1. Two reactant molecules A and BC, on collision course with initial velocities wA and wBC with respect to their
centre-of-mass, react to form AB and C. If the final velocities of the two products are w'AB and w'C, show that
their final translational energy is
Discuss the dynamical consequences of Eq. (7.20) if the atom C acts as a 'spectator', so that the relative
speed of C and BC are equal, w'C = wBC. Under what conditions would you expect 'spectator' dynamics to
convert the initial collision energy, Ec, into product translational energy, product internal energy, or to
promote collision induced dissociation?
[The reader might find it helpful to consult Study Box 7.1 before attempting this question.]
2. Let a bright state |s〉 be described as a linear superposition over zero-order states of a molecule, |n〉, (cf.
Eq. (7.12))
·
The states |n〉 can be thought of as the first tier of directly couple states shown in Figure 7.7(b). After
excitation of the bright state |s〉, the time-dependent wavefunction, Ψs(t), of the system (which includes
the possibility of coupling to a continuum, or quasi-continuum of state - see Figure 7.7(c))
where |l〉 satisfy the time-independent Schrödinger equation, Ĥ|l〉 = El|l〉. Show that the time-dependent
Schrödinger equation,
is satisfied if the coefficients, 
, are given by the expression
is the coefficient at t = 0 of the bright state in eigenstate |l〉, which has eigenvalue El. Hence, show that
the probability of finding the system in state |s〉 at time t is given by
where the phenomenological decay term exp(-γt) has been introduced to allow for the decay of the state to
the continuum, with rate constant γ.
3. The decays and quantum beats discussed in several cases are relatively easy to calculate from the expression
for Ps(t) derived in the previous question.
Consider three eigenstates of energy 3000.000 cm-1, 3000.040 cm-1, and 3000.071 cm-1, and a damping
lifetime of 2000 ps. These values mimic the results for an absorption experiment in butyne, and, of course,
only the differences in the energy matter.596
(a) Using the result from the previous question, show that if damping is ignored (γ = 0) the time-dependent
population in the case of three states can be written
(b) Plot the time evolution for this system if all three eigenstates contain the same amount of bright state
character. Plot the time evolution again but with the 3000 cm-1 eigenstate having 90% of the bright state
character and the other two equal amounts.
(c) Repeat the plots including the damping (γ-1 = 2000 ps). The first case should show a rapid decay almost to
zero but the second does not. How does the short time behaviour compare to that of the same system with
a damping lifetime of 200 ps? Would you be able to distinguish rapid initial decay (perhaps followed by
recurrences) from rapid damping if you observed only the first several hundred picoseconds?
(d) This machinery is useful for examining the effect of adding more states at slightly different energies without
having an explicit damping term. One can test how many levels need to be participating for the time
evolution to look roughly like a single decay with minor recurrences. Generate a random set of energy
levels Em spanning a range of 0.1 cm-1 and a random set of properly normalized coefficients for the levels.
Calculate the time evolution over a period of 50 ns for three levels to confirm that your scheme is producing
a relatively simple beat pattern such as you obtained above. Obtain the evolution for 5, 15, 25, and 50
levels. (Try it a few times to see how the pattern changes. Note that over some long time this finite system
must have recurrences.)
How many levels make the time evolution look like a simple decay? With a maximum energy spread of
0.1 cm-1, what is the fastest decay you can expect? (Consider a two level system with that energy
separation.) Are your calculations consistent with this result?
4. One can obtain the classical RRK expression for the rate constant k(ε) starting with microcanonical transition
state theory and using the classical expression for the number of states of a set of s classical oscillators
having frequencies υi (see, for example, ref. [10]),
where the product runs over all s oscillators. (The expression ε/hυi) is the maximum number of quanta that
can be localized in the i-th oscillator.) The density of states is just ρ(ε) = dN(ε)/dε. Assuming that only
the frequency of the vibration that becomes the reaction coordinate changes, derive the classical RRK
expression
where υ‡ is the frequency along this spatial coordinate. Sketch the dependence of k(ε) on ε and s.
What do you think of the assumption that only the one vibration changes frequency?
5. Trans-diphenylbutadiene undergoes unimolecular photoisomerization in one of its low lying electronically
excited states. The rate constants for isomerization (determine from 'real-time' fluorescence lifetime
measurements) vary with energy in the excited butadiene in the following way:
|
ε/cm-1 |
2000 |
3000 |
4000 |
5000 |
6000 |
7000 |
8000 |
|
k2(ε)/1010s-1 |
0.36 |
1.1 |
1.9 |
2.7 |
3.5 |
4.1 |
4.8 |
(a) Use the data and the expression in question (3) to make a classical RRK estimate of the number of
oscillators, s, and the frequency υ‡, given that the critical energy ε0 = 1100 cm-1.
(b) Implicit in the statistical (free energy flow) assumption, upon which RRK theory is based, is that all
vibrational modes should be active (i.e. vibrational energy is assumed to be distributed randomly among
all the 3N-6 vibrational modes of the molecule). Using your estimate of υ‡ in (a), calculate k2(ε) at a
selection of energies given in the table above, assuming all the modes in diphenylbutadiene are active.
Comment on the answers you obtain.
Above Problems are available as a PDF to print
Solutions to Chapter 7 Problems