Valleys and fall lines on a Riemannian manifold

Rowe, D.J.; Ryman, A.

doi:10.1063/1.525427

Cited by 54 publications

(39 citation statements)

References 6 publications

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“…This work was very influential on our own thinking, though in strict mathematical terms, it is a paraphrase of the earlier result of Rowe [45], who remarked that geodesic fall lines are decoupled. In the language of this review, this means that a path that satisfies the mass and force conditions and is also geodesic is decoupled, since the last condition is equivalent to also satisfying the curvature condition.…”

Section: A Survey Of Literaturementioning

confidence: 99%

“…The application of this method to the theory of large amplitude collective motion will be discussed below. A second body of work to which we are indebted is that of Rowe and collaborators [149,45,150]. The first of these papers contains, as far as we know, the first formulation of the LHA without curvature, associated with a stable local method for obtaining a solution that we have, in effect, utilized in our work.…”

Section: A Survey Of Literaturementioning

confidence: 99%

“…This may be expressed by the condition, choosing the "simplest" K + 1 vector fields, in the first form of the fundamental theorem, 45) where Λ i are a set of point functions that may be identified as Lagrange multipliers. This is because (2.45) is clearly the differential expression of the following constrained variational problem: Find extremal values of X (K) subject to fixed values of X (i) , i = 0, ..., K − 1.…”

Section: Generalized Valley Algorithm For Implementation Of Fundamentmentioning

confidence: 99%

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Self-consistent theory of large-amplitude collective motion: applications to approximate quantization of nonseparable systems and to nuclear physics

2000

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The goal of the present account is to review our efforts to obtain and apply a "collective" Hamiltonian for a few, approximately decoupled, adiabatic degrees of freedom, starting from a Hamiltonian system with more or many more degrees of freedom. The approach is based on an analysis of the classical limit of quantum-mechanical problems. Initially, we study the classical problem within the framework of Hamiltonian dynamics and derive a fully self-consistent theory of large amplitude collective motion with small velocities. We derive a measure for the quality of decoupling of the collective degree of freedom. We show for several simple examples, where the classical limit is obvious, that when decoupling is good, a quantization of the collective Hamiltonian leads to accurate descriptions of the low energy properties of the systems studied. In nuclear physics problems we construct the classical Hamiltonian by means of time-dependent mean-field theory, and we transcribe our formalism to this case. We report studies of a model for monopole vibrations, of 28 Si with a realistic interaction, several qualitative models of heavier nuclei, and preliminary results for a more realistic approach to heavy nuclei. Other * Email: Giu.Dodang@th.u-psud.fr † Email: aklein@walet.physics.upenn.edu ‡ Email: Niels.Walet@umist.ac.uk 1 topics included are a nuclear Born-Oppenheimer approximation for an ab initio quantum theory and a theory of the transfer of energy between collective and non-collective degrees of freedom when the decoupling is not exact. The explicit account is based on the work of the authors, but a thorough survey of other work is included. PACS number(s): 21.60.-n, 21.60.Jz, 21.60.Ev

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Section: A Survey Of Literaturementioning

confidence: 99%

Section: A Survey Of Literaturementioning

confidence: 99%

Section: Generalized Valley Algorithm For Implementation Of Fundamentmentioning

confidence: 99%

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Self-consistent theory of large-amplitude collective motion: applications to approximate quantization of nonseparable systems and to nuclear physics

2000

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“…This definition does not imply that the path will follow a thalweg everywhere, unless it is also a geodesic of the surface [66]. Along a steepest descent path, transverse instabilities may occur [67].…”

Section: The Specific Location Of Transition Statesmentioning

confidence: 95%

Algorithmic tools in the study of semiempirical potential surfaces

Liotard

1992

Int J of Quantum Chemistry

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The most efficient optimization methods implemented in the semiempirical package AMPAC are presented. They concern the minimization of the energy or of the gradient norm by either pseudo-Newton or quadratic procedures, the search for transition states, and the intrinsic reaction coordinate in conjunction with variational transition-state theories. Nonlocal methods such as simulated annealing are also introduced. 0 1992 John Wiley & Sons, Inc.

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“…This is because of the metric dependence of the Hessian matrix [17]. An invariant formulation of GE's is given in [18,19], cf also the textbook [20].…”

Section: Discussionmentioning

confidence: 99%

Gradient extremals and valley floor bifurcations on potential energy surfaces

Quapp

1989

Theoret. Chim. Acta

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Gradient extremals are curves in configuration space defined by the condition that the gradient of the potential energy is an eigenvector of the Hessian matrix. Solutions of a corresponding equation go along a valley floor or along a crest of a ridge, if the norm of the gradient is a minimum, and along a cirque or a cliff or a flank of one of the two if the gradient norm is a maximum. Properties of gradient extremals are discussed for simple 2D model surfaces including the problem of valley bifurcations.

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Valleys and fall lines on a Riemannian manifold

Cited by 54 publications

References 6 publications

Self-consistent theory of large-amplitude collective motion: applications to approximate quantization of nonseparable systems and to nuclear physics

Self-consistent theory of large-amplitude collective motion: applications to approximate quantization of nonseparable systems and to nuclear physics

Algorithmic tools in the study of semiempirical potential surfaces

Gradient extremals and valley floor bifurcations on potential energy surfaces

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