Points In Configuration Space Research Articles

Multiple scaling regimes in simple aging models

We investigate aging in glassy systems based on a simple model, where a point in configuration space performs thermally activated jumps between the minima of a random energy landscape. The model allows us to show explicitly a subaging behavior and multiple scaling regimes for the correlation function. Both the exponents characterizing the scaling of the different relaxation times with the waiting time and those characterizing the asymptotic decay of the scaling functions are obtained analytically by invoking a "partial equilibrium" concept.

Linear-least-squares fitting method for the solution of the time-dependent Schrödinger equation: Applications to atoms in intense laser fields

An alternative theoretical approach for solving the Lime-dependent Schrodinger equation for atoms in an intense laser field is presented. In this method the time-dependent wave function is expanded in a basis set but the expansion coefficients are determined by linear-least-squares fitting of the wave function on discrete mesh points in configuration space, thus avoiding the need of evaluating a large number of matrix elements. We illustrate the method by computing wave functions, above-threshold ionization spectra, and harmonic generation spectra of a model atom and compare the results with those obtained using the split-operator method.

Iterative and direct methods employing distributed approximating functionals for the reconstruction of a potential energy surface from its sampled values

The reconstruction of a function from knowing only its values on a finite set of grid points, that is the construction of an analytical approximation reproducing the function with good accuracy everywhere within the sampled volume, is an important problem in all branches of sciences. One such problem in chemical physics is the determination of an analytical representation of Born–Oppenheimer potential energy surfaces by ab initio calculations which give the value of the potential at a finite set of grid points in configuration space. This article describes the rudiments of iterative and direct methods of potential surface reconstruction. The major new results are the derivation, numerical demonstration, and interpretation of a reconstruction formula. The reconstruction formula derived approximates the unknown function, say V, by linear combination of functions obtained by discretizing the continuous distributed approximating functional (DAF) approximation of V over the grid of sampling. The simplest of contracted and ordinary Hermite-DAFs are shown to be sufficient for reconstruction. The linear combination coefficients can be obtained either iteratively or directly by finding the minimal norm least-squares solution of a linear system of equations. Several numerical examples of reconstructing functions of one and two variables, and very different shape are given. The examples demonstrate the robustness, high accuracy, as well as the caveats of the proposed method. As to the mathematical foundation of the method, it is shown that the reconstruction formula can be interpreted as, and in fact is, frame expansion. By recognizing the relevance of frames in determining analytical approximation to potential energy surfaces, an extremely rich and beautiful toolbox of mathematics has come to our disposal. Thus, the simple reconstruction method derived in this paper can be refined, extended, and improved in numerous ways.

HF dimer: Empirically refined analytical potential energy and dipole hypersurfaces from ab initio calculations

The interaction between two HF molecules as a function of all six internal coordinates is reported as calculated for 3284 selected points in configuration space at counterpoise-corrected explicitly correlated levels and fitted to an analytical expression, which is described in detail. The unweighted rms deviation for all 3284 points is 21 cm−1. Empirical refinements are applied through mixing and scaling of the ab initio data, guided by the comparison of multidimensional nuclear quantum energy levels with experimental data. The resulting semiempirical pair potentials (labeled SC-2.9 and SO-3) contain 67 and 61 freely adjusted parameters and are combined with a four parameter monomer potential of generalized Pöschl–Teller type. Various minimum energy paths and cuts are investigated. Major improvements over earlier HF dimer potentials are demonstrated via multidimensional solutions of the nuclear Schrödinger equation. Comparison with other high level ab initio calculations and with various experimental data reveals very good overall consistency. The new potential suggests strong Coriolis coupling in the librational degrees of freedom. Best estimates of stationary points, of the dimer dissociation energy (De=19.1±0.2 kJ/mol), of the electronic barrier to hydrogen bond exchange (4.2±0.2 kJ/mol), of the electronic barrier to linearity (3.9±0.2 kJ/mol), and of the electronic barrier to hydrogen exchange (175±10 kJ/mol) are inferred. Based on accurate electric dipole functions for the monomer and distortion contributions calculated with a large basis at SCF level, a simple analytical six-dimensional electric dipole hypersurface is presented.

The symmetrized quantum thermal flux operator

Analysis of the symmetrized thermal flux operator leads to explicit expressions for its eigenvalues and eigenfunctions. At any point in configuration space one finds two nonzero eigenvalues of opposite sign. The associated eigenfunctions are L2 integrable. The eigenfunctions and eigenvalues are expressed in terms of the thermal density matrix in the vicinity of the transition state. The positive eigenvalue of the thermal flux operator gives an upper bound to the rate and allows for a formulation of a quantum mechanical variational transition state theory. This new upper bound, though, is only a slight improvement over previous theories.

Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms

We describe a global optimization technique using `basin-hopping' in which the potential energy surface is transformed into a collection of interpenetrating staircases. This method has been designed to exploit the features which recent work suggests must be present in an energy landscape for efficient relaxation to the global minimum. The transformation associates any point in configuration space with the local minimum obtained by a geometry optimization started from that point, effectively removing transition state regions from the problem. However, unlike other methods based upon hypersurface deformation, this transformation does not change the global minimum. The lowest known structures are located for all Lennard-Jones clusters up to 110 atoms, including a number that have never been found before in unbiased searches.

Shortest Paths for a Car-like Robot to Manifolds in Configuration Space

Reeds and Shepp (1990) studied the problem of finding the shortest feasible path for a car-like robot between two points in configuration space. We extend their results to find the shortest feasible path between a point and a manifold in configuration space. Our approach is based on the Lagrange method for optimizing a function while constrained to a manifold. Solving the problem analytically is much faster than numerical dis cretization techniques. In addition to providing insight into the underlying structure of Reeds and Shepp paths, this research has many applications in path planning. Planning algorithms often rely on the notion of clearance from obstacles, and for car-like mobile robots, clearance is closely related to the length of the shortest feasible path to an obstacle. In addition, one may want to bring the robot to a predefined path (such as a skeleton). Skeletonization and potential field methods are two examples of planning paradigms where our algorithm would prove useful.

First-principles calculations of the energy barrier to dislocation motion in Si and GaAs.

The single kink formation and migration energies ${\mathit{F}}_{\mathit{k}}$ and ${\mathit{W}}_{\mathit{m}}$ of 90\ifmmode^\circ\else\textdegree\fi{} glide partial dislocations in Si and GaAs are calculated using an ab initio local density-functional cluster method. Kink migration occurs via a concerted exchange of an atom at a dislocation core with one of its glide plane nearest neighbors. By constraining these atoms to sit in high-energy positions and relaxing a surrounding cluster of atoms, sufficient points in configuration space can be sampled for the energy barrier for the first step in kink pair formation to be estimated. By including an estimate of the elastic energy of the interaction of kink pairs, the single kink formation energy is calculated. It is found that ${\mathit{F}}_{\mathit{k}}$ and ${\mathit{W}}_{\mathit{m}}$ for Si are 0.1 and 1.8 eV, respectively. For the 90\ifmmode^\circ\else\textdegree\fi{} \ensuremath{\alpha} glide partial in GaAs, these quantities are 0.07 and 0.7 eV, respectively, and 0.3 and 1.1 eV for \ensuremath{\beta} partials.

The ergodic limit of multipass absorption for fast wave current drive in tokamaks

In many parameter regimes of interest for fast wave current drive (FWCD) in tokamaks, direct absorption of the fast wave by resonant electrons is a weak process and multipass absorption is an important issue. Although both full wave codes and ray tracing codes have been developed to model FWCD, in the multipass regime these tools are computationally intensive, and yield little insight into the nature of the solutions. In this work, an alternative approach is considered. Based on the wave kinetic equation, a natural limit emerges for the multipass regime, where wave energy density, convected along stochastic ray trajectories, uniformly fills the entire accessible phase space. In this ergodic, weak damping limit, the absorbed power density and corresponding wave-driven current density are readily obtained by calculating the appropriate set of one-dimensional k-space integrals at every point in configuration space. The method is used here to model FWCD on the DIII-D tokamak [R. I. Pinsker and the DIII-D Team, Plasma Physics and Controlled Nuclear Fusion Research 1992 (International Atomic Energy Agency, Vienna, 1993), Vol. 1, p. 683]. An example for reactor-grade plasma parameters is also considered.

Collinear H+H–H reaction. Computer simulation of quasiclassical trajectories

An undergraduate computer project, suitable for a computational physics or applied numerical analysis course, is presented. The equations of motion for the three H atoms of the collinear H+H–H system are set up in the Hamiltonian formalism. They are then solved numerically, using the analytical Porter–Karplus potential energy function and the Runge–Kutta algorithm. The program is used to illustrate several types of motion: nonlinear oscillations of the H2 molecule, atom–molecule inelastic collisions, reactive collisions. Results are displayed either as internuclear distances versus time or as the motion of a point in configuration space.

Hidden variables as configuration points

The «pilot wave» hidden-variables formalism is reformulated in terms of observed points in configuration space, without using the concept of material particles. It is shown that the usual quantum-mechanical predictions can be retrieved in these terms. The question of quantum non-locality is discussed, and the present interpretation is compared with other »no-collapse» formulations.

Nonlinear programming approach to locally constrained variational calculations: He and H− in the Hartree–Fock approximation

Nonlinear programming procedures are employed to carry out constrained variational calculations on the helium atom and the hydride ion in the Hartree–Fock approximation. The inequality constraints imposed to improve the local behavior of the wave function are based on the reduced local energy, evaluated at selected points in configuration space. Improved moments 〈rn〉 are obtained for the helium atom and the hydride ion using this procedure. The calculations provide empirical support for the fact that the reduced local energy does reflect errors in the wave function.

Nonlinear programming approach to locally constrained variational calculations on the ground state of the helium atom

The techniques of nonlinear programming are incorporated into the standard variational method. The set of inequality constraints employed in the nonlinear optimizations is based on the reduced local energy, evaluated at various points of configuration space. These constraints give, indirectly, a means to incorporate the local behaviour of the wavefunction in the variational procedure. A test of the procedure is carried out using a 20-term Hylleraas type wavefunction for the ground state of the helium atom. The impact of the constrained optimization on a variety of expectation values is examined.

Grid representation of rotating triatomics

A grid formulation is presented for the body-fixed (BF) motion of a rotating triatomic molecule. The BF component of the wave function, described in terms of Jacobi coordinates, is discretized on a {Ri×rj×θα} grid. By use of the generalized discrete variable representation (DVR) of Light, Hamilton, and Lill [J. Chem. Phys. 82, 1400 (1985)] one can employ the same θ grid for the different Ω components of the wave function, Ω corresponding to the projection of the total angular momentum J onto the z BF axis. Such a unique grid definition allows one to define adiabatic states, with respect to the θ coordinate, independent of the Ω value considered. These states have been directly computed by means of a distributed 2D DVR, sampling only the relevant points in configuration space. Further reduction of the rovibrational basis set was achieved by defining new adiabatic states, with respect to the Ω component. This formation is applied to the calculation of bound rovibrational states of the HCN molecule for the J=2 value.

Molecular point symmetry and the phase of the electronic wave function: Tools for the prediction of critical points of potential energy surfaces

AbstractBy combining a well‐known theorem of Longuet‐Higgins on the phase changes of electronic wave functions along loops encircling a conical intersection of potential surfaces, and a recently proven “vertical symmetry theorem” on the relative distributions of point symmetry groups and potential surface critical points in configuration space, a new result is obtained, suitable for the prediction of the presence of a critical point within a domain of configurations.

A reaction-path description of the HO2 reactive system

The potential-energy surface of the reactive molecular system HO2 is analysed in terms of reaction path and transversal modes, along the line of arguments initially proposed by Miller et al. This is a complicated case because, for any value of the reaction-path abscissa, the minimum-energy configuration of the system is non-symmetric (i.e. Cs and not C2v or C∞v), and therefore corresponds to several points in configuration space through what can be called ‘transversal isomerization’ (rotation of H around O2). It is shown that the reaction-path description is nevertheless possible in this case, and can be used as a starting point for dynamical investigations of the system. A thorough discussion of the branching of the reaction paths for both the exchange reaction and the isomerization process is presented.

Diffusion in supercooled liquids via normal mode analysis

A theory is developed for diffusion in supercooled liquids. The viewpoint taken is that the dynamics of system, represented by a point in configuration space, consists of long sojourns in the vicinity of the local minima of the total potential, interrupted by relatively infrequent barrier crossings to adjacent minima. An equation due to Zwanzig, plus a new relation for the hopping rate among the local minima, are combined in a theory for the self-diffusion constant D. All the quantities entering the two equations are shown to be derivable from the configuration-averaged densities of vibrational states, 〈ρ(ω)〉, introduced by us previously; the contribution of unstable modes plays a crucial role. Solution of the coupled equations reveals the existence of a critical temperature below which D vanishes. The theory is in excellent agreement with molecular dynamics values for D, over a wide temperature range in which D displays activated behavior, in Lennard-Jones argon at a reduced density of unity.

Orbifolds as configuration spaces of systems with gauge symmetries

In systems like Yang-Mills or gravity theory, which have a symmetry of gauge type, neither phase space nor configuration space is a manifold but rather an orbifold with singular points corresponding to classical states of non-generically higher symmetry. The consequences of these symmetries for quantum theory are investigated. First, a certain orbifold configuration space is identified. Then, the Schrodinger equation on this orbifold is considered. As a typical case, the Schrodinger equation on (double) cones over Riemannian manifolds is discussed in detail as a problem of selfadjoint extensions. A marked tendency towards concentration of the wave function around the singular points in configuration space is observed, which generically even reflects itself in the existence of additional bound states and can be interpreted as a quantum mechanism of symmetry enhancement.

Time-suboptimal quasi-continuous path generation for industrial robots

SUMMARYThe problem of constructing a smooth path with respect to time through N given points in configuration space is considered. Two variants of an algorithm suggested by Paul are presented and evaluated. It is shown that the algorithms suggested in this paper yield in general considerable improvements in two respects: Firstly, the deviation of the resulting path from the given points is reduced markedly and secondly, the overall time needed for the movement is reduced significantly and consequently, is closer to the true minimum-time. The price to be paid for these improvements is a moderate increase of computation time allowing still online use of the algorithm.

Locating stationary paths in functional integrals: An optimization method utilizing the stationary phase Monte Carlo sampling function

A method is presented for determining the stationary phase points for multidimensional path integrals employed in the calculation of finite-temperature quantum time correlation functions. The method can be used to locate stationary paths at any physical time; in the case that t≫βℏ, the stationary points are the classical paths linking two points in configuration space. Both steepest descent and simulated annealing procedures are utilized to search for extrema in the action functional. Only the first derivatives of the action functional are required. Examples are presented first of the harmonic oscillator for which the analytical solution is known, and then for anharmonic systems, where multiple stationary phase points exist. Suggestions for Monte Carlo sampling strategies utilizing the stationary points are made. The existence of many and closely spaced stationary paths as well as caustics presents no special problems. The method is applicable to a range of problems involving functional integration, where optimal paths linking two end points are desired.