Quantum/classical

Gillam M J 1987 Quantum-classical crossover of the transition rate in the damped double well J. Phys. C Solid State Phys. 20 3621... 

Fast P L and Truhlar D G 1998 Variational reaction path algorithm J. Chem. Phys. 109 3721 Billing G D 1992 Quantum classical reaction-path model for chemical reactions Chem. Phys. 161 245... 

Figure 2. Quantum classical cross-sections for the reaction D-I-Ha (r — l,j — 1) DH (v — l,/)-l-H at 1.8-eV total energy as a function of /. The solid line indicates results obtained without including the geometric phase effect. Boxes show the results with geometric phase effect included using either a complex phase factor (dashed) or a vector potential (dotted).

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

Aqvist, J., Warshel, A. Simulation of enzyme reactions using valence bond force fields and other hybrid quantum/classical approaches. Chem. Rev. 93... 

Bala, R, Lesyng, B., McCammon, J.A. Extended Hellmann-Feynman theorem for non-stationary states and its application in quantum-classical molecular dynamics simulations. Chem. Phys. Lett. 219 (1994) 259-266. 

Bornemann, F.A., Nettesheim, R, Schiitte, C. Quantum-classical molecular dynamics as an approximation to full quantum dynamics. J. Chem. Rhys. 105 (1996) 1074-1083. 

Approximation Properties and Limits of the Quantum-Classical Molecular Dynamics Model... 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

This paper is meant as a contribution to systematize the quantum-classical modeling of molecular dynamics. Hence, we are interested in an extended theoretical understanding of the models rather than to further contribute to the bunch of numerical experiments which have been performed on certain models by applying them to particular molecular systems. Thus, we will carefully review the assumptions under which our models are known to approximate the full quantum dynamical (QD) evolution of the system. This knowledge... 

P. Bala, P. Grochowski, B. Lesyng, and J. A. McCammon Quantum-classical molecular dynamics. Models and applications. In Quantum Mechanical Simulation Methods for Studying Biological Systems (M. Fields, ed.). Les Houches, France (1995)... 

Billing, G. D. Quantum-Classical Methods. In Numerical Grid Methods and Their Application to Schrodinger s equation (C. Cerjan, eds.). Kluwer Academics Publishers (1993)... 

Bornemann, F. A., Schiitte, Ch. On the Singular Limit of the Quantum-Classical Molecular Dynamics Model. Preprint SC 97-07 (1997) Konrad-Zuse-Zentrum Berlin. SIAM J. Appl. Math, (submitted)... 

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... 

Numerical Integrators for Quantum-Classical Molecular Dynamics... 

Various kinds of mixed quantum-classical models have been introduced in the literature. We will concentrate on the so-called quantum-classical molecular dynamics (QCMD) model, which consists of a Schrodinger equation coupled to classical Newtonian equations (cf. Sec. 2). 

P. Nettesheim, F. A. Bornemann, B. Schmidt, and Ch. Schiitte An explicit and symplectic integrator for quantum-classical molecular dynamics. Chem. Phys. Lett. 256 (1996) 581-588... 

P. Nettesheim, and S. Reich Symplectic raultiple-time-stepping integrators for quantum-classical molecular dynamics. (1998) (this volume)... 

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

In this paper, we consider the symplectic integration of the so-called Quantum-Classical Molecular Dynamics (QCMD) model. In the QCMD model (see [11, 9, 2, 3, 6] and references therein), most atoms are described by classical mechanics, but an important small portion of the system by quantum mechanics. This leads to a coupled system of Newtonian and Schrddinger equations. 

F.A. Bornemann and Ch. Schiitte. On the singular limit of the quantum-classical molecular dynamics model. Preprint SC 97-07, ZIB Berlin, 1997. Submitted to SIAM J. Appl. Math. 

U. Schmitt and J. Brinkmann. Discrete time-reversible propagation scheme for mixed quantum classical dynamics. Chem. Phys., 208 45-56, 1996. 

Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos method. This permits to take longer time steps than in standard integrators. 

M. Hochbruck and Ch. Lubich. Exponential integrators for quantum-classical molecular dynamics. Tech. Rep., Universitat Tubingen, 1998. In preparation. 

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