The transition dipole moment

Whether or not an absorption band has a large integrated absorption coefficient (and, consequently, can be driven by the surrounding electromagnetic field) depends on a quantity called the transition dipole moment, The underlying classical idea is that, for the molecule to be able to interact with the electromagnetic field and absorb or create a photon of frequency v, it must possess, at least transiently, a dipole oscillating at that frequency. This transient dipole is expressed quantum mechanically as 

The name isosbestic comes from the Greek words for the same and extinguished . 

For a one-electron, one-dimensional system, like (to a good approximation) a carotene molecule, = -ex, so 

For a conjugated hydrocarbon of N carbon atoms and length L=(N— l)/cc. the first excitation energy is from the n orbital with n = to the one above, so 

For N an odd number, this expression evaluates (using mathematical software) to 2eL/n = 0.2eL virtually the same mmierical value is obtcuned when Nis ui even number. This value su ests that the electron migrates through a distcuice of about 20 per cent of the length of the molecule when the trcuisition tcJces place. 

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Before substiUitmg everything back into equation B1.2.6, we define the transition dipole moment between states 1 and 2 to be the integral... 

The molecular dipole moment (not the transition dipole moment) is given as a Taylor series expansion about the equilibrium position... 

C3.4.13)). The dimer has a common ground state and excitation may temrinate in eitlier tire or excited state (see tire solid arrows in figure C3.4.3). The transition dipole moments of tliese transitions are defined as ... 

Using the Condon approximation, the transition dipole moment is taken to be a constant with respect to the nuclear coordinates. Equation (26) then reduces to the familiar expression... 

Because of difficulties in calculating the non-adiabatic conpling terms, this method did not become very popular. Nevertheless, this approach, was employed extensively in particular to simulate spectroscopic measurements, with a modification introduced by Macias and Riera [47,48]. They suggested looking for a symmetric operator that behaves violently at the vicinity of the conical intersection and use it, instead of the non-adiabatic coupling term, as the integrand to calculate the adiabatic-to-diabatic transformation. Consequently, a series of operators such as the electronic dipole moment operator, the transition dipole moment operator, the quadrupole moment operator, and so on, were employed for this purpose [49,52,53,105]. However, it has to be emphasized that immaterial to the success of this approach, it is still an ad hoc procedure. 

Qualitatively, the selection rule for IR absorption for a given mode is that the symmetry of qT ) " must he the same as qT ). Qiianii-talivcly, the transition dipole moment is proportion al to tlie dipole derivative with respect to a given normal mode dp/di. ... 

The derivation of these seleetion rules proeeeds as before, with the following additional eonsiderations. The transition dipole moment s itrans eomponents along the lab-fixed axes must be related to its moleeule-fixed eoordinates (that are determined by the nature of the vibrational transition as diseussed above). This transformation, as given in Zare s text, reads as follows ... 

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ah initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems. 

State averaging gives a wave function that describes the first few electronic states equally well. This is done by computing several states at once with the same orbitals. It also keeps the wave functions strictly orthogonal. This is necessary to accurately compute the transition dipole moments. 

The energies, and Ep of the initial and final states of transitions in equations (178) and (179) are determined by the Cl eigenvalues and the transition dipole moment is obtained by using the Cl eigenvectors, that is. 

Keeping only the linear term, the transition dipole moment is given by... 

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... 

The above results indicate that the selcelion rules are relaxed when the geometry modifications taking place upon pholoexcitalion are considered. Although the transition dipole moment between the ground state and the lowest excited state remains small, the luminescence is no longer entirely quenched by the interchain in-... 

Consider an absorption band in the infra-red spectrum associated with a vibration where the transition dipole moment n for a typical unit of structure makes an angle p with the electric vector E. The absorption of infra-red radiation A is given by... 

The potential energy curves (Fig. 1), the non-adiabatic coupling, transition dipole moments and other system parameters are same as those used in our previous work (18,19,23,27). The excited states 1 B(0 ) and 2 B( rio) are non-adiabatically coupled and their potential energy curves cross at R = 6.08 a.u. The ground 0 X( Eo) state is optically coupled to both the and the 2 R( nJ) states with the transition dipole moment /ioi = 0.25/xo2-The results to be presented are for the cw field e(t) = A Yll=o cos (w - u pfi)t described earlier. However, for IBr, we have shown (18) that similar selectivity and yield may be obtained using Gaussian pulses too. 

The laser parameters should be chosen so that a and p can make the nonadiabatic transition probability V as close to unity as possible. Figure 34 depicts the probability P 2 as a function of a and p. There are some areas in which the probabilty is larger than 0.9, such as those around (ot= 1.20, p = 0.85), (ot = 0.53, p = 2.40), (a = 0.38, p = 3.31), and so on. Due to the coordinate dependence of the potential difference A(x) and the transition dipole moment p(x), it is generally impossible to achieve perfect excitation of the wave packet by a single quadratically chirped laser pulse. However, a very high efficiency of the population transfer is possible without significant deformation of the shape of the wave packet, if we locate the wave packet parameters inside one of these islands. The biggest, thus the most useful island, is around ot = 1.20, p = 0.85. The transition probability P 2 is > 0.9, if... 

Fig. 35). The potential energy curves and the transition dipole moment are taken from [117]. The time evolution of the populations on the ground and excited states is shown in Fig. 36 More than 86% of the initial state is excited to the B state within the period shorter than a few femtoseconds. The integrated total transition probability V given by Eq. (173) is P = 0.879, which is in good agreement with the value 0.864 obtained by numerical solution of the original coupled Schroedinger equations. This means that the population deviation from 100% is not due to the approximation, but comes from the intrinsic reason, that is, from the spread of the wavepacket. Note that the LiH molecule is one of the...

The laser intensities are taken to be the possible lowest. The intensity in case (b) is almost three times larger than the others. This is simply due to the fact that the transition dipole moment exponentially decays from the equilibrium position and also the potential energy difference increases. Note again that the coordinate-dependent level approximation works well. In order to demonstrate the selectivity the time evolution of the wave packets on the excited state are shown in Fig 41. As a measure of the selectivity, we have calculated the target yield by... 

Rh(CO)2 [16]. Such a dicarbonyl should possess two vibration modes. However, only the symmetric mode is observable in the IR spectrum. The asymmetric mode is inaccessible to an IR experiment on a metal surface due to the so-called metal surface selection rule, which prohibits the observation of dipole excitation if the transition dipole moment is oriented parallel to the surface. It should be noted that the observed frequencies fit well to values observed for Rh(CO)2 on technical Rh/Al203 catalysts [35-40] ( 2100 cm ) and Rh(CO)2 on planar TiO2(110) surfaces [41] (2112 cm ). 

The PPP-MO method is capable of calculating not only the magnitude of the dipole moment change on excitation, but it can also predict the direction of the electron transfer. The vector quantity that expresses the magnitude and direction of the electronic transition is referred to as the transition dipole moment. For example, the direction of the transition dipole moment of azo dye 15f as calculated by the PPP-MO method is illustrated in Figure 2.15. 

We have seen that the transition dipole moment occurring upon excitation of a molecule has a distinct orientation with regard to the molecular axis. This orientation can be determined by measuring the absorption of polarized light (oscillating in only one plane) by oriented single crystals,... 

Another important linear parameter is the excitation anisotropy function, which is used to determine the spectral positions of the optical transitions and the relative orientation of the transition dipole moments. These measurements can be provided in most commercially available spectrofluorometers and require the use of viscous solvents and low concentrations (cM 1 pM) to avoid depolarization of the fluorescence due to molecular reorientations and reabsorption. The anisotropy value for a given excitation wavelength 1 can be calculated as... 

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