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We investigate a critically-coupled chain of nonlinear oscillators, whose dynamics displays complex spatiotemporal patterns of activity, including regimes in which glider-like coherent excitations move about and interact. The units in the network are identical simple neural circuits whose dynamics is given by the Wilson-Cowan model and are arranged in space along a one-dimensional lattice with nearest neighbor interactions. The interactions follow an alternating sign rule, and hence the “synaptic matrix” M embodying them is tridiagonal antisymmetric and has purely imaginary (critical) eigenvalues. The model illustrates the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a “critical” synaptic matrix, i.e., having purely imaginary eigenvalues. In order to identify the dynamical underpinnings of these behaviors, we explored a discrete-time coupled-map lattice inspired by our system: the dynamics of the units is dictated by a chaotic map of the interval, and the interactions are given by allowing the critical coupling to act for a finite period τ, thus given by a unitary matrix U = exp ⁡ ( τ 2 M ). It is now explicit that such critical couplings are volume-preserving in the sense of Liouville’s theorem. We show that this map is also capable of producing a variety of complex spatiotemporal patterns including gliders, like our original chain of neural circuits. Our results suggest that if the units in isolation are capable of featuring multiple dynamical states, then local critical couplings lead to a wide variety of emergent spatiotemporal phenomena.We investigate a critically-coupled chain of nonlinear oscillators, whose dynamics displays complex spatiotemporal patterns of activity, including regimes in which glider-like coherent excitations move about and interact. The units in the network are identical simple neural circuits whose dynamics is given by the Wilson-Cowan model and are arranged in space along a one-dimensional lattice with nearest neighbor interactions. The interactions follow an alternating sign rule, and hence the “synaptic matrix” M embodying them is tridiagonal antisymmetric and has purely imaginary (critical) eigenvalues. The model illustrates the interplay of two properties: circuits with a complex internal dynamics, such as multiple stable periodic solutions and period doubling bifurcations, and coupling with a “critical” synaptic matrix, i.e., having purely imaginary eigenvalues. In order to identify the dynamical underpinnings of these behaviors, we explored a discrete-time coupled-map lattice inspired by our system: the dy...

This article shows that the asymptotic quantum mechanical exchange energies and wave functions in a bistable potential can be obtained by computing the complex-valued first-passage time across the potential barrier from a transient state endowed with Smoluchowski boundary conditions, whose probability density is concentrated on one side of the barrier. This interpretation is validated by one of the authors’ previous work on diffusion in a random field, showing that this model yields a diffusion equation equivalent to quantum mechanical equations.

A model of a hard oscillator with analytic solution is presented. Its behavior under periodic kicking, for which a closed form stroboscopic map can be obtained, is studied. It is shown that the general structure of such an oscillator includes four distinct regions; the outer two regions correspond to very small or very large amplitude of the external force and match the corresponding regions in soft oscillators (invertible degree one and degree zero circle maps, respectively). There are two new regions for intermediate amplitude of the forcing. Region 3 corresponds to moderate high forcing, and is intrinsic to hard oscillators; it is characterized by discontinuous circle maps with a flat segment. Region 2 (low moderate forcing) has a certain resemblance to a similar region in soft oscillators (noninvertible degree one circle maps); however, the limit set of the dynamics in this region is not a circle, but a branched manifold, obtained as the tangent union of a circle and an interval; the topological structure of this object is generated by the finite size of the repelling set, and is therefore also intrinsic to hard oscillators.

Preface 1. Introduction 2. Characterization of the SOC state 3. Systems exhibiting SOC 4. Computer models 5. The search for a formalism 6. Is it SOC or is it not? Appendices.

Short‐range interatomic interactions in singlet He3 and He4 are discussed in terms of a single determinant of doubly occupied nonorthogonal localized orbitals through the formalism described in Paper I [V. Magnasco and G. F. Musso, J. Chem. Phys. 60, 3744 (1974)]. Ab‐initio calculations using a minimal set of Slater 1s orbitals are reported for several nuclear configurations. Nonadditive terms can be very well represented as resulting from penetration of pairs of nonorthogonal localized electron groups into the core provided by nuclei and ``unperturbed'' electrons of all other groups. Even in the small overlap region it is essential to include all the penetration effects, and this is easily done by the present formalism.

The separation of the electronic energy into Coulombic and penetration components [V. Magnasco, G. F. Musso, and R. McWeeny, J. Chem. Phys. 47, 4617 (1967)] in the case of a (closed‐shell) single determinant wavefunction of nonorthogonal localized spin bond orbitals (BO) is studied on the basis of a convenient density matrix formulation. The main correction that results when overlap is admitted consists of two terms, the first describing the energy of the density ``perturbation'' in the field of the ``unperturbed'' one‐electron operator F0, the second the Coulomb‐exchange self‐interaction of the density ``perturbation.'' When the groups interacting at short range are atoms or molecules, compact closed formulas for the interaction energies can be derived, and the connection with first‐order perturbation theory is discussed. The present formalism is suitable for analyzing both nonadditivity of short‐range intermolecular forces and the nature of barriers to internal rotation in molecules.

Ab‐initio calculations for eclipsed and staggered C2H6 and for several conformers of H2O2 are performed using a single determinant of doubly occupied nonorthogonal localized bond orbitals constructed from a minimal set of Slater AO's. The energies calculated with optimized BO's closely follow the pattern of the full SCF calculations. A component analysis of the rotation barriers is given along the lines suggested in Paper I [V. Magnasco and G. F. Musso, J. Chem. Phys. 60, 3744 (1974)]. The barrier in ethane (2.67 kcal) is primarily due to the penetration of nonorthogonal electron pairs belonging to vicinal C–H bonds into the core provided by nuclei and unperturbed electrons of all other groups. Although the minimal set is not suited to account for the small trans barrier in H2O2 both in BO and full SCF calculations, the calculated cis barrier in H2O2 (13.79 kcal) is seen to have a nature similar to that found in ethane, except for an enhanced importance of the Coulombic contribution. Delocalization effects are found to be of little importance in determining the conformational behavior of both molecules.

With a Huckel‐type Hamiltonian it can be shown that in a molecule described by a one‐determinant wavefunction of localized nonorthogonal bond orbitals the intramolecular interaction energy turns out to be additive between pairs of different bonds to second order in the intergroup overlap. The possibility of building up the conformational energy of a molecule simply by adding contributions from all pairwise interactions between vicinal bonds has been examined for ethane, propane, and butane. The numerical results obtained within a crude first approximation to the matrix elements of the energy operator are highly encouraging.

A self‐consistent group calculation of the orientation dependence of short‐range intermolecular forces in the H2–H2 system has been carried out using strong orthogonal Weinbaum‐like functions constructed from a set of four symmetrically orthogonalized 1s orbitals. The one‐configuration approximation to the wavefunction describing the electronic ground state of the composite system has been improved by admitting localized excitations on the two molecules. The resulting intermolecular energy is for some relative orientations slightly better and for others worse than that obtained in the corresponding nonorthogonal calculation, but the orientation dependence of the interaction proves to be definitely worst in the orthogonal case. Only about 60%of the ``barrier'' obtained with the one‐configuration nonorthogonal SCF—GF calculation is accounted for by the corresponding strong orthogonal wavefunction. The resolution of the one‐configuration interaction into Coulomb and penetration components after the distortio...

In a preceding paper a one‐configuration wavefunction of strong orthogonal group functions was found to give a rather poor description of the orientation dependence of short‐range intermolecular forces in H2–H2. As an alternative to the necessity of relaxing the strong orthogonality condition, it is suggested that the introduction of a small amount of the charge‐transfer state H2−–H2+ describing one‐electron transfer between the two molecules should improve the interaction energy and its change with the relative molecular orientation. Ab initio calculations in a region of intermolecular separations ranging from two to four times the internuclear equilibrium distance of an isolated H2 molecule show that less than 1% of such a charge‐transfer state gives interaction energies which are in substantial agreement with those obtained with a complete configurational interaction.

Short‐range intermolecular forces in the H2–H2 system have been approached by a rigorous quantum‐mechanical calculation using a wavefunction consisting of one configuration of nonorthogonal group functions constructed from a minimal basis set of 1s Slater orbitals. The need of avoiding, with restricted wavefunctions, the forced orthogonalization of the atomic‐orbital basis, which amounts to the relaxation of the strong orthogonality condition in a group‐function approach, appears to be essential in order to give the correct angular dependence of the intermolecular energy. About 98% of the potential `barrier' resulting from the minimal‐basis‐set calculation with complete configuration interaction is accounted for by nonorthogonal group functions which allow only for intramolecular correlation. The even simpler one‐determinant wavefunction constructed from SCF bond orbitals gives numerical results within 5% to 10% of the reference value for the intersection and within 3% to 5% for the energy difference betw...

The valence‐bond theory of molecular structure in its rigorous formulation is applied to investigate the orientation dependence of the short‐range interaction energy for two H2 molecules in their ground state. The electronic wavefunction is extended to include all the singlet valence‐bond (VB) structures arising from the minimal basis set of four 1s Slater orbitals; namely, two covalent, twelve singly polar, and six doubly polar structures having S=0 and MS=0. The occurring three‐ and four‐center molecular integrals have been computed by numerical integration to six significant figures. The resulting intermolecular energy and the electron distribution within the system are compared with those obtained in a preceding paper (Part I) using a wavefunction restricted to nine VB structures drawn from the same set of atomic orbitals after symmetrical orthogonalization. The present unrestricted calculation results not only in a much lower intermolecular energy than that obtained in I, but also in a substantially ...

Uniformly localized molecular orbitals (LMO's) have been obtained from one‐determinant wavefunctions of minimal‐basis‐set LCAO‐SCF‐MO's for CH4, NH3, H2O, H2CO, HCN, C2H6, C2H4, C2H2, and B2H6. The localization criterion is based on the simultaneous maximization in all the available (doubly filled) orbitals of local electron populations which are required be to localized around atoms or between pairs (sometimes triplets) of atoms in the molecule. The well‐defined inner shell, lone pair, and localized bond orbitals resulting for polyatomic molecules appear to fit well into the usual concepts of conventional chemistry. As in the case of diatomics discussed in a previous publication, the uniformly localized orbitals obtained in the present paper prove to be very close to the corresponding LMO's resulting from the rather cumbersome method of maximizing the sum of the orbital self‐repulsion energies.

Hylleraas variational calculations of the second‐order energy for the ground state of H2+ have been performed in the modified form of Murrell–Shaw–Musher–Amos (MS–MA) perturbation theory including exchange described in Paper I. Simple first‐order trial functions consisting in a single optimized Slater function for each multipole are found to be effective in representing nonexpanded multipole components of the interaction energy. The modulated form of the theory includes a large part of the spherical component of the polarization function already in first order, leaving dipole polarization as the dominant second‐order contribution.

Starting with the Hartree–Fock wave function for the ground state, the C6,C8, and C10 dispersion coefficients are calculated for He2. (AIP)

In connection with the modulated form of Murrell–Shaw–Musher–Amos (MS–MA) perturbation theory, some properties of nonlinear minimization of first and second‐order energy functionals with respect to STO exponents are examined. A general formulation for the system of equations for the orbital exponents is deduced, which can be conveniently used in application to specific problems.