Single Saddle Point Research Articles

Theory of the collective behavior of two-dimensional periodic ensembles of dipole-coupled magnetic nanoparticles

The magnetic behavior of two-dimensional (2D) periodic ensembles of dipole-coupled nanoparticles (NPs) is investigated theoretically by considering all possible structural organizations in 2D Bravais lattices, namely, square, triangular, rectangular, rhombic, and oblique lattices. The different interaction-energy landscapes (ELs) are characterized by determining the local minima that define the stable and metastable magnetic configurations as well as the transition states connecting them. The topology of the resulting ergodic ensemble of stationary states is analyzed from both local and energy perspectives by calculating the corresponding kinetic networks and disconnectivity graphs. For all lattices, the magnetic orders of the ground-state and low-lying metastable configurations are identified, including the elementary relaxation processes between them. A remarkable profound dependence of the collective magnetic behavior on the structural arrangement of the NPs is revealed. Square and triangular nanostructures are extremely good structure seekers, showing starlike kinetic networks and disconnectivity graphs with palm-tree form. Their continuously degenerate ground states are throughout-reaching hubs connected to nearly all excited magnetic configurations over a single first-order saddle point. Rhombic ensembles have double-funnel ELs in which the time-inversion-related ground states play the role of hubs with exceedingly high connectivity densities. These systems need to undergo very few elementary transitions with small downward energy barriers to relax from any configuration towards one of the ground states. However, the energy barriers between the two funnels are quite large and, furthermore, they increase as the number of particles in the unit cell is increased. Therefore, ergodicity breaking is expected in the thermodynamic limit. Finally, the rectangular and oblique lattices show contrasting ground-state orders, the former consists of an antiferromagnetic alternation of head-to-tail chains of spins while the latter is ferromagnetic. Nevertheless, the ELs of these nanostructures are qualitatively very similar. In both cases, all excited magnetic configurations consist of independent flipping of the chains of spins that are formed along the direction of the shorter Bravais vector. The whole energy spectrum of metastable magnetic configurations can thus be mapped to a one-dimensional Ising model. As a result, the kinetic network of stationary points is latticelike and the disconnectivity graphs have a willow-tree form. In conclusion, the collective magnetic behaviors of nanostructures having different point-group symmetries are contrasted and related by varying the parameters of the lattices that interpolate between them.

Dynamics and decay rates of a time-dependent two-saddle system.

The framework of transition state theory (TST) provides a powerful way for analyzing the dynamics of physical and chemical reactions. While TST has already been successfully used to obtain reaction rates for systems with a single time-dependent saddle point, multiple driven saddles have proven challenging because of their fractal-like phase space structure. This paper presents the construction of an approximately recrossing-free dividing surface based on the normally hyperbolic invariant manifold in a time-dependent two-saddle model system. Based on this, multiple methods for obtaining instantaneous (time-resolved) decay rates of the underlying activated complex are presented and their results discussed.

Absolute instabilities in the spatially developing Kuroshio Extension

Satellite observations have long revealed to us a spatially growing Kuroshio Extension (KEx), but its underlying dynamics is yet to be studied. With a normal mode model of absolute/convective instability, it is found that the mean zonal jet is unstable at all the sections in the downstream region (east of 154 °E). In each of the resulting complex dispersion relation diagrams there lies a single saddle point associated with a positive temporal growth rate; that is to say, the mean jet is absolutely unstable, implying that KEx favors self-sustained oscillations. By calculation the absolute instability wave has a period increasing from about 27 days to 72 days, and a slightly decreasing wavelength from 360 km to 250 km, as longitude increases from 154 °E to 174 °E, agreeing with those inferred from the wavelet power spectra and Hovmöller diagram of the satellite observations. As KEx travels downstream, the associated eigen-structure of the perturbation velocity changes from a surface trapped mode to a mode with components maximized in the vertical interior. This study shows that at least a portion of the KEx intraseasonal variability is of intrinsic origin, and may be predictable with the absolute/convective instability theory.

Квантово-химическое исследование механизмов реакций бензамида и бензолсульфонамида с хлорангидридом 3-нитробензолсульфокислоты в газовой фазе

Quantum-chemical simulation of mechanisms of 3-nitrobenzenesulfonic acid chloride interactions with benzoic and benzenesulfonic acids amides in the gas phase was carried out by calculating the three-dimensional potential energy surfaces of these reactions (DFT//B3LYP/6-311G(d,p) level). It was found that in both of the processes considered, a single route can be realized containing a single saddle point and starting as an axial attack of the nucleophile. Further approach of the reagent molecules proceeds with a decrease in the angle of nucleophilic attack to ≈ 130o in the reaction transition state and ≈ 100o – in the reaction product – sulfonamide. It was shown that the studied reactions proceed according to the bimolecular concerted mechanism of nucleophilic substitution SN2, which implies the formation of a single transition state along the reaction pathway. It was found that the geometric structure of the reaction centers in the transition states of the processes is intermediate between the trigonal bipyramid and the tetragonal pyramid, which is explained by the change in the angle of nucleophilic attack when the reagent molecules approach each other. It was found that in benzamide sulfonylation reaction, a cyclic transition state is formed, in which the forming and loosening bonds lie in the same plane, and the H-Cl distance corresponds to the length of the hydrogen bond. In benzenesulfonamide reaction with 3-nitrobenzenesulfonyl chloride, the transition state is not cyclic. The activation energies of the reactions are calculated; they were 155 kJ/mol in the benzamide sulfonylation reaction and 150 kJ/mol in the process with the participation of benzenesulfonic acid amide. The closeness of the obtained values is associated with the similar structure of the amide and sulfamide groups containing electrophilic centers near the amino groups. A significant difference in the rate constants of the studied reactions, which was found earlier, when they occur in aqueous dioxane, is explained by the features of –CONH2 and –SO2NH2 groups specific solvation and the contribution of the entropy factor to the reaction rate: the cyclic transition state of the benzamide reaction with 3-nitrobenzenesulfonyl chloride is more ordered in comparison with a non-cyclic transition state of the reaction with benzenesulfonamide participation, which can promote faster occurence of the first process.

Квантово-химическое исследование механизмов сульфонилирования гидразидов бензойной и бензолсульфоновой кислот в газовой фазе

Quantum-chemical simulation of the mechanisms of 3-nitrobenzenesulfonyl chloride interaction with benzhydrazide (RHF/6-31G(d)) and benzenesulfohydrazide (DFT//B3LYP/6 311G(d,p)) in the gas phase was carried out. Three-dimensional potential energy surfaces of these processes are calculated in the coordinates of the angle of attack of the nucleophile and the distance between the reacting molecules. It has been established that in the both cases considered, reactions can proceed in the gas phase along a single route, through a single saddle point corresponding to a single transition state; processes begin as an axial attack of nucleophile, which subsequently proceeds with a decrease in the attack angle as the reagents molecules approach each other. It was shown that the both studied processes proceed in accordance with the bimolecular concerted mechanism of nucleophilic substitution SN2, which involves the formation of a single transition state in a reaction pathway and the absence of intermediates on it. Scanning the internal coordinate of benzhydrazide reaction with 3-nitrobenzenesulfonyl chloride made it possible to confirm the reaction route and mechanism of the process pointed out and to clarify the structure of its products and reagents. It was found that the geometric structure of the reaction center in the reactions transition states is medium between the trigonal-bipyramidal and tetragonal-pyramidal, which is due to the change in the nucleophilic attack angle when the reagents molecules approach each other. It was found that in reactions involving hydrazides a “synchronous” transition state is formed in which a new S-N bond is formed simultaneously with the loosening of the old S-Cl bond. The activation energies of the reactions are calculated; they amounted to 173 and 113 kJ/mol, respectively. The high values obtained are explained by the fact that the simulation was carried out for processes occurring in a gas hase. It was shown that the decrease in the activation energy of the reaction involving benzenesulfohydrazide as compared to the benzhydrazide reaction is due to a decrease in steric hindrances during nucleophilic attack created by the lone electron pair of the benzenesulfohydrazide secondary amino group as compared to the benzhydrazide molecule. The calculated values of charges on the nitrogen atoms of the –NH– groups in the hydrazides molecules indicate a weakening of the α-effect upon the transition from benzenesufohydrazide to benzhydrazide.

Homogeneous transitions during inflation: A description in quantum cosmology

The usual description of inflationary fluctuations uses the framework of quantum field theory (QFT) in curved spacetime, in which quantum fluctuations are superimposed on a classical background spacetime. Even for large fluctuations, such as those envisioned during a regime of eternal inflation, this framework is frequently used. In the present work we go one step beyond this description by quantising both the scalar field and the scale factor of the universe. Employing the Lorentzian path integral formulation of semi-classical gravity we restrict to a simplified minisuperspace setting by considering homogeneous transitions. This approach allows us to determine the dominant geometry and inflaton evolution contributing to such amplitudes. We find that for precisely specified initial scale factor and inflaton values (and uncertain momenta), two distinct saddle point geometries contribute to the amplitude, leading to interference effects. However, when the momenta of both scale factor and inflaton are specified with sufficient certainty, only a single saddle point is relevant and QFT in curved spacetime is applicable. In particular we find that for inflaton transitions up the potential, meaningful results are only obtained when the initial uncertainty in the inflaton value is large enough, allowing the dominant evolution to be a complexified slow-roll solution \emph{down} from a comparatively unlikely position higher up in the potential.

Fast Stokesian dynamics

We present a new method for large scale dynamic simulation of colloidal particles with hydrodynamic interactions and Brownian forces, which we call fast Stokesian dynamics (FSD). The approach for modelling the hydrodynamic interactions between particles is based on the Stokesian dynamics (SD) algorithm (J. Fluid Mech., vol. 448, 2001, pp. 115–146), which decomposes the interactions into near-field (short-ranged, pairwise additive and diverging) and far-field (long-ranged many-body) contributions. In FSD, the standard system of linear equations for SD is reformulated using a single saddle point matrix. We show that this reformulation is generalizable to a host of particular simulation methods enabling the self-consistent inclusion of a wide range of constraints, geometries and physics in the SD simulation scheme. Importantly for fast, large scale simulations, we show that the saddle point equation is solved very efficiently by iterative methods for which novel preconditioners are derived. In contrast to existing approaches to accelerating SD algorithms, the FSD algorithm avoids explicit inversion of ill-conditioned hydrodynamic operators without adequate preconditioning, which drastically reduces computation time. Furthermore, the FSD formulation is combined with advanced sampling techniques in order to rapidly generate the stochastic forces required for Brownian motion. Specifically, we adopt the standard approach of decomposing the stochastic forces into near-field and far-field parts. The near-field Brownian force is readily computed using an iterative Krylov subspace method, for which a novel preconditioner is developed, while the far-field Brownian force is efficiently computed by linearly transforming those forces into a fluctuating velocity field, computed easily using the positively split Ewald approach (J. Chem. Phys., vol. 146, 2017, 124116). The resultant effect of this field on the particle motion is determined through solution of a system of linear equations using the same saddle point matrix used for deterministic calculations. Thus, this calculation is also very efficient. Additionally, application of the saddle point formulation to develop high-resolution hydrodynamic models from constrained collections of particles (similar to the immersed boundary method) is demonstrated and the convergence of such models is discussed in detail. Finally, an optimized graphics processing unit implementation of FSD for mono-disperse spherical particles is used to demonstrated performance and accuracy of dynamic simulations of $O(10^{5})$ particles, and an open source plugin for the HOOMD-blue suite of molecular dynamics software is included in the supplementary material.

Global modes with multiple saddle points

Significant progress has been made towards understanding the global stability of slowly-developing shear flows. The WKBJ theory developed by Patrick Huerre and his co-authors has proved absolutely central, with the result that both the linear and the nonlinear stability of a wide range of flows can now be understood in terms of their local absolute/convective instability properties. In many situations, the local absolute frequency possesses a single dominant saddle point in complex X-space (where X is the slow streamwise coordinate of the base flow), which then acts as a single wavemaker driving the entire global linear dynamics. In this paper we consider the more complicated case in which multiple saddles may act as the wavemaker for different values of some control parameter. We derive a frequency selection criterion in the general case, which is then validated against numerical results for the linearized third-order Ginzburg–Landau equation (which possesses two saddle points). We believe that this theory may be relevant to a number of flows, including the boundary layer on a rotating disk and the eccentric Taylor–Couette–Poiseuille flow.

Period doubling and Blazhko modulation in BL Herculis hydrodynamic models

We present the hydrodynamic BL Herculis-type models which display a long-term modulation of pulsation amplitudes and phases. The modulation is either strictly periodic or it is quasi-periodic, with the modulation period and modulation pattern varying from one cycle to the other. Such behaviour has not been observed in any BL Her variable so far, however, it is a common property of their lower luminosity siblings - RR Lyrae variables showing the Blazhko effect. These models provide a support for the recent mechanism proposed by Buchler & Kollath to explain this still mysterious phenomenon. In their model, a half-integer resonance that causes the period doubling effect, discovered recently in the Blazhko RR Lyrae stars, is responsible for the modulation of the pulsation as well. Although our models are more luminous than is appropriate for RR Lyrae stars, they clearly demonstrate, through direct hydrodynamic computation, that the mechanism can indeed be operational. Of great importance are models which show quasi-periodic modulation - a phenomenon observed in Blazhko RR Lyrae stars. Our models coupled with the analysis of the amplitude equations show that such behaviour may be caused by the dynamical evolution occurring in the close proximity of the unstable single periodic saddle point.

A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows

In this paper we introduce and analyze new mixed finite element schemes for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. The methods are based on a non-standard mixed approach in which the velocity, the pressure, and the pseudostress are the original unknowns. However, we use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown, which yields a twofold saddle point operator equation as the resulting dual-mixed variational formulation. In addition, a suitable augmented version of the latter showing a single saddle point structure is also considered. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are well-posed. In particular, we show that Raviart–Thomas elements of order k ⩾ 0 for the pseudostress, and piecewise polynomials of degree k for the velocity and its gradient, ensure the well-posedness of the associated Galerkin schemes. Moreover, we prove that any finite element subspace of the square integrable tensors can be employed to approximate the velocity gradient in the case of the augmented formulation. Then, we derive a reliable and efficient residual-based a posteriori error estimator for each scheme. Finally, we provide several numerical results illustrating the good performance of the resulting mixed finite element methods, confirming the theoretical properties of the estimator, and showing the behaviour of the associated adaptive algorithms.

Misfit dislocation dipoles in wire composite solids

Developed in this note is a theoretical model describing the mobility of a misfit screw dislocation dipole in a wire composite consisting of a stiff cylindrical substrate covered by a soft co-axial cylindrical film. A critical value of the film thickness, which is a function of the parameter measuring the stiffness of the film with respect to the substrate, is identified. It is observed that: (i) there exist two equilibrium positions of the misfit dislocation dipole (one stable and the other one unstable) when the film is thicker than the critical value; (ii) the two equilibrium positions of the misfit dislocation dipole converge to a single saddle point equilibrium position which is neither stable nor unstable when the thickness of the film is at the critical value; (iii) there exists no equilibrium position of the misfit dislocation dipole when the thickness of the film is below the critical value. These features could be useful to the design of wire composites and to the dislocation-related plasticity analysis.

Transition state theory with Tsallis statistics

We discuss the rate of an elementary chemical reaction. We use the reaction path and especially its saddle point on the potential energy surface. The reaction path connects reactant and product of a reaction over the transition state (TS). Usually, the TS is assumed near or at the single saddle point of the reaction path. By means of comparison of the statistics of states at the reactant and at the TS, one can estimate the reaction rate by the Eyring theory. We propose to use the Tsallis statistics at the TS, a statistics of seldom accidents. Thus, we propose to generalize the well-known Boltzmann-Gibbs statistics, which is the limiting case of the Tsallis statistics. We use features of this nonextensive thermostatistics. The basic properties of the statistics are used to derive (approximated) partition functions, and they are applied on reaction rates. The approximation includes a factorization of the partition functions. The theory is applied to HCN isomerization to HNC, and to the reaction H(2) + CN --> H + HCN. It allows an accordance with experimental estimations of the reaction rates.

Multi-saddle-point approximation for pulse propagation in resonant tunneling

Poles of the scattering matrix in the complex energy plane play an important role for pulse propagation through a resonant tunneling system. Therefore, we employ a special variant of the method of steepest descent to calculate the transmitted pulse. In this variant, the parameters defining the transmission resonances and the shape of the incident pulse are taken into account for the evaluation of the saddle points. As a result, we obtain a multi-saddle-point approximation (MSA) with a number of saddle points which are specific for a given combination of pulse and resonance. This is in contrast to the usual variant of the saddle point approximation [single-saddle-point approximation (SSA)] in which only the phase of the free time evolution operator is made stationary, leading to a single saddle point in the resonant tunneling problem. We apply our multi-saddle-point method to an incident Gaussian pulse comparing with the results of the SSA and the analytical solution. Apart from quantitative advantages of the MSA, in general, it is found that in contrast to the SSA, the MSA is capable of describing the center of the transmitted pulse for a narrow resonance.

Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods

Full geometry optimizations within given molecular symmetries have been performed on the chair and boat forms involved in the Cope rearrangement of 1,5-hexadiene using highly correlated methods (MR-CISD and MR-AQCC) and extended basis sets. These optimizations are based on analytic gradient procedures developed within the COLUMBUS program system and are the first ones for the Cope rearrangement, which have been carried out at a multireference, post-CASSCF level. By comparison of MR-CISD and MR-AQCC results, the importance of size-extensivity corrections is clearly demonstrated. Computed energetic stabilities with respect to 1,5-hexadiene and bis-allyl are in good agreement with experimental data. Only a single saddle point for the chair form of C2h symmetry is found in contrast to CASSCF results, but in agreement with CASPT2N and MRMP2 investigations. Thus, we confirm these previous results showing that the mechanism of the Cope rearrangement of 1,5-hexadiene is concerted via an aromatic transition state.

Transition-state Ensemble in Enzyme Catalysis: Possibility, Reality, or Necessity?

Proteins are not rigid structures; they are dynamic entities, with numerous conformational isomers (substates). The dynamic nature of protein structures amplifies the structural variation of the transition state for chemical reactions performed by proteins. This suggests that utilizing a transition state ensemble to describe chemical reactions involving proteins may be a useful representation. Here we re-examine the nature of the transition state of protein chemical reactions (enzyme catalysis), considering both recent developments in chemical reaction theory (Marcus theory for SN2 reactions), and protein dynamics effects. The classical theory of chemical reactions relies on the assumption that a reaction must pass through an obligatory transition-state structure. The widely accepted view of enzymatic catalysis holds that there is tight binding of the substrate to the transition-state structure, lowering the activation energy. This picture, may, however, be oversimplified. The real meaning of a transition state is a surface, not a single saddle point on the potential energy surface. In a reaction with a “loose” transition-state structure, the entire transition-state region, rather than a single saddle point, contributes to reaction kinetics. Consequently, here we explore the validity of such a model, namely, the enzymatic modulation of the transition-state surface. We examine its utility in explaining enzyme catalysis. We analyse the possibility that instead of optimizing binding to a well-defined transition-state structure, enzymes are optimized by evolution to bind efficiently with a transition-state ensemble, with a broad range of activated conformations. For enzyme catalysis, the key issue is still transition state (ensemble) stabilization. The source of the catalytic power is the modulation of the transition state. However, our definition of the transition state is the entire transition-state surface rather just than a single well-defined structure. This view of the transition-state ensemble is consistent with the nature of the protein molecule, as embodied and depicted in the protein energy landscape of folding, and binding, funnels.

Two-terminal conductance fluctuations in the integer quantum Hall regime

Motivated by recent experiments on the conductance fluctuations in mesoscopic integr quantum Hall systems, we consider a model in which the Coulomb interactions are incorporated into the picture of edge-state transport through a single saddle-point. The occupancies of `classical' localised states in the two-dimensional electron system change due to the interactions between electrons when the gate voltage on top of the device is varied. The electrostatic potential between the localised states and the saddle-point causes fluctuations of the saddle-point potential and thus fluctuations of the transmission probability of edge states. This simple model is studied numerically and compared with the observation.

A new functional form for global potentials of floppy molecules

A new functional form for the global potential of floppy molecules is given. It is based on a polynomial switching function which allows for a simple, yet flexible means to switch between two minima and a single saddle point separating them. The method is illustrated for a model ArOH( 2Σ +) van der Waals system, based in part on ab initio calculations of Werner.

Holes in the probability density of strongly colored noise driven systems

A qualitative change in the topology of the joint probability densityP(e,x), which occurs for strongly colored noise in multistable systems, has recently been observed first by analog simulation (F. Moss and F. Marchesoni,Phys. Lett. A 131:322 (1988)) and confirmed by matrix continued fraction methods (Th. Leiber and H. Riskin, unpublished), and by analytic theory (P. Hanggi, P. Jung, and F. Marchesoni,J. Stat. Phys., this issue). Systems studied were of the classx=−∂U(x)/∂x+e(t,τ), whereU(x) is a multistable potential and e(t, τ) is a colored, Gaussian noise of intensityD, for which 〈e〉=0, and 〈e(t) e(s)〉=(D/τ)exp(−∥t−s∥/τ). When the noise correlation time τ is smaller than some critical value τ0, which depends onD, the two-dimensional densityP(e,x) has the usual topology [P. Jung and H. Risken,Z. Phys. B 61:367 (1985); F. Moss and P. V. E. McClintock,Z. Phys. B 61:381 (1985)]: a pair of local maxima ofP(e,x), which correspond to a pair of adjacent local minima ofU(x), are connected by a single saddle point which lies on thex axis. When τ>τ0, however,the single saddle disappears and is replaced by a pair of off-axis saddles. A depression, or hole, which is bounded by the saddles and the local maxima thus appears. The most probable trajectory connecting the two potential wells therefore does not pass through the origin for τ>τ0, but instead must detour around the local barrier. This observation implies that successful mean-first-passage-time theories of strongly colored noise driven systems must necessarily be two dimensional (Hanggiet al.). We have observed these holes for several potentialsU(x): (1)a “soft,” bistable potential by analog simulation (Moss and Marchesoni); (2) a periodic potential [Th. Leiber, F. Marchesoni, and H. Risken,Phys. Rev. Lett. 59:1381 (1987)] by matrix continued fractions; (3) the usual “hard,” bistable potential,U(x)=−ax 2/2+bx 4/4, by analog simulations only; and (4) a random potential for which the forcingf(x)=−∂U(x)/∂x is an approximate Gaussian with nonzero correlation length, i.e., colored spatiotemporal noise, by analog simulation. There is a critical curve τ0(D) in the τ versusD plane which divides the two topological behaviors. For a fixed value ofD, this curve is shifted toward larger values of τ0 for progressively weaker barriers between the wells. Therefore, strong barriers favor the observation of this topological transformation at smaller values of τ. Recently, an analytic expression for the critical curve, valid asymptotically in the small-D limit, has been obtained (Hanggiet al.).

An investigation of the structure of a laminar non-premixed flame in an unsteady vortical flow

Although organized vortical motions have been observed in a variety of turbulent flames, there have been only a few experimental studies which have directly examined the structure of a flame imbedded in an unsteady vortical flow. In the present work, this problem is investigated using a laminar co-flowing jet flame in which a periodic vortical motion is induced by acoustic excitation of the fuel stream. A variety of optical planar imaging techniques are employed to define the instantaneous flame structure. Particle tracking is used to obtain the instantaneous two-dimensional velocity field. These measurements are combined to follow the evolution of the flame in the unsteady velocity field over one cycle of the excitation. The flame is observed to break up into a series of axisymmetric flamelets which convect and distort under the influence of buoyancy and the vortical flow. When the velocity field is viewed in a frame of reference moving with the flamelets the flow pattern is seen to consist of a pair of saddle points on either side of the flow centerline near the flame tip and a single saddle point on the centerline near the base of the flame. This latter saddle point is also evident in images of light scattered from TiO 2 particles added to the fuel stream. Entrained fluid is fed into a large axisymmetric vortex surrounding the base of the flame. Because of heat release, the flame acts as an effective volume source in the flow. A high strain rate exists on the centerline at the base of the flame, and planar laser-induced fluorescence measurements of the concentration of OH show local extinction of the flame.

Semiclassical description of a two-dimensional electron in a strong magnetic field and an external potential

We study the effects of tunneling between the classical trajectories of a two-dimensional electron in a strong magnetic field and a slowly varying external potential on the quantum energy levels of the electron and on the phonon-induced hopping rate of localized states. The tunneling is important when the classical orbits pass within a few magnetic lengths of a saddle point of the potential. We discuss first the case of a double-well potential with a single saddle point. For energies well below the potential energy at the saddle point, the spectrum is of the form one expects for two wells very distant from one another. For energies well above the energy at the saddle point, the spectrum is similar to that of a single-well potential. The spectrum evolves smoothly from the former to the latter in the energy interval between these two limits. In addition, we study the spectrum for orbits coupled at two saddle points, and show that our technique can be applied to any number of orbits coupled at any number of saddle points. The method may be used to find the energy spectrum of an electron in a slowly varying random potential. We also calculate the tunnel splitting of states of a symmetric double well, and the phonon-induced hopping rate between localized states of an asymmetric double well. In both cases, we find that these quantities fall off as Gaussians in the distance of closest approach of the classical trajectories to the saddle point.