The Selection Rules for Electronic Transitions

Electronic transitions between energy levels in organic molecules are governed by some compulsions, known as selection rules. 

An electronic transition proceeds more rapidly when the wave functions of the initial and final states closely resemble each other, i.e., the transition is allowed. In the jt 71 transition, ti and ti orbitals occupy the same regions of space and so the overlap between them is large. Thus, the ti ti is allowed process. In n TI transitions, the n and ti orbitals are perpendicular to each other and so they overlap to much smaller extent. Therefore, this transition is forbidden. In practice, this n ti transition is weakly allowed due to coupling interaction between vibrational and electronic motion in the molecule. 

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An important feature of photodetachment methods is that the selection rule for electronic transition is A5 = +1. Therefore, photodetachment of (doublet) negative ions... 

The selection rules for electronic transitions are not as clear-cut as in the case of vibration and rotation. In the case of molecules consisting of relatively light nuclei, which is the case for many molecules of tropospheric interest, the selection rules... 

Thus a transition between two given electronic states shows many bands, each such band corresponding to a different pair of initial and final vibrational states under high resolution, each band shows many closely spaced lines, each such line corresponding to a different pair of initial and final rotational states. (The electronic spectra of molecules are called band spectra, whereas the electronic spectra of atoms are called line spectra.) Consider the selection rules for electronic transitions. The electric di-... 

As we have seen in Chapter 11, the energy levels of atoms and ions, depending on the relative role of various intra-atomic interactions, are classified with the quantum numbers of different coupling schemes (11.2)— (11.5) or their combinations. Therefore, when calculating electron transition quantities, the accuracy of the coupling scheme must be accounted for. The latter in some cases may be different for initial and final configurations. Then the selection rules for electronic transitions are also different. That is why in Part 6 we presented expressions for matrix elements of electric multipole (Ek) transitions for various coupling schemes. 

Experimentally, the oscillator strength is given by the integrated intensity (area) under the absorption band, while theoretically it is given by the square of the transition moment integral J gM s. dr. This leads to the selection rules for electronic transitions when jTgMTedi is nonzero, there is absorption intensity and the transition is allowed when this integral is required to be zero, the transition is forbidden. ... 

A further distinction between (n, n ) and n, tt ) excited states results from the selection rules for electronic transitions which, as can be seen in the orbital diagram above, classify the former as partially forbidden — essentially because of the poor spatial overlap between n and n orbitals, and the latter as allowed. In consequence molar extinction coefficients for typical n- n transitions lie in the range 10 — 10 M cm whereas those for n n transitions range from 10 —10 cm, a matter... 

The chapter is organized as follows. In Section 2 the symmetry properties of CNTs and the selection rules for electronic transitions are described. In Section 3 the form of P tensor is determined, together with the several state-to-state contributions coming from electrons inside the ID. In Section 4 the principles for estimation of the magnirnde of p are reported together with the results of a calculation for several CNT topologies. 

XANES spectra of different systems have been interpreted with the band structure aproximation As an example for a transition metal we discuss here palladium absorption edges. The comparison between the K and Lj edge of Pd metal with the theoretical band approach is shown in Fig. 21. We can observe that the K and Li edges present the same spectral features and therefore contain identical information. In fact, the selection rule for electronic transitions selects the same I = 1 projected density of states. Because the L, edge occurs at lower energy a better instrumental energy resolution is obtained and the structures are better resolved. 

Although it is out of the scope of this book to discuss the selection rules for electronic transitions in detail, it may be worthwhile mentioning an important principle proposed by Franck and Condon. According to the Franck-Condon principle, transitions from the vibrational levels of one electronic state to the vibrational levels of another occur so rapidly that the positions and velocities of the nuclei have no time to change. That is, the nuclei remain fixed in position for the duration of the electronic rearrangement,... 

From the properties of the 6-j symbol in (11), the selection rules for electron transitions in this scheme are obtained [8],... 

Unfortunately, for electronic transitions, gross selection rules are not as straightforward to define. Therefore, we will consider the selection rules for electronic transitions as they arise in the discussion of the material. The electronic spectrum of the hydrogen atom, for example, has a relatively simple selection rule. The electronic spectrum of the benzene molecule, as a counter-example, follows more complex rules. 

These integrals are similar to those involved in the determination of the selection rules for electronic transitions in the hydrogen atom, and are zero unless M = M / = J =t 1. From the recursion formulas 4-85, we find for the squares of the pertfnent integrals ... 

To complete our sketch of atomic spectral phenomena the effects of electric and magnetic fields and the selection rules for electronic transition will be noted. 

Beyond such electronic symmetry analysis, it is also possible to derive vibrational and rotational selection rules for electronic transitions that are El allowed. As was done in the vibrational spectroscopy case, it is conventional to expand i j (R) in a power series about the equilibrium geometry of the initial electronic state (since this geometry is more characteristic of the molecular structure prior to photon absorption) ... 

To investigate the spectra of diatomic molecules, we need the selection rules for radiative transitions. We now investigate the electric-dipole selection rules for transitions between vibration-rotation levels belonging to the same 2 electronic state. (Transitions in which the electronic state changes will be considered in Chapter 7.)... 

We now consider radiative transitions foi which both v and J change, but the electronic state does not these transitions give the vibration-rotation spectra of diatomic molecules. The selection rules for these transitions were found in Section 4.4 to be ( 2 states only)... 

Tirana afld comparison with (9.189) shows that the group-theory selection rules for electronic transitions are the same as for vibrational transitions, except that we must consider the symmetry species of the electronic wave functions, rather than the vibrational wave functions. One complication is that the molecular geometry may change on electronic excitation in this case, we use the point group of lower symmetry to classify the wave functions and determine the selection rules. 

Selection rules for electronic transitions with the participation of core electrons are similar to those for transitions when the core is left unchanged, with the exception of the selection rules following from the CFP with one detached electron. Then seniority quantum numbers t ,- and v- of the subshells, between which the electron is jumping , must be changed by unity, i.e. At ,- = 1, At - = 1. 

Between the A and B regions, a radioffequency field was applied to induce fine-structure transitions within the v" = 1 level of the ground electronic state, split by the nuclear hyperfine interaction. The selection rules for these transitions, which ranged in frequency from 360 to 7700 MHz, were A J = 1, AF = 0, 1. They were detected through resonant changes in the fluorescence intensity an example of a radioffequency double resonance line is shown in figure 11.53. The observed spectrum involved N values from 1 to 27. 

Group theory is also used prior to calculations to determine whether a quantum-mechanical integral of the type /i j, op. % dt is different from zero or not. This is important in such areas as selection rules for electronic transitions, chemical reactions, infrared and Raman spectroscopy, and other spectroscopies. 

Thus fer in chis chapccr we have seen electronic spectra of four complexes [TOHjOU (Hg. 11.8). find (Crier ) . (Cr(oxy>-, and (CrF.f (Rg. 11.13). Casual inspection of these examples reveals that the number of absorptions varies. At the heart of the interpretation of electronic spectra is the question of how many absorptions are expected for a given complex. Answering this question requires an accurate energy level diagram for the complex of interest as well as familiarity with the selection rules governing electronic transitions. 

Each electron in an atom is defined by four quantum numbers n, l, m and s. The principal quantum number (n) defines the shell for example, the K shell as 1, L shell as 2 and the M shell as 3. The angular quantum number (/) defines the number of subshells, taking all values from 0 to (n — 1). The magnetic quantum number (m) defines the number of energy states in each subshell, taking values —l, 0 and +1. The spin quantum number fsj defines two spin moments of electrons in the same energy state as + and —f The quantum numbers of electrons in K, L and M shells are listed in Table 6.1. Table 6.1 also gives the total momentum (J), which is the sum of (7 + s). No two electrons in an atom can have same set quantum numbers (n, /, m, s). Selection rules for electron transitions between two shells are as follows ... 

Dominicis and Fantoni present a method for the computation of the electronic first hyperpolarizability of chiral carbon nano-tubes (CNTs). The CNT eigenstates are computed by an algorithm reported by Damnjanovic et al. They discuss the symmetry properties of CNTs and selection rules for electronic transitions and demonstrate that the use of symmetry reveals the state-to-state transitions, which contribute to the first hyperpolarizability of CNTs. The latter is related to particular state-to-state transitions. The principles for predicting the magnitude of the first hyperpolarizability and its relation to the topology of CNTs are also discussed. 

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