Computing the Density Matrix in Electronic Structure Theory on Graphics Processing Units.

The self-consistent solution of a Schrödinger-like equation for the density matrix is a critical and computationally demanding step in quantum-based models of interatomic bonding. This step was tackled historically via the diagonalization of the Hamiltonian. We have investigated the performance and accuracy of the second-order spectral projection (SP2) algorithm for the computation of the density matrix via a recursive expansion of the Fermi operator in a series of generalized matrix-matrix multiplications. We demonstrate that owing to its simplicity, the SP2 algorithm [Niklasson, A. M. N. Phys. Rev. B2002, 66, 155115] is exceptionally well suited to implementation on graphics processing units (GPUs). The performance in double and single precision arithmetic of a hybrid GPU/central processing unit (CPU) and full GPU implementation of the SP2 algorithm exceed those of a CPU-only implementation of the SP2 algorithm and traditional matrix diagonalization when the dimensions of the matrices exceed about 2000 × 2000. Padding schemes for arrays allocated in the GPU memory that optimize the performance of the CUBLAS implementations of the level 3 BLAS DGEMM and SGEMM subroutines for generalized matrix-matrix multiplications are described in detail. The analysis of the relative performance of the hybrid CPU/GPU and full GPU implementations indicate that the transfer of arrays between the GPU and CPU constitutes only a small fraction of the total computation time. The errors measured in the self-consistent density matrices computed using the SP2 algorithm are generally smaller than those measured in matrices computed via diagonalization. Furthermore, the errors in the density matrices computed using the SP2 algorithm do not exhibit any dependence of system size, whereas the errors increase linearly with the number of orbitals when diagonalization is employed.

Surface and interstitial Ti diffusion at the rutile TiO(2)(110) surface.

Diffusion of Ti through the TiO(2)(110) rutile surface plays a key role in the growth and reactivity of TiO(2). To understand the fundamental aspects of this important process, we present an analysis of the diffusion of Ti ad-species at the stoichiometric TiO(2)(110) surface using complementary computational methodologies of density functional theory corrected for on-site Coulomb interactions (DFT + U) and a charge equilibration (QEq) atomistic potential to identify minimum energy pathways. We find that diffusion of Ti from the surface to subsurface (and vice versa) follows an interstitialcy exchange mechanism, involving exchange of surface Ti with the 6-fold coordinated Ti below the bridging oxygen rows. Diffusion in the subsurface between layers also follows an interstitialcy mechanism. The diffusion of Ti is discussed in light of continued attempts to understand the re-oxidation of non-stoichiometric TiO(2)(110) surfaces.

A grid-based Bader analysis algorithm without lattice bias.

A computational method for partitioning a charge density grid into Bader volumes is presented which is efficient, robust, and scales linearly with the number of grid points. The partitioning algorithm follows the steepest ascent paths along the charge density gradient from grid point to grid point until a charge density maximum is reached. In this paper, we describe how accurate off-lattice ascent paths can be represented with respect to the grid points. This improvement maintains the efficient linear scaling of an earlier version of the algorithm, and eliminates a tendency for the Bader surfaces to be aligned along the grid directions. As the algorithm assigns grid points to charge density maxima, subsequent paths are terminated when they reach previously assigned grid points. It is this grid-based approach which gives the algorithm its efficiency, and allows for the analysis of the large grids generated from plane-wave-based density functional theory calculations.