Large scale electronic structure calculations require modern high performance computing (HPC) resources and, as important, mature HPC applications that can make efficient use of those. Real-space grid-based applications of Density Functional Theory (DFT) using the Projector Augmented Wave method (PAW) can give the same accuracy as DFT codes relying on a plane wave basis set but exhibit an improved scalability on distributed memory machines. The projection operations of the PAW Hamiltonian are known to be the performance critical part due to their limitation by the available memory bandwidth. We investigate on the utility of a 3D factorizable basis of Hermite functions for the localized PAW projector functions which allows to reduce the bandwidth requirements for the grid representation of the projector functions in projection operations. Additional on-the-fly sampling of the 1D basis functions eliminates the memory transfer almost entirely. For an quantitative assessment of the expected memory bandwidth savings we show performance results of a first implementation on GPUs. Finally, we suggest a PAW generation scheme adjusted to the analytically given projector functions.
The idea of using the SHO basis has probably not received attention as the related basis set has been found not to converge as rapidly as a set of only Gaussian-type orbitals (GTOs) contracted over a set of different spread parameters @math @cite_10 when used as a basis set for quantum chemistry calculations @cite_16 . Note that the subset of radial SHO states with @math are GTOs.
@cite_16 @cite_10
2078146328 2148424525 2085454684 1988689259
Various contracted Gaussian basis sets for atoms up to Kr are presented which have been determined by optimizing atomic self‐consistent field ground state energies with respect to all basis set parameters, i.e., orbital exponents and contraction coefficients. We describe a complete set of algorithms for ab initio molecular simulations based on numerically tabulated atom-centered orbitals (NAOs) to capture a wide range of molecular and materials properties from quantum-mechanical first principles. The full algorithmic framework described here is embodied in the Fritz Haber Institute “ab initio molecular simulations” (FHI-aims) computer program package. Its comprehensive description should be relevant to any other first-principles implementation based on NAOs. The focus here is on density-functional theory (DFT) in the local and semilocal (generalized gradient) approximations, but an extension to hybrid functionals, Hartree–Fock theory, and MP2 GW electron self-energies for total energies and excited states is possible within the same underlying algorithms. An all-electron full-potential treatment that is both computationally efficient and accurate is achieved for periodic and cluster geometries on equal footing, including relaxation and ab initio molecular dynamics. We demonstrate the construction of transferable, hierarchical basis sets, allowing the calculation to range from qualitative tight-binding like accuracy to meV-level total energy convergence with the basis set. Since all basis functions are strictly localized, the otherwise computationally dominant grid-based operations scale as O(N) with system size N. Together with a scalar-relativistic treatment, the basis sets provide access to all elements from light to heavy. Both low-communication parallelization of all real-space grid based algorithms and a ScaLapack-based, customized handling of the linear algebra for all matrix operations are possible, guaranteeing efficient scaling (CPU time and memory) up to massively parallel computer systems with thousands of CPUs. We describe a new method to truncate and recouple basis functions in general variational calculations based on a direct‐product representation of multidimensional wave functions. The method is presented for molecular vibrations; however, the procedure is quite general and can be used in any basis set expansion method. The direct‐product Hamiltonian matrix H is decomposed into a block diagonal matrix H0 plus a remainder H1 . A new subset of basis functions is obtained by diagonalizing H0 . This subset of basis functions is shown to be eigenfunctions of a Hamiltonian in a reduced dimensionality space, ‘‘dressed’’ by the remaining degrees of freedom. These dressed eigenfunctions are then augmented by the component of the original direct‐product basis in which H0 is diagonal. The new basis is recoupled using an energy selection criterion, yielding a substantial reduction in the size of the final full Hamiltonian matrix. The method also suggests a generalization of the vibrational self‐consistent field method,... In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the acc...