Elias Rudberg

Calculations of two-photon charge-transfer excitations using Coulomb-attenuated density-functional theory

In this work, we show that an implementation of Coulomb-attenuated density-functional theory leads to considerably better prospects than hitherto for modeling two-photon absorption cross sections for charge-transfer species. This functional, which corrects for the effect of poor asymptotic dependence of commonly used functionals, essentially brings down the widely different results for larger charge-transfer species between Hartree-Fock and density-functional theory (DFT)-B3LYP into a closer range. The Coulomb-attenuated functional, which retains the best aspects of the Hartree-Fock and DFT-B3LYP methods, proves to be very promising for further modeling design of multiphoton materials with technical applications.

Kohn-Sham Density Functional Theory Electronic Structure Calculations with Linearly Scaling Computational Time and Memory Usage.

We present a complete linear scaling method for hybrid Kohn-Sham density functional theory electronic structure calculations and demonstrate its performance. Particular attention is given to the linear scaling computation of the Kohn-Sham exchange-correlation matrix directly in sparse form within the generalized gradient approximation. The described method makes efficient use of sparse data structures at all times and scales linearly with respect to both computational time and memory usage. Benchmark calculations at the BHandHLYP/3-21G level of theory are presented for polypeptide helix molecules with up to 53 250 atoms. Threshold values for computational approximations were chosen on the basis of their impact on the occupied subspace so that the different parts of the calculations were carried out at balanced levels of accuracy. The largest calculation used 307 204 Gaussian basis functions on a single computer with 72 GB of memory. Benchmarks for three-dimensional water clusters are also included, as well as results using the 6-31G** basis set.

Density matrix purification with rigorous error control

Density matrix purification, although being a powerful tool for linear scaling construction of the density matrix in electronic structure calculations, has been limited by uncontrolled error accumulation. In this article, a strategy for the removal of small matrix elements in density matrix purification is proposed with which the forward error can be rigorously controlled. The total forward error is separated into two parts, the error in eigenvalues and the error in the occupied invariant subspace. We use the concept of canonical angles to measure and control differences between exact and approximate occupied subspaces. We also analyze the conditioning of the density matrix construction problem and propose a method for calculation of interior eigenvalues to be used together with density matrix purification.

Difficulties in applying pure Kohn–Sham density functional theory electronic structure methods to protein molecules

Self-consistency-based Kohn-Sham density functional theory (KS-DFT) electronic structure calculations with Gaussian basis sets are reported for a set of 17 protein-like molecules with geometries obtained from the Protein Data Bank. It is found that in many cases such calculations do not converge due to vanishing HOMO-LUMO gaps. A sequence of polyproline I helix molecules is also studied and it is found that self-consistency calculations using pure functionals fail to converge for helices longer than six proline units. Since the computed gap is strongly correlated to the fraction of Hartree-Fock exchange, test calculations using both pure and hybrid density functionals are reported. The tested methods include the pure functionals BLYP, PBE and LDA, as well as Hartree-Fock and the hybrid functionals BHandHLYP, B3LYP and PBE0. The effect of including solvent molecules in the calculations is studied, and it is found that the inclusion of explicit solvent molecules around the protein fragment in many cases gives a larger gap, but that convergence problems due to vanishing gaps still occur in calculations with pure functionals. In order to achieve converged results, some modeling of the charge distribution of solvent water molecules outside the electronic structure calculation is needed. Representing solvent water molecules by a simple point charge distribution is found to give non-vanishing HOMO-LUMO gaps for the tested protein-like systems also for pure functionals.

Hartree-Fock calculations with linearly scaling memory usage

We present an implementation of a set of algorithms for performing Hartree-Fock calculations with resource requirements in terms of both time and memory directly proportional to the system size. In particular, a way of directly computing the Hartree-Fock exchange matrix in sparse form is described which gives only small addressing overhead. Linear scaling in both time and memory is demonstrated in benchmark calculations for system sizes up to 11 650 atoms and 67 204 Gaussian basis functions on a single computer with 32 Gbytes of memory. The sparsity of overlap, Fock, and density matrices as well as band gaps are also shown for a wide range of system sizes, for both linear and three-dimensional systems.

Heisenberg Exchange in Dinuclear Manganese Complexes : A Density Functional Theory Study

This work presents a systematic investigation of the performance of broken symmetry density functional theory for the evaluation of Heisenberg exchange constants. We study dinuclear Mn(IV)-Mn(IV) complexes with bis(μ-oxo), bis(μ-oxo)(μ-carboxylato), and tris(μ-oxo) cores for this purpose, as these are of fundamental biological interest as well as being potential precursors for molecular magnets based on manganese complexes, the so-called Mn12 magnets. The obtained results indicate that quantitative agreement with available experimental data for the Heisenberg exchange constants can be achieved for most of the investigated complexes but also that there are significant failures for some compounds. We evaluate factors influencing the accuracy of obtained results and examine effects of different mappings between broken symmetry and Heisenberg Hamiltonian states in an attempt to formulate a reliable recipe for the evaluation of magnetic coupling in these complexes. An assessment of the bonding situation in the molecular system under investigation is found crucial in choosing the appropriate scheme for evaluation of the Heisenberg exchange constants.

Assessment of density matrix methods for linear scaling electronic structure calculations

Purification and minimization methods for linear scaling computation of the one-particle density matrix for a fixed Hamiltonian matrix are compared. This is done by considering the work needed by each method to achieve a given accuracy in terms of the difference from the exact solution. Numerical tests employing orthogonal as well as non-orthogonal versions of the methods are performed using both element magnitude and cutoff radius based truncation approaches. It is investigated how the convergence speed for the different methods depends on the eigenvalue distribution in the Hamiltonian matrix. An expression for the number of iterations required for the minimization methods studied is derived, taking into account the dependence on both the band gap and the chemical potential. This expression is confirmed by numerical tests. The minimization methods are found to perform at their best when the chemical potential is located near the center of the eigenspectrum. The results indicate that purification is considerably more efficient than the minimization methods studied even when a good starting guess for the minimization is available. In test calculations without a starting guess, purification is more than an order of magnitude more efficient than minimization.

Bringing about matrix sparsity in linear-scaling electronic structure calculations.

The performance of linear‐scaling electronic structure calculations depends critically on matrix sparsity. This article gives an overview of different strategies for removal of small matrix elements, with emphasis on schemes that allow for rigorous control of errors. In particular, a novel scheme is proposed that has significantly smaller computational overhead compared with the Euclidean norm‐based truncation scheme of Rubensson et al. (J Comput Chem 2009, 30, 974) while still achieving the desired asymptotic behavior required for linear scaling. Small matrix elements are removed while ensuring that the Euclidean norm of the error matrix stays below a desired value, so that the resulting error in the occupied subspace can be controlled. The efficiency of the new scheme is investigated in benchmark calculations for water clusters including up to 6523 water molecules. Furthermore, the foundation of matrix sparsity is investigated. This includes a study of the decay of matrix element magnitude with distance between basis function centers for different molecular systems and different methods. The studied methods include Hartree–Fock and density functional theory using both pure and hybrid functionals. The relation between band gap and decay properties of the density matrix is also discussed.

Chunks and Tasks: A programming model for parallelization of dynamic algorithms

We propose Chunks and Tasks, a parallel programming model built on abstractions for both data and work. The application programmer specifies how data and work can be split into smaller pieces, chunks and tasks, respectively. The Chunks and Tasks library maps the chunks and tasks to physical resources. In this way we seek to combine user friendliness with high performance. An application programmer can express a parallel algorithm using a few simple building blocks, defining data and work objects and their relationships. No explicit communication calls are needed; the distribution of both work and data is handled by the Chunks and Tasks library. This makes efficient implementation of complex applications that require dynamic distribution of work and data easier. At the same time, Chunks and Tasks imposes restrictions on data access and task dependencies that facilitate the development of high performance parallel back ends. We discuss the fundamental abstractions underlying the programming model, as well as performance, determinism, and fault resilience considerations. We also present a pilot C++ library implementation for clusters of multicore machines and demonstrate its performance for irregular block-sparse matrix-matrix multiplication.

Rotations of occupied invariant subspaces in self-consistent field calculations

In this article, the self-consistent field (SCF) procedure as used in Hartree–Fock and Kohn–Sham calculations is viewed as a sequence of rotations of the so-called occupied invariant subspace of the potential and density matrices. Computational approximations are characterized as erroneous rotations of this subspace. Differences between subspaces are measured and controlled by the canonical angles between them. With this approach, a first step is taken toward a method where errors from computational approximations are rigorously controlled and threshold values are directly related to the accuracy of the current trial density, thus eliminating the use of ad hoc threshold values. Then, the use of computational resources can be kept down as much as possible without impairment of the SCF convergence.