Integral evaluation algorithms and their implementation

Three different algorithms for the calculation of many center electron-repulsion integrals are discussed, all of which are considered to be economic in terms of the number of arithmetic operations. The common features of the algorithms are as follows: Cartesian Gaussian functions are used, integrals are calculated by blocks (a block being defined as the set of integrals obtainable from four given exponents on four given centers), and functions may be adopted to R(3). Adaption to molecular point group symmetry is not considered. Tables are given showing the minimum number of operations for a selection of block types allowing one to identify the theoretically most economic, and the corresponding salient features. Comments concerning the computer implementations are also given both on sealar and vector processors. In particular, the Cyber 205 is considered, a vector processor on which we have implemented what we believe to be the most efficient algorithm.     

Erratum: On the numerical stability of two widely used PLS algorithms

The above article (DOI: 10.1002/cem.1112) was published online on 14 February 2008. An error was subsequently identified: the captions for Figures 1 and 2 were omitted; they should read as follows: Figure 1. Orthogonality criterion (θ_A) for the octane data as a function of number of components (A) calculated using the standard PLS algorithm and SIMPLS. Figure 2. Orthogonality criterion (θ_A) for the wines data as a function of number of components (A) calculated using the standard PLS algorithm and SIMPLS.     

On the numerical stability of two widely used PLS algorithms

Ideally, the score vectors numerically computed by an orthogonal scores partial least squares (PLS) algorithm should be orthogonal close to machine precision. However, this is not ensured without taking special precautions. The progressive loss of orthogonality with increasing number of components is illustrated for two widely used PLS algorithms, i.e., one that can be considered as a standard PLS algorithm, and SIMPLS. It is shown that the original standard PLS algorithm outperforms the original SIMPLS in terms of numerical stability. However, SIMPLS is confirmed to perform much better in terms of speed. We have investigated reorthogonalization as the special precaution to ensure orthogonality close to machine precision. Since the increase of computing time is relatively small for SIMPLS, we therefore recommend SIMPLS with reorthogonalization. Copyright © 2008 John Wiley & Sons, Ltd.     

Spectral implementation of some quantum algorithms by one- and two-dimensional nuclear magnetic resonance

Quantum information processing has been effectively demonstrated on a small number of qubits by nuclear magnetic resonance. An important subroutine in any computing is the readout of the output. "Spectral implementation" originally suggested by Z. L. Madi, R. Bruschweiler, and R. R. Ernst [J. Chem. Phys. 109, 10603 (1999)], provides an elegant method of readout with the use of an extra "observer" qubit. At the end of computation, detection of the observer qubit provides the output via the multiplet structure of its spectrum. In spectral implementation by two-dimensional experiment the observer qubit retains the memory of input state during computation, thereby providing correlated information on input and output, in the same spectrum. Spectral implementation of Grover's search algorithm, approximate quantum counting, a modified version of Berstein-Vazirani problem, and Hogg's algorithm are demonstrated here in three- and four-qubit systems.     

On generalized Borgen plots II: The line‐moving algorithm and its numerical implementation

Borgen plots are geometric constructions that represent the set of all nonnegative factorizations of spectral data matrices for three‐component systems. The classical construction by Borgen and Kowalski (Anal. Chim. Acta 174, 1‐26 (1985)) is limited to nonnegative data and results in nonnegative factorizations. The new approach of generalized Borgen plots allows factors with small negative entries. This makes it possible to construct Borgen plots for perturbed or noisy spectral data and stabilizes the computation. In the first part of this paper, the mathematical theory of generalized Borgen plots has been introduced. This second part presents the line‐moving algorithm for the construction of generalized Borgen plots. The algorithm is justified, and the implementation in the FACPACK software is validated.     

Spatial updating grand canonical Monte Carlo algorithms for fluid simulation: generalization to continuous potentials and parallel implementation

Spatial updating grand canonical Monte Carlo algorithms are generalizations of random and sequential updating algorithms for lattice systems to continuum fluid models. The elementary steps, insertions or removals, are constructed by generating points in space either at random (random updating) or in a prescribed order (sequential updating). These algorithms have previously been developed only for systems of impenetrable spheres for which no particle overlap occurs. In this work, spatial updating grand canonical algorithms are generalized to continuous, soft-core potentials to account for overlapping configurations. Results on two- and three-dimensional Lennard-Jones fluids indicate that spatial updating grand canonical algorithms, both random and sequential, converge faster than standard grand canonical algorithms. Spatial algorithms based on sequential updating not only exhibit the fastest convergence but also are ideal for parallel implementation due to the absence of strict detailed balance and the nature of the updating that minimizes interprocessor communication. Parallel simulation results for three-dimensional Lennard-Jones fluids show a substantial reduction of simulation time for systems of moderate and large size. The efficiency improvement by parallel processing through domain decomposition is always in addition to the efficiency improvement by sequential updating.     

Highly efficient and exact method for parallelization of grid‐based algorithms and its implementation in DelPhi

The Gauss–Seidel (GS) method is a standard iterative numerical method widely used to solve a system of equations and, in general, is more efficient comparing to other iterative methods, such as the Jacobi method. However, standard implementation of the GS method restricts its utilization in parallel computing due to its requirement of using updated neighboring values (i.e., in current iteration) as soon as they are available. Here, we report an efficient and exact (not requiring assumptions) method to parallelize iterations and to reduce the computational time as a linear/nearly linear function of the number of processes or computing units. In contrast to other existing solutions, our method does not require any assumptions and is equally applicable for solving linear and nonlinear equations. This approach is implemented in the DelPhi program, which is a finite difference Poisson–Boltzmann equation solver to model electrostatics in molecular biology. This development makes the iterative procedure on obtaining the electrostatic potential distribution in the parallelized DelPhi several folds faster than that in the serial code. Further, we demonstrate the advantages of the new parallelized DelPhi by computing the electrostatic potential and the corresponding energies of large supramolecular structures. © 2012 Wiley Periodicals, Inc.     

Organic mixed-valence compounds: a playground for electrons and holes

Mixed-valence (MV) compounds are excellent model systems for the investigation of basic electron-transfer (ET) or charge-transfer (CT) phenomena. These issues are important in complex biophysical processes such as photosynthesis as well as in artificial electronic devices that are based on organic conjugated materials. Organic MV compounds are effective hole-transporting materials in organic light emitting diodes (OLEDs), solar cells, and photochromic windows. However, the importance of organic mixed-valence chemistry should not be seen in terms of the direct applicability of these species but the wealth of knowledge about ET phenomena that has been gained through their study. The great variety of organic redox centers and spacer moieties that may be combined in MV systems as well as the ongoing refinement of ET theories and methods of investigation prompted enormous interest in organic MV compounds in the last decades and show the huge potential of this class of compounds. The goal of this Review is to give an overview of the last decade in organic mixed valence chemistry and to elucidate its impact on modern functional materials chemistry.     

Self-trapping vs. non-trapping of electrons and holes in organic insulators: polyethylene

We show, by electronic structure based molecular dynamics simulations, that an extra electron injected in crystalline polyethylene should fall spontaneously into a self-trapped state, a shallow donor with a large novel distortion pattern involving a pair of trans-gauche defects. Parallel calculations show instead that a hole will remain free and delocalized. We trace the difference of behavior to the intrachain nature of the hole, as opposed to the interchain one of the electron, and argue that applicability of this concept could be more general. Thus electrons (but not holes) should tend to self-trap in saturated organic insulators, but not for example in aromatic insulators, where both carriers are intrachain.     

Improvements of DRISM calculations: symmetry reduction and hybrid algorithms

We present a symmetry-reduced version of the dielectrically-consistent reference interaction site model (DRISM) equation and an adaptation of the Labík-Malijevsky-Vonka hybrid algorithm for its numerical solution. This approach is used for the calculation of site-site correlation functions of water, acetone and a water-acetone mixture. Compared to the traditional Picard iteration of non-reduced DRISM theories, savings of more than 90% in computational time are obtained. The resulting site-site pair-correlation functions are in reasonable agreement with computer simulations.     

Vectorization of quantum chemical algorithms: MC-SCF procedures

Examples of the types of algorithmic modification needed to achieve substantial enhancement for quantum chemical codes on vector processors such as the Cyber 205 are presented. Specific examples include matrix transformations, 4-index integral transformations, and general multiconfiguration–self-consistent-field (MC -SCF ) codes.     

Grid-based energy density analysis: implementation and assessment

Grid-based energy density analysis (grid-EDA) that decomposes the total energy into atomic energies by a space-partitioning function is proposed. The kinetic energy, nuclear attraction, and exchange-correlation functional are evaluated on grid points and are split into atomic contributions. To reduce numerical errors in the conventional scheme of numerical integration, the electronic Coulomb and HF exchange interactions are evaluated by the pseudospectral method, which was first applied to an ab initio method by Friesner [Chem. Phys. Lett. 116, 39 (1985)], and are decomposed into atomic contributions. Grid-EDA using the pseudospectral method succeeds in ensuring less than 1 kcalmol error in total energies for small molecules and providing reliable atomic energy contributions for the problematic lithium cluster, which exhibits a strong basis-set dependence for Mulliken-type EDA. Also, site-dependent atomization energies are estimated by grid-EDA for cluster models such as Li(48), C(41)H(60), and Mg(32)O(32). Grid-EDA reveals that these models imitate crystal environments reasonably because atomization energies estimated from the inner atoms of the models are close to the experimental cohesive energies.     

Fermi and Coulomb holes of molecules in excited states

Correlation holes of electrons with the same (Fermi hole) and different (Coulomb hole) spins in the ground (X¹Σ⁺), first (A¹Σ⁺) and second (B¹II) excited states of LiH were constructed from full configuration interaction (CI ) wave functions. It was found that the shapes of both the Fermi and Coulomb holes in these states are dependent on the location of the reference electron. When the reference electron is chosen to be close to the Li nucleus, the Fermi correlation results in a large negative hole for all three states. However, the A¹Σ⁺ excited state is further characterized by displaying a second hole around the H nucleus, and in the B¹II state, the hole is elongated along the molecular axis. Coulomb correlation shows up strongly in the A¹Σ⁺ state and, in addition, there is clearly correlation of electrons at the two nuclei. These features of the correlation holes were compared with those from a two-Slater-determinant model wave function. The Hartree, Fermi, and Coulomb screening potentials in these states were also studied in the light of possible modeling of the correlation functionals for the excited states. © 1995 John Wiley & Sons, Inc.     

A microfluidic-based hydrodynamic trap: design and implementation

We report an integrated microfluidic device for fine-scale manipulation and confinement of micro- and nanoscale particles in free-solution. Using this device, single particles are trapped in a stagnation point flow at the junction of two intersecting microchannels. The hydrodynamic trap is based on active flow control at a fluid stagnation point using an integrated on-chip valve in a monolithic PDMS-based microfluidic device. In this work, we characterize device design parameters enabling precise control of stagnation point position for efficient trap performance. The microfluidic-based hydrodynamic trap facilitates particle trapping using the sole action of fluid flow and provides a viable alternative to existing confinement and manipulation techniques based on electric, optical, magnetic or acoustic force fields. Overall, the hydrodynamic trap enables non-contact confinement of fluorescent and non-fluorescent particles for extended times and provides a new platform for fundamental studies in biology, biotechnology and materials science.     

Spatial symmetry holes in many-electron atoms and molecules

When a many-electron system has a spatial symmetry, it is shown that there exist spatial symmetry holes, which imply that two or more electrons are prohibited from being at certain spatial positions simultaneously. Inversion holes, rotation holes, and reflection holes, which result from inversion, twofold rotation, and reflection symmetries, respectively, are discussed in detail. The electron-electron counterbalance hole reported in literature is a particular case of the inversion hole. The spatial symmetry holes are illustrated for simple atoms and molecules.     

Multicomponent analysis: Comparison of various graphical and numerical methods

The performance of several graphical (zero-crossing and derivative quotient spectra with standardized divisor) and numerical methods (MULTIC and PLS) for the resolution of binary and ternary mixtures of species is compared. Numerical methods were found to be specially suited to multicomponent analysis, particularly for mixtures containing more than two analytes with highly overlapped spectra. The results obtained by using the compared methods to analyse various synthetic mixtures of acetylsalicylic acid, caffeine and thiamine were quite consistent and errors in the simultaneous quantification of the analytes amounted to less than 5% in all instances.     

Correlated one-particle method: numerical results

In a previous paper a correlated one-particle method was formulated, where the effective Hamiltonian was composed of the Fock operator and a correlation potential. The objective was to define a correlated one-particle theory that would give all properties that can be obtained from a one-particle theory. The Fock-space coupled-cluster method was used to construct the infinite-order correlation potential, which yields correct ionization potentials (IP's) and electron affinities (EA's) as the negative of the eigenvalues. The model, however, was largely independent of orbital choice. To exploit the degree of freedom of improving the orbitals, the Brillouin-Brueckner condition is imposed, which leads to an effective Brueckner Hamiltonian. To assess its numerical properties, the effective Brueckner Hamiltonian is approximated through second order in perturbation. Its eigenvalues are the negative of IP's and EA's correct through second order, and its eigenfunctions are second-order Brueckner orbitals. We also give expressions for its energy and density matrix. Different partitioning schemes of the Hamiltonian are used and the intruder state problem is discussed. The results for ionization potentials, electron affinities, dipole moments, energies, and potential curves are given for some sample molecules.     

Particles and holes in the unitary group method

Based on the definition for complementary Gel'fand states, we proved the simple relationship between the matrix elements of particle states and those of hole states by unitary calculus.     

RAFTing down under: Tales of missing radicals,fancy architectures,and mysterious holes

This highlight describes recent developments in reversible addition–fragmentation transfer (RAFT) polymerization. Succinct coverage of the RAFT mechanism is supplemented by details of synthetic methodologies for making a wide range of architectures ranging from stars to combs, microgels, and blocks. In addition, RAFT reactions in different media such as emulsion and ionic liquids receive attention. Finally, a specific example of a novel material design is briefly introduced, whereas polymers prepared via RAFT are adopted for microporous/honeycomb membrane design. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 365–375, 2003