Mixed quantum-classical method

Haug, K., Metiu, H. A test of the possibility of calculating absorption spectra by mixed quantum-classical methods. J. Chem. Phys. 97 (1992) 4781-4791... 

Lyne, P. Mixed quantum/classical methods, in Computational Biochemistry Biophysics, Becker, O. Mackerell Jr., A. Roux, B. Watanabe, M., Eds., N.Y., New York, 2001. 

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

By definition, a mixed quantum-classical method treats the various degrees of freedom (DoF) of a system on a different dynamical footing—for example, quantum mechanics for the electronic DoF and classical mechanics for the... 

The mapping approach outlined above has been designed to furnish a well-defined classical limit of nonadiabatic quantum dynamics. The formalism applies in the same way at the quantum-mechanical, semiclassical (see Section VIII), and quasiclassical level, respectively. Most important, no additional assumptions but the standard semiclassical and quasi-classical approximations are needed to get from one level to another. Most of the established mixed quantum-classical methods such as the mean-field-trajectory method or the surface-hopping approach do invoke additional assumptions. The comparison of the mapping approach to these formulations may therefore (i) provide insight into the nature of these additional approximation and (ii) indicate whether the conceptual virtues of the mapping approach may be expected to result in practical advantages. 

In the last few years we have witnessed the successful development of several methods for the numerical solution of multi-dimensional quantum Hamiltonians Monte Carlo methods centroid methods,mixed quantum-classical methods, and recently a revival of semiclassical methods. We have developed another approach to this problem, the exponential resummation of the evolution operator. - The rest of this Section will explain briefly this method. 

We have examined the proton transfer reaction AH-B A -H+B in liquid methyl chloride, where the AH-B complex corresponds to phenol-amine. The intermolecular and the complex-solvent potentials have a Lennard-Jones and a Coulomb component as described in detail in the original papers. There have been other quantum studies of this system. Azzouz and Borgis performed two calculations one based on centroid theory and another on the Landau-Zener theory. The two methods gave similar results. Hammes-Schiffer and Tully used a mixed quantum-classical method and predicted a rate that is one order of magnitude larger and a kinetic isotope effect that is one order of magnitude smaller than the Azzouz-Borgis results. 

The focus of this chapter is exploration of the ability of mixed quantum classical approaches to capture the effects of interference and coherence in the approximate dynamics used in these different mixed quantum classical methods. As outlined below, the expectation values of computed observables are fundamentally non-equilibrium properties that are not expressible as equilibrium time correlation functions. Thus, the chapter explores the relationship between the approximations to the quantum dynamics made in these different approaches that attempt to capture quantum coherence. 

The main goal in the development of mixed quantum classical methods has as its focus the treatment of large, complex, many-body quantum systems. While applications to models with many realistic elements have been carried out [10,11], here we test the methods and algorithms on the spin-boson model, which is the standard test case in this field. In particular, we focus on the asymmetric spin-boson model and the calculation of off-diagonal density matrix elements, which present difficulties for some simulation schemes. We show that both of the methods discussed here are able to accurately and efficiently simulate this model. 

In Fig. 2 we compare results using e = 0.4 for the two mixed quantum-classical methods outlined in this chapter with exact results obtained from MCTDH wavepacket dynamics calculations. To make a reliable comparison the approximate finite temperature calculations were performed at very low temperatures (/ = 25), though a product of ground state wave functions for the independent harmonic oscillator modes could have been used to make the initial conditions identical to those used in the MCTDH calculations. 

Our simulations are based on well-established mixed quantum-classical methods in which the electron is described by a fully quantum-statistical mechanical approach whereas the solvent degrees of freedom are treated classically. Details of the method are described elsewhere [27,28], The extent of the electron localization in different supercritical environments can be conveniently probed by analyzing the behavior of the correlation length R(fih/2) of the electron, represented as polymer of pseudoparticles in the Feynman path integral representation of quantum mechanics. Using the simulation trajectories, R is computed from the mean squared displacement along the polymer path, R2(t - t ) = ( r(f) - r(t )l2), where r(t) represents the electron position at imaginary time t and 1/(3 is Boltzmann constant times the temperature. 

G.D. Billing, Mixed Quantum-classical Methods, Encyclopedea of computational Chemistry, 1998. 

In recent work [16, 17] we presented a new mixed quantum-classical method, which we call LAND-Map (Linearized approach to non-adiabatic dynamics in the mapping formulation), for calculating correlation functions. The method couples the linearization ideas put forth by various workers [18-26] with the mapping description of non-adiabatic transitions [27-31]. 

By definition, a mixed quantum-classical method treats the various degrees of freedom (DoF) of a system on a different djmamical footing, e.g. quantum mechanics for the electronic DoF and classical mechanics for the nuclear DoF. As was discussed above, some of the problems with these methods are related to inconsistencies inherent in this mixed quantum-classical ansatz. To avoid these problems, recently a conceptually different way to incorporate quantum mechanical DoF into a semiclassical or quasiclassical theory has been proposed, the so-called mapping approach. " In this formulation, the problem of a classical treatment of discrete DoF such as electronic states is bypassed by transforming the discrete quantum variables to continuous variables. In this section we briefly introduce the general concept of the mapping approach and discuss the quasiclassical implementation of this method as well as applications to the three models introduced above. The semiclassical version of the mapping approach is discussed in Sec. 7. 

Nonadiabatic dynamics is a quantum phenomenon which occurs in systems that interact sufficiently strongly with their environments to cause a breakdown of the Born-Oppenheimer approximation. Nonadiabatic transitions play significant roles in many chemical processes such as proton and electron transfer events in solution and biological systems, and in the response of molecules to radiation fields and their subsequent relaxation. Since the bath in which the quantum dynamics of interest occurs often consists of relatively heavy molecules, its evolution can be modeled by classical mechanics to a high degree of accuracy. This observation has led to the development of mixed quantum-classical methods for nonadiabatic processes. 

Linear and nonlinear infrared spectroscopy are powerful tools for probing the structure and vibrational dynamics of molecular systems." In order to take full advantage of them, however, accurate models and methods are required for simulating and interpreting spectra. A common approach for modeling spectra is based on computing optical response functions (ORFs)." Unfortunately, exact calculations of quantum-mechanical ORFs are not feasible for most systems of practical interest due to the large number of DOF. Instead, mixed quantum-classical methods ean provide suitable alternatives." " " ... 

Even in view of all these limitations, excited-state non-adiabatic dynamics simulations based on mixed quantum-classical methods constitute a fascinating and active research field providing essential information on the nature of molecular phenomena. 

Carbocation Stabilities Comparison of Theory and Experiment Force Fields A General Discussion Mixed Quantum-Classical Methods Molecular Mechanics Conjugated Systems. 

Integrating the Classical Equations off Motion Mixed Quantum-Classical Methods Trajectory Simulations off 1 Introduction 402... 

AMI AMBER A Program for Simulation of Biological and Organic Molecules CHARMM The Energy Function and Its Parameterization Combined Quantum Mechanics and Molecular Mechanics Approaches to Chemical and Biochemical Reactivity Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Divide and Conquer for Semiempirical MO Methods Electrostatic Catalysis Force Fields A General Discussion Force Fields CFF GROMOS Force Field Hybrid Methods Hybrid Quantum Mechanical/Molecular Mechanical (QM/MM) Methods Mixed Quantum-Classical Methods MNDO MNDO/d Molecular Dynamics Techniques and Applications to Proteins OPLS Force Fields Parameterization of Semiempirical MO Methods PM3 Protein Force Fields Quantum Mechanical/Molecular Mechanical (QM/MM) Coupled Potentials Quantum Mecha-nics/Molecular Mechanics (QM/MM) SINDOI Parameterization and Application. 

Combined Quantum Mechanical and Molecular Mechanical Potentials Divide and Conquer for Semiempirical MO Methods Hybrid Methods Hybrid Quantum Mecha-nical/Molecular Mechanical (QM/MM) Methods Mixed Quantum-Classical Methods Quantum Mechanical/Mole-cular Mechanical (QM/MM) Coupled Potentials Quantum Mechanics/Molecular Mechanics (QM/MM). 

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