Electronic Transition Moments

There is no strict selection rule for vibration in the case of electronic transitions (i.e. between two different electronic states). The Condon principle states that the f .-variation of the (< / fL matrix elements can in general be neglected... 

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

A related measure of the intensity often used for electronic spectroscopy is the oscillator strengdi,/ This is a dimensionless ratio of the transition intensity to tliat expected for an electron bound by Hooke s law forces so as to be an isotropic hanuonic oscillator. It can be related either to the experimental integrated intensity or to the theoretical transition moment integral ... 

Equation (Bl.1,1) for the transition moment integral is rather simply interpreted in the case of an atom. The wavefiinctions are simply fiinctions of the electron positions relative to the nucleus, and the integration is over the electronic coordinates. The situation for molecules is more complicated and deserves discussion in some detail. 

Here each < ) (0 is a vibrational wavefiinction, a fiinction of the nuclear coordinates Q, in first approximation usually a product of hamionic oscillator wavefimctions for the various nomial coordinates. Each j (x,Q) is the electronic wavefimctioii describing how the electrons are distributed in the molecule. However, it has the nuclear coordinates within it as parameters because the electrons are always distributed around the nuclei and follow those nuclei whatever their position during a vibration. The integration of equation (Bl.1.1) can be carried out in two steps—first an integration over the electronic coordinates v, and then integration over the nuclear coordinates 0. We define an electronic transition moment integral which is a fimctioii of nuclear position ... 

This last transition moment integral, if plugged into equation (B 1.1.2). will give the integrated intensity of a vibronic band, i.e. of a transition starting from vibrational state a of electronic state 1 and ending on vibrational level b of electronic state u. 

The electronic transition moment of equation (B1.1.5) is related to the intensity that the transition would have if the nuclei were fixed in configuration Q, but its value may vary with that configuration. It is often usefiil to expand Pi CQ) as a power series in the nonnal coordinates, Q. ... 

Equation (B1.1.10) and equation (B1.1.11) are the critical ones for comparing observed intensities of electronic transitions with theoretical calculations using the electronic wavefiinctions. The transition moment integral... 

If one of the components of this electronic transition moment is non-zero, the electronic transition is said to be allowed if all components are zero it is said to be forbidden. In the case of diatomic molecules, if the transition is forbidden it is usually not observed unless as a very weak band occurring by magnetic dipole or electric quadnipole interactions. In polyatomic molecules forbidden electronic transitions are still often observed, but they are usually weak in comparison with allowed transitions. 

Here, (ATp) is the pth component of the electronic transition moment from state a to b,. and E j-are the energy... 

Since the vibrational eigenstates of the ground electronic state constitute an orthonomial basis set, tire off-diagonal matrix elements in equation (B 1.3.14) will vanish unless the ground state electronic polarizability depends on nuclear coordinates. (This is the Raman analogue of the requirement in infrared spectroscopy that, to observe a transition, the electronic dipole moment in the ground electronic state must properly vary with nuclear displacements from... 

The fitting parameters in the transfomi method are properties related to the two potential energy surfaces that define die electronic resonance. These curves are obtained when the two hypersurfaces are cut along theyth nomial mode coordinate. In order of increasing theoretical sophistication these properties are (i) the relative position of their minima (often called the displacement parameters), (ii) the force constant of the vibration (its frequency), (iii) nuclear coordinate dependence of the electronic transition moment and (iv) the issue of mode mixing upon excitation—known as the Duschinsky effect—requiring a multidimensional approach. 

The transition between levels coupled by the oscillating magnetic field B corresponds to the absorption of the energy required to reorient the electron magnetic moment in a magnetic field. EPR measurements are a study of the transitions between electronic Zeeman levels with A = 1 (the selection rule for EPR). 

Figure C3.2.13. Orientation in a photoinitiated electron transfer from dimetliylaniline (DMA) solvent to a coumarin solute (C337). Change in anisotropy, r, reveals change in angle between tire pumped and probed electronic transition moments. From [46],...

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

The electronic transition intensity is proportional to Rg, the square of the electronic transition moment R, where... 

If vibrations are excited in either the lower or the upper electronic state, or both, the vibronic transition moment corresponding to the electronic transition moment Rg in Equation (7.115), is given by... 

The Franck-Condon approximation (see Section 7.2.5.3) assumes that an electronic transition is very rapid compared with the motion of the nuclei. One important result is that the transition moment for a vibronic transition is given by... 

The three bands in Figure 9.46 show resolved rotational stmcture and a rotational temperature of about 1 K. Computer simulation has shown that they are all Ojj bands of dimers. The bottom spectmm is the Ojj band of the planar, doubly hydrogen bonded dimer illustrated. The electronic transition moment is polarized perpendicular to the ring in the — Ag, n — n transition of the monomer and the rotational stmcture of the bottom spectmm is consistent only with it being perpendicular to the molecular plane in the dimer also, as expected. 

By expanding the electronic transition moment operator ft(Q) as a Taylor series in the nuclear normal coordinates about the equilibrium configuration Q0. one obtains ... 

The mixing coefficients a and b in (4.10) depend upon the efficiency of the spin-orbit coupling process, parameterized by the so-called spin-orbit coupling coefficient A (or for a single electron). As A O, so also do a or b. Spin-orbit coupling effects, especially for the first period transition elements, are rather small compared with either Coulomb or crystal-field effects, so the mixing coefficients a ox b are small. However, insofar that they are non-zero, we might write a transition moment as in Eq. (4.11). 

A mistake often made by those new to the subject is to say that The Laporte rule is irrelevant for tetrahedral complexes (say) because they lack a centre of symmetry and so the concept of parity is without meaning . This is incorrect because the light operates not upon the nuclear coordninates but upon the electron coordinates which, for pure d ox p wavefunctions, for example, have well-defined parity. The lack of a molecular inversion centre allows the mixing together of pure d and p ox f) orbitals the result is the mixed parity of the orbitals and consequent non-zero transition moments. Furthermore, had the original statement been correct, we would have expected intensities of tetrahedral d-d transitions to be fully allowed, which they are not. 

Equation 17 can be viewed as the general form of a sum rule for an arbitrary one-electron operator O expressed in terms of the square of the transition moment of the operator and its excitation energies. 

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