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Books on the topic 'Approximation du potentiel'

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Published: 4 June 2021
Last updated: 11 February 2022

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1

Bartόk-Pártay, Albert. The Gaussian Approximation Potential. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-14067-9.

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2

Arakelian, N., P. M. Gauthier, and G. Sabidussi, eds. Approximation, Complex Analysis, and Potential Theory. Dordrecht: Springer Netherlands, 2001. http://dx.doi.org/10.1007/978-94-010-0979-9.

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3

The Gaussian approximation potential: An interatomic potential derived from first principles quantum mechanics. Heidelberg: Springer, c2010., 2010.

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4

Baumeister, Kenneth J. Combining comparison functions and finite element approximations in CFD. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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5

Limit theorems of polynomial approximation with exponential weights. Providence, R.I: American Mathematical Society, 2008.

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6

G, Vaĭnikko, ed. Periodic integral and pseudodifferential equations with numerical approximation. Berlin: Springer, 2002.

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7

Rossetti, Cesare. On the bound states of power law center potentials. Bologna: Editrice Compositori, 1989.

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8

N, Demkov I͡U. Zero-range potentials and their applications in atomic physics. New York: Plenum Press, 1988.

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9

1969-, Michel Volker, ed. Multiscale potential theory: With applications to geoscience. Boston: Birkhäuser, 2004.

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10

Burglind, Jöricke, ed. The uncertainty principle in harmonic analysis. Berlin: Springer-Verlag, 1994.

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11

Saff, E. B., Douglas Patten Hardin, Brian Z. Simanek, and D. S. Lubinsky. Modern trends in constructive function theory: Conference in honor of Ed Saff's 70th birthday : constructive functions 2014, May 26-30, 2014, Vanderbilt University, Nashville, Tennessee. Providence, Rhode Island: American Mathematical Society, 2016.

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12

N, Arakelian, Gauthier Paul M, and Sabidussi Gert, eds. Approximation, complex analysis, and potential theory. Dordrecht: Kluwer Academic, 2001.

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13

N, Arakelian, Gauthier Paul M, and Sabidussi Gert, eds. Approximation, complex analysis, and potential theory. Dordrecht: Kluwer Academic, 2001.

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14

Morawetz, Klaus. Multiple Impurity Scattering. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0005.

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Furnished with basic ideas about the scattering on a single impurity, the motion of a particle scattered by many randomly distributed impurities is approached. In spite of having a single particle only, this system already belongs to many-body physics as it combines randomising effects of high-angle collisions with mean-field effects due to low-angle collisions. The averaged wave function leads to the Dyson equation. Various approximations are systematically introduced and discussed ranging from Born, averaged T-matrix to coherent potential approximation. The effective medium and the effective mass as wave function renormalisations are discussed and the various approximations are accurately compared.
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15

Bartók-Pártay, Albert. The Gaussian Approximation Potential: An Interatomic Potential Derived from First Principles Quantum Mechanics. Springer, 2011.

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16

Bartók-Pártay, Albert. The Gaussian Approximation Potential: An Interatomic Potential Derived from First Principles Quantum Mechanics. Springer, 2012.

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17

Zeitlin, Vladimir. Primitive Equations Model. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198804338.003.0002.

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The chapter gives the foundations of modelling of large-scale atmospheric and oceanic motions and presents the ‘primitive equations’ (PE) model. After a concise reminder on general fluid mechanics, the main hypotheses leading to the PE model are explained, together with the tangent-plane (so-called f and beta plane) approximations, and ‘traditional’ approximation to the hydrodynamical equations on the rotating sphere. PE are derived in parallel for the ocean and for the atmosphere. It is then shown that, with a judicious choice of the vertical coordinate, the ‘pseudo-height’, in the atmosphere, these two sets of equations are practically equivalent. The main properties of PE are derived and the key concepts of wave–vortex dichotomy, and of slow and fast motions, are explained. The essential notion of potential vorticity is introduced and its conservation by fluid masses is demonstrated. Inertia–gravity waves are explained and their properties presented. Limitations of the hydrostatic hypothesis are demonstrated.
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18

(Adapter), Gert Sabidussi, Norair Arakelian (Editor), and Paul M. Gauthier (Editor), eds. Approximation, Complex Analysis, and Potential Theory (NATO SCIENCE SERIES: II: Mathematics, Physics and. Springer, 2001.

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19

(Adapter), Gert Sabidussi, Norair Arakelian (Editor), and Paul M. Gauthier (Editor), eds. Approximation, Complex Analysis, and Potential Theory (NATO SCIENCE SERIES: II: Mathematics, Physics and. Springer, 2001.

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20

Saranen, Jukka, Gennadi Vainikko, and J. Saranen. Periodic Integral & Pseudodifferential Equations with Numerical Approximation. Springer, 2001.

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21

Birkhauser, Willi Freeden, and Volker Michel. Multiscale Potential Theory. Birkhäuser Boston, 2004.

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22

Horing, Norman J. Morgenstern. Random Phase Approximation Plasma Phenomenology, Semiclassical and Hydrodynamic Models; Electrodynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0010.

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Chapter 10 reviews both homogeneous and inhomogeneous quantum plasma dielectric response phenomenology starting with the RPA polarizability ring diagram in terms of thermal Green’s functions, also energy eigenfunctions. The homogeneous dynamic, non-local inverse dielectric screening functions (K) are exhibited for 3D, 2D, and 1D, encompassing the non-local plasmon spectra and static shielding (e.g. Friedel oscillations and Debye-Thomas-Fermi shielding). The role of a quantizing magnetic field in K is reviewed. Analytically simpler models are described: the semiclassical and classical limits and the hydrodynamic model, including surface plasmons. Exchange and correlation energies are discussed. The van der Waals interaction of two neutral polarizable systems (e.g. physisorption) is described by their individual two-particle Green’s functions: It devolves upon the role of the dynamic, non-local plasma image potential due to screening. The inverse dielectric screening function K also plays a central role in energy loss spectroscopy. Chapter 10 introduces electromagnetic dyadic Green’s functions and the inverse dielectric tensor; also the RPA dynamic, non-local conductivity tensor with application to a planar quantum well. Kramers–Krönig relations are discussed. Determination of electromagnetic response of a compound nanostructure system having several nanostructured parts is discussed, with applications to a quantum well in bulk plasma and also to a superlattice, resulting in coupled plasmon spectra and polaritons.
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23

Fox, Raymond. The Use of Self. Oxford University Press, 2011. http://dx.doi.org/10.1093/oso/9780190616144.001.0001.

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This monograph presents recent advances in neural network (NN) approaches and applications to chemical reaction dynamics. Topics covered include: (i) the development of ab initio potential-energy surfaces (PES) for complex multichannel systems using modified novelty sampling and feedforward NNs; (ii) methods for sampling the configuration space of critical importance, such as trajectory and novelty sampling methods and gradient fitting methods; (iii) parametrization of interatomic potential functions using a genetic algorithm accelerated with a NN; (iv) parametrization of analytic interatomic potential functions using NNs; (v) self-starting methods for obtaining analytic PES from ab inito electronic structure calculations using direct dynamics; (vi) development of a novel method, namely, combined function derivative approximation (CFDA) for simultaneous fitting of a PES and its corresponding force fields using feedforward neural networks; (vii) development of generalized PES using many-body expansions, NNs, and moiety energy approximations; (viii) NN methods for data analysis, reaction probabilities, and statistical error reduction in chemical reaction dynamics; (ix) accurate prediction of higher-level electronic structure energies (e.g. MP4 or higher) for large databases using NNs, lower-level (Hartree-Fock) energies, and small subsets of the higher-energy database; and finally (x) illustrative examples of NN applications to chemical reaction dynamics of increasing complexity starting from simple near equilibrium structures (vibrational state studies) to more complex non-adiabatic reactions. The monograph is prepared by an interdisciplinary group of researchers working as a team for nearly two decades at Oklahoma State University, Stillwater, OK with expertise in gas phase reaction dynamics; neural networks; various aspects of MD and Monte Carlo (MC) simulations of nanometric cutting, tribology, and material properties at nanoscale; scaling laws from atomistic to continuum; and neural networks applications to chemical reaction dynamics. It is anticipated that this emerging field of NN in chemical reaction dynamics will play an increasingly important role in MD, MC, and quantum mechanical studies in the years to come.
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24

Raff, Lionel, Ranga Komanduri, Martin Hagan, and Satish Bukkapatnam. Neural Networks in Chemical Reaction Dynamics. Oxford University Press, 2012. http://dx.doi.org/10.1093/oso/9780199765652.001.0001.

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This monograph presents recent advances in neural network (NN) approaches and applications to chemical reaction dynamics. Topics covered include: (i) the development of ab initio potential-energy surfaces (PES) for complex multichannel systems using modified novelty sampling and feedforward NNs; (ii) methods for sampling the configuration space of critical importance, such as trajectory and novelty sampling methods and gradient fitting methods; (iii) parametrization of interatomic potential functions using a genetic algorithm accelerated with a NN; (iv) parametrization of analytic interatomic potential functions using NNs; (v) self-starting methods for obtaining analytic PES from ab inito electronic structure calculations using direct dynamics; (vi) development of a novel method, namely, combined function derivative approximation (CFDA) for simultaneous fitting of a PES and its corresponding force fields using feedforward neural networks; (vii) development of generalized PES using many-body expansions, NNs, and moiety energy approximations; (viii) NN methods for data analysis, reaction probabilities, and statistical error reduction in chemical reaction dynamics; (ix) accurate prediction of higher-level electronic structure energies (e.g. MP4 or higher) for large databases using NNs, lower-level (Hartree-Fock) energies, and small subsets of the higher-energy database; and finally (x) illustrative examples of NN applications to chemical reaction dynamics of increasing complexity starting from simple near equilibrium structures (vibrational state studies) to more complex non-adiabatic reactions. The monograph is prepared by an interdisciplinary group of researchers working as a team for nearly two decades at Oklahoma State University, Stillwater, OK with expertise in gas phase reaction dynamics; neural networks; various aspects of MD and Monte Carlo (MC) simulations of nanometric cutting, tribology, and material properties at nanoscale; scaling laws from atomistic to continuum; and neural networks applications to chemical reaction dynamics. It is anticipated that this emerging field of NN in chemical reaction dynamics will play an increasingly important role in MD, MC, and quantum mechanical studies in the years to come.
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25

Demkov, Yu N., and V. N. Ostrovskii. Zero-Range Potentials and Their Applications in Atomic Physics (Physics of Atoms and Molecules). Springer, 1988.

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26

Horing, Norman J. Morgenstern. Non-Equilibrium Green’s Functions: Variational Relations and Approximations for Particle Interactions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0009.

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Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.
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27

Roca-Royes, Sònia. Rethinking the Epistemology of Modality for Abstracta. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198792161.003.0012.

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This chapter is an exploration of the sort of epistemology available to explain our de re modal knowledge about abstract entities. The thesis suggested—in a first approximation to the issue—is somewhat provocative: as modal epistemologists, we don’t have much work to do; instead, the work is down to ontologists. The chapter first motivates the thesis by relying on a conception of abstract objects that makes the thesis a rather plausible one. The chapter then goes on to consider some potential concerns and it concludes that, while their treatment imposes some refinements and qualifications, the thesis stands as it is.
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28

Henriksen, Niels Engholm, and Flemming Yssing Hansen. Introduction to Condensed-Phase Dynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805014.003.0009.

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This chapter discusses chemical reactions in solution; first, how solvents modify the potential energy surface of the reacting molecules and second, the role of diffusion. As a first approximation, solvent effects are described by models where the solvent is represented by a dielectric continuum, focusing on the Onsager reaction-field model for solvation of polar molecules. The reactants of bimolecular reactions are brought into contact by diffusion, and the interplay between diffusion and chemical reaction that determines the overall reaction rate is described. The solution to Fick’s second law of diffusion, including a term describing bimolecular reaction, is discussed. The limits of diffusion control and activation control, respectively, are identified. It concludes with a stochastic description of diffusion and chemical reaction based on the Fokker–Planck equation, which includes the diffusion of particles interacting via a potential U(r).
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29

Havin, Victor, and Burglind Jöricke. The Uncertainty Principle in Harmonic Analysis (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics). Springer, 1997.

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30

Saha, Prasenjit, and Paul A. Taylor. Nuclear Fusion in Stars. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198816461.003.0006.

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This chapter enumerates some of the many nuclear reactions in stars. It focuses on a general principle: nuclear fusion requires overcoming the Coulomb barrier between nuclei, which is possible through the relatively infrequent process of quantum tunnelling. The tunnelling probability depends on the atomic num-bers and mass numbers of the nuclei involved, and also on their relative speed. These translate into steep and interesting temperature dependencies for nuclear reactions. Analytic approximations yield a rate that is almost vanishingly rare, yet within the incredibly large number of potential interactions within stars quantum tunnelling can still provide the underlying ignition of nuclear reactions.
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31

Dyall, Kenneth G., and Knut Faegri. Introduction to Relativistic Quantum Chemistry. Oxford University Press, 2007. http://dx.doi.org/10.1093/oso/9780195140866.001.0001.

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This book provides an introduction to the essentials of relativistic effects in quantum chemistry, and a reference work that collects all the major developments in this field. It is designed for the graduate student and the computational chemist with a good background in nonrelativistic theory. In addition to explaining the necessary theory in detail, at a level that the non-expert and the student should readily be able to follow, the book discusses the implementation of the theory and practicalities of its use in calculations. After a brief introduction to classical relativity and electromagnetism, the Dirac equation is presented, and its symmetry, atomic solutions, and interpretation are explored. Four-component molecular methods are then developed: self-consistent field theory and the use of basis sets, double-group and time-reversal symmetry, correlation methods, molecular properties, and an overview of relativistic density functional theory. The emphases in this section are on the basics of relativistic theory and how relativistic theory differs from nonrelativistic theory. Approximate methods are treated next, starting with spin separation in the Dirac equation, and proceeding to the Foldy-Wouthuysen, Douglas-Kroll, and related transformations, Breit-Pauli and direct perturbation theory, regular approximations, matrix approximations, and pseudopotential and model potential methods. For each of these approximations, one-electron operators and many-electron methods are developed, spin-free and spin-orbit operators are presented, and the calculation of electric and magnetic properties is discussed. The treatment of spin-orbit effects with correlation rounds off the presentation of approximate methods. The book concludes with a discussion of the qualitative changes in the picture of structure and bonding that arise from the inclusion of relativity.
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32

Akemann, Gernot. Random matrix theory and quantum chromodynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797319.003.0005.

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This chapter was originally presented to a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian unitary ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions, and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in quantum chromodynamics with three colors in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered, including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. This chapter also provides some open random matrix problems.
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33

Kanduč, M., A. Schlaich, E. Schneck, and R. R. Netz. Interactions between biological membranes: theoretical concepts. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198789352.003.0012.

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In this chapter we review the various types of generic (non-specific) forces acting between lipid membranes in an aqueous environment and discuss the underlying mechanisms, with particular focus on the competing roles of enthalpic and entropic contributions. The interaction free energy (or interaction potential) is typically the result of a subtle interplay of several, often antagonistic contributions with comparable magnitude. First, we will briefly introduce the underlying physics of various kinds of surface–surface interactions, starting with theories of van der Waals and undulation interactions, covering electrostatics, depletion, and order–parameter fluctuation effects as well. We then turn our attention to a strong and universal repulsive force at small membrane–membrane separations, namely the hydration interaction. It has been under debate and investigation for decades and is not well captured by continuum approximations, thus here we will mainly rely on atomistic simulation techniques.
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34

Horing, Norman J. Morgenstern. Retarded Green’s Functions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0005.

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Chapter 5 introduces single-particle retarded Green’s functions, which provide the probability amplitude that a particle created at (x, t) is later annihilated at (x′,t′). Partial Green’s functions, which represent the time development of one (or a few) state(s) that may be understood as localized but are in interaction with a continuum of states, are discussed and applied to chemisorption. Introductions are also made to the Dyson integral equation, T-matrix and the Dirac delta-function potential, with the latter applied to random impurity scattering. The retarded Green’s function in the presence of random impurity scattering is exhibited in the Born and self-consistent Born approximations, with application to Ando’s semi-elliptic density of states for the 2D Landau-quantized electron-impurity system. Important retarded Green’s functions and their methods of derivation are discussed. These include Green’s functions for electrons in magnetic fields in both three dimensions and two dimensions, also a Hamilton equation-of-motion method for the determination of Green’s functions with application to a 2D saddle potential in a time-dependent electric field. Moreover, separable Hamiltonians and their product Green’s functions are discussed with application to a one-dimensional superlattice in axial electric and magnetic fields. Green’s function matching/joining techniques are introduced and applied to spatially varying mass (heterostructures) and non-local electrostatics (surface plasmons).
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35

Anderson, James A. The Brain Works by Logic. Oxford University Press, 2018. http://dx.doi.org/10.1093/acprof:oso/9780199357789.003.0007.

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Brains and computers were twins separated at birth. In 1943, it was known that action potentials were all or none, approximating TRUE or FALSE. In that year, Walter Pitts and Warren McCulloch wrote a paper suggesting that neurons were computing logic functions and that networks of such neurons could compute any finite logic function. This was a bold and exciting large-scale theory of brain function. Around the same time, the first digital computer, the ENIAC, was being built. The McCulloch-Pitts work was well known to the scientists building ENIAC. The connection between them appeared explicitly in a report by John von Neumann on the successor to the ENIAC, the EDVAC. It soon became clear that biological brain computation was not based on logic functions. However, this idea was believed by many scientists for decades. A brilliant wrong theory can sometimes cause trouble.
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36

Horing, Norman J. Morgenstern. Interacting Electron–Hole–Phonon System. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0011.

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Chapter 11 employs variational differential techniques and the Schwinger Action Principle to derive coupled-field Green’s function equations for a multi-component system, modeled as an interacting electron-hole-phonon system. The coupled Fermion Green’s function equations involve five interactions (electron-electron, hole-hole, electron-hole, electron-phonon, and hole-phonon). Starting with quantum Hamilton equations of motion for the various electron/hole creation/annihilation operators and their nonequilibrium average/expectation values, variational differentiation with respect to particle sources leads to a chain of coupled Green’s function equations involving differing species of Green’s functions. For example, the 1-electron Green’s function equation is coupled to the 2-electron Green’s function (as earlier), also to the 1-electron/1-hole Green’s function, and to the Green’s function for 1-electron propagation influenced by a nontrivial phonon field. Similar remarks apply to the 1-hole Green’s function equation, and all others. Higher order Green’s function equations are derived by further variational differentiation with respect to sources, yielding additional couplings. Chapter 11 also introduces the 1-phonon Green’s function, emphasizing the role of electron coupling in phonon propagation, leading to dynamic, nonlocal electron screening of the phonon spectrum and hybridization of the ion and electron plasmons, a Bohm-Staver phonon mode, and the Kohn anomaly. Furthermore, the single-electron Green’s function with only phonon coupling can be rewritten, as usual, coupled to the 2-electron Green’s function with an effective time-dependent electron-electron interaction potential mediated by the 1-phonon Green’s function, leading to the polaron as an electron propagating jointly with its induced lattice polarization. An alternative formulation of the coupled Green’s function equations for the electron-hole-phonon model is applied in the development of a generalized shielded potential approximation, analysing its inverse dielectric screening response function and associated hybridized collective modes. A brief discussion of the (theoretical) origin of the exciton-plasmon interaction follows.
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37

Henriksen, Niels E., and Flemming Y. Hansen. Theories of Molecular Reaction Dynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805014.001.0001.

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This book deals with a central topic at the interface of chemistry and physics—the understanding of how the transformation of matter takes place at the atomic level. Building on the laws of physics, the book focuses on the theoretical framework for predicting the outcome of chemical reactions. The style is highly systematic with attention to basic concepts and clarity of presentation. Molecular reaction dynamics is about the detailed atomic-level description of chemical reactions. Based on quantum mechanics and statistical mechanics or, as an approximation, classical mechanics, the dynamics of uni- and bimolecular elementary reactions are described. The first part of the book is on gas-phase dynamics and it features a detailed presentation of reaction cross-sections and their relation to a quasi-classical as well as a quantum mechanical description of the reaction dynamics on a potential energy surface. Direct approaches to the calculation of the rate constant that bypasses the detailed state-to-state reaction cross-sections are presented, including transition-state theory, which plays an important role in practice. The second part gives a comprehensive discussion of basic theories of reaction dynamics in condensed phases, including Kramers and Grote–Hynes theory for dynamical solvent effects. Examples and end-of-chapter problems are included in order to illustrate the theory and its connection to chemical problems. The book has ten appendices with useful details, for example, on adiabatic and non-adiabatic electron-nuclear dynamics, statistical mechanics including the Boltzmann distribution, quantum mechanics, stochastic dynamics and various coordinate transformations including normal-mode and Jacobi coordinates.
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