The Molecular Eigenstates

As mentioned previously for the case in which the set of coupled states is a finite small number, the projection P takes the form (in terms of diagonalized projections) 

The variable n as well as its range depends on the type of problem to be considered. The projection operator on the other states is 

For the case in which the set of closely coupled states is a finite number, the 

If the number of closely coupled states is two, then the above matrix element reduces to 

It often happens that in studying a given transition, we find a single predominant contribution. This takes place if we stipulate that the molecular eigenstates are very sparse in the sense that 

---

The second term is the covalent bond energy that arises from occupying the molecular eigenstates, n, with electrons, namely... 

A correct representation of the molecular eigenstates, zero-order, Born-Oppenheimer states,6... 

The energy levels En, corresponding to the molecular eigenstates, are now obtained from the solution of the eigenvalue problem... 

The expansion coefficients representing the weights of the zero-order state molecular eigenstate 0n can be displayed in the form6... 

The initial state of the excited system has been represented as a superposition of the (time-independent) molecular eigenstates, each of which is a superposition of BO basis functions. The decay process is then described in terms of the time evolution of the amplitudes of the molecular eigenstates. The general theory of quantum mechanics implies that the decay of the state (10-4) will exhibit interference effects. 

The properties of the metastable state resulting from the simple excitation process considered above can be further elucidated by considering the nature of the expansion of the molecular eigenstates in terms of the zero-order states (eq. (2-18)). Equation (10-4) can be rewritten in the form... 

We now consider a pump-probe experiment from a time-domain perspective. The coherent superposition of exact molecular eigenstates constructs, for a short time, a zeroth-order state. Zeroth-order states are often physically intuitive solutions to a simpler Hamiltonian Ho, and can give a picture of the basic dynamics of the problem. The full Hamiltonian is then given by H = Ho + V. Suppose we choose to expand the molecular eigenstates in a complete zeroth-order basis of Ho that we denote by n)... 

Thus the hamiltonian (2.15) couples the electronic excitations to the vibrations by linear terms in and by quadratic terms in A. The molecular eigenstates of (2.15) are the vibronic states they are different from tensorial products of electronic excitations and undressed vibrations. Even for this simple intramolecular effect, we cannot, when moving to the crystal, consider excitonic and vibrational motions as independent. 

Fig. 1. The molecular energy level model used to discuss radiationless transitions in polyatomic molecules. 0O, s, and S0,S are vibronic components of the ground, an excited, and a third electronic state, respectively, in the Born-Oppenheimer approximation. 0S and 0 and 0j are assumed to be allowed, while transitions between j0,j and the thermally accessible 00 are assumed to be forbidden. The f 0n are the molecular eigenstates...

From theoretical discussions involving the molecular eigenstates picture questions have arisen as to whether particular quantum mechanical interference effects can be observed by the use of suitably monochromatic radiation for excitation of the molecules 13>. (See Sect. 7.) Of course, it is also necessary to settle the controversies as to whether the BO or molecular eigenstates are correct, and if the former is indeed correct, which particular version of the BO approximation is to be employed for the calculation of nonradiative decay rates. 

The state s is not really a physically accessible state, since in order to prepare the molecule initially in radiation with particular coherence between different frequencies which correspond to all the observed spectral lines from the molecular states which contain this zeroth order state initial state cj>0. The molecular eigenstates should then have unit quantum yields at very low pressure, a result which is not inconsistent with experimental extrapolations to zero pressure. This small molecule limit is depicted schematically in Fig. 4a. 

Fig. 4. (a) A schematic representation of the small molecule limit. The states are the same as those represented in Fig. 1. The molecular eigenstates approximately diagonalize the effective molecular Hamiltonian (20), and each carries only a portion of the original oscillator strength to s can therefore decay radiatively to

In the small molecule limit, the physical situation corresponds more closely to that described by the molecular eigenstates 10 I. Each molecular eigenstate contains some admixture of the zeroth order states, the vibrationally cold ... 

Under these conditions the singlet amplitude is distributed according to a Lorentzian distribution over the molecular eigenstates. Exciting with a broad (white) laser (or at least with a laser that completely spans the interaction width), one then sees in the fluorescence first the Fourier transform of the Lorentzian distribution, that is, an exponential decay. The density of /c> was, however, not taken to be so high as to dilute the singlet amplitude effectively to zero. It was taken to be intermediate, which meant that each ME still had enough radiative probability so as to radiate independently,... 

The molecular eigenstates will be denoted in this section by n>, m>,..., where... 

The most commonly used experimental techniques probe molecules in the frequency domain rather than in the time domain. As emphasized recently (1) the increased level of detail provided by frequency-domain methods produces a more complete picture of the vibrational energy redistribution process, invalidating frequently made claims that time domain techniques, being more direct, are somehow superior. The molecular eigenstate spectra provided by high-resolution experiments currently provide the most complete picture of molecular dynamics. Of course, the frequency-domain and time-domain viewpoints are complementary and we frequently obtain enhanced understanding by considering both viewpoints. 

---