Dynamics Quantum

We solve for J and round the result to an integer to give the rotational quantum number. The vibrational energy is given by 

The extension of the trajectory calculations to a system with any number of atoms is straightforward except for the quantization of the vibrational and rotational states of the molecules. For a molecule with three different principal moments of inertia, there does not exist a simple analytical expression for the quantized rotational energy. This is only the case for molecules with some symmetry like a spherical top molecule, where all moments of inertia are identical, and a symmetric top, where two moments of inertia are identical and different from the third. For the vibrational modes, we may use a normal coordinate analysis to determine the normal modes (see Appendix E) and quantize those as for a one-dimensional oscillator. 

After having discussed the approximate quasi-classical dynamics, we return (see Section 1.1) now to exact quantum dynamics.9 The Schrodinger equation for motion of the atomic nuclei is given by Eq. (1.10)  

9In order to fully appreciate the content of this section, a good background in quantum mechanics is required see also Appendix F. 

We recall some basic results of quantum dynamics [3], First, the state of the system and the time evolution can be expressed in a generalized (Dirac) notation, which is often very convenient. The state at time t is specified by x(t)) with the representations x(-Rjf) = (R x t)) and x P,t) = (P x(t)) in coordinate and momentum space, respectively. Probability is a concept that is inherent in quantum mechanics. (R x(t)) 2 is the probability density in coordinate space, and (-P x(f) 2 is H e same quantity in momentum space. The time evolution (in the Schrodinger picture) can be expressed as 

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Dennison coupling produces a pattern in the spectrum that is very distinctly different from the pattern of a pure nonnal modes Hamiltonian , without coupling, such as (Al.2,7 ). Then, when we look at the classical Hamiltonian corresponding to the Darling-Deimison quantum fitting Hamiltonian, we will subject it to the mathematical tool of bifiircation analysis [M]- From this, we will infer a dramatic birth in bifiircations of new natural motions of the molecule, i.e. local modes. This will be directly coimected with the distinctive quantum spectral pattern of the polyads. Some aspects of the pattern can be accounted for by the classical bifiircation analysis while others give evidence of intrinsically non-classical effects in the quantum dynamics. 

Warren W S, Rabitz H and Dahleh M 1993 Coherent control of quantum dynamics the dream is alive Science 259 1581... 

The foundations of the modem tireory of elementary gas-phase reactions lie in the time-dependent molecular quantum dynamics and molecular scattering theory, which provides the link between time-dependent quantum dynamics and chemical kinetics (see also chapter A3.11). A brief outline of the steps hr the development is as follows [27],... 

There are two different aspects to these approximations. One consists in the approximate treatment of the underlying many-body quantum dynamics the other, in the statistical approach to observable average quantities. An exlmistive discussion of different approaches would go beyond the scope of this introduction. Some of the most important aspects are discussed in separate chapters (see chapter A3.7. chapter A3.11. chapter A3.12. chapter A3.131. 

Although the field of gas-phase kinetics remains hill of challenges it has reached a certain degree of maturity. Many of the fiindamental concepts of kinetics, in general take a particularly clear and rigorous fonn in gas-phase kinetics. The relation between fiindamental quantum dynamical theory, empirical kinetic treatments, and experimental measurements, for example of combustion processes [72], is most clearly established in gas-phase kmetics. It is the aim of this article to review some of these most basic aspects. Details can be found in the sections on applications as well as in the literature cited. 

Gi-op A, Wilke S and Scheffler M 1995 6-dimensional quantum dynamics of adsorption and desorption of H2 at Pd(IOO)-steering and steric effects Phys.Rev. Lett. 75 2718... 

Luntz A C and Harris J 1991 CH dissociation on metals—a quantum dynamics model Surf. Sc/. 258 397... 

Hase W L (ed) 1998 Comparisons of Classical and Quantum Dynamics (Adv. in Classical Trajectory Methods III) (Greenwich, CT JAI Press)... 

Balint-Kurti G G, Dixon R N and Marston C C 1992 Grid methods for solving the Schrodinger equation and time-dependent quantum dynamics of molecular photofragmentation and reactive scattering processes/of. Rev. Phys. Chem. 11 317—44... 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

Detailed analyses of the above experiments suggest that the apparent steps in k E) may not arise from quantized transition state energy levels [110.111]. Transition state models used to interpret the ketene and acetaldehyde dissociation experiments are not consistent with the results of high-level ab initio calculations [110.111]. The steps observed for NO2 dissociation may originate from the opening of electronically excited dissociation chaimels [107.108]. It is also of interest that RRKM-like steps in k E) are not found from detailed quantum dynamical calculations of unimolecular dissociation [91.101.102.112]. More studies are needed of unimolecular reactions near tln-eshold to detennine whether tiiere are actual quantized transition states and steps in k E) and, if not, what is the origin of the apparent steps in the above measurements of k E). 

As in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrence of IVR in polyatomic molecules, depends both on the Flamiltonian and the initial conditions, i.e. the initial quantum mechanical state I /(tQ)). We focus here on the time-dependent aspects of IVR, and in this case such initial conditions always correspond to the preparation, at a time of superposition states of molecular (spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occurs if these levels involve at least two distinct... 

Quack M 1995 Molecular femtosecond quantum dynamics between less than yoctoseconds and more than days experiment and theory Femtosecond Chemistry ed J Manz and L Woeste (Weinheim Verlag Chemie) pp 781-818... 

Quack M 1993 Molecular quantum dynamics from high resolution spectroscopy and laser chemistry J. Mol. Struct. 292 171-95... 

Maynard A T, Wyatt R E and lung C 1995 A quantum dynamical study of CH overtones in fluoroform. 

Wyatt R E, lung C and Leforestier C 1992 Quantum dynamics of overtone relaxation in benzene. II. Sixteen-mode model for relaxation from CH(v = 3) J. Chem. Phys. 97 3477-86... 

Minehardt T A, Adcock J D and Wyatt R E 1999 Quantum dynamics of overtone relaxation in 30-mode benzene a time-dependent local mode analysis for CH(v = 2) J. Chem. Phys. 110 3326-34... 

Luckhaus D 2000 6D vibrational quantum dynamics generalized coordinate discrete variable representation and (a)diabatic contraction J. Chem. Phys. 113 1329—47... 

Quack M and Suhm M A 1998 Spectroscopy and quantum dynamics of hydrogen fluoride clusters Advances in Moiecuiar Vibrations and Coiiision Dynamics, Voi. Hi Moiecuiar dusters ed J Bowman and Z Bai (JAI Press) pp 205—48... 

Quack M 1992 Time dependent intramolecular quantum dynamics from high resolution spectroscopy and laser chemistry Time Dependent Quantum Molecular Dynamics Experiment and Theory. Proc. NATO ARW 019/92 (NATO ASI Ser. Vol 299) ed J Broeckhove and L Lathouwers (New York Plenum) pp 293-310... 

Quack M, Stohner J and Sutcliffe E 1985 Time-dependent quantum dynamics of the picosecond... 

Marquardt R, Quack M and Thanopoulos I 2000 Dynamical chirality and the quantum dynamics of bending vibrations of the CH chromophore In methane Isotopomers J. Phys. Chem. A 104 6129—49... 

A completely different approach, in particular for fast imimolecular processes, extracts state-resolved kinetic infomiation from molecular spectra without using any fomi of time-dependent observation. This includes conventional line-shape methods, as well as the quantum-dynamical analysis of rovibrational overtone spectra [18, 33, 34 and 35]. 

The approach is ideally suited to the study of IVR on fast timescales, which is the most important primary process in imimolecular reactions. The application of high-resolution rovibrational overtone spectroscopy to this problem has been extensively demonstrated. Effective Hamiltonian analyses alone are insufficient, as has been demonstrated by explicit quantum dynamical models based on ab initio theory [95]. The fast IVR characteristic of the CH cliromophore in various molecular environments is probably the most comprehensively studied example of the kind [96] (see chapter A3.13). The importance of this question to chemical kinetics can perhaps best be illustrated with the following examples. The atom recombination reaction... 

Quack M, Sutcliffe E, Hackett P A and Rayner D M 1986 Molecular photofragmentation with many infrared photons. Absolute rate parameters from quantum dynamics, statistical mechanics, and direct measurement Faraday Discuss. Chem. Soc. 82 229-40... 

Quack M and Stohner J 1993 Femtosecond quantum dynamics of functional groups under coherent infrared multiphoton excitation as derived from the analysis of high-resolution spectra J. Rhys. Chem. 97 12 574-90... 

Ciccotti G and Ferrario M 1998 Constrained and nonequilibrium molecular dynamics Classical and Quantum Dynamics In Condensed Phase Simulations ed B J Berne, G Ciccotti and D F Coker (Singapore World Scientific) pp 157-77... 

Berne B J, Ciccotti G and Coker D F (ed) 1998 Classical and Quantum Dynamics In Condensed Phase S/mu/af/ons (Singapore World Scientific)... 

The classical counterpart of resonances is periodic orbits [91, 95, 96, 97 and 98]. For example, a purely classical study of the H+H2 collinear potential surface reveals that near the transition state for the H+H2 H2+H reaction there are several trajectories (in R and r) that are periodic. These trajectories are not stable but they nevertheless affect strongly tire quantum dynamics. A study of tlie resonances in H+H2 scattering as well as many other triatomic systems (see, e.g., [99]) reveals that the scattering peaks are closely related to tlie frequencies of the periodic orbits and the resonance wavefiinctions are large in the regions of space where the periodic orbits reside. 

Often a degree of freedom moves very slowly for example, a heavy-atom coordinate. In that case, a plausible approach is to use a sudden approximation, i.e. fix that coordinate and do reduced dimensionality quantum-dynamics simulations on the remaining coordinates. A connnon application of this teclmique, in a three-dimensional case, is to fix the angle of approach to the target [120. 121] (see figure B3.4.14). 

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

The preceding sections were concerned with the description of molecular motion. An ambitious goal is to proceed further and influence molecular motion. This lofty goal has been at the centrepiece of quantum dynamics in the past decade and is still under intense investigation [182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193 and 194]. Here we will only describe some general concepts and schemes. 

Truhlar D G, Schwenke D W and Kouri D J 1990 Quantum dynamics of chemical reactions by converged algebraic variational calculations J. Phys. Chem. 94 7346... 

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