Parity-allowed/forbidden transitions

Calculations for Ceo in the LDA approximation [62, 60] yield a narrow band (- 0.4 0.6 eV bandwidth) solid, with a HOMO-LUMO-derived direct band gap of - 1.5 eV at the X point of the fee Brillouin zone. The narrow energy bands and the molecular nature of the electronic structure of fullerenes are indicative of a highly correlated electron system. Since the HOMO and LUMO levels both have the same odd parity, electric dipole transitions between these levels are symmetry forbidden in the free Ceo moleeule. In the crystalline solid, transitions between the direct bandgap states at the T and X points in the cubic Brillouin zone arc also forbidden, but are allowed at the lower symmetry points in the Brillouin zone. The allowed electric dipole... 

Both absorption and emission spectra have been recorded for a variety of octahedral chromium(III) complexes. For the systems of interest here, A/B 2. Inspection of Figure 2 leads to the expectation of three spin-allowed, parity-forbidden transitions between the iA2g and the other quartet states and two spin- and parity-forbidden transitions between the iA2g and the 2Eg and 2T2g states. Aqueous solutions of Cr(H20)s3+ display three bands with e 15 at 17,400, 24,500, and 38,000 cm-1, assigned respectively to the transitions iA2g- iT2g,... 

By reference to the MO scheme in Fig. 2, the bands at X < 400 nm have been assigned to sharp and intense parity-allowed transitions between occupied (bonding) and empty (antibonding) MOs. Such excitations include Au(HOMO)— fJg(LUMO + 1) and hg—> Optical transitions between the HOMO(Au) and LUMO(f/u), which are electric dipole forbidden, occur via excitation of a vibronic state with appropriate parity symmetry and account for the broad and low intensity band at X > 400 nm. 

Table 2 lists the selection rules for beta decay the entry A means that for die indicated spin and parity change llie transition is allowed 1, means that it is first forbidden II, second forbidden. .. 

When the initial and final states in p decay have opposite parities, decay by an allowed transition cannot occur. However, such decays can occur, albeit with reduced probability compared to the allowed transition. Such transitions are called forbidden transitions even though they do occur. The forbidden transitions can be classified by the spin and parity changes (and the corresponding observed values of log ft) as in Table 8.2. 

Fig. 3). Note that optical transitions between the two configurations are parity allowed as electric-dipole transitions. The spin-selection rule is relaxed by spin-orbit coupling, the more so the higher the principal quantum number is. Due to selection rules on AJ, the transitions 1S0-3F0 and 1S0-3P2 remain strongly forbidden. The emission is due to the 3P0,i >1 o transition. Whether 3P0 or 3Pj is the initial level depends on their energy difference and the temperature.

Relaxation of the rules can occur, especially since the selection rules apply strongly only to atoms that have pure Russell-Saunders (I-S) coupling. In heavy atoms such as lanthanides, the Russell-Saunders coupling is not entirely valid as there is the effect of the spin-orbit interactions, or so called j mixing, which will cause a breakdown of the spin selection rule. In lanthanides, the f-f transitions, which are parity-forbidden, can become weakly allowed as electric dipole transitions by admixture of configurations of opposite parity, for example d states, or charge transfer. These f-f transitions become parity-allowed in two-photon absorptions that are g g and u u. These even-parity transitions are forbidden for one photon but not for two photons, and vice versa for g u transitions [46],... 

Magnetic dipole transitions play a role in the luminescence of some lanthanide ions, specially Eu +, when the local symmetry deviates little from inversion symmetry. They are parity-allowed between states ofthe3d or4f configurations but have a low probability. They are subject to selection rules AL = A5" = 0 and AJ = 0, 1 (0 0 forbidden). 

The intrinsic E A2 luminescence (R-line) has a lower intensity, as typical for strictly octahedral symmetry, than the Cr(III) luminescence close to inverted sites l This is not surprising, since the parity-forbidden transitions become partially allowed when the high symmetry is lowered. The R-line intensity increases when Cr(III) is... 

Since the experimental discovery of TPA, multiphoton excitation has become a popular tool in the photochemical sciences to determine the excitation energy of states with parity forbidden transition [3-54], Transitions that are parity forbidden by one-photon (OP) excitation can thus become allowed by two-photon (TP) excitation. TP excitation spectroscopy localizes the energetic position of TP excited states, which cannot be observed by OP excitation. These pioneering works confirmed many quantum chemical studies predicting the existence of TP excited states and therefore experimentally completed the pattern of electronic transitions in organic compounds. In general, TP excitation had been mainly limited to academic interest until the end of the 1980s [2-24, 26-45, 47-52, 55-69]. 

In the above discussion of the electronic structure of the donor levels, the electron spin has been neglected. It has been, however, proven necessary to introduce the spin-orbit coupling to explain the observation of parity-forbidden transitions for donors with relatively deep ls(Ai) ground states. Using the double group representation of Td, it is found (see Table B.4 of appendix B) that the simple representations Ai and E transform into the T6 and Tg double representations, respectively and that T2 transforms into T7 + Eg. Electric-dipole transitions are symmetry-allowed between A (Tg) and the two T2 (Ty) and T2 (r8) levels. 

The Ch-related donor spectra differ on that point as several parity-forbidden transitions are observed. They start with symmetry-allowed transitions from the Is ground state to the valley-orbit split Is excited states, and are supplemented with 2s (T2) and 3s (T2) lines and Fano resonances within the photoionization spectrum. This is shown in Fig. 6.13 for Se°. Compared to group-V donors, this extends the energy span of the Ch°-related spectra to the ionization energy of the Is (T2) level (35-40 meV in isolated chalcogens) and it can even increase to 40-48 meV when singlet-triplet spin-forbidden transitions are observed. 

In natSi samples diffused with 34S, an IS of —0.14meV (—1.1cm-1) with respect to 32 07S (natural S) has been measured for the parity-allowed transitions of the 34S+ spectrum by Forman [64] and a comparable value (—0.175meV or -1.41cm-1) has been reported by Steger et al. [233]. The sign of IS, which is in agreement with the model of [132] is the opposite of the above-reported one for the parity-forbidden 1 s (T2) T7 and I g transitions of S and Se+, which is related to different vibronic coupling effects. 

Transition probabilities in fact provide a particularly severe test of atomic calculations, because they are rather sensitive to the wavefunctions of both levels involved in the transition and to the approximations used, especially when electron correlations and relativistic effects are considered. On a more fundamental level, measurements of transition probabilities are also being used to explore the non-conservation of parity predicted by the unified theories of the weak and electromagnetic interaction here, highly forbidden transitions must be tested for an admixture of allowed transition probabilities [8,9]. 

Transitions occur mainly by an electric dipole mechanism. Such transitions are allowed if the initial and final states are made up of orbitals of opposite parity (A/ =1,3,... / orbital angular momentum quantum number) and if the spin remains unchanged AS = 0) (Laporte rules, see Ligand Field Theory Spectra. However parity-forbidden transitions can occur as a result of mixing with states of opposite parity. Mixing of states by the crystal field requires that the cationic site lacks an inversion center. If the site is centrosymmetric, transitions can nevertheless be observed owing to vibronic coupling. Their probability is low and increases with temperature. 

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