Chapter 7: Quasidissipative Behavior in Isolated Molecular Systems

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I. INTRODUCTION

The flow of probability from a metastable quantum state into a dense set of

"receptor" states has been discussed by many authors. In fact, the basic theory of metastable state decay may be found in most textbooks on Quantum Mechanics (see, e.g., Refs. 1-3) and dates back to the seminal work of Wigner and Weisskopf.

⁴

The standard treatments of a decaying metastable state are based on certain phys- ically motivated pole approximations which simplify the energy dependence of the resolvent operator (see, e.g., Ref. 5). The time-dependent behavior of the decaying state is then obtained via inverse Laplace transformation of the simplified integral equation. In this way, the approximate time-dependent probability of the initial state is found to obey the exponential decay law P(t) ... exp(-rt), having a decay rate r which is given by the "Golden Rule" expression

¹,2

(1.1)

Here, (JVF

²⁾

is the average of the absolute square of the matrix elements coupling the initial state to the receptor states,

⁶

p(Ei) is the density of receptor states at the energy Ei of the initially prepared state, and n equals unity. For short times, this result agrees, of course, with that obtained from time-dependent perturbation theory (see, e.g., Refs. 1-3).

In the field of intramolecular dynamics, the decay of probability of initally

prepared quantum states in "large" molecules is a subject of great interest. For

example, there is an extensive literature on the dynamics of electronic radiationless

transitions

⁷

in molecular systems (i.e., redistribution of electronic energy). The

theoretical approach to these problems is also based largely on approximate resol-

vent operator approaches (see, e.g., Ref. 7), and results similar to those mentioned

above have been derived. More specifically, Golden Rule exponential decay rates

are predicted for initially prepared states in large molecules which undergo elec-

tronic radiationless transitions and have large densities of states.

⁷•8

The theory of

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electronic radiationless transitions has been rather successful in explaining experi- mental trends and results (see, e;g., Ref. 7).

The phenomenon of intramolecular vibrational energy redistribution (IVR) is perhaps more subtle than that for electronic energy redistribution because, in the former case, the coupling mechanisms are more diverse and not as well character- ized as in the latter case. Nevertheless, "real time" experiments which probe the process of IVR in large molecules have suggested that exponential decay of initially prepared states is the norm at moderate to high energies (see, e.g., Refs. 9 and 10).

Theoretical studies,

¹¹- 14

many of which are based on the same formalism devel- oped to treat electronic radiationless transitions,

¹¹•12

also predict that exponential decay of the initial state is the most probable behavior in large molecules. It is perhaps useful then to ask the following question: Given that the probability of an initially prepared state in a large molecule is expected to decay exponentially, can a quantum dynamical theory be formulated which not only predicts this behavior, but also allows one to calculate the IVR decay rate more accurately than is given by the Golden Rule? It is the purpose of this chapter to demonstrate that, upon consideration of the basic phenomenology of IVR processes, it is indeed possible to derive a somewhat more flexible and general formula for the decay rate.

The theoretical approach used in this chapter represents an extension of the

theory developed in Chapter 5 and is intended to be complementary to those ap-

proaches based on resolvent operator equations.

^{1 - 5 ,7,i2}

The present approach is

based on the adiabatic approximation, which has been related by previous authors

to certain pole approximations for the resolvent (see, e.g., Refs. 15 and 16). This

chapter is organized as follows: In Sec. II, the basic phenomenology of IVR in large

molecules is discussed and the present theory is derived. In Sec. III, the relationship

between average coupling matrix elements for different basis sets is examined, and

the relevance of those results to the theory of Sec. II is indicated. A discussion of

the results is given in Sec. IV, and

condudi~g

remarks appear in Sec. V.

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II.THEORY

It is first assumed that there exists a zeroth-order orthonormal basis set which captures the essential features of the molecular (i.e., nuclear) motions of a large molecule in a given Born-Oppenheimer electronic state. An example of such a basis set would be the harmonic normal mod.es of the molecule, and the present approach does not preclude a treatment of molecular rotations. As usual, the zeroth-order basis {j<pi)} satisfies the time-independent Schrodinger equation for the zeroth-order Hamiltonian Ho:

Ho

^{l1n·) -}

E·

I,,....\

l r i - i IY*f '

r

_\

₂ ....

^{1 \ }}

where E1 is the zeroth-order energy of the state l<pi)· The coupling term V in the total Hamiltonian H (= Ho + V) introduces couplings between the zeroth-order states.

It is also assumed that one of the zeroth-order states carries all of the oscillator strength for a radiative transition from the ground state. Therefore, this is the state that is prepared by an appropriate pulsed laser excitation and is the state that subsequently undergoes IVR (cf. discussion in Ref. 9 by Bloembergen and Zewail). The coupling between the zeroth-order states is responsible for the flow of probability from the initially prepared state. By virtue of that coupling, the initial state interacts with a set of other states, which interact with even more states, and so on. This "tier" structure of the basis states is suggested to give rise to the basic phenomenology of intramolecular vibrational (and rotational} energy flow .

¹⁷

A tier structure has been invoked, for example, in the numerical analyses of IVR in benzene

¹³

and hydrocarbon chains,

¹⁴

and is not inconsistent with experimental results on anthracene and t-stilbene at higher energies,

⁹

p-difiuorobenzene,

IO,ll

and tetramethyldioxetane.

¹⁸

Most of the earlier approximate resolvent operator treatments

¹²

of IVR are also based on vario~s simple tier models.

¹⁹

In order to exploit the basic phenomenology of IVR, the basis set is partitioned

into three unique sets: The inital state (denoted by "P"), those states which are

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coupled to the initial state (denoted by "Q"), and the remaining states (denoted by "Q' "). It is assumed that the Q' -space contains many states since the present analysis is concerned with the large molecule limit. If the general time-dependent wavefunction J'l!(t)) is expanded as

¹⁵

N

Jw(t)) = exp(-i (H) t) L bi(t) l'Pi) , (2.2)

i=l

where (H) equals the expectation value of the energy for the initial state, then the coupled differential equations for the amplitiudes bi(t) are given by

i :t ^{bp(t) - yPQbQ(t) (2.3)}

i ~bQ(t) ^{- yQPbP(t) +} (HQ - (H) lQ)bQ(t) + yQQ'bQ' (t) (2.4) dt

i ~bQ' (t) yQ'QbQ (t) + (HQ' - (H) 1 Q')bQ' (t) . (2.5) dt

In these equations, the previously mentioned partitioning scheme has been imple- mented, and vector-matrix notation has been used. (The superscripts identify the different subspaces.) The l's are identity matrices, the h's contain the state am- plitudes, the H's contain the different Hamiltonian matrix sub-blocks, and the V's contain the coupling matrices between the different subspaces. The dimensions of the P, Q, and Q' subspaces are given by the numbers Np, N

^Q,

and N

^Q',

respec- tively. For simplicity, Np equals unity in the present analysis, but this is not a crucial restriction.

It is now assumed that there are so many states in the Q' -manifold that the derivatives dbQ' / dt satisfy the approximate relation

(2.6)

This approximation is based on the assumption that the probability will flow slowly

and uniformly into the Q' states, and hence the individual Q' state probabilities

will have magnitudes of the order l/Nq1. Similar approximations have been used

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to eliminate the continuum state's contribution to the density matrix equations describing the multiphoton ionization of atoms (see, e.g., Ref. 20). It is also noted here that the physical basis for this approximation differs somewhat from that used in Chapter 5 and Ref. 15.

By virtue of the approximation given in Eq. (2.6), the Q' amplitudes are found from Eq. (2.5} to be

(2.7) If the states in the Q' subspace are thought of as being "prediagonalized" (i.e., the matrix HQ' is prediagonalized), then Eq. (2.7) simplifies to

(2.8)