Model Order Reduction Algorithm for Estimating the Absorption Spectrum.

The ab initio description of the spectral interior of the absorption spectrum poses both a theoretical and computational challenge for modern electronic structure theory. Due to the often spectrally dense character of this domain in the quantum propagator's eigenspectrum for medium-to-large sized systems, traditional approaches based on the partial diagonalization of the propagator often encounter oscillatory and stagnating convergence. Electronic structure methods which solve the molecular response problem through the solution of spectrally shifted linear systems, such as the complex polarization propagator, offer an alternative approach which is agnostic to the underlying spectral density or domain location. This generality comes at a seemingly high computational cost associated with solving a large linear system for each spectral shift in some discretization of the spectral domain of interest. In this work, we present a novel, adaptive solution to this high computational overhead based on model order reduction techniques via interpolation. Model order reduction reduces the computational complexity of mathematical models and is ubiquitous in the simulation of dynamical systems and control theory. The efficiency and effectiveness of the proposed algorithm in the ab initio prediction of X-ray absorption spectra is demonstrated using a test set of challenging water clusters which are spectrally dense in the neighborhood of the oxygen K-edge. On the basis of a single, user defined tolerance we automatically determine the order of the reduced models and approximate the absorption spectrum up to the given tolerance. We also illustrate that, for the systems studied, the automatically determined model order increases logarithmically with the problem dimension, compared to a linear increase of the number of eigenvalues within the energy window. Furthermore, we observed that the computational cost of the proposed algorithm only scales quadratically with respect to the problem dimension.

Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT.

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires half of the memory compared with the nonsymmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the nonsymmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but the algorithm may require more matrix-vector multiplications in practice. We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.

Combinatorial × computational × cheminformatics (C3) approach to characterization of congeneric libraries of organic pollutants.

Congeners are molecules based on the same carbon skeleton but are different by the number of substituents and/or a substitution pattern. Examples are 1-chloronaphthalene, 1,4-dichloronaphthalene, and 1,3,8-trichloronaphthalene. Various persistent organic pollutants (POPs) exist in the environment as families of congeners. Very large numbers of possible congeners make their experimental characterization and risk assessment unfeasible. Computational high-throughput and quantitative structure-property relationship (QSPR) modeling has been limited by the lack of tools and approaches facilitating analysis of such POP families. We present a comprehensive approach that enables modeling of extremely large congeneric libraries. The approach involves three steps: (1) combinatorial generation of a library of congeners, (2) quantum chemical characterization of each structure at the PM6 semiempirical level to obtain molecular descriptors, and (3) analysis of the information generated in step 2. In steps 1-3, we employ combinatorial, computational, and cheminformatics techniques, respectively. Therefore, this hybrid approach is named "Combinatorial × Computational × Cheminformatics", or just abbreviated as C(3) (or C-cubed) approach. We demonstrate the usefulness of this approach by generating and characterizing Br- and Cl-substituted congeneric families of 23 typical POPs. The analysis of the resulting set of 1 840 951 congeners that includes Cl-, Br-, and mixed Br/Cl-substituted species, proves that, based on structural similarities defined by the molecular descriptors' values, the existing QSPR models developed originally for Cl- and Br-substituted congeners can be applied also to mixed Br/Cl-substituted ones. Thus, the C(3) approach may serve as a tool for exploring structural applicability domains of the existing QSPR models for congeneric sets.

On enumeration of congeners of common persistent organic pollutants.

Congeners are molecules based on the same carbon skeleton but different by the number of substituents and/or a substitution pattern. Various Persistent Organic Pollutants (POPs) exist in the environment as families of halogen substituted congeners and/or their hydroxyl and methoxy substituted derivatives. Numbers of possible congeners resulting from substitution of a parent POP molecule with only one type of chemical group are generally available. At the same time, numbers of mixed-substituent congeners have not been counted and presented yet, although there is an increasing interest in such as is the increasing number of research articles presenting results on already identified Cl-/Br-mixed type congeners and/or their HO-/CH(3)O-mixed metabolites. We have enumerated and counted possible mixed-substituent congeners of common POPs. This article presents the obtained numbers for congener families of benzene, naphthalene, biphenyl, diphenyl ether, dibenzo-p-dioxin, dibenzofuran, anthracene, pyrene and others and obtained by substitution of up to five chemical group types.

The parallelization of SPIDER on distributed-memory computers using MPI.

We describe the strategies and implementation details we employed to parallelize the SPIDER software package on distributed-memory parallel computers using the message passing interface (MPI). The MPI-enabled SPIDER preserves the interactive command line and batch interface used in the sequential version of SPIDER, thus does not require users to modify their existing batch programs. We show the excellent performance of the MPI-enabled SPIDER when it is used to perform multi-reference alignment and 3-D reconstruction operations on a number of different computing platforms. We point out some performance issues when the MPI-enabled SPIDER is used for a complete 3-D projection matching refinement run, and propose several ways to further improve the parallel performance of SPIDER on distributed-memory machines.

Unified 3-D structure and projection orientation refinement using quasi-Newton algorithm.

We describe an algorithm for simultaneous refinement of a three-dimensional (3-D) density map and of the orientation parameters of two-dimensional (2-D) projections that are used to reconstruct this map. The application is in electron microscopy, where the 3-D structure of a protein has to be determined from a set of 2-D projections collected at random but initially unknown angles. The design of the algorithm is based on the assumption that initial low resolution approximation of the density map and reasonable guesses for orientation parameters are available. Thus, the algorithm is applicable in final stages of the structure refinement, when the quality of the results is of main concern. We define the objective function to be minimized in real space and solve the resulting nonlinear optimization problem using a Quasi-Newton algorithm. We calculate analytical derivatives with respect to density distribution and the finite difference approximations of derivatives with respect to orientation parameters. We demonstrate that calculation of derivatives is robust with respect to noise in the data. This is due to the fact that noise is annihilated by the back-projection operations. Our algorithm is distinguished from other orientation refinement methods (i) by the simultaneous update of the density map and orientation parameters resulting in a highly efficient computational scheme and (ii) by the high quality of the results produced by a direct minimization of the discrepancy between the 2-D data and the projected views of the reconstructed 3-D structure. We demonstrate the speed and accuracy of our method by using simulated data.