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Stability Analysis of Turing Patterns Generated by the Schnakenberg Model
We consider the following Schnakenberg model on the interval (−1, 1): ut = D1u − u + vu2 in (−1, 1), vt = D2v + B − vu2 in (−1, 1), u (−1) = u (1) = v (−1) = v (1) = 0, where D1 > 0, D2 > 0, B>0. We rigorously show that the stability of symmetric N−peaked steady-states can be reduced to computing two matrices in terms of the diffusion coefficients D1,D2 and the number N of peaks. These matrices and their spectra are calculated explicitly and sharp conditions for linear stability are derived. The results are verified by some numerical simulations
Heats of formation of perchloric acid, HClO, and perchloric anhydride, ClO. Probing the limits of W1 and W2 theory
The heats of formation of HClO and ClO have been determined to
chemical accuracy for the first time by means of W1 and W2 theory. These
molecules exhibit particularly severe degrees of inner polarization, and as
such obtaining a basis-set limit SCF component to the total atomization energy
becomes a challenge. (Adding high-exponent functions to a standard
basis set has an effect on the order of 100 kcal/mol for ClO.) Wilson's
aug-cc-pV(n+d)Z basis sets represent a dramatic improvement over the standard
aug-cc-pVnZ basis sets, while the aug-cc-pVnZ+2d1f sequence converges still
more rapidly. Jensen's polarization consistent basis sets still require
additional high-exponent functions: for smooth convergence we suggest the
\{aug-pc1+3d,aug-pc2+2d,aug-pc3+d,aug-pc4\} sequence. The role of the tight
functions is shown to be an improved description of the Cl (3d) Rydberg
orbital, enhancing its ability to receive back-bonding from the oxygen lone
pairs. In problematic cases like this (or indeed in general), a single
SCF/aug-cc-pV6Z+2d1f calculation may be preferable over empirically motivated
extrapolations. Our best estimate heats of formation are HClO(g)1 kcal/mol and ClO(g)2 kcal/mol, the largest source of
uncertainty being our inability to account for post-CCSD(T) correlation
effects. While G2 and G3 theory have fairly large errors, G3X theory reproduces
both values to within 2 kcal/mol.Comment: J. Mol. Struct. (THEOCHEM), in press (WATOC'05 special issue
Long-range three-body atom-diatom potential for doublet Li
An accurate long-range {\em ab initio} potential energy surface has been
calculated for the ground state lithium trimer in the frozen diatom
approximation using all electron RCCSD(T). The {\em ab initio} energies are
corrected for basis set superposition error and extrapolated to the complete
basis limit. Molecular van der Waals dispersion coefficients and three-body
dispersion damping terms for the atom-diatomic dissociation limit are presented
from a linear least squares fit and shown to be an essentially exact
representation of the {\em ab initio} surface at large range
Mesa-type patterns in the one-dimensional Brusselator and their stability
The Brusselator is a generic reaction-diffusion model for a tri-molecular
chemical reaction. We consider the case when the input and output reactions are
slow. In this limit, we show the existence of -periodic, spatially bi-stable
structures, \emph{mesas}, and study their stability. Using singular
perturbation techniques, we find a threshold for the stability of mesas.
This threshold occurs in the regime where the exponentially small tails of the
localized structures start to interact. By comparing our results with Turing
analysis, we show that in the generic case, a Turing instability is followed by
a slow coarsening process whereby logarithmically many mesas are annihilated
before the system reaches a steady equilibrium state. We also study a
``breather''-type instability of a mesa, which occurs due to a Hopf
bifurcation. Full numerical simulations are shown to confirm the analytical
results.Comment: to appear, Physica
The Gierer-Meinhardt system on a compact two-dimensional Riemannian Manifold: Interaction of Gaussian curvature and Green's function
In this paper, we rigorously prove the existence and stability of single-peaked patterns for the singularly perturbed Gierer-Meinhardt system on a compact two-dimensional Riemannian manifold
without boundary which are far from spatial homogeneity. Throughout the paper we assume that the activator diffusivity is small enough. We show that for a threshold ratio of the activator diffusivity and the inhibitor diffusivity, the Gaussian curvature and the Green's function interact. A convex combination of the Gaussian curvature and the Green's function together with their derivatives are linked to the peak locations and the o(1) eigenvalues. A nonlocal eigenvalue problem (NLEP) determines the O(1) eigenvalues which all have negative part in this case.RGC of Hong Kon
Potential energy surface of the 2A' Li2+Li doublet ground state
The lowest doublet electronic state for the lithium trimer (2A') is
calculated for use in three-body scattering calculations using the valence
electron FCI method with atomic cores represented using an effective core
potential. It is shown that an accurate description of core-valence correlation
is necessary for accurate calculations of molecular bond lengths, frequencies
and dissociation energies. Interpolation between 2A' ab initio surface data
points in a sparse grid is done using the global interpolant moving least
squares method with a smooth radial data cutoff function included in the
fitting weights and bivariate polynomials as a basis set. The Jahn-Teller
splitting of the 2E' surface into the 2A1 and 2B2 states is investigated using
a combination of both CASSCF and FCI levels of theory. Additionally,
preliminary calculations of the 2A'' surface are also presented using second
order spin restricted open-shell Moller-Plesset perturbation theory.Comment: 7 pages, 5 figure
Core repulsion effects in alkali trimers
The present paper is related to a talk presented during the Symposium on
Coherent Control and Ultracold Chemistry held during the Sixth Congress of the
International Society for Theoretical Chemical Physics (ISTCP-VI, July 2008).
The talk was entitled "Electronic structure properties of alkali dimers and
trimers. Prospects for alignment of ultracold molecules". Here we report on the
electrostatic repulsion forces of the ionic cores at short separation, involved
when the potential energy surfaces of alkali trimers are calculated with a
quantum chemistry approach based on effective large-core potentials for ionic
core description. We demonstrate that such forces in the triatomic molecule can
be obtained as the sum of three pairwise terms. We illustrate our results on
the lowest electronic states of Cs, which are computed for the first time
within a full configuration interaction based on a large Gaussian basis set. As
a preliminary section, we also propose a brief introduction about the
importance of alkali trimer systems in the context of cold and ultracold
molecules
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