Intensity of electronic transitions

A rough measure of the intensity of an electronic transition is provided by the maximum value of the molar extinction coefficient. A physically more meaningful quantity is the total area under the absorption band, given by the integral /cdi , or the oscillator strength 

In order to obtain a theoretical expression for the oscillator strength, perturbation theory may be used to treat the interaction between electromagnetic radiation and the molecule. Since an oscillating field is a perturbation that varies in time, time-dependent perturbation theory has to be used. Thus, the Hamiltonian of the perturbed system is = 0) + 

It turns out that the probability of a transition from a state 0 with total wave function into a state with wave function is proportional to the 

When the origin of coordinates is chosen to lie within a molecule of ordinary size, the length r, is much smaller than A for UV light and light of longer wavelengths. Therefore, K-r 1, and the expansion of the exponential e 0 into an infinite series converges rapidly 

The dominant contribution to the squared matrix element of the interaction operator t) stems from the first term in Equation (1.29), unity. The oscillator strength of the transition — is then 

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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... 

Intensities of Electronic Transitions in Molecular Spectra. IV. Cyclic Dienes and Hyperconjugation. J. chem. Physics 7, 339 (1939). 

Up to now the four-orbital model-based theory of porphyrin absorption electronic spectra ( ) has not taken into account the influence of NH-tautomerism in non-symmetrical porphyrins on the positions and intensities of electronic transitions in the visible region. The theoretical consideration of this problem involves solving the fundamental question of the absolute orientation of electronic transition oscillators for each tautomer. First of all. 

The induced magnetic dipole moment has transformation properties similar to rotations Rx, Rt, and Rz about the coordinate axes. These transformations are important in deducing the intensity of electronic transitions (selection rules) and the optical rotatory strength of electronic transitions respectively. If P and /fare the probabilities of electric and magnetic transitions respectively, then... 

The intensities of electronic transitions vary greatly. The area (FL) under the absorption band, when e is plotted against wave number v, is directly proportional (Eq. 23-6) to a dimensionless quantity called the oscillator strength/. 

In Ln3+ ions, the 4f orbitals are radially much more contracted than the d orbitals of transition metals, to the extent that the filled 5s and 5p orbitals largely shield the 4f electrons from the ligands. The result is that vibronic coupling is much weaker in Ln3+ compounds than in transition-metal compounds, and hence the intensities of electronic transitions are much lower. As many of... 

The radiative lifetime may be calculated from the Einstein B coefficient as determined from the integrated absorption spectrum. The absolute intensity of electronic transitions is usually determined from the absorption spectrum since for emission it is difficult to determine the number of molecules in the excited state. The parameter measured experimentally is the absorption coefficient, kv, which is defined by the relation... 

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... 

Transition moments and intensities of electronic transitions may also be solvent dependent due to solvent effects on the wave functions of the various states. Large effects are to be expected especially for weak transitions... 

At the same time as Pople was developing his adaptation of Roothaan s theory, other related lines of thought were being pursued on both sides of the Atlantic. A particularly troublesome question at that time was how far is it possible to describe the excited states of molecules by single determinant wave functions, or small numbers of such determinants In other words, how far is it necessary to invoke interaction between excited configurations in order to account quantitatively for the energies and intensities of electronic transitions ... 

The energies and intensities of electronic transitions are modified by the overlap of the atomic orbitals and the symmetry and composition of molecular orbitals. A detailed examination of the electronic structure of different organosulfur compounds and their complexes in different coordination environments would be useful to interpret their NEXAFS spectra. This information is also useful in the interpretation of the NEXAFS spectra of unknown compounds in the natural systems. 

The oscillating part of the secondary electron spectrum fine structure in the expression obtained is determined by two interference terms resulting from scattering of secondary electrorrs of final and intermediate states (the latter are due to the second-order process only). Here intensities of oscillating terms are determined by the amplitudes and intensities of electron transitions in the atom ionized. In this section we make estimations of these values within the framework of the simple hydrogen like model using the atomic unit system as in the preceding section. This section s content is based on papers [20,22,29-31,33,35,37,45-47]. 

MuUiken, R. S. (1939) Intensities of Electronic Transitions in Molecular Spectra II. Charge-Transfer Spectra. The Journal of Chemical Physics, 7(1), 20-34. 

When applied to spectroscopy, the response methods, also called Green s function methods or Liouville methods, yield the energies and the intensities of electronic transitions directly. Starting with the generaUzed Schrodinger equation. 

Because the intensities of electronic transitions vary over a very wide range, the preparation of samples for UV-vis spectra determination is highly concentration dependent. Intense absorption can result from the high molecular extinction coefficients found in many organic chro-mophores. The sanrpling of these materials requires very dilute solutions (on the order of 10 —10 M). These solutions can be conveniently obtained by the technique of serial dilution. In this method a sample of the material to be analyzed is accurately weighed, dissolved in the chosen solvent, and... 

The basic information in an electronic spectrum consists of the number and positions (energies) of bands and their intensities. The number of bands depends on the number of orbitals to which electronic transitions can occur and the selection rules governing such transitions. Selection rules are not absolute, and the ways in which they can be relaxed are responsible for much of the great variation in intensities of electronic transitions. The relevance of all these factors to transition-metal complexes, usually studied in solution, is described in Section 9.6. 

To understand how Nature makes use of colored materials in such important processes as photosynthesis and vision, we need to know the factors that control the intensity of electronic transitions and the shapes of absorption bands. 

The final step in our procedure is to determine the relative intensities of electronic transitions from the ground state vibrational levels of the states (pj and relative intensities of these transitions may be found by calculating the first-order corrections to the states. If the transition to associated with this level is essentially unchanged in intensity by small value of [313]. The 0-1... 

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