A simple nonrecursive form of the tensor decomposition in $d$ dimensions is presented. It does not inherently suffer from the curse of dimensionality, it has asymptotically the same number of parameters as the canonical decomposition, but it is stable and its computation is based on low-rank approximation of auxiliary unfolding matrices. The new form gives a clear and convenient way to implement all basic operations efficiently. A fast rounding procedure is presented, as well as basic linear algebra operations. Examples showing the benefits of the decomposition are given, and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19-dimensional operator.

Tensor-Train Decomposition

(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

We review recent progress in developing potential energy and dipole moment surfaces for polyatomic systems with up to 10 atoms. The emphasis is on global linear least squares fitting of tens of thousands of scattered ab initio energies using a special, compact fitting basis of permutationally invariant polynomials in Morse-type variables of all the internuclear distances. The computational mathematics underlying this approach is reviewed first, followed by a review of the practical approaches used to obtain the data for the fits. A straightforward symmetrization approach is also given, mainly for pedagogical purposes. The methods are illustrated for potential energy surfaces for , (H2O)2 and CH3CHO. The relationship of this approach to other approaches is also briefly reviewed.

Permutationally invariant potential energy surfaces in high dimensionality

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

http://sma.epfl.ch/~anchpcommon/publications/tensorsurvey.pdf

A literature survey of low-rank tensor approximation techniques

Nuclear quantum effects influence the structure and dynamics of hydrogen-bonded systems, such as water, which impacts their observed properties with widely varying magnitudes. This review highlights the recent significant developments in the experiment, theory, and simulation of nuclear quantum effects in water. Novel experimental techniques, such as deep inelastic neutron scattering, now provide a detailed view of the role of nuclear quantum effects in water's properties. These have been combined with theoretical developments such as the introduction of the principle of competing quantum effects that allows the subtle interplay of water's quantum effects and their manifestation in experimental observables to be explained. We discuss how this principle has recently been used to explain the apparent dichotomy in water's isotope effects, which can range from very large to almost nonexistent depending on the property and conditions. We then review the latest major developments in simulation algorithms and theory that have enabled the efficient inclusion of nuclear quantum effects in molecular simulations, permitting their combination with on-the-fly evaluation of the potential energy surface using electronic structure theory. Finally, we identify current challenges and future opportunities in this area of research.

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Nuclear Quantum Effects in Water and Aqueous Systems: Experiment, Theory, and Current Challenges

/pdf/multidimensional-quantum-dynamics-36abnlpfx5.pdf

Multidimensional Quantum Dynamics

Introduction THEORY The Road to MCTDH Basic MCTDH Theory Integration Schemes Preparation of the Initial Wavepacket Analysis of the Propagated Wave Packet MCTDH for Density Operators Computing Eigenstates by Relaxation and Improved Relaxation Iterative Diagonalzation of Operators Correlation Discrete Variable Represenation Potential Representations (potfit) Kinetic Energy Operators EXTENSION TO NEW AREAS Direct Dynamics with Quantum Nuclei Multilayer Formulation of the Multiconfiguration Time-Dependent Hartree Theory Shared Memory Parallelization of the Multiconfiguration Time-Dependent Hartree Method Strongly Driven Few-Fermion Systems - MCTDHF The Multiconfigurational Time-Dependent Hartree Method for Identical Particles and Mixtures Thereof APPLICATIONS Multidimensional Non-Adiabatic Dynamics MCTDH Calculation of Flux Correlation Functions: Rates and Reaction Probabilities for Polyatomic Chemical Reactions Reactive and Non-Reactive Scattering of Molecules From Surfaces Intramolecular Vibrational Energy Redistribution and Infrared Spectroscopy Open System Quantum Dynamics with Discretized Environments Proton Transfer and Hydrated Proton in Small Water Systems Laser-Driven Dynamics and Quantum Control of Molecular Wavepackets Polyatomic Dynamics of Dissociative Electron Attachment to Water Using MCTDH Ultracold Few-Boson Systems in Traps

Multidimensional quantum dynamics : MCTDH theory and applications

The infrared absorption spectrum of the protonated water dimer (H5O2+) is simulated in full dimensionality (15 dimensional) in the spectral range of 0-4000 cm(-1). The calculations are performed using the multiconfiguration time-dependent Hartree (MCTDH) method for propagation of wavepackets. All the fundamentals and several overtones of the vibrational motion are computed. The spectrum of H5O2+ is shaped to a large extent by couplings of the proton-transfer motion to large amplitude fluxional motions of the water molecules, water bending and water-water stretch motions. These couplings are identified and discussed, and the corresponding spectral lines are assigned. The large couplings featured by H5O2+ do not hinder, however, to describe the coupled vibrational motion by well defined simple types of vibration (stretching, bending; etc.) based on well defined modes of vibration, in terms of which the spectral lines are assigned. Comparison of our results to recent experiments and calculations on the system is given. The reported MCTDH IR spectrum is in very good agreement to the recently measured spectrum by Hammer et al. [J. Chem. Phys. 122, 244301 (2005)].

Full dimensional (15-dimensional) quantum-dynamical simulation of the protonated water dimer. II. Infrared spectrum and vibrational dynamics.

In the context of studies of the intramolecular vibrational-energy redistribution (IVR) in molecules with the aid of the multiconfiguration time-dependent Hartree (MCTDH) method, we simulate the dynamics of the selective population of vibrational levels in H 2 CS in the presence of an external time-dependent field. The molecule is described by means of six valence polyspherical coordinates. The potential-energy surface (PES), the dipole moment function, and the kinetic energy are of direct-product form in these coordinates and hence perfectly adapted to MCTDH. In order to obtain the eigenstates and the corresponding transition moments of the system, recent developments of the MCTDH improved relaxation method, for which a first comprehensive description is given here, are exploited.

Calculation and selective population of vibrational levels with the Multiconfiguration Time-Dependent Hartree (MCTDH) algorithm

The infrared absorption spectrum of the protonated water dimer (H5O2+) is simulated in full dimensionality (15D) in the spectral range 0-4000 cm-1. The calculations are performed using the Multiconfiguration Time-Dependent Hartree (MCTDH) method for propagation of wavepackets. All the fundamentals and several overtones of the vibrational motion are computed. The spectrum of H5O2+ is shaped to a large extent by couplings of the proton-transfer motion to large amplitude fluxional motions of the water molecules, water bending and water-water stretch motions. These couplings are identified and discussed, and the corresponding spectral lines assigned. The large couplings featured by H5O2+ do not hinder, however, to describe the coupled vibrational motion by well defined simple types of vibration (stretching, bending, etc.) based on well defined modes of vibration, in terms of which the spectral lines are assigned. Comparison of our results to recent experiments and calculations on the system is given. The reported MCTDH IR-spectrum is in very good agreement to the recently measured spectrum by Hammer et al. [JCP, 122, 244301, (2005)].

Fabien Gatti

Papers

Multidimensional Quantum Dynamics

Multidimensional quantum dynamics : MCTDH theory and applications

Full dimensional (15-dimensional) quantum-dynamical simulation of the protonated water dimer. II. Infrared spectrum and vibrational dynamics.

Calculation and selective population of vibrational levels with the Multiconfiguration Time-Dependent Hartree (MCTDH) algorithm

Full dimensional (15D) quantum-dynamical simulation of the protonated water-dimer II: infrared spectrum and vibrational dynamics