Transition dipole moment integrals

At this point, it is of interest to discuss the relationship between MO theory and the intensity of electronic transitions. The oscillator strength of an electronic absorption band is proportional to the square of the transition dipole moment integral, ( /gM I/e) where /G and /E are the ground- and excited-state wave functions, and r is the dipole moment operator. In a one-electron approximation, (v(/G r v(/E) 2= K Mrlvl/fe) 2> where v /H and /fe are the two MOs involved in the one-electron promotion v /H > v / ,. Metal-ligand covalency results in MO wave... 

Combination and difference bands Besides overtones, anharmonicity also leads to the appearance of combination bands and difference bands in the IR spectrum of a polyatomic molecule. In the harmonic case, only one vibration may be excited at a time (the transition dipole moment integral vanishes when the excited state is given by a product of more than one Hermite polynomial corresponding to different excited vibrations). This restriction is relaxed in the anharmonie case and one photon can simultaneously excite two different fundamentals. A weak band appears at a frequency approximately equal to the sum of the fundamentals involved. (Only approximately because the final state is a new one resulting from the anharmonie perturbation to the potential energy mixing the two excited state vibrational wave functions.)... 

Using eqn [61] for a fundamental vibration Qp it can be seen that the band intensity is proportional to the following transition dipole moment integral ... 

The IR selection rule depends on the fact that the transition dipole moment integral must be nonzero in order to observe the transition as an IR absorption band. This means that the integrand for Md must have Aj symmetry. 

In the main text we introduced the selection rules for IR spectroscopy via the transition dipole moment integral. This appendix gives a little more detail on the origin of the selection rules, with explicit formulae for the vibrational wavefunctions. This also allows a more complete explanation of the observation that absorption due to transitions involving neighbouring levels (e.g. n = 0 to n = 1) are more easily observed than overtones which involve transitions to higher levels in the ladder of vibrational states. 

Using Eq. 8.35, the transition dipole moment integral is given by the relation... 

If a diatomic molecule is represented as a rigid rotor, the transition dipole moment integral for a rotational transition is... 

When transitions are observed between vibrational energy levels, infrared radiation is emitted or absorbed. The vibrational selection rules are derived in the Bom-Oppenheimer approximation by evaluating the transition dipole moment integral... 

In order for a given normal mode of a polyatomic molecule to give rise to a vibrational band (be infrared active ), the transition dipole moment integral for the two vibrational wave functions of the normal modes must be nonzero. This integral can be studied by group theory. However, it is often possible by inspection of the normal modes to identify those that modulate the dipole moment of the molecule. 

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

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...

Equations (4.30) and (4.31) have been developed and dehned within a time-dependent framework. These equations are identical to Eqs. (35) and (32), respectively, of Ref. 80. They differ only in that a different, more appropriate, normalization has been used here for the continuum wavefunction and that the transition dipole moment function has not been expanded in terms of a spherical harmonic basis of angular functions. All the analysis given in Ref. 80 continues to be valid. In particular, the details of the angular distributions of the various differential cross sections and the relationships between the various possible integral and differential cross sections have been described in that paper. 

It is sufficient to determine the quantity Rxk in the second order with respect to the CC photon interaction. We further assume that the optical preparation of the excited state by the applied field E is short compared to the emission process and, finally, we neglect anti-resonant contributions. When calculating F(u> t) we also have to perform a summation with respect to the transversal polarization and a solid angle integration. Introducing dm = dmem where eTO is the unit vector pointing in the direction of the transition dipole moment one gets... 

Consider now the average over the M quantum number where all components are Z polarized. The relevant integrals in Eq. (53) would then have q = 0. In accord with this equation the entire M dependence, within the transition dipole moments, is contained in a single 3-j symbol. The property of the 3-j coefficients with respect to M — —M transformation is given by [85]... 

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