Mixed quantum-classical

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. 

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. 

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... 

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). 

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. 

In the mixed quantum-classical molecular dynamics (QCMD) model (see [11, 9, 2, 3, 5] and references therein), most atoms are described by classical mechanics, but an important small portion of the system by quantum mechanics. The full quantum system is first separated via a tensor product ansatz. The evolution of each part is then modeled either classically or quan-tally. This leads to a coupled system of Newtonian and Schrbdinger equations. 

MIXED QUANTUM-CLASSICAL CALCULATIONS IN BIOLOGICAL SYSTEMS... 

Mixed Quantum-Classical Calculations in Biological Systems... 

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. 

Jansen, G., Colonna, F. and Angyan, J. G. Mixed quantum-classical calculations on the water molecule in liquid phase Influence of a polarizable environment on electronic properties, Int.J. Quantum Chem, in press (1995),... 

Mixed Quantum/Classical Approach Single Chromophore... 

Mixed Quantum/Classical Approach Coupled Chromophores and the Time Averaging Approximation... 

H2O, where N/2 molecules have N OH stretch chromophores. In this case we need to label the transition dipoles and frequencies by an index i that runs from 1 to N. In addition, in general these chromophores interact, with couplings (in frequency units) coy. In this case the above mixed quantum/classical formula can be generalized to [95 98]... 

From a theoretical perspective, since the designation of the lab-fixed axes is arbitrary, what is relevant is the relative orientation of the polarizations of the excitation and scattered light. Thus the line shape for excitation light polarized along axis p, and scattered light polarized along axis q (p or q denote X, Y, or Z axes in the lab frame) is called Ipq(co). When p = q this is lyy, and when p q this is IVH. Mixed quantum/classical formulae for Ipq(co) are identical to those for the IR spectmm, except mPi is replaced by apqP which is the pq tensor element of the transition polarizability for chromophore i. Thus we have, for example [6],... 

Within the mixed quantum/classical approach, at each time step in a classical molecular dynamics simulation (that is, for each configuration of the bath coordinates), for each chromophore one needs the transition frequency and the transition dipole or polarizability, and if there are multiple chromophores, one needs the coupling frequencies between each pair. For water a number of different possible approaches have been used to obtain these quantities in this section we begin with brief discussions of each approach to determine transition frequencies. For definiteness we consider the case of a single OH stretch chromophore on an HOD molecule in liquid D2O. 

A number of researchers [15, 38 40, 43, 113, 124 126, 128, 146] have used mixed quantum/classical models, mostly as described in Section III.A, to calculate vibrational line shapes for this system, and several are in fair agreement with experiment. Here we describe our latest work involving approaches discussed in Section III.C. Our theoretical line shapes are calculated as briefly described in previous sections and in published work [98]. From an MD simulation of SPC/E heavy water, we determine the electric field on each putative H atom. We then use electric field maps to determine the transition frequency and dipole derivative. The orientational contribution to mp(t) we... 

Theoretical calculations for ultrafast neat water spectroscopy are difficult to perform and difficult to interpret (because of the near-resonant OH stretch coupling). One classical calculation of the 2DIR spectrum even preceded the experiments [163] Torii has calculated the anisotropy decay [97], finding reasonable agreement with the experimental time scale. Mixed quantum/ classical calculations of nonlinear spectroscopy for many coupled chromo-phores is a daunting task. We developed the TAA for linear spectroscopy, and Jansen has very recently extended it to nonlinear spectroscopy [164]. We hope that this will allow for mixed quantum/classical calculations of the 2DIR spectrum for neat water and that this will provide the context for a molecular-level interpretation of these complex but fascinating experiments. 

In this section we focused our attention to a rationalization of the factors determining the stereoselectivity through ab initio mixed quantum/classical (QM/MM) Car-Parrinello molecular dynamic simulations. We have used the same basic computational approach used in Section 3 to explore the potential energy surface of the reaction. Since the catalyst system, 1, is relatively large, we have used the combined QM/MM model system B as shown in Figure 3 and described in subsections 2.1 and 3.1. 

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