Continuum oscillator strength transitions

The oscillator strength of the transition in eh, obtained by the dipole length formulation, is 1.1 the experimental value is >0.7. This overestimation seems to be common among all continuum models. No information on line shape is obtained in this simple model. 

Fig. 30. The total differential oscillator strength for benzene including the structure factor but neglecting any vibrational effects. The cross-hatched portion of the figure represents the transition to a continuum orbital of e2ll symmetry, while the remainder is for a transition to an elu orbital. The positions of higher ionization thresholds are indicated.

Fig. 5 Electronic transitions of [Re(Cl)(CO)3(bpy)] calculated in vacuum (a), acetonitrile (b), simulated absorption spectra in vacuum (c, dashed) and acetonitrile (c, full), and experimental absorption spectrum measured in acetonitrile (d). It follows that successful TD-DFT simulation of the absorption spectrum and singlet CT states requires using hybrid functionals and continuum dielectric models for the solvent [11, 33], Calculation TD-DFT G03/PBE0, CPCM for MeCN. Simulation All calculated transitions included. Gaussian shapes (jwhm = 0.4 eV cm- ) of the absorption bands are assumed. Band areas are proportional to calculated oscillator strengths. Simulated using the GaussSum software. Reprinted with permission from [33]...

Combining the idea of solvent-induced changes in molecular structure with the concept of a solvent continuum around the solvatochromic molecule, a micro-structural model of solvatochromism has been developed by Dahne et al., which reproduces, qualitatively correctly and quantitatively satisfactorily, the solvatochromic behavior of simple merocyanine dyes [95b], The results obtained with this model for 5-(dimethylamino)penta-2,4-dienal are in good agreement with the solvent-dependent experimental data such as transition energies, oscillator strengths, r-electron densities, and r-bond energies [95b] cf. also [326, 327],... 

Table 38 Calculated excitation energies (in eV), oscillator strength, and transition dipole moment (in D) for the coumarin derivative Cl in different media. Two different polarizable-continuum models (SCRF-S and PCM) and four different density functionals (CAMB3LYP, mCAMB3LYP, PBEO, and B3LYP) have been used. In all cases, the same size of the basis set has been used, except for the B3LYP calculations in vacuum, for which the second and third entry list results for a smaller and a larger basis set, respectively. All results are from ref. 92 ...

In the preceding sections, we have assumed that an absorption line has a Lorentzian shape. If this is not true, then the linewidth cannot be defined as the full width at half maximum intensity. Transitions from the ground state of a neutral molecule to an ionization continuum often have appreciable oscillator strength, in marked contrast to the situation for ground state to dissociative continuum transitions. The absorption cross-section near the peak of an auto-ionized line can be significantly affected by interference between two processes (1) direct ionization or dissociation, and (2) indirect ionization (autoionization) or indirect dissociation (predissociation). The line profile must be described by the Beutler-Fano formula (Fano, 1961) ... 

The rare gas excimer lasers are based on bound-continuum transitions from an excited diatomic species to its dissociative ground state. The observed continuum emission is a superposition of the Franck-Condon factors from the vibrational levels of the upper state. Thus these molecular dissociation lasers display relatively broad fluorescence as a consequence of the steeply repulsive ground-state potential, and there is always a population inversion on such transitions. However, the net gain is significantly lower than that for a bound-bound transition because of the distribution of oscillator strength over the broad fluorescence band. Figure 1 illustrates schematic potential energy curves for such transitions in the excimer and exciplex lasers. 

Why do the LDA spectra look so similar to the exact one Is this just a coincidence Returning to Figure 3, we notice that the LDA (or a GGA) potential runs almost exactly parallel to the true potential for r 2, i.e., where most of the density is. Thus, the scattering orbitals of the LDA potential, with transition energies between 0.6 and 0.9 h, almost match exactly the Rydberg orbitals of the exact KS potential with the same energyWhen defined carefully, i.e., when we use phase space factors for the continuum relative to bound states, the oscillator strengths for both the LDA and exact KS... 

The fruitfly of TDDFT benchmarks is the tt —> tt transition in benzene. This transition occurs at about 5 eV in a ground-state LDA calculation, and ALDA shifts it correctly to about 7 Unfortunately, this value lies in the LDA continuum, which starts at about 6.5 eV This shift is an example of the same general phenomenon we saw above for He, where LDA has pushed some oscillator strength into the continuum, but its overall spectra remains about right. 

Figure 2 Empirical curve of growth for solar Fe I and Ti I lines. The y-axis is the equivalent width (line strength relative to the continuum) and the x-axis is based on the oscillator strength of the transition.

The summation extends over all levels (including the continuum) which are accessible from level by electric dipole transitions. If is an excited level, induced emission to lower levels may occur, which diminishes the effective absorption. The corresponding oscillator strengths f. j with are therefore negative. 

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