Transition-moment

The intensity of the absorption line depends on the oscillator strength, which in turn depends on the transition moment squared (see Chapter 7). The electronic part of the transition moment for x-polarized light may be written as 

Notice that according to Equation 3.50, % k transitions are forbidden for z-poIarized light. Both ( )i and 4)3 are antisymmetric at reflection in the xy-plane, and so is the function z. The product ( )i ( )a is thus synunetric, the integrand ( )i z 4)3 is antisynunetric, and the integral is equal to zero. This means that the transition is forbidden. 

The spectrum for naphthalene (C,oHg) is shown in Eigure 3.14. Spectra of other cyclic aromatic systems resemble the naphthalene spectrum, and this has stimulated development of theories, for example, those of Platt and Gouterman. 

The expressions for the orbital energy and orbitals for cyclic systems were derived in Equations 3.34 and 3.35. The orbitals are given by 

We may interpret tf as a quantum number tf(HOMO) = (N/2 - l)/2. We first add and subtract degenerate orbitals with the same tf and obtain real wave functions  

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Equation (A3.13.17) is a simple, usefiil fomuila relating the integrated cross section and the electric dipole transition moment as dimensionless quantities, in the electric dipole approximation [10, 100] ... 

Hollenstein H, Marquardt R, Quack M and Suhm M A 1994 Dipole moment function and equilibrium structure of methane In an analytical, anharmonic nine-dimenslonal potential surface related to experimental rotational constants and transition moments by quantum Monte Carlo calculations J. Chem. Phys. 101 3588-602... 

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... 

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

If we can use only the zero-order tenn in equation (B 1.1.7) we can remove the transition moment from the integral and recover an equation hrvolving a Franck-Condon factor ... 

Transition intensities are detennined by the wavefiinctions of the initial and final states as described in the last sections. In many systems there are some pairs of states for which tire transition moment integral vanishes while for other pairs it does not vanish. The temi selection rule refers to a simnnary of the conditions for non-vanishing transition moment integrals—hence observable transitions—or vanishing integrals so no observable transitions. We discuss some of these rules briefly in this section. Again, we concentrate on electric dipole transitions. 

The applications to selection rules work as follows. Intensities depend on the values of the transition moment integral of equation (Bl.l.lT... 

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... 

In turn, an expression for is obtained, which, in the frequency domain, consists of a numerator containing a product of (.s + 1) transition moment matrix elements and a denominator of. s complex energy... 

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 operator in the first temi goes as r, and is thus proportional to the optical dipole transition moment... 

The second temi is proportional to the optical quadnipole transition moment, and so on. For small values of momentum transfer, only the first temi is significant, thus... 

Whiting E E, Schadee A, Tatum J B, Hougen J T and Nicholls R W 1980 Recommended conventions for defining transition moments and intensity factors in diatomic molecular spectra J. Moiec. Spectrosc. 80 249-56... 

Piper L G and Cowles L M 1986 Einstein coefficients and transition moment variation for the NC(A S -X n) transition J. Chem. Phys. 85 2419-22... 

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 Einstein coefficients are related to the wave functions j/ and of the combining states through the transition moment R , a vector quantity given by... 

The transition intensity is proportional to the square of the transition moment, which is given by... 

Apart from depending on the numerical value of the square of the transition moment of Equation (5.13), which varies relatively little with J, intensities depend on the population of the lower state of a transition. The population A/ of the Jth level relative to Aq is obtained from Boltzmann s distribution law. Equation (2.11) gives... 

The transition moment (Equation 2.13) for a transition between lower and upper states with vibrational wave functions and j/[ respectively is given by... 

By analogy with Equation (6.6) the vibrational Raman transition moment is given by... 

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

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