Symmetric stretching wavefunction

In Fig. 11, the system is unstable above E = 7.8. The corresponding symmetric stretch wavefunctions and trajectories are delocalized by resonance with the asymmetric stretch. 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross-sections. Using the method of spectral quantization (SQ), it is possible to extract a Seigert state wavefunction from time-dependent quantum wavepackets using the Fourier relation Eq. (21). The state obtained in this way for J = 0 is shown in Fig. 7 this state is localized in the collinear F — H — D arrangement with 3-quanta of excitation in the asymmetric stretch mode, and 0-quanta of excitation in the bend and symmetric stretch modes. If the state pictured in Fig. 7 is used as an initial (prepared) state in a wavepacket calculation, one observes pure... 

The SQ method extracts resonance states for the J = 25 dynamics by using the centrifugally-shifted Hamiltonian. In Fig. 20, the SQ wavefunc-tion for a trapped state at Ec = 1.2 eV is shown. The wavefunction has been sliced perpendicular to the minimum energy path and is plotted in the symmetric stretch and bend normal mode coordinates. As anticipated, the wavefunction shows a combination of one quanta of symmetric stretch excitation and two quanta of bend excitation. The extracted state is barrier state (or quantum bottleneck state) and not a Feshbach resonance. 

Fig. 23.8 Probability distribution Nn x,y z) 2 for the intrashell wavefunction N = n = 6 in the x = 0 plane corresponding to the collinear arrangement rn = rj +r2. The axes have a quadratic scale to account for the wave propagation in coulombic systems, where nodal distances increase quadratically. The fundamental orbit (-----------) (as) as well as the symmetric stretch motion (------) (ss) along the Wannier ridge are overlayed on the figure (from...

Before leaving the discussion of this area, let us consider a specific chemical example. The water molecule has C2V symmetry, hence its normal vibrational modes have A, Ai, B, or B2 symmetry. The three normal modes of H2O are pictorially depicted in Fig. 6.3.1. From these illustrations, it can be readily seen that the atomic motions of the symmetric stretching mode, iq, are symmetric with respect to C2, bending mode, i>2, also has A symmetry. Finally, the atomic motions of the asymmetric stretching mode, V3, is antisymmetric with respect to C2 and This example demonstrates all vibrational modes of a molecule must have the symmetry of one of the irreducible representations of the point group to which this molecule belongs. As will be shown later, molecular electronic wavefunctions may be also classified in this manner. 

A direct test of the vibrationally adiabatic approximation for H + H2 has also been made (Bowman et al.. 1973). This test was done by projecting accurate wavefunctions on the vibrationally adiabatic functions for zero curvature, and measuring deviations of the resulting probability weight from unity. The symmetric stretch motion was found to be adiabatic to within 10 % for total energies between 0-51 and 0-72 eV, but adiabaticity was lost at lower and higher energies. 

The pump pulse leaves about half the population still in the ground electronic state. This population mostly remains in the initial ground vibrational state, but some low vibrational states (up to 2 quanta for bending, 3 for symmetric stretch) are excited due to the Rabi oscillation. These low energy (< 0.7 eV) components were projected out of the wavefunction when plotting for clarity of presentation. 

It is, of course, possible to extend this calculation to obtain, in closed analytical form, the first excited polyad, Vj = 2. The result is shown schematically in Fig. 34. In particular, we notice the direct coupling between pairs of stretching modes in light of the selection rule (for Ub = const) A(u -I-Uj) = 0 and Au, Au = 0, 1. This means that the (initially degenerate) stretches 100), 010) now mix and split under the effect of Mj2. Due to the symmetry under bond exchange, we obtain either symmetric or antisymmetric wavefunctions, as discussed for the one-dimensional case. The difference here is the presence of the bending mode, which is also involved in the coupling scheme induced by the Majorana operator. We can see in both Fig. 34 and Eq. (4.45) that... 

In computing the bound-state wavefunction we continue to use the body-fixed coordinates defined in 3.3. We choose a fixed orientation angle 0, neglect the and terms of equation (1), and use a coupled basis set of harmonic oscillators to solve the resulting bound-state Schrodinger equation which depends on and r only. The energy levels obtained in this way are callede, (e) and correspond approximately to those for the symmetric and asymmetric stretch normal coordinates. The wavefunction obtained from this calculation is < ), (R,r 6). Once a... 

Fig. 3.1 shows that the H-I-H2 system is not strictly separable. The wave-functions depicted can not be described as a product of one-dimensional wavefunctions in the symmetric and antisymmetric stretch. This can be seen most clearly in the lowest panel. Thus, the assumption of separability is very helpful in the interpretation of flux eigenstates but not accurate enough to be not used in numerical calculations. The computation has to treat the full nonseparable Hamiltonian of the system. 

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