Parallel absorption and emission transition moments

Taking into account the excitation probability, i.e. cos 0, the number of excited molecules whose transition moment is oriented within angles 0 and 0 + AO, and tj) and (j + d, is proportional to cos 9 sin OAOAtj). The fraction of molecules oriented in this direction is 

The denominator, which is proportional to the total number of excited molecules, can be calculated by setting x = cos 0, hence dx = —sin 9A9, and its value is 4re/3. Equation (5.21) then becomes 

It is then possible to calculate the average of cos 6 over all excited molecules 

The difference between the theoretical value of the emission anisotropy in the absence of motions fundamental anisotropy) and the experimental value limiting anisotropy) deserves particular attention. The limiting anisotropy can be determined either by steady-state measurements in a rigid medium (in order to avoid the effects of Brownian motion), or time-resolved measurements by taking the value of the emission anisotropy at time zero, because the instantaneous anisotropy can be written in the following form  

It should first be noted that the measurement of emission anisotropy is difficult, and instrumental artefacts such as large cone angles of the incident and/or observation beams, imperfect or misaligned polarizers, re-absorption of fluorescence, optical rotation, birefringence, etc., might be partly responsible for the difference between the fundamental and limiting anisotropies. 

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When the absorption and emission transition moments are parallel, 6a = 6e, the common value being denoted 6 hence cos2 6a = cos2 0E = cos2 6. Before excitation, the number of molecules whose transition moment is oriented within angles 6 and 8 + dd, and and 0 I delementary surface on a sphere whose radius is unity, i.e. 2n sin dddd< > (Figure 5.6). 

Fig. 5.6. The fraction of molecules whose absorption and emission transition moments are parallel and oriented in a direction within the elementary solid angle. This direction is defined by angles 8 and .

Fig. 5.7. Definition of angles and yr when the absorption and emission transition moments are not parallel.

Fig. 5.9. Rotational motions inducing depolarization of fluorescence. The absorption and emission transition moments are assumed to be parallel.

We have considered spherical molecules so far, but it should be noted that isotropic rotations can also be observed in the case of molecules with cylindrical symmetry and whose absorption and emission transition moments are parallel and oriented along the symmetry axis. In fact, any rotation around this axis has no effect on the fluorescence polarization. Only rotations perpendicular to this axis have an effect. A typical example is diphenylhexatriene whose transition moment is very close to the molecular axis (see Chapter 8). 

The possible fluorescence anisotropies r thus range from - 0.2 (for perpendicular absorption and emission transition moments, c = 0) to a maximum of +0.4 (for parallel transition moments, c = 1). Complete depolarization (r = 0) occurs only when the absorption and emission moments are separated by the magic angle 54.7°, for which (3c - 1) = 0. This example illustrates the sensitivity of the anisotropy to chromophore architecture in the protein this emerges as a consequence of the well-defined chromoprotein structure. 

Assume that absorption and emission transition moments are parallel (the effect of electronic delocalization will be treate xlateg). In the afeye expressions 0= 0, 3=3 and the quantities 20 02 2 ... 

Tables 1.1, 1.2, and 1.4, respectively) confirms the parallel orientation of the absorption and emission transition dipole moment for a single chromophore. 

The fluorescence depolarization technique for mobility and ordering is based on the fact that the probability of absorption and emission is directional. Light polarized along a certain axis will preferably excite molecules oriented with their transition dipole moment in the same direction. The probability varies with cos 0, where 0 is the angle between the transition dipole moment and the electric field vector of the light. Emission of a photon obeys the same cos 0 (28) rule. That means that a molecule oriented with its transition dipole moment along the Z-axis will be likely to emit a photon with the same polarization. In the depolarization technique, polarizers are used to quantify the intensity of the parallel (ly) and perpendicular (Ij.) components to the original direction of polarization. 

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