implemented in codes such asVASP, ABINIT, and many others. However, since the PW method uses a globalFourier basis, with strictly uniform resolution at all points in space, itsuffers from substantial inefficiencies in calculations involving atoms withlocalized states, such as first-row and transition-metal atoms, and requiressignificant nonlocal communications, which limit parallel efficiency.Real-space methods such as finite-differences and finite-elements havepartially addressed both resolution and parallel-communications issues but havebeen plagued by one key disadvantage relative to PW: excessive number ofdegrees of freedom needed to achieve the required accuracies. We present areal-space partition of unity finite element (PUFE) method to solve theKohn-Sham equations of density functional theory. In the PUFE method, we buildthe known atomic physics into the solution process using partition-of-unityenrichment techniques in finite element analysis. The method developed hereinis completely general, applicable to metals and insulators alike, andparticularly efficient for deep, localized potentials, as occur in calculationsat extreme conditions of pressure and temperature. Full self-consistentKohn-Sham calculations are presented for LiH, involving light atoms, and CeAl,involving heavy atoms with large numbers of atomic-orbital enrichments. We findthat the new PUFE approach attains the required accuracies with substantiallyfewer degrees of freedom, typically by an order of magnitude or more, than thePW method. We compute the equation of state of LiH and show that the computedlattice constant and bulk modulus are in excellent agreement with reference PWresults, while requiring an order of magnitude fewer degrees of freedom toobtain.

Partition of unity finite element method for quantum mechanical materials calculations