Selection rules electronic transitions

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... 

For each L value, 2(2L- -1) electrons can be allocated 2 for L = 0, 6 for L=l, 10 for L = 2, 14 for L = 3, 18 for L = 4 and the remaining 10 for L = 5. The latter energy level would thus be only partly filled and the lowest energy absorption transition (selection rule AL=1) would involve an electron promotion from L = 4 to L — 5. The calculated wavelength from this model is 398 nm, which is in surprisingly good agreement with the experimental value, 404 nm (ref. 143). 

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. 

Molecular UV-vis spectroscopy is prevalent in the more advanced chemistry curriculum for undergraduates. It appears in Organic Chemistry in the analysis of organic compounds, and it can also be applied to Physical (or Quantum) Chemistry courses in discussions of molecular orbitals, electronic transitions between these orbitals, and also transition selection rules and microstates. It is also relevant to Inorganic Chemistry, as it is investigated in terms of transition metal complex color, crystal field theory, and molecular orbital diagrams and electronic transitions for a variety of inorganic compounds. 

Fig. 6.6a can be understood with the help of Eq. (6.28). which shows us a model of the phenomena taking place. At room temperature, most of the molecules (Boltzmann law) are in their ground electronic and vibrational states k = 0, v = 0). IR quanta are unable to change quantum number k, but they have sufficient eneigy to change v and 7 quantum numbers. Fig. 6.6a shows what in fact has been recorded. From the transition selection rules (see above), we have An — 1 — 0=1 and either the transitions of the kind AJ = (7 + 1) — 7 = +l (what is known as the R branch, right side of the spectrum) or of the kind AJ = 7— (7 + 1) = —1 (the P branch, left side). 

The fourth integral governs the electronic orbital selection rule. It has to do with the symmetries of the electronic wave functions in the electronic GS and ES. In order for the electronic transition to be orbitally allowed, the triple direct product must contain the totally symmetric IRR for the point group of the... 

When the Ln ion is situated at a centrosymmetric site (i.e., with an inversion center), the pure electronic transitions between 4 levels are ED forbidden [10]. Magnetic dipole transitions (which are up to 10 times weaker than ED transitions) may then be allowed between states of the same parity in the solid if (8) is satisfied, since the magnetic dipole operator, Fq, is of even parity. The only way to destroy the centrosymmetry of Ln " and permit an ED transition between two electronic states is by motions of odd (ungerade) vibrations so that the electronic spectra of Ln " at an inversion center of a crystal are vibronic (vibrational-electronic) in nature. The transition selection rules then become ... 

An interesting example of a many-electron spectrum is that of He, in which the shown low-energy transitions involve orbital jumps of one of the two electrons. For this case our one-electron atomic selection rules (A/ = 1, Ay = 0, 1) hold for the electron involved in the transition. The He electronic spectrum resembles... 

Strong El transitions in many-electron atoms are observed only when one electron changes its orbital quantum numbers for this electron, the selection rule A/= +1 must be obeyed (cf. our discussion following Eq. 2.12). To appreciate this, we recall that spatial wavefunctions in many-electron atoms may be expressed (Section 2.3) in terms of products (1, 2,..., p) = i(i)(j>2 2) 4>pip) of one-electron orbitals 0i(l), 2(2),. .., (j>p p Since the pertinent electric dipole operator is p= — eSr, the El transition moment from electronic state 2,. .., p) to state 2,. .., p) = i(l)( 2(2). .. 0p(p)... 

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. 

Fig. 19. Composite two-photon excitation spectrum of the 4f ->5d transition in 0.003% in CaF, at 6K. The transition is studied by monitoring the 5d— 4f, no phonon transition occurring at 313.1 nm. As noted in text this transition is normally two-photon forbidden because of parity selection rules, however, odd crystal-fields components admix parity to make the transitions partially allowed. The pure electronic transition of the state is labeled as 0 other excitations, 1 to 12, are identified as phonon or normal mode excitations of the lattice which couple to the pure transition. Selection rules for assisted transitions follow selection rules which differ from the one-photon case. After Gayen and Hamilton (1982).

The value of this energy spacing between the groups of d-orbitals can be calculated (19B) from the spectra of the ions and the free atoms. In fact the visible spectra of the simple complexes of the transdtion metal ions is largely due to excitation of these d-electrons. The extinction coefficients are often very small since the transitions are forbidden by symmetry and electron pairing selection rules. 

Generally, the room temperature emission spectra of Ln species show incompletely resolved stmcture within the peaks. However, an advantageous attribute of luminescent Ln complexes is the dependence of this emission spectral form on the specific coordination environment of the ion. This sensitivity arises from the selection rules associated with intraconfigurational (4f-4f) electronic transitions the selection rules for forced electric dipole transitions are relaxed due to 5d and 4/orbital mixing. In reality the majority of the complexes included for discussion here are non-centrosymmetric, low symmetry species and the relative intensities of the 4/-4/transitions are generally determined by the induced electric dipole transition selection rules. It should also be noted that visibly emissive Eu also possesses a magnetic dipole transition, F, whose intensity is relatively independent of the coordination environment [1,9]. 

Electronic spectra are almost always treated within the framework of the Bom-Oppenlieimer approxunation [8] which states that the total wavefiinction of a molecule can be expressed as a product of electronic, vibrational, and rotational wavefiinctions (plus, of course, the translation of the centre of mass which can always be treated separately from the internal coordinates). The physical reason for the separation is that the nuclei are much heavier than the electrons and move much more slowly, so the electron cloud nonnally follows the instantaneous position of the nuclei quite well. The integral of equation (BE 1.1) is over all internal coordinates, both electronic and nuclear. Integration over the rotational wavefiinctions gives rotational selection rules which detemiine the fine structure and band shapes of electronic transitions in gaseous molecules. Rotational selection rules will be discussed below. For molecules in condensed phases the rotational motion is suppressed and replaced by oscillatory and diflfiisional motions. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

Most stable polyatomic molecules whose absorption intensities are easily studied have filled-shell, totally synuuetric, singlet ground states. For absorption spectra starting from the ground state the electronic selection rules become simple transitions are allowed to excited singlet states having synuuetries the same as one of the coordinate axes, v, y or z. Other transitions should be relatively weak. 

Often it is possible to resolve vibrational structure of electronic transitions. In this section we will briefly review the symmetry selection rules and other factors controlling the intensity of individual vibronic bands. 

The selection rule for vibronic states is then straightforward. It is obtained by exactly the same procedure as described above for the electronic selection rules. In particular, the lowest vibrational level of the ground electronic state of most stable polyatomic molecules will be totally synnnetric. Transitions originating in that vibronic level must go to an excited state vibronic level whose synnnetry is the same as one of the coordinates, v, y, or z. 

One of the consequences of this selection rule concerns forbidden electronic transitions. They caimot occur unless accompanied by a change in vibrational quantum number for some antisynnnetric vibration. Forbidden electronic transitions are not observed in diatomic molecules (unless by magnetic dipole or other interactions) because their only vibration is totally synnnetric they have no antisymmetric vibrations to make the transitions allowed. 

A very weak peak at 348 mn is the 4 origin. Since the upper state here has two quanta of v, its vibrational syimnetry is A and the vibronic syimnetry is so it is forbidden by electric dipole selection rules. It is actually observed here due to a magnetic dipole transition [21]. By magnetic dipole selection rules the A2- A, electronic transition is allowed for light with its magnetic field polarized in the z direction. It is seen here as having about 1 % of the intensity of the syimnetry-forbidden electric dipole transition made allowed by... 

The transition between levels coupled by the oscillating magnetic field B corresponds to the absorption of the energy required to reorient the electron magnetic moment in a magnetic field. EPR measurements are a study of the transitions between electronic Zeeman levels with A = 1 (the selection rule for EPR). 

At this stage, we would like to mention that the model, without the vector potential, is constructed in such a way that it obeys certain selection rules, namely, only the even —> even and the odd —> odd transitions are allowed. Thus any deviation in the results from these selection rules will be interpreted as a symmetry change due to non-adiabatic effects from upper electronic states. 

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