Evaluation of the transition dipole moments

Typically, the life-time of the lowest excited state in polymers is 10 ° — 10 s. 

This behaviour after photoexcitation is encapsulated by the following empirical rules (Birks 1970). 

However, in some polymers the lowest excited singlet state is not dipole connected to the ground state, and in those cases there are only nonradiative transitions to the ground state. 

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The evaluation of the transition dipole moment between the triplet and singlet wave functions considered so far... 

The use of the dipole length formula is based on the evaluation of the transition dipole moment... 

Evidently, the evaluation of optically important parameters, such as nd Tj, depends on the evaluation of the transition dipole moments, (0 /( J). In calculating the dipole moments a considerable simplification arises if we adopt the Franck-Condon principle. This is discussed in the next section. 

In order to evaluate the matrix elements of the dipole moment operator in Eq. (24), it is convenient to separate out the geometrical aspects of the problem from the dynamical parameters. To that end, it is convenient to decompose the LF scalar product of the transition dipole moment d with the polarization vector of the probe laser field e in terms of the spherical tensor components as [40]... 

This effective interaction Hamiltonian operator is difficult to evaluate as it requires the knowledge of the transition dipole moments between the essential states and all the non-essential states. For a general molecular system, there is an infinity of such states, including bound states but also the various continua. In practice however, one can further approximate the second term of Eq. (6.51) by neglecting the (ro)vibrational structure of the non-essential electronic states. This term can then be expressed as a function of static electronic polarizabilities, that can be computed using standard electronic structure methods available in the major quantum chemistry program packages. 

One of the most familiar uses of dipole derivatives is the calculation of infrared intensities. To relate the intensity of a transition between states with vibrational wavefunctions i/r and jfyi it is necessary to evaluate the transition dipole moment... 

Remembering that each of these is further split by the hyperfine interaction, there are obviously several possible transitions among these four energy levels. To find out which are important, we must evaluate the transition dipole moment matrix elements, (i Sx j), since the absorption intensity is proportional to the square of these matrix elements. The operator Sx can be written ... 

It is the last term diat accounts for differences in absorption probabilities. This term is the expectation value of the dipole moment operator (see Section 9.1.1) evaluated over different determinants. Its expectation value is referred to as the transition dipole moment. 

The factor in (6.67) that multiplies the integral (6.73) contains the derivatives of the dipole-moment components with respect to the normal coordinate Qk, evaluated at the equilibrium configuration. We conclude that a radiative infrared transition in which the vibrational quantum number of the A th normal mode changes by one is forbidden unless the Acth mode has a change in dipole moment associated with it. The value of the equilibrium dipole moment de is irrelevant for infrared transitions of a polyatomic molecule. 

The intermolecular term has the same general form as the absorption cross section in the case of direct photodissociation, namely the overlap of a set of continuum wavefunctions with outgoing free waves in channel j, a bound-state wavefunction, and a coupling term. For absorption cross sections, the coupling between the two electronic states is given by the transition dipole moment function fi (R,r, 7) whereas in the present case the coupling between the different vibrational states n and n is provided by V (R, 7) = dVi(R, r, 7)/dr evaluated at the equilibrium separation r = re. 

In order to evaluate quantitatively the orientation of vibrational modes from the dichroic ratio in molecular films, we assume a uniaxial distribution of transition dipole moments in respect to the surface normal, (z-axis in Figure 1). This assumption is reasonable for a crystalline-like, regularly ordered monolayer assembly. An alternative, although more complex model is to assume uniaxial symmetry of transition dipole moments about the molecular axis, which itself is tilted (and uniaxially symmetric) with respect to the z-axis. As monolayers become more liquid-like, this may become a progressively more valid model (8,9). We define < > as the angle between the transition dipole moment M and the surface normal (note that 0° electric field of the evenescent wave (2,10), in the ATR experiment are given by equations 3-5 (8). 

The evaluation of the transition moment is straightforward now. Even though the energy difference between a5 B and ll A2 is small and the Tx level is strongly perturbed, the dipole transition to the ground state is forbidden as long as the molecule remains planar because of the dipole selection rules this transition would require an operator of A2 symmetry, but x, y, and z transform like B1, B2, and A, respectively. The transition may gain some intensity due to second-order spin-vibronic interactions, however. 

Symmetry Considerations Evaluation of transition densities gives a clearer pattern for the amplitudes of OP and TP optical transitions. The transition dipole moment Mnm between the ground state i//n(r) and the excited state i/t (r) is given by the matrix element of the dipole moment operator Jl (light wave), Eq. (22) ... 

An alternative means for the experimental evaluation of //da is by means of electroabsorption spectroscopy. " The electric field dependence of the absorption spectrum can be used to obtain the difference in the ground state and excited state dipole moments, A/UdaI- This may be combined with the transition dipole moment (in Debyes), l gel = 16[Cda/(108 x 10 da)]. to obtain (neglecting vibronic and nonresonance overlap contributions), ... 

Absolute intensities of 57 rovibrational lines in the V2 and V4 band were measured, and the transition dipole moments were evaluated [23]. 

The transition dipole moments may be evaluated using the explicit expressions for as described in Appendix F. Below we summarize the results of... 

Let the permanent dipole moment of the molecules in the solid be m. We are concerned with the evaluation of the transition moment (4.5) or we have to calculate the sum ... 

The approximation inherent in Equation [5b] means that the nuclear kinetic operator, -hyi) d ldQ ), has no influence on the electronic wavefunction, namely, the terms, -h d g/dQ)ei[S/dQi) and (-fi2/2)(a2 aQ2), which operate on the electronic part of the wavefunction, are omitted. This implies that infinitesimal changes in the nuclear configuration do not afiect the electronic wavefunction, g. This approximation is, however, not good enough in evaluations of the magnetic dipole transition moments. The disturbance to the electronic state caused by changes in the nuclear configuration has to be taken... 

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