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Fronts from complex two-dimensional dispersal kernels: theory and application to Reid's paradox

Bimodal dispersal probability distributions with characteristic distances differing by several orders of magnitude have been derived and favorably compared to observations by Nathan [Nature (London) 418, 409 (2002)]. For such bimodal kernels, we show that two-dimensional molecular dynamics computer simulations are unable to yield accurate front speeds. Analytically, the usual continuous-space random walks (CSRWs) are applied to two dimensions. We also introduce discrete-space random walks and use them to check the CSRW results (because of the inefficiency of the numerical simulations). The physical results reported are shown to predict front speeds high enough to possibly explain Reid's paradox of rapid tree migration. We also show that, for a time-ordered evolution equation, fronts are always slower in two dimensions than in one dimension and that this difference is important both for unimodal and for bimodal kernels

Additional compact formulas for vibrational dynamic dipole polarizabilities and hyperpolarizabilities

Compact expressions, complete through second order in electrical and/or mechanical anharmonicity, are given for the dynamic dipole vibrational polarizability and dynamic first and second vibrational hyperpolarizabilities. Certain contributions not previously formulated are now included

Critical thoughts on computing atom condensed Fukui functions

Different procedures to obtain atom condensed Fukui functions are described. It is shown how the resulting values may differ depending on the exact approach to atom condensed Fukui functions. The condensed Fukui function can be computed using either the fragment of molecular response approach or the response of molecular fragment approach. The two approaches are nonequivalent; only the latter approach corresponds in general with a population difference expression. The Mulliken approach does not depend on the approach taken but has some computational drawbacks. The different resulting expressions are tested for a wide set of molecules. In practice one must make seemingly arbitrary choices about how to compute condensed Fukui functions, which suggests questioning the role of these indicators in conceptual density-functional theory

Critical analysis and extension of the Hirshfeld atoms in molecules

The computational approach to the Hirshfeld [Theor. Chim. Acta 44, 129 (1977)] atom in a molecule is critically investigated, and several difficulties are highlighted. It is shown that these difficulties are mitigated by an alternative, iterative version, of the Hirshfeld partitioning procedure. The iterative scheme ensures that the Hirshfeld definition represents a mathematically proper information entropy, allows the Hirshfeld approach to be used for charged molecules, eliminates arbitrariness in the choice of the promolecule, and increases the magnitudes of the charges. The resulting "Hirshfeld-I charges" correlate well with electrostatic potential derived atomic charges

Selected configuration interaction with truncation energy error and application to the Ne atom

Selected configuration interaction (SCI) for atomic and molecular electronic structure calculations is reformulated in a general framework encompassing all CI methods. The linked cluster expansion is used as an intermediate device to approximate CI coefficients BK of disconnected configurations (those that can be expressed as products of combinations of singly and doubly excited ones) in terms of CI coefficients of lower-excited configurations where each K is a linear combination of configuration-state-functions (CSFs) over all degenerate elements of K. Disconnected configurations up to sextuply excited ones are selected by Brown's energy formula, ΔEK=(E-HKK)BK2/(1-BK2), with BK determined from coefficients of singly and doubly excited configurations. The truncation energy error from disconnected configurations, Δdis, is approximated by the sum of ΔEKS of all discarded Ks. The remaining (connected) configurations are selected by thresholds based on natural orbital concepts. Given a model CI space M, a usual upper bound ES is computed by CI in a selected space S, and EM=E S+ΔEdis+δE, where δE is a residual error which can be calculated by well-defined sensitivity analyses. An SCI calculation on Ne ground state featuring 1077 orbitals is presented. Convergence to within near spectroscopic accuracy (0.5 cm-1) is achieved in a model space M of 1.4× 109 CSFs (1.1 × 1012 determinants) containing up to quadruply excited CSFs. Accurate energy contributions of quintuples and sextuples in a model space of 6.5 × 1012 CSFs are obtained. The impact of SCI on various orbital methods is discussed. Since ΔEdis can readily be calculated for very large basis sets without the need of a CI calculation, it can be used to estimate the orbital basis incompleteness error. A method for precise and efficient evaluation of ES is taken up in a companion paper

Anharmonicity contributions to the vibrational second hyperpolarizability of conjugated oligomers

Restricted Hartree-Fock 6-31G calculations of electrical and mechanical anharmonicity contributions to the longitudinal vibrational second hyperpolarizability have been carried out for eight homologous series of conjugated oligomers - polyacetylene, polyyne, polydiacetylene, polybutatriene, polycumulene, polysilane, polymethineimine, and polypyrrole. To draw conclusions about the limiting infinite polymer behavior, chains containing up to 12 heavy atoms along the conjugated backbone were considered. In general, the vibrational hyperpolarizabilities are substantial in comparison with their static electronic counterparts for the dc-Kerr and degenerate four-wave mixing processes (as well as for static fields) but not for electric field-induced second harmonic generation or third harmonic generation. Anharmonicity terms due to nuclear relaxation are important for the dc-Kerr effect (and for the static hyperpolarizability) in the σ-conjugated polymer, polysilane, as well as the nonplanar π systems polymethineimine and polypyrrole. Restricting polypyrrole to be planar, as it is in the crystal phase, causes these anharmonic terms to become negligible. When the same restriction is applied to polymethineimine the effect is reduced but remains quantitatively significant due to the first-order contribution. We conclude that anharmonicity associated with nuclear relaxation can be ignored, for semiquantitative purposes, in planar π-conjugated polymers. The role of zero-point vibrational averaging remains to be evaluated

Linear response functions for a vibrational configuration interaction state

Linear response functions are implemented for a vibrational configuration interaction state allowing accurate analytical calculations of pure vibrational contributions to dynamical polarizabilities. Sample calculations are presented for the pure vibrational contributions to the polarizabilities of water and formaldehyde. We discuss the convergence of the results with respect to various details of the vibrational wave function description as well as the potential and property surfaces. We also analyze the frequency dependence of the linear response function and the effect of accounting phenomenologically for the finite lifetime of the excited vibrational states. Finally, we compare the analytical response approach to a sum-over-states approach

Basis set superposition error-counterpoise corrected potential energy surfaces : application to hydrogen peroxide ... X (X=F-, Cl-, Br-, Li+, Na+) complexes

Møller-Plesset (MP2) and Becke-3-Lee-Yang-Parr (B3LYP) calculations have been used to compare the geometrical parameters, hydrogen-bonding properties, vibrational frequencies and relative energies for several X- and X+ hydrogen peroxide complexes. The geometries and interaction energies were corrected for the basis set superposition error (BSSE) in all the complexes (1-5), using the full counterpoise method, yielding small BSSE values for the 6-311 + G(3df,2p) basis set used. The interaction energies calculated ranged from medium to strong hydrogen-bonding systems (1-3) and strong electrostatic interactions (4 and 5). The molecular interactions have been characterized using the atoms in molecules theory (AIM), and by the analysis of the vibrational frequencies. The minima on the BSSE-counterpoise corrected potential-energy surface (PES) have been determined as described by S. Simón, M. Duran, and J. J. Dannenberg, and the results were compared with the uncorrected PES

The relevance of the Laplacian of intracule and extracule density distributions for analyzing electron-electron interactions in molecules

A topological analysis of intracule and extracule densities and their Laplacians computed within the Hartree-Fock approximation is presented. The analysis of the density distributions reveals that among all possible electron-electron interactions in atoms and between atoms in molecules only very few are located rigorously as local maxima. In contrast, they are clearly identified as local minima in the topology of Laplacian maps. The conceptually different interpretation of intracule and extracule maps is also discussed in detail. An application example to the C2H2, C2H4, and C2H6 series of molecules is presented

Simple finite field nuclear relaxation method for calculating vibrational contribution to degenerate four-wave mixing

A simple extended finite field nuclear relaxation procedure for calculating vibrational contributions to degenerate four-wave mixing (also known as the intensity-dependent refractive index) is presented. As a by-product one also obtains the static vibrationally averaged linear polarizability, as well as the first and second hyperpolarizability. The methodology is validated by illustrative calculations on the water molecule. Further possible extensions are suggested

Simple finite field method for calculation of static and dynamic vibrational hyperpolarizabilities: curvature contributions

In the static field limit, the vibrational hyperpolarizability consists of two contributions due to: (1) the shift in the equilibrium geometry (known as nuclear relaxation), and (2) the change in the shape of the potential energy surface (known as curvature). Simple finite field methods have previously been developed for evaluating these static field contributions and also for determining the effect of nuclear relaxation on dynamic vibrational hyperpolarizabilities in the infinite frequency approximation. In this paper the finite field approach is extended to include, within the infinite frequency approximation, the effect of curvature on the major dynamic nonlinear optical processes

A systematic and feasible method for computing nuclear contributions to electrical properties of polyatomic molecules

An analytic method to evaluate nuclear contributions to electrical properties of polyatomic molecules is presented. Such contributions control changes induced by an electric field on equilibrium geometry (nuclear relaxation contribution) and vibrational motion (vibrational contribution) of a molecular system. Expressions to compute the nuclear contributions have been derived from a power series expansion of the potential energy. These contributions to the electrical properties are given in terms of energy derivatives with respect to normal coordinates, electric field intensity or both. Only one calculation of such derivatives at the field-free equilibrium geometry is required. To show the useful efficiency of the analytical evaluation of electrical properties (the so-called AEEP method), results for calculations on water and pyridine at the SCF/TZ2P and the MP2/TZ2P levels of theory are reported. The results obtained are compared with previous theoretical calculations and with experimental values

Determination of vibrational polarizabilities and hyperpolarizabilities using field-induced coordinates

An analytical set of field-induced coordinates is defined and is used to show that the vibrational degrees of freedom required to completely describe nuclear relaxation polarizabilities and hyperpolarizabilities is reduced from 3N-6 to a relatively small number. As this number does not depend upon the size of the molecule, the process provides computational advantages. A method is provided to separate anharmonic contributions from harmonic contributions as well as effective mechanical from electrical anharmonicity. The procedures are illustrated by Hartree-Fock calculations, indicating that anharmonicity can be very important

Field-induced coordinates for the determination of dynamic vibrational nonlinear optical properties

Three conjugated organic molecules that span a range of polarity and valence-bond/charge transfer characteristics were studied. It was found that dispersion can be insignificant, and that adequate treatment can be achieved with frequency-dependent field-induced vibrational coordinates (FD-FICs)