Li Jingrui, Woywod Clemens, Vallet Valérie, Meier Christoph
Theoretical Chemistry, Department of Chemistry, Technical University Munich, D-85747 Garching, Germany.
J Chem Phys. 2006 May 14;124(18):184105. doi: 10.1063/1.2196408.
The dynamics of two coupled oscillators can become quite complex if anharmonic potential energy functions are employed. This type of system therefore represents a good model for an investigation of the performance of mixed quantum-classical methods. In this work, the motion of two coupled particles with a mass ratio of one to ten is studied with three different mixed quantum-classical methods in the presence of anharmonic potential terms for a comparison with exact quantum mechanical calculations. The mixed quantum-classical approaches include the multitrajectory Ehrenfest, the mixed quantum-classical Bohmian (MQCB), and the so-called coupled Schrödinger equations (CSE) formalisms. The analysis shows that while the description of a weakly anharmonic system by the Ehrenfest and MQCB schemes is accurate if proper sampling techniques are applied, both approximations break down rapidly if the anharmonic terms are increased. The performance of the simple CSE prescription, which corresponds to a reduction of the full two-dimensional wave function to two one-dimensional wave functions representing two quantum oscillators coupled via the potential energy in a classical fashion, decreases if the width of the initial wave packet is enlarged. The dependence of the CSE method on the diffuseness of the initial wave packet is therefore opposite to that of the MQCB method, which is more accurate for wide wave packets. Overall, the multitrajectory Ehrenfest ansatz is found to be most successful in reproducing the exact quantum results.
如果采用非简谐势能函数，两个耦合振子的动力学行为会变得相当复杂。因此，这类系统是研究混合量子 - 经典方法性能的一个很好的模型。在这项工作中，研究了质量比为1比10的两个耦合粒子在存在非简谐势项情况下的运动，使用三种不同的混合量子 - 经典方法，并与精确的量子力学计算进行比较。这些混合量子 - 经典方法包括多轨迹埃伦费斯特方法、混合量子 - 经典玻姆方法（MQCB）以及所谓的耦合薛定谔方程（CSE）形式。分析表明，虽然如果应用适当的采样技术，埃伦费斯特方法和MQCB方法对弱非简谐系统的描述是准确的，但如果增加非简谐项，这两种近似都会迅速失效。简单的CSE方法，即把完整的二维波函数简化为两个一维波函数，这两个一维波函数以经典方式通过势能耦合表示两个量子振子，其性能会随着初始波包宽度的增大而降低。因此，CSE方法对初始波包扩散度的依赖性与MQCB方法相反，MQCB方法对于宽波包更准确。总体而言，发现多轨迹埃伦费斯特假设在重现精确量子结果方面最为成功。
