Volume 131, Issue 4, 28 July 2009

We report a detailed numerical investigation of the relative abundance of periodic and chaotic oscillations in phase diagrams for the Belousov–Zhabotinsky (BZ) reaction as described by a nonpolynomial, autonomous, threevariable model suggested by Györgyi and Field [Nature (London)355, 808 (1992)]. The model contains 14 parameters that may be tuned to produce rich dynamical scenarios. By computing the Lyapunov spectra, we find the structuring of periodic and chaotic phases of the BZ reaction to display unusual global patterns, very distinct from those recently found for gas and semiconductor lasers, for electric circuits, and for a few other familiar nonlinear oscillators. The unusual patterns found for the BZ reaction are surprisingly robust and independent of the parameter explored.
 ARTICLES

 Theoretical Methods and Algorithms

Nonmetallic electronegativity equalization and pointdipole interaction model including exchange interactions for molecular dipole moments and polarizabilities
View Description Hide DescriptionThe electronegativity equalization model (EEM) has been combined with a pointdipole interaction model to obtain a molecular mechanics model consisting of atomic charges, atomic dipole moments, and twoatom relay tensors to describe molecular dipole moments and molecular dipoledipole polarizabilities. The EEM has been phrased as an atomatom chargetransfer model allowing for a modification of the chargetransfer terms to avoid that the polarizability approaches infinity for two particles at infinite distance and for long chains. In the present work, these shortcomings have been resolved by adding an energy term for transporting charges through individual atoms. A Gaussian distribution is adopted for the atomic charge distributions, resulting in a damping of the electrostatic interactions at short distances. Assuming that an interatomic exchange term may be described as the overlap between two electronic charge distributions, the EEM has also been extended by a shortrange exchange term. The result is a molecular mechanics model where the difference of charge transfer in insulating and metallic systems is modeled regarding the difference in bond length between different types of system. For example, the model is capable of modeling charge transfer in both alkanes and alkenes with alternating double bonds with the same set of carbon parameters only relying on the difference in bond length between carbon and bonds. Analytical results have been obtained for the polarizability of a long linear chain. These results show that the model is capable of describing the polarizability scaling both linearly and nonlinearly with the size of the system. Similarly, a linear chain with an end atom with a high electronegativity has been analyzed analytically. The dipole moment of this model system can either be independent of the length or increase linearly with the length of the chain. In addition, the model has been parametrized for alkane and alkene chains with data from density functional theory calculations, where the polarizability behaves differently with the chain length. For the molecular dipole moment, the same two systems have been studied with an aldehyde end group. Both the molecular polarizability and the dipole moment are well described as a function of the chain length for both alkane and alkene chains demonstrating the power of the presented model.

Gas content of binary clathrate hydrates with promoters
View Description Hide DescriptionWe develop a methodology to calculate the gas storage capacity of binary hydrates stabilized by promoters. This model utilizes equilibrium experimental data of both hydrates (i.e., the hydrate of pure promoter and the binary hydrate of gas and promoter) in order to calculate the gas occupancy in the small cavities that are not occupied by the promoter. The new approach, although based on the traditional van der Waals–Platteeuw theory, has two significant advantages. The hypothetical state of an empty hydrate for the calculation of the chemical potential of water is replaced by the hydrate of the pure promoter whose properties can be determined through simple thermodynamic calculations. In addition, the computational difficulties related to the complete occupancy of the large cavities by the promoter are removed. The proposed methodology is applied to calculate the gas storage capacity of hydrates of two energycarrier gases (methane and hydrogen) stabilized by tetrahydrofuran. Excellent agreement is observed between the proposedmodel predictions and published experimental values for the gas content of hydrates.

On the Kohn–Sham density response in a localized basis set
View Description Hide DescriptionWe construct the Kohn–Sham density response function in a previously described basis of the space of orbital products. The calculational complexity of our construction is for a molecule of atoms and in a spectroscopic window of frequency points. As a first application, we use to calculate the molecular spectra from the Petersilka–Gossmann–Gross equation. With as input, we obtain the correct spectra with an extra computational effort that grows also as and, therefore, less steeply in than the complexity of solving Casida’s equations. Our construction should be useful for the study of excitons in molecular physics and in related areas where is a crucial ingredient.

A selection rule for molecular conduction
View Description Hide DescriptionConditions for transmission of a conjugated molecular conductor are derived within the source and sink potential approach in terms of numbers of nonbonding levels of four graphs: The molecular graph and the three vertexdeleted subgraphs obtained by removing one or both contact vertices. For all bipartite and most nonbipartite , counting nonbonding levels gives a simple necessary and sufficient condition for conduction at the Fermi level. The exceptional case is where is nonbipartite and all four graphs have the same number of nonbonding levels; then, an auxiliary requirement involving tail coefficients of the four characteristicpolynomials must also be checked.

Relative abundance and structure of chaotic behavior: The nonpolynomial Belousov–Zhabotinsky reaction kinetics
View Description Hide DescriptionWe report a detailed numerical investigation of the relative abundance of periodic and chaotic oscillations in phase diagrams for the Belousov–Zhabotinsky (BZ) reaction as described by a nonpolynomial, autonomous, threevariable model suggested by Györgyi and Field [Nature (London)355, 808 (1992)]. The model contains 14 parameters that may be tuned to produce rich dynamical scenarios. By computing the Lyapunov spectra, we find the structuring of periodic and chaotic phases of the BZ reaction to display unusual global patterns, very distinct from those recently found for gas and semiconductor lasers, for electric circuits, and for a few other familiar nonlinear oscillators. The unusual patterns found for the BZ reaction are surprisingly robust and independent of the parameter explored.

Quantum chemistry of quantum dots: Effects of ligands and oxidation
View Description Hide DescriptionWe report Gaussian basis set density functional theory(DFT) calculations of the structure and spectra of several colloidal quantum dots(QDs) with a core , that are either passivated by trimethylphosphine oxide ligands, or unpassivated and oxidized. From the ground state geometry optimization results we conclude that trimethylphosphine oxide ligands preserve the wurtzite structure of the QDs. Evaporation of the ligands may lead to surface reconstruction. We found that the number of twocoordinated atoms on the nanoparticle’s surface is the critical parameter defining the optical absorptionproperties. For wurtzitederived QD this number is maximal among all considered QDs and the optical absorptionspectrum is strongly redshifted compared to QDs with threefold coordinated surface atoms. According to the timedependent DFT results, surface reconstruction is accompanied by a significant decrease in the linear absorption.Oxidation of QDs destroys the perfection of the QDsurface, increases the number of twocoordinated atoms and results in the appearance of an infrared absorption peak close to 700 nm. The vacant orbitals responsible for this near infrared transition have strong Se–O antibonding character. Conclusions of this study may be used in optimization of engineered nanoparticles for photodetectors and photovoltaic devices.

How to tell when a model Kohn–Sham potential is not a functional derivative
View Description Hide DescriptionA model exchangecorrelation potential constructed with Kohn–Sham orbitals should be a functional derivative of some density functional. Several necessary conditions for a functional derivative are discussed including: (i) minimization of the totalenergy expression by the groundstate solution of the Kohn–Sham equations, (ii) path independence of the van Leeuwen–Baerends line integral, and (iii) net zero force and zero torque on the density. A number of existing model potentials are checked for these properties and it is found that most of the potentials tested are not functional derivatives. Physical properties obtained from potentials that have no parent functionals are ambiguous and, therefore, should be interpreted with caution.

Can shortrange hybrids describe longrangedependent properties?
View Description Hide DescriptionLongrangecorrected hybrids, which incorporate all of the longrange exact exchange interaction, improve performance for a host of molecular properties. The longrange portion of exact exchange is both computationally and formally problematic in solids, and screened hybrids therefore eliminate it. While screened hybrids give similar results to their parent global hybrids for many molecular properties, one may worry that they perform poorly for those properties that are improved by the longrangecorrection procedure. In this paper, we show that at least for the Heyd–Scuseria–Ernzerhof (HSE) screened hybrid, this is not the case; for many properties improved by longrangecorrection, screened hybrids and global hybrids deliver essentially the same results. We suggest that this is because screened hybrids and global hybrids have fundamentally the same manyelectron selfinteraction error. We also introduce some small revisions to our computational implementation of the HSE screened hybrid, and we recommend these revisions for future applications of HSE.

Polarization of onedimensional periodic systems in a static electric field: Sawtooth potential treatment revisited
View Description Hide DescriptionVarious periodic piecewise linear potentials for extracting the electronic response of an infinite periodic system to a uniform electrostatic field are examined. It is shown that discontinuous potentials, such as the sawtooth, cannot be used for this purpose. Continuous triangular potentials can be successfully employed to determine both even and oddorder (hyper)polarizabilities, as demonstrated here for the first time, although the permanent dipole moment of the corresponding long finite chain remains out of reach. Moreover, for typical highly polarizable organic systems, the size of the repeated unit has to be much larger than that of the finite system in order to obtain convergence with respect to system size. All results are illustrated both through extensive model calculations and through ab initio calculations on poly and oligoacetylenes.

Exactexchange kernel of timedependent density functional theory: Frequency dependence and photoabsorption spectra of atoms
View Description Hide DescriptionIn this work we have calculated excitation energies and photoionization cross sections of Be and Ne in the exactexchange (EXX) approximation of timedependent density functional theory (TDDFT). The main focus has been on the frequency dependence of the EXX kernel and on how it affects the spectrum as compared to the corresponding adiabatic approximation. We show that for some discrete excitation energies the frequency dependence is essential to reproduce the results of timedependent HartreeFock theory. Unfortunately, we have found that the EXX approximation breaks down completely at higher energies, producing a response function with the wrong analytic structure and making innershell excitations disappear from the calculated spectra. We have traced this failure to the existence of vanishing eigenvalues of the KohnSham noninteracting response function. Based on the adiabatic TDDFT formalism we propose a new way of deriving the Fano parameters of autoionizing resonances.

An improved treatment of spectator mode vibrations in reduced dimensional quantum dynamics: Application to the hydrogen abstraction reactions , , , and
View Description Hide DescriptionA method for projecting chemical reaction surface coordinates from a Hessian in curvilinear internal coordinates has recently been developed. Here we introduce a modification to this approach which allows for analytical evaluation of the necessary coordinate derivatives, thus reducing the number of ab initio calculations required. We apply this method to the determination of spectator mode frequencies and zeropoint energies for the series of hydrogen abstraction reactions, , H, D, . Comparison of these frequencies with those obtained using rectilinear coordinates allows us to examine how the mass of X affects the coordinate sensitivity of the spectator modes. We carry out twodimensional quantum reactive scattering calculations for these reactions to highlight instances where the choice of coordinates may have a significant impact on the evaluated thermal rate constants.

Quasienergy formulation of damped response theory
View Description Hide DescriptionWe present a quasienergybased formulation of damped response theory where a common effective lifetime parameter has been introduced for all excited states in terms of complex excitation energies. The introduction of finite excited state lifetimes leads to a set of (complex) damped response equations, which have the same form to all orders in the perturbation. An algorithm is presented for solving the damped response equations in Hartree–Fock theory and Kohn–Sham density functional theory. The use of the quasienergy formulation allows us to obtain directly the computationally simplest expressions for damped response functions by applying a set of response parameter elimination rules, which minimize the total number of damped response equations to be solved. In standard response theory broadened absorption spectra are obtained by ad hoc superimposing lineshape functions onto the absorption stick spectra, whereas an empirical lineshape function common to all excitations is an integrated part of damped response theory. By superimposing the lineshape functions inherent in damped response theory onto the stick spectra of standard response theory, we show that the absorption spectra obtained in standard and damped response theory calculations are identical. We demonstrate that damped response theory may be applied to obtain absorption spectra in all frequency ranges, also those that are not readily addressed using standard response theory. This makes damped response theory an effective tool, e.g., for determining absorption spectra for large molecules, where the density of the excited states may be very high, and where standard response theory therefore is not applicable in practice. A thorough comparison is given between our formulation of damped response theory and the formulation by Norman et al. [J. Chem. Phys.123, 194103 (2005)].

VARICELLA: A variablecell direct space method for structure determination from powder diffraction data
View Description Hide DescriptionA direct space method for structure determination from powderdiffraction data is proposed. Employing a hybrid Monte Carlo algorithm for generating the random conformations of a flexible molecular model, and sampling in a modified multicanonical statistical ensemble, it allows for variable cell parameters during an iterative search process. The acceptancerejection criterion involves both a disagreement factor between the calculated and the experimental diffraction profiles and a modified crystal energy so that the space of tentative solutions can be widely explored while maintaining some physical meaningfulness of the proposals. Allowing the cell to be variable requires the zero shift to be treated as an optimizing parameter; this, in turn, requiring the disagreement factor to be based on the Fourier transform of the spectrum. The algorithm is presented in both a serial and a parallel version, the latter presenting several advantages, such as the possibility to probe different structures at a time while keeping them far from each other in the space defined by suitable order parameters. The method is built up and carefully tested by using, as a case study, a crystal of 3ethyl 2,3exodisyndiotactic norbornene heptamer recently determined by single crystal xray diffraction techniques.

Charge conservation in electronegativity equalization and its implications for the electrostatic properties of fluctuatingcharge models
View Description Hide DescriptionAn analytical solution of fluctuatingcharge models using Gaussian elimination allows us to isolate the contribution of charge conservation effects in determining the charge distribution. We use this analytical solution to calculate dipole moments and polarizabilities and show that charge conservation plays a critical role in maintaining the correct translational invariance of the electrostaticproperties predicted by these models.

Quantum Monte Carlo ground state energies for the atoms Li through Ar
View Description Hide DescriptionAllelectron quantum Monte Carloenergies are reported for the ground state of the atoms Li to Ar. The present work is mainly focused on the atoms Na to Ar as well as in those that have a stronger multiconfiguration nature, i.e., Be, B, and C and Mg, Al, and Si. Explicitly correlated wave functions with a single configuration model function times a Jastrow factor are employed for all of the atoms studied. The accuracy obtained for the atoms Na to Ar is similar to that reached for the atoms Li to Ne. In addition, a restricted multiconfiguration expansion has been employed for the atoms Be, B, and C and Mg, Al, and Si obtaining accurate results. Near degeneracy and the effect of other configurations are systematically analyzed for these systems, at both variational and diffusion Monte Carlo levels.

Frozen Gaussian series representation of the imaginary time propagator theory and numerical tests
View Description Hide DescriptionThawed Gaussian wavepackets have been used in recent years to compute approximations to the thermal density matrix. From a numerical point of view, it is cheaper to employ frozen Gaussian wavepackets. In this paper, we provide the formalism for the computation of thermal densities using frozen Gaussian wavepackets. We show that the exact density may be given in terms of a series, in which the zeroth order term is the frozen Gaussian. A numerical test of the methodology is presented for deep tunneling in the quartic double well potential. In all cases, the series is observed to converge. The convergence of the diagonal density matrix element is much faster than that of the antidiagonal one, suggesting that the methodology should be especially useful for the computation of partition functions. As a by product of this study, we find that the density matrix in configuration space can have more than two saddle points at low temperatures. This has implications for the use of the quantum instanton theory.

A Hirshfeld interpretation of the charge, spin distribution, and polarity of the dipole moment of the open shell nitrogen halides: NF, NCl, and NBr
View Description Hide DescriptionWe calculated the dipole moment function for the ground state of the open shell molecules, NF, NCl, and NBr, and analyzed it in terms of the charge and spin distribution and the induced atomic dipoles using the Hirshfeld partitioning of the electron density. The smallest dipole moment obtains with NF, in which the atoms have the largest difference in electronegativity, while the dipole moments in NCl and NBr are and , respectively. All dipoles have the polarity. In the system spin electrons flow from N to the halogen while spin electrons flow in the opposite direction and interestingly from both the and the systems of the halogen to the system of N. In NF the number of spins lost by F is essentially equal to the number of spins gained and the atomic charges are essentially 0. The small dipole in NF is the result of a slight imbalance in the induced atomic dipoles. For NCl and NBr the halogen loses more spins than it gains spins resulting in the polarity . It is interesting that at equilibrium N gained electrons in the system while the halogen lost electrons relative to the separated atoms. This however is not back donation in the usual sense because the electrons gained by N have spin while those lost by the halogen have spin. Detailed examination of the spin flow shows that the excess electrons in the system of N come from an intraatomic transfer from the N system. The induced atomic dipole moments essentially cancel at all internuclear separations and the polarity of the dipole moment accurately reflects the molecular charge distribution.

Explicitly correlated combined coupledcluster and perturbation methods
View Description Hide DescriptionCoupledcluster singles and doubles (CCSD) or coupledcluster singles, doubles, and triples (CCSDT) with noniterative, perturbation corrections for higherorder excitations have been extended to include the basis functions that explicitly depend on interelectronic distances in the wave function expansions with the aim of dramatically accelerating the basisset convergence of correlation energies. The extension has been based on the socalled R12 (or F12) scheme and applied to a secondorder triples correction to CCSD , a secondorder triples and quadruples correction to CCSD , a thirdorder triples correction to CCSD , and a secondorder quadruples correction to CCSDT . A simplified R12 treatment suggested by Fliegl et al. [J. Chem. Phys.122, 084107 (2005)] has been combined with some of these methods, introducing and . The CCSD(T)R12 method has also been developed as an approximation to . These methods have been applied to dissociation of hydrogen fluoride and double dissociation of water. For the molecules at their equilibrium geometries, molecular properties predicted by these methods converge extremely rapidly toward the completecorrelation, completebasisset limits with respect to the cluster excitation rank, perturbation order, and basisset size. Although the R12 scheme employed in this work does not improve the basisset convergence of connected triples or quadruples corrections, the basisset truncation errors in these contributions have roughly the same magnitude as small residual basisset truncation errors in the connected singles and doubles contributions even in the dissociation of hydrogen fluoride. In the double dissociation of water, the basisset truncation errors in the connected triples contribution can be a few times as great as those in the connected singles and doubles contributions.

Correlationinduced corrections to the band structure of boron nitride: A wavefunctionbased approach
View Description Hide DescriptionWe present a systematic study of the correlationinduced corrections to the electronic band structure of zincblende BN. Our investigation employs an ab initio wavefunctionbased local Hamiltonian approach which offers a rigorous theoretical framework for the calculation of the polarization and local charge redistribution effects around an extra electron or hole placed into the conduction or valence bands of semiconducting and insulating materials. Moreover, electron correlations beyond relaxation and polarization can be readily incorporated. The electron correlation treatment is performed on finite clusters. In conducting our study, we make use of localized Wannier functions and embedding potentials derived explicitly from prior periodic Hartree–Fock calculations. The onsite and nearestneighbor charge relaxations bring corrections of several eV to the Hartree–Fock band gap. Additional corrections are caused by longrange polarization effects. In contrast, the dispersion of the Hartree–Fock bands is marginally affected by electron correlations. Our final result for the fundamental gap of zincblende BN compares well with that derived from soft xray experiments at the B and N edges.

Exact rate calculations by trajectory parallelization and tilting
View Description Hide DescriptionA sampling procedure to compute exactly the rate of activated processes arising in systems at equilibrium or nonequilibrium steady state is presented. The procedure is a generalization of the method proposed in [A. Warmflash et al., J. Chem. Phys.127, 154112 (2007); A. Dickson et al., J. Chem. Phys.130, 074104 (2009)] in which one performs simulations restricted into cells by using a reinjection rule at the boundaries of the cells which is consistent with the exact probability fluxes through these boundaries. Our generalization uses results from transition path theory which indicate how to tilt the dynamics to calculate reaction rates.