High Order Corrections to the Time-Dependent Born-Oppenheimer Approximation I: Smooth Potentials

1986 ◽  
Vol 124 (3) ◽  
pp. 571 ◽  
Author(s):  
George A. Hagedorn
Keyword(s):  
High Order ◽  
Time Dependent ◽  
Oppenheimer Approximation

Introduction

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
State Level ◽  
Boltzmann Distribution ◽  
Theoretical Description ◽  
Time Dependent ◽  
Nuclear Dynamics ◽  
Molecular Reaction ◽  
Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

scholarly journals Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1113
Author(s):  
Isaías Alonso-Mallo ◽  
Ana M. Portillo
Keyword(s):  
Numerical Experiments ◽  
Splitting Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Order ◽  
Time Dependent ◽  
Boundary Values ◽  
Lipschitz Continuous

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

scholarly journals High Order Semi-implicit Multistep Methods for Time-Dependent Partial Differential Equations

2021 ◽  
Author(s):  
Giacomo Albi ◽  
Lorenzo Pareschi
Keyword(s):  
Multistep Methods ◽  
General Setting ◽  
Reaction Diffusion ◽  
Strong Stability ◽  
High Order ◽  
Iterative Solvers ◽  
Time Dependent ◽  
Linear Multistep Methods ◽  
Advection Diffusion Equation ◽  
Separation Of Scales

AbstractWe consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great flexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-diffusion equation and in the setting of strong stability preserving (SSP) methods. Our findings are demonstrated on several examples, including nonlinear reaction-diffusion and convection-diffusion problems.

scholarly journals A high order splitting method for time-dependent domains

2008 ◽  
Vol 197 (51-52) ◽  
pp. 4763-4773 ◽  
Author(s):  
Tormod Bjøntegaard ◽  
Einar M. Rønquist
Keyword(s):  
Splitting Method ◽  
High Order ◽  
Time Dependent

High-order harmonic generation with a single-color mid-infrared laser field by frequency-chirping

2019 ◽  
Vol 33 (13) ◽  
pp. 1950122 ◽  
Author(s):  
Yunhui Wang ◽  
Dandan Song ◽  
Qiang Zuo ◽  
Hong Wu ◽  
Zhihong Yang
Keyword(s):  
Harmonic Generation ◽  
Laser Field ◽  
Infrared Laser ◽  
Pulse Generation ◽  
High Order ◽  
Time Dependent ◽  
Mid Infrared ◽  
High Order Harmonic ◽  
Isolated Attosecond Pulse ◽  
Frequency Chirping

By numerically solving the time-dependent Schrödinger equation for helium atoms in a single mid-infrared laser field, we explore the frequency-chirping effect of laser field on high-order harmonic and isolated attosecond pulse generation. One or two ultrabroad supercontinuum harmonic plateaus can be controlled through modulating the laser field frequency by a small time-dependent signal. Under the best chirping condition, an ultrashort 2.2 as pulse can be obtained by Fourier transformation with the bandwidth of 782 eV. Furthermore, we explain the harmonic generation physical mechanisms by classical ionizing and returning energy maps and time–frequency analyzes.

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