Showing papers on "Configuration space published in 1980"
TL;DR: In this article, a modified version of the Smith-Whitten symmetric hyperspherical coordinate system is presented for mapping the internal configurations of a three body system to points in a 3D configuration space.
Abstract: A procedure for mapping the internal configurations of a three body system to points in a three dimensional ’’configuration space’’ is presented. The mapping is based on a modified version of the Smith–Whitten symmetric hyperspherical coordinate system. Several years ago, Kuppermann developed a mapping procedure based on a different hyperspherical coordinate system. Transformation equations relating these two mappings are derived. A comparison shows that the present method produces maps identical to Kuppermann’s method. The coordinate axes in configuration space, however, are rotated by 90 degrees so that the z axis is coincident with the axis of kinematic rotation.
224 citations
TL;DR: The theory of quantum topology as mentioned in this paper has been used for the analysis of molecular topology and its morphogenesis, and it is shown that the quantum mechanical partitioning of a system into subsystems coincides with the topological partitioning: both are defined by a set of zero flux surfaces.
Abstract: In this paper we review and exemplify a new and rigorous approach to the problem of molecular structure and its morphogenesis: the theory of quantum topology. The basis for this approach is provided by the topology of the total charge density in a given molecular system. The essential observation is that the only local maxima of a ground state distribution occur at the positions of the nuclei. The nuclei are therefore identified as point attractors of the gradient vector field of the charge density. The associated basins partition the molecular system into atomic fragments. Each atom is a stable structural unit defined as the union of an attractor and its basin. The common boundary of two neighbouring atomic fragments, the interatomic surface, contains a particular critical point, which generates a pair of gradient paths linking the two neighbouring attractors. The union of this pair of gradient paths and their endpoints is called a bond path. The network of bond paths defines a molecular graph of the system.
Having defined a unique molecular graph for any molecular geometry, the total configuration space is partitioned into a finite number of regions. Each region is associated with a particular structure defined as an equivalence class of molecular graphs. A chemical reaction in which chemical bonds are broken and/or formed is therefore a trajectory in configuration space which must cross one of the boundaries between two neighbouring structural regions. These boundaries form the catastrophe set of the system which, like a phase diagram in thermodynamics, denotes the points of “balance” between neighbouring structures. A general analysis of the structural changes in an ABC type system is given in detail together with specific examples of all possible structural elements in a molecular system.
The properties of the topologically defined atoms and their temporal changes are identified within a general formulation of subspace quantum mechanics. It is shown that the quantum mechanical partitioning of a system into subsystems coincides with the topological partitioning: both are defined by the same set of “zero flux” surfaces. Consequently the total energy, or any other property, is partitioned into additive atomic contributions.
We show that, in general, a definite structure can be assigned to a given molecular system. Quantum mechanically this structure is associated with an open neighbourhood of the most probable nuclear geometry. Finally we generalize the notion of molecular structure to non-isolated molecules and, in contrast to recent work by Woolley, we conclude that molecular structure exists in spite of intermolecular interactions and not as a result of them.
175 citations
01 Jan 1980
TL;DR: In this paper, a geometrical approach to the nonlinear solvable equations, based on the study of the "groups of motion" of special infinite-dimensional manifolds called "symplectic Kahler manifolds", is suggested.
Abstract: A geometrical approach to the nonlinear solvable equations, based on the study of the “groups of motion” of special infinite-dimensional manifolds called “symplectic Kahler manifolds”, is suggested. This approach is constructive, tensorial and simple in its ideas. It allows to recover the equations obtained through the socalled AKNS approach, together with some other examples. It leads to conjecture a possible “integrability condition” for infinite-dimensional systems, and to hope to be able to give a geometrical explanation of the so-called “spectral transform method”.
161 citations
TL;DR: In this article, a general procedure is given for decomposing N -particle configuration space into orbits of a kinematical collective group and a smooth transversal, and it is shown that with center-of-mass motion removed, the configuration space can be considered as an orbit of the group GL + (3, R ) × SO (N − 1).
54 citations
TL;DR: Improved canonical variational theory (ICVT) as mentioned in this paper provides a bound on the classical mechanical equilibrium reaction rate, which involves a more accurate treatment of the threshold region, and is illustrated and tested by applications to classical collinear reaction rates in several systems.
Abstract: A new variational theory of chemical reaction rates, called improved canonical variational theory (ICVT), provides a bound on the classical mechanical equilibrium reaction rate. As compared to canonical variational theory (the method of free-energy curves) it involves a more accurate treatment of the threshold region. Classically the method consists of eliminating all contributions to the generalized transition-state theory rate constant from energies below the energetic reaction threshold and then estimating the contribution from the remaining truncated canonical ensemble by variational optimization of a generalized transition state dividing surface in configuration space. The ICVT is illustrated and tested by applications to classical collinear reaction rates in several systems. Tests of the unified statistical theory for two new reactions are also provided.
54 citations
TL;DR: In this paper, the space groups of the 100 possible rotation functions which do not involve cubic crystallographic symmetry have been derived, and all of them belong to one of 16 basic space groups, but many possess additional translational symmetry.
Abstract: Space groups of the 100 possible rotation functions which do not involve cubic crystallographic symmetry have been derived. All 100 belong to one of 16 basic space groups, but many possess additional translational symmetry. Asymmetric units for Eulerian coordinates and for 0÷, 0_ coordinates are tabulated.
47 citations
TL;DR: In this paper, the Faddeev-type equations in configuration space are formulated using splines and the method of orthogonal collocation, and Triton observables and wave-function probabilities are calculated for s-wave NN interaction models of Malfliet and Tjon and the tensor force model of Reid.
Abstract: The formulation of Faddeev-type equations in configuration space is discussed. Numerical solutions are obtained using splines and the method of orthogonal collocation. Triton observables and wave-function probabilities are calculated for s-wave NN interaction models of Malfliet and Tjon and the tensor force model of Reid. Comparison with previously published triton results is made; our full five-channel results for the Reid soft-core potential are in excellent agreement with those obtained by Afnan and Birrell using separable expansion methods.
45 citations
TL;DR: In this paper, mode damping techniques that allow a controlled "walk" on the energy hypersurface both within the orbital and the configuration space are introduced, which is particularly crucial in cases where there is strong coupling between the orbit and configuration space, e.g., the second 1Σ+g state of C2.
Abstract: Mode damping techniques that allow a controlled ’’walk’’ on the energy hypersurface both within the orbital and the configuration space are introduced. The use of these techniques is particularly crucial in cases where there is strong coupling between the orbital and the configuration space, e.g., the second 1Σ+g state of C2. The mode damping technique allows us to use a set of single configuration Hartree–Fock orbitals as an initial guess of orbitals and then coverge in very few iterations.
37 citations
TL;DR: In this paper, it was shown that if a system admits only local lagrangians for a configuration space Q, then under certain conditions, it admits a global lagrangian when Q is enlarged to a suitable U(1) bundle over Q. Conditions under which a symplectic form is derivable from a Lagrangian are also found.
29 citations
TL;DR: In this article, the inelastic body-frame Sudden Rotational Approximation (SRA) and the canonical extension thereof to exchange reactions of light-heavy light systems are given.
29 citations
TL;DR: In this paper, the structural stability of resistive nonlinear n-ports is investigated from a geometric point of view using recent tools from differential topology, and structural stability is defined as the persistence of the configuration space under small perturbations of the internal resistor constitutive relations.
Abstract: This paper presents several general properties of resistive nonlinear n -ports from a geometric point of view using recent tools from differential topology. The geometric approach is coordinate-free and hence the results of the paper do not depend on the particular choice of a tree, a loop matrix, a cutset matrix, a set of independent variables, etc. Firstiy, a classification is given of resistive n -ports into logical categories such as weakly regular n -ports, regular n -ports, completely regular n -ports, and universally regular n -ports, etc. Transversality of the internal resistor constitutive relations and the Kirchhoff space plays an important role in this paper. Secondly, a structural stability result is given. In this paper, structural stability means the persistence of the configuration space under small perturbations of the internal resistor constitutive relations. Essentially the result asserts that a resistive n -port is structurally stable if and only if the internal resistor constitutive relations are transversal to the Kirchhoff space. Thirdly, two basic perturbation techniques are given which guarantee the transversality of the internal resistor constitutive relations and the Kirchhoff space. The first technique involves element perturbations, i.e., perturbations of the internal resistor constitutive relations. The second technique involves network perturbations, i.e, by augmenting extra ports to an original n -port. Lastly, coordinate-free definitions of reciprocity and anti-receprocity are given in terms of exterior product and symmetric product of two tensors, respectively, and then some of their properties are investigated.
TL;DR: For local variational systems (like a charged particle in the field of a Dirac monopole) a quantum mechanically well-defined action (Q.M.W.D.A.) can be introduced iff the system is prequantizable in the Kostant-Souriau sense.
Abstract: For local variational systems (like a charged particle in the field of a Dirac monopole) a quantum mechanically well-defined action (Q.M.W.D.A.) can be introduced iff the system is prequantizable in the Kostant-Souriau sense. If the configuration space is multiply connected (as in the Bohm-Aharonov experiment), different expressions for the classical action may emerge; they are quantum mechanically equivalent (Q.M.E.) iff the corresponding prequantizations are equivalent. In both cases the situation depends on the behaviour of the non integrable phase factor of Wu and Yang.
01 Jan 1980
TL;DR: In this paper, the authors consider a smooth connected manifold M and a physical system which is based on M (configuration space), i.e., one can localize the system in a sufficiently large class of regions U M and observe how the localization region moves.
Abstract: Consider a smooth connected manifold M and a physical system which is “based” on M (configuration space), i.e. one can localize the system in a sufficiently large class of regions U M and observe how the localization region moves. The properties of such a system will depend on the geometry of M. Its description may be based in a natural manner on those objects on M, which are characteristic for its structure, like compactly supported functions, n-forms, vector fields,jets, etc. Special examples are Hamiltonian systems [1], non-relativistic quantum systems on homogeneous spaces [2], [3] and Maxwell fields on manifolds [4], [5].
TL;DR: In this article, a spin-statistics theorem for composites consisting of electrically and magnetically charged particles is proved in the framework of a nonrelativistic theory, taking as the classical configuration space aU(1) bundle over the space of physical configurations, and as the quantum Hilbert space the homogeneous square integrable functions on that bundle.
Abstract: The present paper states and proves an asymptotic spin-statistics theorem for composites consisting of electrically and magnetically charged particles. We work in the framework of a nonrelativistic theory, taking as the classical configuration space aU(1) bundle over the space of physical configurations, and as the quantum hilbert space the homogeneous square integrable functions on that bundle. The theorems are proved using a formalism we develop here for treating “gauge spaces” —U(1) bundles with connections; in particular, two products related to tensor products of vector bundles prove to be extremely useful in displaying the structure of the gauge spaces that naturally arise in this theory.
TL;DR: In this article, a nonrelativistic gravitating fermions described rigorously by Thomas-Fermi theory is shown to exist for the microcanonical free energy, and for the entropy divided by log|∧|.
Abstract: For a system of (infinitely many) nonrelativistic gravitating fermions described rigorously by Thomas-Fermi theory, a nontrivial limit of infinite configuration volume |∧| is shown to exist for the microcanonical free energy, and for the entropy divided by log|∧|. It can be calculated explicitly using the scaling behaviour of the (ground state). Thomas-Fermi equation and shows a phase transition at zero energy. For all (possible) negative energies, the heat capacity of the infinitely extended system is negative and a nonzero fraction of the particles is in the condensed phase.
TL;DR: In this article, configuration space equations for the description of reactive pair dynamics are derived from the pair phase space kinetic equations derived in the preceding paper, which are generalizations of the conventional Smoluchowski equations.
Abstract: Configuration space equations for the description of reactive pair dynamics are obtained from the pair phase space kinetic equations derived in the preceding paper. These configuration space equations are generalizations of the conventional ’’sink’’ Smoluchowski equations. In addition to displaying the modifications in Smoluchowski equation descriptions due to velocity relaxation effects, computationally tractable expressions for the diffusion and friction tensors are obtained. The structure of the rate kernel which follows from the phase space kinetic equation is also examined.
TL;DR: In this paper, a solution procedure for the Kramers' equation for the distribution function underlying the motion of a pair of dipoles compelled to rotate about an axis through their common center under the influence of their mutual dipole-dipole interaction may be derived.
Abstract: It is indicated how Kramers' equation for the distribution function (in configuration-angular velocity space) underlying the motion of a pair of dipoles compelled to rotate about an axis through their common centre under the influence of their mutual dipole-dipole interaction may be derived. A solution procedure (analogous to that originally given by Brinkman in a discussion of translational brownian movement) for the equation is then set up, whereby the velocity-dependent portion of the distribution function is expressed in terms of two dimensional Hermite polynomials, whence a hierarchy of equations for the Laplace transform of the distribution function in configuration space is obtained. The lowest order approximation to the solution of the hierarchy is taken. This equation is then used to calculate the ensemble averages appropriate to the decay of electric polarization of an assembly of pairs of dipoles (rotating under the influence of their dipole-dipole interaction) following on the removal of a uni...
01 Jan 1980
TL;DR: In this paper, the configuration space Cm(M) of m identical particles moving on a manifold M and the cohomology groups Hq(Cm(m), Z) may be calculated, and compute Ha(C3(Rn, Z)
Abstract: We define the configuration space Cm(M) of m identical particles moving on a manifold M and give several examples We indicate how the cohomology groups Hq(Cm(M), Z) may be calculated, and compute Ha(C3(Rn), Z)
TL;DR: In this article, a procedure has been outlined for generating configuration space basis states of many-particle systems in which the maximal occupancy of any single particle orbital is arbitrary, and an efficient computer program has been developed for determining the matrix elements of the generators of the unitary group U(n) over configuration spaces basis states.
Abstract: A procedure has been outlined for generating configuration space basis states of many-particle systems in which the maximal occupancy of any single-particle orbital is arbitrary. An efficient computer program has been developed for determining the matrix elements of the generators of the unitary group U(n) over configuration space basis states. Computer times for generating these matrix elements have been presented for specific examples.
TL;DR: In this paper, a semi-classical, WKB-type formalism is introduced to treat systems with potential energy surfaces that are multi-dimensional in configuration space, are non-paraboloidal and possess several minima and that arise from several electronic states.
Abstract: A semi-classical, WKB-type formalism is introduced to treat systems with potential energy surfaces that are multi-dimensional in configuration space, are non-paraboloidal and possess several minima and that arise from several electronic states. Working rules are provided for the use of the formalism, which is applied in the sequel to this paper.
TL;DR: In this paper, a state space theory for time-varying state trajectories is presented, which is based on the approach of Schnure and Steinberger, and it is shown that the Hilbert space is the direct integral integral integral of the state spaces.
TL;DR: In this article, the number of nodal cells (defined by the nodal surfaces in configuration space) is a conserved quantity for a quantum-mechanical system, which is useful in finding the correspondence diagrams of the energy levels for continuously related hamiltonians.
TL;DR: The existence, number and type of the constants of motion in non-relativistic mechanics are examined for different set-ups of Newton's equations in configuration space, the Noetherian symmetries in the Lagrangian formulation, the Hamiltonian formulation and the Schrodinger equation in quantum theory as discussed by the authors.
Abstract: The existence, number and type of the constants of motion in non-relativistic mechanics are examined for different set-ups of Newton's equations in configuration space, the Noetherian symmetries in the Lagrangian formulation, the Hamiltonian formulation and the Schrodinger equation in quantum theory and are found to be equivalent and exhaustive, as already known. Time-dependent constants are shown to be arbitrary, but nevertheless amenable to the general symmetry methods of Katzin, Levine (1977) and Mariwalla (1979).
TL;DR: In this paper, it is shown that the Gibbs states of an infinite classical statistical system correspond to the states of a reservoir at infinity, and that the configuration space of such a reservoir can be thought of as a generalized projective limit of configuration spaces of remote reservoirs.
Abstract: The Gibbs states of an infinite classical statistical system correspond to the states of “reservoir at infinity”. It is shown that its configuration space can be thought of as a generalized projective limit of configuration spaces of remote reservoirs. This notion of projective limit is defined and it is noted that it can also be used e.g. for proofs of the existence of Gibbs states in the thermodynamic limit and their decomposition into pure phase. A similar approach to (nonperturbative) Euclidean quantum field theory is suggested and connections with the concept of renormalizability are found.
TL;DR: In this paper, a nonrelativistic wave equation for the relative motion of composite particles is derived from a given, microscopic Schr\"odinger equation for two composite particles, the theory is identical to resonating group theory.
Abstract: A nonrelativistic wave equation for the relative motion of composite particles is derived from a given, microscopic Schr\"odinger equation. For two composite particles, the theory is identical to resonating group theory. For three and more composite particles it differs by the use of distortion functions with a continuous degree of freedom which allows a correct asymptotic description of the system. The dynamical equation, formally, is a multichannel Schr\"odinger equation with a matrix of nonlocal and energy-dependent interactions. The internal degrees of freedom of the composite particles and the Pauli principle are incorporated into the interaction. The relative motion wave function of the composite particles is a probability density amplitude in the asymptotic region only. In regions where the composite particles interact, the relative motion function is not a probability density amplitude and may even have a certain degree of off-shell ambiguity. In all regions of configuration space, however, the probability density of particles is defined by the microscopic wave function. The theory is formulated for nuclear fragments but is valid also for atoms and particles consisting of quarks.
TL;DR: In this paper, it is shown that point splitting can be used to regularize arbitrary Feynman amplitudes and Taylor subtraction terms of the type found in the Zimmermann formulation of renormalized perturbation theory.
Abstract: It is shown that point splitting may be used to regularize arbitrary Feynman amplitudes and Taylor subtraction terms of the type found in the Zimmermann formulation (BPHZ) of renormalized perturbation theory. Point splitting refers to the Fourier transformation of the Feynman integrand with respect to the internal (loop) momenta. It is shown that this has the effect in configuration space of breaking open the circuits of the diagram. The regularization may be removed, interchangeably, before or after the propagator epsilon is taken to zero in the sense of distribution theory. This result is proven elsewhere but is mentioned here for its importance. The result may make the method useful in establishing the unitarity of BPHZ. Two appendices contain a review of circuit‐based graph theory and a demonstration that an arbitrary circuit based graph may sometimes have no realization as a graph.
TL;DR: In this article, the topological properties of the configuration space in the SU2 Yang-Mills theory are discussed, and the authors show that all closed paths in the physical configuration space, which in the temporal gauge correspond to continuous paths connecting a field configuration Aia(x) with a gauge transform thereof, can be classified with the help of an "angular variable" by a winding number, which is a nonsingular, single-valued functional of the potentials.
Abstract: We discuss the topological properties of the configuration space in theSU2 Yang-Mills theory, consisting of all the field configurations for which the classical Hamiltonian is bounded, using a language which is familiar from the study of two- or three-dimensional spaces with a nontrivial topology Assuming that the physically relevant configurations consist of all potentials approaching a pure gauge at spacial infinity, we show that all closed paths in the physical configuration space, which in the temporal gauge correspond to continuous paths connecting a field configurationAia(x) with a gauge transform thereof, can be classified with the help of an «angular variable» by a winding number This «angular variable» is a nonsingular, single-valued functional of the potentials, but a multivalued functional of the gauge invariants built from these As a consequence, inequivalent representations for the canonical momenta exist which give rise to different θ-worlds
01 Jan 1980
TL;DR: In this paper, a multichannel scattering system composed of spinless, distinguishable, nonrelativistic particles, each with configuration space ℝ3 and interacting pairwise by local potentials, was considered.
Abstract: We consider a multichannel scattering system composed of N ⩾ 2 spinless, distinguishable, nonrelativistic particles, each with configuration space ℝ3 and interacting pairwise by local potentials V ij (1 ⩽ i < j ⩽ N) consisting of long-range and short-range parts Each V ij can be chosen, roughly, with the same degree of generality as in ALSHOLM [3], when the configuration space in the latter reference is taken as ℝ3 Using techniques similar to those of ALSHOLM [2,3], we have proved for the class of potentials considered that suitable modified wave operators μ ± α exist and have a generalized intertwining property for each channel α such that the corresponding bound states have a mild decay property of infinity For the present class of V ij 's, this property is known to be possessed by those bound states corresponding to eigenvalues of the discrete spectrum of the pertinent cluster Hamiltonian, or even to arbitrary nonthreshold eigenvalues if in addition the V ij 's are dilatation analytic We have also proved that the usual range-orthogonality property of the μ ± α holds under the conditions stated below The results of this paper can be readily generalized to the case when the single-particle configuration space is ℝv(v ⩾ 1)