Collinear

Figure Al.6.26. Stereoscopic view of ground- and excited-state potential energy surfaces for a model collinear ABC system with the masses of HHD. The ground-state surface has a minimum, corresponding to the stable ABC molecule. This minimum is separated by saddle points from two distmct exit chaimels, one leading to AB + C the other to A + BC. The object is to use optical excitation and stimulated emission between the two surfaces to steer the wavepacket selectively out of one of the exit chaimels (reprinted from [54]).

Pack R T 1976 Simple theory of diffuse vibrational structure in continuous UV spectra of polyatomic molecules. I. Collinear photodissociation of symmetric triatomics J. Chem. Phys. 65 4765... 

Figure A3.7.1. Two-dimensional contour plot for direct collinear reaction A + BC —> AB + C. Transition state is indicated by J.

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

The interplay between favourable reactivity at a collinear geometry and electrostatic forces favouring a T-shaped geometry leads to a bent geometry at the transition state. 

Let us continue with the atom-diatom collinear collision model, this time allowing for the possibility of the reaction A -r BC —> AB -i- C. We first introduce mass-scaled coordinates, as these are especially convenient to describe rearrangements, using... 

Kaye J A and Kuppermann A 1988 Mass effect in quantum-mechanical collision-induced dissociation in collinear reactive atom diatomic molecule collisions Chem. Phys. 125 279-91... 

These limitations have recently been eliminated using solid-state sources of femtosecond pulses. Most of the femtosecond dye laser teclmology that was in wide use in the late 1980s [11] has been rendered obsolete by tliree teclmical developments the self-mode-locked Ti-sapphire oscillator [23, 24, 25, 26 and 27], the chirped-pulse, solid-state amplifier (CPA) [28, 29, 30 and 31], and the non-collinearly pumped optical parametric amplifier (OPA) [32, 33 and 34]- Moreover, although a number of investigators still construct home-built systems with narrowly chosen capabilities, it is now possible to obtain versatile, nearly state-of-the-art apparatus of the type described below Ifom commercial sources. Just as home-built NMR spectrometers capable of multidimensional or solid-state spectroscopies were still being home built in the late 1970s and now are almost exclusively based on commercially prepared apparatus, it is reasonable to expect that ultrafast spectroscopy in the next decade will be conducted almost exclusively with apparatus ifom conmiercial sources based around entirely solid-state systems. 

Figure B2.1.2 Modified Michelson interferometer for non-collinear intensity autocorrelation. Symbols used rl, r2, retroreflecting mirror pair mounted on a translation stage bs, beamsplitter x, nonlinear crystal pint, photomultiplier Pibe.

Figure B2.1.3 Output of a self-mode-locked titanium-sapphire oscillator (a) non-collinear intensity autocorrelation signal, obtained with a 100 pm p-barium borate nonlinear crystal (b) intensity spectrum.

Eqnation (B3.4.1) is general and applies to both scattering and bonnd state spectroscopy. Scattering will be considered first. For shnplicity, the discnssion rises the collinear model for the A -l- BC AB -l- C reaction (i.e. assuming all particles lie on a line). This model is easy to visualize and embodies most elements of tlnee-dimensional (3D) scattering of larger molecules. 

After removal of centre-of-mass motion, there are two independent distances which need to be considered for a collinear problem, - r, where fg and denote the positions of A and B on the line) and... 

The close-coupling approach works readily and simply if the reaction is purely melastic . The method can also be made to work very simply for a single product arrangement (as in collinear reactions), by using a twisted coordinate system, most conveniently reaction path coordinates [37, 38 and 39] as shown in figure B3.4.3. 

Figure B3.4.3. A schematic figure showing, for the DH2 collinear system, a reaction-path coordmate Q coimecting continuously the reactants and the single products asymptote. Also shown are the cuts denoting the coordinate perpendicular to Q.

For example, for the collinear reaction A+BC this would be the probability that if initially the diatom BC is in a vibrational state then after the reaction a diatom AB is fonned (in any product vibrational state). In... 

Figure B3.4.8. The correlation fimction c(t) = (ii1q vii(0) as a function of time for photodissociation m a collinear (or tliree-dimensional) polyatomic case. There are tliree relevant time scales T, which measures how rapidly the initial wavefimction dephases T2, which measures how long it takes this mitial wavefimction to regroup and which measures how long the wavefimction takes to Teak to other degrees of freedom. In practice, photodissociation experiments may yield spectra which are more blurred, and/or are not...

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. 

Baer M 1975 Adiabatic and diabatic representations for atom-molecule collisions treatment of the collinear arrangement Chem. Rhys. Lett. 35 112... 

D Mello M, Duneczky C and Wyatt R E 1988 Recursive generation of individual S-matrix elements application to the collinear H + H2 reaction Chem. Phys. Lett. 148 169... 

Sadeghi R and Skodje R T 1995 Barriers, thresholds and resonances—spectral quantization of the transition state for the collinear D + H2 reaction J. Chem. Phys. 102 193... 

Figure 2. Wavepacket dynamics of the H + H H2 + H scattering reaction, shown as snapshots of the density (wave packet amplitude squard) at various times, The coordinates, in au, are described in Figure la, and the wavepacket is initially moving to describe the H atom approaching the H2 molecule. The density has been integrated over the angular coordinate, The PES is plotted for the collinear interaction geometry, 0 180, ...

To examine our assumption regarding the dependence of xe(9,0) on q, we consider the well-known (collinear) conical intersection of the C2H molecule formed by the two lowest states, namely, the A and the states... 

Finally, in Sec. 5, the theoretical results are illustrated by applying two adaptive schemes to the collinear photo dissociation of ArHCl. 

Fig. 1. Total energy (in kj/mol) versus time (in fs) for different integrators for a collinear collision of a classical particle with a harmonic quantum oscillator (for details see [2]). Dashed line Nonsymplectic scheme. Dotted Symplectic integrator of first order. Solid PICKABACK (symplectic, second order).

Fig. 2. Collinear ArHCl-systera with the Jacobi-coordinates used.

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