Mapped Finite Element Discrete Variable Representation

Efficient numerical solver for the SchrSdinger equation is very important in physics and chemistry. The finite element discrete variable representation （FE-DVR） was first proposed by Rescigno and Mc-Curdy [Phys. Rev. A 62, 032706 （2000）] for solving quantum-mechanical scattering problems. In this work, an FE-DVR method in a mapped coordinate was proposed to improve the efficiency of the original FE-DVR method. For numerical demonstration, the proposed approach is applied for solving the electronic eigenfunctions and eigenvalues of the hydrogen atom and vibrational states of the electronic state 3E＋ of the Cs2 molecule which has long-range interaction potential. The numerical results indicate that the numerical efficiency of the original FE-DVR has been improved much using our proposed mapped coordinate scheme.

Numerical Convergence of the Sinc Discrete Variable Representation for Solving Molecular Vibrational States with a Conical Intersection in Adiabatic Representation

Within the Born-Oppenheimer(BO)approximation,nuclear motions of a molecule are often envisioned to occur on an adiabatic potential energy surface(PES).However,this single PES picture should be reconsidered if a conical intersection(CI)is present,although the energy is well below the CI.The presence of the CI results in two additional terms in the nuclear Hamiltonian in the adiabatic presentation,i.e.,the diagonal BO correction(DBOC)and the geometric phase(GP),which are divergent at the CI.At the same time,there are cusps in the adiabatic PESs.Thus usually it is regarded that there is numerical difficulty in a quantum dynamics calculation for treating CI in the adiabatic representation.A popular numerical method in nuclear quantum dynamics calculations is the Sinc discrete variable representation(DVR)method.We examine the numerical accuracy of the Sinc DVR method for solving the Schrodinger equation of a two dimensional model of two electronic states with a CI in both the adiabatic and diabatic representation.The results suggest that the Sinc DVR method is capable of giving reliable results in the adiabatic representation with usual density of the grid points,without special treatment of the divergence of the DBOC and the GP.The numerical uncertainty is not worse than that after the introduction of an arbitrary vector potential for accounting the GP,whose accurate form usually is not easy to obtain.

Integrable Properties Associated with a Discrete Three-by-Three Matrix Spectral Problem

Based on a new discrete three-by-three matrix spectral problem, a hierarchy of integrable lattice equations with three potentials is proposed through discrete zero-curvature representation, and the resulting integrable lattice equation reduces to the classical Toda lattice equation. It is shown that the hierarchy possesses a HamiItonian structure and a hereditary recursion operator. Finally, infinitely many conservation laws of corresponding lattice systems are obtained by a direct way.

MILP formulations for highway petrol station replenishment in initiative distribution mode

To investigate highway petrol station replenishment in initiative distribution mode,this paper develops a mixed-integer linear programming(MILP)model with minimal operational costs that includes loading costs,unloading costs,transport costs and the costs caused by unpunctual distribution.Based on discrete representation,the working day is divided into equal time intervals,and the truck distribution process is decomposed into a pair of tasks including driving,standby,rest,loading and unloading.Each truck must execute one task during a single interval,and the currently executing task is closely related to the preceding and subsequent tasks.By accounting for predictive time-varying sales at petrol stations,real-time road congestion and a series of operational constraints,the proposed model produces the optimal truck dispatch,namely,a detailed task assignment for all trucks during each time interval.The model is tested on a real-world case of a replenishment system comprising eight highway petrol stations,one depot,one garage and eight trucks to demonstrate its applicability and accuracy.

Hermiticity of Hamiltonian Matrix using the Fourier Basis Sets in Bond-Bond-Angle and Radau Coordinates

In quantum calculations a transformed Hamiltonian is often used to avoid singularities in a certain basis set or to reduce computation time. We demonstrate for the Fourier basis set that the Hamiltonian can not be arbitrarily transformed. Otherwise, the Hamiltonian matrix becomes non-hermitian, which may lead to numerical problems. Methods for cor- rectly constructing the Hamiltonian operators are discussed. Specific examples involving the Fourier basis functions for a triatomic molecular Hamiltonian （J=0） in bond-bond angle and Radau coordinates are presented. For illustration, absorption spectra are calculated for the OC10 molecule using the time-dependent wavepacket method. Numerical results indicate that the non-hermiticity of the Hamiltonian matrix may also result from integration errors. The conclusion drawn here is generally useful for quantum calculation using basis expansion method using quadrature scheme.

Hamiltonian System of New Nonlinear Lattice Equations

A 3-dimensional Lie algebra sμ（3） is obtained with the help of the known Lie algebra. Based on the sμ（3）, a new discrete 3 × 3 matrix spectral problem with three potentials is constructed. In virtue of discrete zero curvature equations, a new matrix Lax representation for the hierarchy of the discrete lattice soliton equations is acquired. It is shown that the hierarchy possesses a Hamiltonian operator and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.

NEW SYMPLECTIC MAPS: INTEGRABILITY AND LAX REPRESENTATION

New family of integrable symplectic maps are reduced from the Toda hierarchy via constraint for a higher flow of the hierarchy in terms of square eigenfunctions.Their integrability and Lax representation are deduced systematically from the discrete zero curvature representation of the Toda hierarchy. Also a discrete zero curvature representation for the Toda hierarchy with sources is presented.

RESTRICTED FLOWS OF A HIERARCHYOF INTEGRABLE DISCRETE SYSTEMS

The restricted flows of a hierarchy of discrete integrable systems are presented. It is shownthat integrals of motion and Lax representation for restricted flows as well as a discrete zerocurvature representation for the hierarchy with sources can be deduced from the zero-curvaturerepresentation for the hierarchy. Also it is studied how the hierarchy is related to the AKNShierarchy in the continuous limit.