Franck-Condon factors wavefunctions

Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

The last factor, the square of the overlap integral between the initial and final vibrational wavefunctions, is called the Franck-Condon factor for this transition. 

The variations in efficiency (rate) of radiationless transitions result from differences in the Franck-Condon factor, visualised by superimposing the vibrational wavefunctions, / (or /2 - the probability distributions), of the initial and final states. We will consider three cases illustrated in Figure 5.2. 

In the quantum mechanical description (in continuation of Box 2.2), the wavefunction can be described by the product of an electronic wavefunction VP and a vibrational wavefunction / (the rotational contribution can be neglected), so that the probability of transition between an initial state defined by ViXa and a final state defined by TQ/b is proportional to electron coordinates, this expression can be rewritten as the product of two terms < f i M vP2> 2 Franck-Condon factor. Qualitatively, the transition occurs from the lowest vibrational state of the ground state to the vibrational state of the excited state that it most resembles in terms of vibrational wavefunction. 

The function G in eq 1 is the Franck-Condon factor which accounts for the contribution of nuclear degrees of freedom and represents the thermal average of the overlap integrals between nuclear wavefunctions with respect to conservation of energy, and is given by (2, 3, 8, 9)... 

Distance The affects of electron donor-acceptor distance on reaction rate arises because electron transfer, like any reaction, requires the wavefunctions of the reactants to mix (i.e. orbital overlap must occur). Unlike atom transfer, the relatively weak overlap which can occur at long distances (> 10 A) may still be sufficient to allow reaction at significant rates. On the basis of work with both proteins and models, it is now generally accepted that donor-acceptor electronic coupling, and thus electron transfer rates, decrease exponentially with distance kji Ve, exp . FCF where v i is the frequency of the mode which promotes reaction (previously estimated between 10 -10 s )FCF is a Franck Condon Factor explained below, and p is empirically estimated to range from 0.8-1.2 with a value of p 0.9 A most common for proteins. 

If the equilibrium position of the excited state C is located outside the configurational coordinate curve of the ground state, the excited state intersects the ground state in relaxing from B to C, leading to a nonradiative process. As described above, the shape of an optical absorption or emission spectrum is decided by the Franck-Condon factor and also by the electronic population in the vibrational levels at thermal equilibrium. For the special case where both ground and excited states have the same angular frequency, the absorption probability can by calculated with harmonic oscillator wavefunctions in a relatively simple form ... 

Notice that as one moves to higher vf values, the energy spacing between the states (Evf -Evf-i) decreases this, of course, reflects the anharmonicity in the excited state vibrational potential. For the above example, the transition to the vf = 2 state has the largest Franck-Condon factor. This means that the overlap of the initial state s vibrational wavefunction Xvi is largest for the final state s %vf function with vf = 2. 

Figure 2.1 Schematic representation of the ground and electronic excited potential energy surfaces (PESs) and the corresponding absorption spectra of the parent molecule, resulting from the reflection of different initial wavefunctions on a directly dissociative PES (a) absorption from a vibrationless ground state consists of a broad continuum and (b) absorption from a vibrationally excited state shows that extended regions are accessed, leading to a structured spectrum with intensities of the features being dependent on the Franck-Condon factors. Reproduced with permission from Ref. [34]. Reproduced by permission of lOP Publishing.

The Gj(t) functions of Eq. (15) have been calculated by Lin [60] when summing over Franck-Condon factors obtained from all possible (infinite) wavefunctions in the harmonic oscillator approximation. These Gj(t) are rather complicated functions of the frequencies arf, co and reduced masses M j, M which are attributed to the corresponding normal coordinates Qf and Q j. They are collected in parameters describing the frequency relation ft2 and the potential minimum shift Aj of the excited state with respect to the ground state... 

If the electronic transition is allowed, is nonzero and the first term dominates the expression. This term can be viewed as a product of the electronic transition moment and the vibrational overlap integral, (v /v, v /v ), connecting the two vibrational wavefunctions, /v, in electronic states e and e". The Franck-Condon factors, which are the square of the vibrational overlap integral, determine the intensity distribution among the vibrational bands. The relative intensities of the band members within a vibrational progression is, therefore, given by the ratio of the Franck-Condon factors. If, through a symmetry restriction, the transition moment M°e vanishes, as in the present case, the band activity in the spectrum comes from the second term. When Qk is a nontotally symmetric... 

If really good wavefunctions can be employed, then the results are convincing. Wolniewicz,175 with very accurate wavefunctions for H2, has calculated transition probabilities for the B-X,C-X and E,F-B systems. He has even considered individual vibrational and rotational lines and has shown that owing to significant variation of the electronic moments with intemuclear distance, the use of Franck-Condon factors is not permissible. 

As one other example illustrating the power of symbolic mathematics programs. Fig. 6 shows a Mathematica calculation of Franck-Condon factors for the I2 B<— X electronic absorption spectrum studied in Exp. 39. These factors are the squares of the overlap integrals of the vibrational wavefunctions for the lower (v") and upper (v ) vibrational levels involved in a transition ... 

It is found that the lifetimes change as v and J vary in the upper state. For example, Capelle and Broida found that the lifetime decreased from 1420 to 690 ns as v decreased from 40 to 21, a drop that can be accounted for by more favorable overlap of ground and excited state wavefunctions (Franck-Condon factors) for lower v levels. The effect of a change in rotational state is less Castano, Martinez, and Martinez found for the v = 25 level that the lifetime decreased from 745 ns to 625 ns as J increased from 0 to 106. Such a shortening of the lifetime is consistent with enhanced predissociation at higher rotational levels due to bond lengthening by the increased centrifugal force. 

The quantity < f m > e has been termed Raman overlap (Myers and Mathies, 1987). It is the product of the modulus of a time-dependent Franck-Condon factor between the final state and the initial wavefunction propagated on the electronic surface, i. e. I < f m > I and a damping function e which decreases exponentially with time. 

Fluorescence is defined simply as the electric dipole tranation from an excited electronic state to a lower state, usually the ground state, of the same multiplicity. Mathematically, the probability of an electric-dipole induced electronic transition between specific vibronic levels is proportional to R f where Rjf, the transition moment integral between initial state i and final state f is given by Eq. (1), where represents the electronic wavefunction, the vibrational wavefunctions, M is the electronic dipole moment operator, and where the Born-Oppenheimer principle of parability of electronic and vibrational wavefunctions has been invoked. The first integral involves only the electronic wavefunctions of the stem, and the second term, when squared, is the familiar Franck-Condon factor. 

In a vibrational state n = 0, the maximum of probability for the intemuclear distance R is near the center of the potential well. For all higher values vibrational states, maxima of probability occur near the two turning points of the potential— where the total energy equals the potential energy. These correspond on the diagram to the endpoints of the horizontal dashes inside the potential curves. Transitions can actually occur to several excited vibrational levels in the vicinity of u. The intensity of a transition between the levels v and v of the ground (gnd) and excited (ex) electronic states, respectivley, depends on the Franck-Condon factor, the overlap of the two vibrational wavefunctions ... 

The quantity (FC) is the Franck-Condon factor it is a sum of products of overlap integrals of the vibrational and solvation wavefunctions of the reactants with those of the products, suitably weighted by Boltzmann factors. The value of the Franck-Condon factor may be expressed analytically by considering the effective potential energy curves, of both the initial and the final states, as a function of their nuclear configurations. Relatively simple relationships can be derived if the appropriate curves are harmonic with identical force constants. Under these conditions ... 

Franck-Condon factors. As an application of the raising and lowering operator formalism we next calculate the Franck Condon factor in a model of shifted harmonic potential surfaces. Franck-Condon factors are absolute square overlap integrals between nuclear wavefunctions associated with different electronic potential... 

Let us consider the last point. The reader is already familiar with two important implications of the timescale separation between electronic and nuclear motions in molecular systems One is the Bom-Oppenheimer principle which provides the foundation for the concept of potential energy surfaces for the nuclear motion. The other is the prominent role played by the Franck-Condon principle and Franck-Condon factors (overlap of nuclear wavefunctions) in the vibrational structure of molecular electronic spectra. Indeed this principle, stating that electronic transitions occur at fixed nuclear positions, is a direct consequence of the observation that electronic motion takes place on a timescale short relative to that of the nuclei. 

The new ingredients are Franck-Condon factors of the initial (final) state proton vibronic wavefunctions(Zfn)- e initial states are summed over the... 

Approximate deperturbed curves can be derived from unperturbed vibrational levels far from the energy of the curve crossing region. The overlap factor between vibrational wavefunctions is calculable numerically. (Note that a Franck-Condon factor is the absolute magnitude squared of the overlap factor.) From Eq. (3.3.5) and the experimental value of an initial trial... 

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