Overlap integrals transitions

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 first term in this expansion, when substituted into the integral over the vibrational eoordinates, gives ifj(Re) , whieh has the form of the eleetronie transition dipole multiplied by the "overlap integral" between the initial and final vibrational wavefunetions. The if i(Rg) faetor was diseussed above it is the eleetronie El transition integral evaluated at the equilibrium geometry of the absorbing state. Symmetry ean often be used to determine whether this integral vanishes, as a result of whieh the El transition will be "forbidden". 

The quantity J dr is called the vibrational overlap integral, as it is a measure of the degree to which the two vibrational wave functions overlap. Its square is known as the Franck-Condon factor to which the intensity of the vibronic transition is proportional. In carrying out the integration the requirement that r remain constant during the transition is necessarily taken into account. 

The solution of the spin-boson problem with arbitrary coupling has been discussed in detail by Leggett et al. [1987]. The displacement of the equilibrium positions of the bath oscillators in the transition results in the effective renormalization of the tunneling matrix element by the bath overlap integral... 

The chemical reactions through cyclic transition states are controlled by the symmetry of the frontier orbitals [11]. At the symmetrical (Cs) six-membered ring transition state of Diels-Alder reaction between butadiene and ethylene, the HOMO of butadiene and the LUMO of ethylene (Scheme 18) are antisymmetric with respect to the reflection in the mirror plane (Scheme 24). The symmetry allows the frontier orbitals to have the same signs of the overlap integrals between the p-or-bital components at both reaction sites. The simultaneous interactions at the both sites promotes the frontier orbital interaction more than the interaction at one site of an acyclic transition state. This is also the case with interaction between the HOMO of ethylene and the LUMO of butadiene. The Diels-Alder reactions occur through the cyclic transition states in a concerted and stereospecific manner with retention of configuration of the reactants. 

The frontier orbital interaction is forbidden by the symmetry for the dimerization of ethylenes throngh the rectangular transition state. The HOMO is symmetric and the LUMO is antisymmetric (Scheme 25a). The overlap integrals have the opposite signs at the reaction sites. The overlap between the frontier orbitals is zero even if each overlap between the atomic p-orbitals increases. It follows that the dimerization cannot occur throngh the fonr-membered ring transition states in a concerted and stereospecfic manner. 

The intensity of a vibronic transition depends upon the square of the overlap integral of the vibrational wave functions,... 

As discussed in Chapter 1, the probability of a nonradiative transition is proportional to the square of the vibrational overlap integral J xiXa drv ... 

In carbonyls and heterocycles possessing both n-> tt and 77 - 77 excited states one would expect S - T n. transitions to be 102-103 times more efficient than S -> T . transitions/1425 We saw in Section 5.6a that the contribution of the overlap integral (J X1X2 dTo) in the transition probability is greater for transitions occurring between states of different type (n, 77 77,77 ) than for transitions between states of the same type (ir, tt -n,... 

When the system approaches the transitional configuration two effects take place (1) the coefficient CL increases and at Q = Q it may be of the order of, or even greater than, the coefficient CA (CL CA), and (2) the distance between the ligand L and atom B decreases with increasing Q at a fixed distance R between atoms A and B, leading to an exponential increase of the overlap integral (0l 0b)-... 

FRET is a nonradiative process that is, the transfer takes place without the emission or absorption of a photon. And yet, the transition dipoles, which are central to the mechanism by which the ground and excited states are coupled, are conspicuously present in the expression for the rate of transfer. For instance, the fluorescence quantum yield and fluorescence spectrum of the donor and the absorption spectrum of the acceptor are part of the overlap integral in the Forster rate expression, Eq. (1.2). These spectroscopic transitions are usually associated with the emission and absorption of a photon. These dipole matrix elements in the quantum mechanical expression for the rate of FRET are the same matrix elements as found for the interaction of a propagating EM field with the chromophores. However, the origin of the EM perturbation driving the energy transfer and the spectroscopic transitions are quite different. The source of this interaction term... 

In these terms, the electronic integrals such as (Mge)° and (Mge) a are constrained by the symmetry of the electronic states. While term I involves Frank-Condon overlap integrals, terms II and HI involve integrals of the form i Qa v) in the harmonic approximation, the integrals of this type obey the selection rule v = i + 1. Keeping these considerations in mind, we will next discuss how terms I, II and in contribute to distinct vibrational transitions. 

The selection rules appropriate for a shake-up transition are of the monopole type2, 76. The intensity of a shake-up peak depends on the overlap integral between the lower state molecular orbital from which the electron is excited (in the neutral molecule) and the upper state molecular orbital to which the electron is excited (in the core-ionized molecule). Consequently one expects transitions of the type au au, ag " ag> 7T nu, and irg - ng with g u and u - g transitions forbidden. 

The probability of a transition v" v is determined by the Franck-Condon factor, which is proportional to the squared overlap integral of both vibrational eigenfunctions in the upper and lower state. 

If the transition probability, P-, is taken to be proportional to i- where p,yis the orbital overlap integral for the dipole transition from state i (neutral molecule) to state/(ion), then. 

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