Computation of Transition Moments

The use of the dipole length formula is based on the evaluation of the transition dipole moment 

The two methods only give identical results for oscillator strengths in the complete Cl limit. In SCI calculations one has in general 

Multiply excited configurations are not only important for calculating excitation energies but also for calculating intensities and polarization direc- 

FIguK 1.23. Spectral characteristics of the alkylidenecyclopropene 14 calculated as a function of the number of configurations used in the Cl procedure a) excitation energies and b) oscillator strengths, using dipole length (—) and dipole velocity ( ) expressions (by permission from Downing et al., 1974). 

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Table 52, reproduced from Kotzian et al. s work, compares the INDO/SCI energy separations with experiment for several electronic states of LaO. It is evident from table 52 that the INDO/S-CI method predicts the energy separations for these compounds reasonably well. Kotzian et al. also computed the transition moments for the observed A<->C, B<- X, C<- X, D<- X and F<- X systems. 

The quantities, e, m, x and p- are the charge, mass, position, and momentum of the j th electron, respectively Zje, M/, ft/ and P , are the charge, mass, position, and momentum of the Zth nucleus, respectively and c is the speed light. The transition may be between electronic states, typically the ground state and an electronically excited state, or it may be between different vibrational levels of the ground state. The theoretical description of electronic CD and VCD37.38 therefore reduces to the task of evaluating the transition moments by q. [2] in a computationally viable manner. Because the inherent complexities involved, reasonably reliable evaluation of transition moments has only been available relatively recently for small molecules. 

As discussed in previous sections, the expression of the intensity for any ionization process always involves a continuum orbital, which may describe the photoelectron in XPS or the secondary emitted electron in Auger decay. In the one-center model the problem is overcome by approximating the molecular continuum orbital by an atomic continuum orbital and finally recurring to available or more easily computable atomic transition moments and two-electron integrals. We have also discussed how the multicenter character of the continuum orbitals can be correctly described by... 

Computed transition moments to the highest electronic state obtained in a given symmetry are generally overestimated considerably (because the computed wavefunction of the highest state picks up all contributions that are not compatible with the description of the lower states), and thus n -F 1 electronic states of a given symmetry should be computed if transition moments are desired between n states. 

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. 

Intensities for electronic transitions are computed as transition dipole moments between states. This is most accurate if the states are orthogonal. Some of the best results are obtained from the CIS, MCSCF, and ZINDO methods. The CASPT2 method can be very accurate, but it often requires some manual manipulation in order to obtain the correct configurations in the reference space. 

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. 

The quality of the ) states has been tested through their energy and also their transition moment. Moreover from the natural orbitals and Mulliken populations analysis, we have determined the predominant electronic configuration of each ) state and its Rydberg character. Such an analysis is particularly interesting since it explains the contribution of each ) to the calculation of the static or dynamic polarizability it allows a better understanding in the case of the CO molecule the difficulty of the calculation and the wide range of published values for the parallel component while the computation of the perpendicular component is easier. In effect in the case of CO ... 

The parameterized, analytical representations of fi, ., fiy, fifi determined in the fitting are in a form suitable for the calculation of the vibronic transition moments V fi V") (a—O, +1), that enter into the expression for the line strength in equation (21). These matrix elements are computed in a manner analogous to that employed for the matrix elements of the potential energy function in Ref. [1]. 

As detailed in Section 2, we have derived and programmed the expression for line strengths of individual rotation-vibration transitions of XY3 molecules the line strengths depend on the vibronic transition moments entering into equation (70). With the theory of Section 2, we can simulate rotation-vibration absorption spectra of XY3 molecules. In computing the transition wavenumbers, line strengths, and intensities we use rovibronic wavefunctions generated as described in Ref. [1]. 

Step 7 is likely to be the most computationally demanding step of an MCD calculation based on transition moments. Step 5 will take very little computer time as will step 4 if a functional that does not include exact exchange is used and step 8 will be quick if the influences of the -tensor and ZFS are neglected or not needed. Steps 2 and 5 will take more time and steps 1, 4 (with a hybrid functional the magnetic CPKS equations must be solved), and 8 (if the g-tensor or ZFS is calculated) may take a significant amount of computer time. Step 3 will consume a large proportion of the computational time of an MCD calculation and may be the most expensive step under some circumstances. 

From Eq. (72) we see that the contribution to the MCD intensity from the perturbation to the transition density can be identified with the MCD due to the mixing of the excited state J with other excited states. The remainder of the MCD intensity from terms and spin-orbit-induced C terms is due to the perturbation of the integrals over the electric dipole moment operator (Eq. 52). The perturbed integrals thus include the contribution to the MCD from the mixing of excited states with the ground state. The perturbed integrals are written in terms of unperturbed orbitals (Eqs. 53 and 54) rather than unperturbed states or transition densities as this form is much easier to compute. With some further effort the contribution to the MCD from the perturbed integrals can also be analyzed in terms of transitions. 

The intensity of absorption for an electronic transition is the probability of absorption between two given energy states. It can be theoretically computed by using the expression forthe transition moment integral. The transition moment integral MU1 between the ground and the first excited state is expressed as... 

The X and Y are thus expressed in terms of the profiles of the up and their inverse down (r v ) transitions. The spectral moments can thus be written as a simple combination of the moments of the up and down transitions, which may be computed from the induced dipole components and the interaction potential. Furthermore, the function X satisfies the (old) detailed balance condition, Eq. 6.72, and is conveniently represented by the successful BC or K0 models. A simple choice for Y could be (co/A) r (ft)) where A is a constant to be specified and r (ci)) is another model function T which satisfies Eq. 6.72. In other words, according to Eq. 6.75, the ro to vibrational profiles K can be represented by the familiar model functions whose parameters may be defined with the help of the associated moment expressions. 

A calculation of the oscillator strength of the y band was made by Erkovich.130 Using low-pressure absorption spectra, he obtained a value of0.043. This result is about 20 times as great as from other measurements, and the method used has been severely criticized by Penner.342 Erkovich and Pisarevskii,131 using a modification of Erkovich s method, calculated the electronic transition moments for the / and y systems. Because their computations were, as before, based on low-pressure absorption spectra, Penner s criticism still applies. 

These results and also the results for the excitation energies will of course be modified by dynamical correlation effects, which will be accounted for by means of MR-CI calculations based on the CASSCF orbitals for each state. The transition moments will be computed using the CASSI method, where the two A are allowed to interact. The final results can be expected to be accurate to within 0.1 eV for the excitation energies and 10% for the transition moments. The study of the CCCN system is not yet finished. Hopefully some result can be presented in the lectures. 

If computations of orientation and transition moment direction are to be meaningfully correlated with the observations, it is clear that factors of the type just discussed must be carefully taken into consideration. 

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