Quasi-classical dynamics

Essential features of the nuclear motion associated with chemical reactions can be described by classical mechanics. The special features of quantum mechanics cannot, of course, be properly described but some aspects like quantization can, in part, be taken into account by a simple procedure which basically amounts to a proper assignment of 

A theoretical determination of the rate constant for a chemical reaction requires a calculation of the reaction cross-section based on the dynamics of the collision process between the reactant molecules. We shall develop a general relation, based on classical dynamics, between reaction probabilities that can be extracted from the dynamics of the collision process and the phenomenological reaction cross-section introduced in Chapter 2. That is, we give a recipe for how to calculate the reaction cross-section in accord with the general definition in Eq. (2.7). 

Since the outcome of the collision only depends on the relative motion of the reactant molecules, we begin with an elimination of the center-of-mass motion of the system. From classical mechanics it is known that the relative translational motion of two atoms may be described as the motion of one pseudo-atom , with the reduced mass fj, = rri nif)/(m + mB), relative to a fixed center of force. This result can be generalized to molecules by introducing proper relative coordinates, to be described in detail in Section 4.1.4. 

The internal (vibrational and rotational) motion of molecule A is the same as that of the pseudo-molecule, while the center of mass of molecule B is at the fixed center of force. The force from the center of force on the pseudo-molecule is determined as the force between A and B, with A at the position of the pseudo-molecule and B at the position of the center of force. The scattering geometry is illustrated in Fig. 4.1.1. The pseudo-molecule moves with a velocity v = va — vB relative to the fixed center of force. We have drawn a line through the force center parallel to v that will be convenient to use as a reference in the specification of the scattering geometry. In addition to the internal quantum states of the pseudo-molecule and velocity v, the impact parameter b and angle j are used to specify the motion of the molecule. 

Several classical trajectories may result from such a collision process, as sketched in the figure. What makes the manifold of trajectories possible are the internal states i and j of the colliding molecules. To make that evident, let us first consider a situation where there are no internal states of the molecules and where the interaction potential only depends on the distance between the molecules, like for two hard spheres. Then there will only be one trajectory possible for a given b, j , v, because the initial conditions for the deterministic classical equations of motion are completely specified. This will not be the case when the molecules have internal degrees of freedom, even if 

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Classical dynamics is studied as a special case by analyzing the Ehrenfest theorem, coherent states (16) and systems with quasi classical dynamics like the rigid rotor for molecules (17) and the oscillator (18) for various particle systems and for EM field in a laser. 

Figure 3.5. 2D dissociation probability S0 (= S) as a function of translational energy and vibrational state v for H2 (D2) dissociation on a PES similar to (but not identical) to that of Figure 3.4(a). (a) Quantum dissociation probabilities plotted logarithmically, (b) Dotted lines are results of quasi-classical dynamics and solid lines are from quantum dynamics. From Ref. [222]. 

The activated dissociation of H2 (D2) on Cu(l 11) and other single crystal Cu surfaces has played a special role in the development of reactive gas-surface dynamics. Early experiments and theory by Cardillo and collaborators [217-219] first demonstrated the power of molecular beam techniques to probe activated adsorption and the theoretical methodology developed by them (6D quasi-classical dynamics on a model PES) only differs from modem treatments in the use of DFT based PES. 

Figure 3.26. (a) The experimental dissociation probability S for N2 on Ru(0001) plotted logarithmically vs. the incident normal energy E = En for three different N2 vibrational temperatures as noted in the legend. The squares varied both En and Tv simultaneously. From Ref. [244]. (b) First principles predictions of the logarithim of the dissociation probability at two vibrational temperatures as noted in the legend. The solid points are from 3D (Z, R, q) quasi-classical dynamics and the open points are from 6D quasi-classical dynamics. The latter are from Ref. [27]. 

Figure 3.28. N2 vibrational state distribution in associative desorption from Ru(0001). (a) Observed in experiment. From Ref. [126]. (b) From 3D (Z, R, q) first principles quasi-classical dynamics, with the solid triangles pointing upward being adiabatic dynamics and the squares from molecular dynamics with electronic frictions also from DFT. Based on the PES and frictions of Ref. [68]. The open triangles pointing downward are the results of 6D first principles adiabatic quasi-classical dynamics from Ref. [253].

There has been a long history in theoretical efforts to understand H + H/Cu(lll) and its isotopic analogs because it represents the best studied prototype of an ER/HA reaction. These have evolved from simple 2D collinear quantum dynamics on model PES [386] to 6D quasi-classical dynamics on PES fit to DFT calculations [380,387,388], and even attempts to include lattice motion on ER/HA reactions [389]. These studies show that there is little reflection of incident H because of the deep well and energy scrambling upon impact, i.e., a % 1. Although some of the... 

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) ... 

Carpenter, B. K. Bimodal distribution of lifetimes for an intermediate from a quasi-classical dynamics simulation, J. Am. Chem. Soc. 1996,118, 10329-10330. 

In principle, any electronic structure method that can produce a gradient and Hessian can be used in quasi-classical dynamics. However, density functional theory (DFT)-based methods are, in general (at the time of writing), limited to ground-state surfaces. CASSCF-based dynamics calculations can be used for excited-state computations, and we will focus our discussion on this method. In this case, the size of the molecule that can be studied is limited by the size of the active space (Section 2.2.1) at present, no more than 10 active elec-... 

The mixed-state Ehrenfest dynamics has problems after leaving the region of the nonadiabatic event. When the surfaces are sufficiently close in energy, the semi-classical Ehrenfest dynamics is switched on. Away from the degeneracy, the population on a single surface is recovered by reverting to the single-state quasi-classical dynamics. 

In the full-quantum dynamics method, the distribution of nuclear positions is accounted for in nuclear wavepacket form, that is, by a function that defines the distribution of momenta of each atom and the distribution of the position in the space of each atom. In classical and semi-classical or quasi-classical dynamics methods, the wavepacket distribution is emulated by a swarm of trajectories. We now briefly discuss how sampling can generate this swarm. 

Esposito, R, Capitelli, M. Gorse, C. (2000). Quasi-classical dynamics and vibrational kinetics in N2 (v) — N system, Chem. Phys. 257 193-202. 

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