Product arrangement channels

D. J. Tannor Some years ago we looked at the photodissociation of the ozone molecule [/ Am. Chem. Soc. Ill, 2772 (1989)]. Ozone (O3) can be viewed in much the same way as Naa, in that it has an initial wavepacket with three lobes, one located in each of three symmetrically equivalent wells (see Fig. 1). In the excited state, at the same locations, the potential has saddle points. As a result, each of the component wavepackets bifurcates, and there is a change of alliances The left portion of one wavepacket joins up with the right portion of another wavepacket, and they tunnel into the same product arrangement channel. 

Here n = m, q where q = 1,2... denotes the product arrangement channel a denotes the remaining quantum numbers other than the energy. 

Further simplifications upon CS involve approximations to the rotational motions. Here there are two schools of thought as to how best to do this. One school argues that since the rotational periods are usually slow compared to vibrational periods, it makes sense to use the rotational sudden approximation wherein the atom-diatom orientation angle is fixed for motion in the reagent and product arrangement channels >The other school argues that since rotational motion correlates into bend motion along the reaction path, and the bend is only weakly coupled by curvature to reaction path motions, while at the same time the bend frequency is comparable to the other perpendicular modes near the reaction bottleneck, it is more... 

Here contains only molecular attributes whereas Fj contains all aspects of the preparation including the magnitudes and phases or the electric field and initially prepared coherent state x (0)>. Experimental control over these parameters allows manipulation of the magnitude of P(q E). In particular, variation of the magnitudes and phases of Fjj, through the cj and the fields, allows one to maximize or minimize a selected final product channel. The case of two product arrangement channels (q=l,2 ) is discussed below. 

Unfortunately, however, the surface eigenfunctions and eigenvalues are themselves often quite difficult to obtain. The fundamental reason for this, which should be clear from Fig. 1.2c, is that the reactant and product arrangement channels are confined to smaller and smaller regions of the available... 

Now, we are in a position to present the relevant extended approximate BO equation. For this purpose, we consider the set of uncoupled equations as presented in Eq. (53) for the = 3 case. The function icq, that appears in these equations are the eigenvalues of the g matrix and these are coi = 2 (02 = —2, and CO3 = 0. In this three-state problem, the first two PESs are u and 2 as given in Eq. (6) and the third surface M3 is chosen to be similar to M2 but with D3 = 10 eV. These PESs describe a two arrangement channel system, the reagent-arrangement defined for R 00 and a product—anangement defined for R —00. 

The different groupings of the atoms are referred to as arrangement channels (or simply channels) we label the reactant channel by a, whereas the different product... 

Of particular interest is the probability of being in a complete subspace of state denoted by the label q that is, all m associated with a fixed q, where n = (m, q). greatest concern in chemistry is the case where q labels the chemical identity (i the arrangement channel) of the product of a chemical reaction hence below often explicitly refer to q in this manner. However, it should be clear that the th applies to any other q chosen from the set of n quantum numbers. 

Thus, in this case, the ratio of product into various arrangement channels controlled entirely by the time delay (td — tx) between pulses. 

Since the Hamiltonian is symmetric with respect to exchange of the two hydrogen nuclei, for reactions involving identical nuclei (X-I-H2/D2) the wavefimctions must be either symmetric or antisymmetric with respect to this operation. For the basis functions in the reactant channel, the symmetry affects only the rotational wavefimction of the H2 moiety, which has the symmetry (-ly. Thus, only even (para-Rj) or odd prtho-RT) values of j need be included in the expansion in eq. 2. For the product arrangement, we use the exchange-symmetrized basis functions [49]... 

Figure 4 shows the dependence on final rotational state of the C1-hH2(v=0, j =0)—> HCl(v=0jO DCSs at two different coUision energies. We observe that the DCSs all have a very similar shape. Likely, this implies that the overaU shape of the DCS is determined by the reactive probabdity (opacity) as a function of impact parameter and by an overall deflection function (h), while the product final state distribution is governed by the shape of the PES in the exit region of the arrangement channels. We remark also that the determination of fully final-state resolved DCS does not present any particular difficulty for our time-independent method, but would not be feasible with some of the time-dependent methods which have been applied recently to the X-hH2 reactions. [7, 16]... 

Here E,n,q > is the continuum state which correlates with, the asymptotic product state E,n,q° > consisting of products in arrangement channel q ( q=l, 2,. ..) and internal quantum states labeled by n. The principal quantity of interest is usually the probability of decay into final arrangement channel q, given, to within an overall normalization factor, by ... 

In the first section, the potential energy channel connecting reactant and product arrangements is cut into several sectors. On the one-dimensional cuts of the potential energy channels taken at a fixed value (the sector midpoint) of the propagation coordinate vibrational eigenfunctions and eigenvalues as well as other quantities independent from both the... 

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