Inverse problem of quadratic time-dependent Hamiltonians

Using an algebraic approach, it is possible to obtain the temporal evolution wave function for a Gaussian wavepacket obeying the quadratic time-dependent Hamiltonian(QTDH). However, in general, most of the practical cases are not exactly solvable, for we need general solutions of the Riccatti equations which are not generally known. We therefore bypass directly solving for the temporal evolution wave function, and study its inverse problem. We start with a particular evolution of the wave-packet, and get the required Hamiltonian by using the inverse method. The inverse approach opens up a new way to find new exact solutions to the QTDH. Some typical examples are studied in detail. For a specific timedependent periodic harmonic oscillator, the Berry phase is obtained exactly.

Quasi-Exactly Solvable Time-Dependent Hamiltonians

A generalized method which helps to find a time-dependent Schr&#214dinger equation for any static potential is established. We illustrate this method with two examples. Indeed, we use this method to find the time-dependent Hamiltonian of quasi-exactly solvable Lamé equation and to construct the matrix 2 × 2 time-dependent polynomial Hamiltonian.

Low-Storage Runge-Kutta Method for Simulating Time-Dependent Quantum Dynamics

A wide range of quantum systems are time-invariant and the corresponding dynamics is dic- tated by linear differential equations with constant coefficients. Although simple in math- ematical concept, the integration of these equations is usually complicated in practice for complex systems, where both the computational time and the memory storage become limit- ing factors. For this reason, low-storage Runge-Kutta methods become increasingly popular for the time integration. This work suggests a series of s-stage sth-order explicit Runge- Kutta methods specific for autonomous linear equations, which only requires two times of the memory storage for the state vector. We also introduce a 13-stage eighth-order scheme for autonomous linear equations, which has optimized stability region and is reduced to a fifth-order method for general equations. These methods exhibit significant performance improvements over the previous general-purpose low-stage schemes. As an example, we ap- ply the integrator to simulate the non-Markovian exciton dynamics in a 15-site linear chain consisting of perylene-bisimide derivatives.

Testing the validity of the Ehrenfest theorem beyond simple static systems: Caldirola–Kanai oscillator driven by a time-dependent force

The relationship between quantum mechanics and classical mechanics is investigated by taking a Gaussian-type wave packet as a solution of the Schr o¨dinger equation for the Caldirola–Kanai oscillator driven by a sinusoidal force. For this time-dependent system, quantum properties are studied by using the invariant theory of Lewis and Riesenfeld. In particular,we analyze time behaviors of quantum expectation values of position and momentum variables and compare them to those of the counterpart classical ones. Based on this, we check whether the Ehrenfest theorem which was originally developed in static quantum systems can be extended to such time-varying systems without problems.

Collective Hamiltonian for Multi-O（4） Model

The collective Bamiltonian up to the fourth order for multi-O（4） model is derived based on the self-consistent collective-coordinate （SCC） method, which is formulated in the framework of the time-dependent Hartree-Bogoliubov （TDHB） theory. The validity of the collective Hamiltonian is checked in the two special cases of the multi-O（4） model： the case where the number of the shells is equal to one （a single j-shell case）, and the case where the Hartree-Bogoliubov equilibrium point is spherical （the spherical case）. The collective Hamiltonian constitutes a good starting point to study nuclear shape coexistence.

经典正则变换与量子u变换的关系

对阻尼谐振子的含时哈密顿用经典正则变换和量子u变换两种方法进行变换 ,论证了两种变换间的对应关系。

Time-dependent relativistic density functional study of Yb and YbO

The low-lying electronic states of Yb and YbO are investigated by using time-dependent relativistic density functional theory,which is based on the newly developed exact two-component Hamiltonian resulting from symmetrized elimination of the small component.The nature of the excited states is analyzed by using the full molecular symmetry.The calculated results support the previous experimental assignment of the ground and excited states of YbO.

An Exact Propagator for Solving the Triatomic Reactive Schrodinger Equation

The exact short time propagator, in a form similar to the Crank-Nicholson method but in the spirit of spectrally transformed Hamiltonian, was proposed to solve the triatomic reactive time-dependent schrodinger equation. This new propagator is exact and unconditionally convergent for calculating reactive scattering processes with large time step sizes. In order to improve the computational efficiency, the spectral difference method was applied. This resulted the Hamiltonian with elements confined in a narrow diagonal band. In contrast to our previous theoretical work, the discrete variable representation was applied and resulted in full Hamiltonian matrix. As examples, the collision energy-dependent probability of the triatomic H＋H2 and O＋O2 reaction are calculated. The numerical results demonstrate that this new propagator is numerically accurate and capable of propagating the wave packet with large time steps. However, the efficiency and accuracy of this new propagator strongly depend on the mathematical method for solving the involved linear equations and the choice of preconditioner.

Effect of Environment on Photo-detachment Dynamics of Halide Ions： A Model Approach

A two dimensional model approach for the photodetachment dynamics of closed shell an-ionic systems in presence of external light field have been proposed in the context of polar environmental media. The effects of strong coupling between the solvent polarization and the extra charge in the system were studied by a simple model. The electronic states of con-cerned halide ions are represented by a two dimensional model Hamiltonian with a potential V（x,y）=-V0e^-σ（x^2＋y^2）. The time dependent Fourier grid Hamiltonian method have been used to follow the detachment process with fairly high intensities of light. The environmental effects on the dynamics are sought to be modeled by two different ways. The first one was the presence of polar solvents which perturb the energy levels of anionic systems by changing the effective potential surface and the second one was allowing the fluctuation of the well depth randomly to mimic the system in a more realistic view point. The average detachment rate constant is calculated as a function of important parameters of the used light field to explain the effects of solvent field on the dynamical behavior of dipole bound anionic system at least in a qualitative way.

Application of the SCC method to the multi-O(4) model:The collective Hamiltonian

The collective Hamiltonian up to the fourth order for a multi-O(4) model is derived for the first time based on the self-consistent collective-coordinate(SCC) method,which is formulated in the framework of the time-dependent Hartree-Bogoliubov(TDHB) theory.This collective Hamiltonian is valid for the spherical case where the HB equilibrium point of the multi-O(4) model is spherical as well as for the deformed case where the HB equilibrium points are deformed.Its validity is tested numerically in both the spherical and deformed cases.Numerical simulations indicate that the low-lying states of the collective Hamiltonian and the transition amplitudes among them mimic fairly well those obtained by exactly diagonalizing the Hamiltonian of the multi-O(4) model.The numerical results for the deformed case imply that the "optimized RPA boundary condition" is also valid for the well-known η*,η expansion around the unstable HB point of the multi-O(4) model.All these illuminate the power of the SCC method.

Theory and algorithms for the excited states of large molecules and molecular aggregates

This project aims to attack the frontiers of electronic structure calculations on the excited states of large molecules and molecular aggregates by developing novel theoretical and computational methods. The methodology development is especially based on the time-dependent density functional theory (TDDFT) and valence bond (VB) theory, and is expected to be computationally effective and accurate as well. Research works on the following related subjects will be performed: (1) The analytical energy-derivative approaches for electronically excited state within TDDFT will be developed to reduce bypass the computational costs in the calculation of molecular excited-state properties. (2) The ab initio methods for electronically excited state based on VB theory and hybrid TDDFT-VB method will be developed to overcome the limitations of current TDDFT in simulating photophysics and photochemistry. (3) For larger aggregates, neither ab initio methods nor TDDFT is applicable. We intend to build the effective model Hamiltonian by developing novel theoretical and computational methods to calculate the involved microscopic physical parameters from the first-principles methods. The constructed effective Hamiltonian is then used to describe the excitonic states and excitonic dynamics of the natural or artificial photosynthesized systems, organic or inorganic photovoltaic cell. (4) The condensed phase environment is taken into account by combining the developed theories and algorithms based on TDDFT and VB with the polarizable continuum solvent models (PCM), molecular mechanism (MM), classical electrodynamics (ED) or molecular dynamics (MD) theory. (5) Highly efficient software packages will be designed and developed.