Spin-forbidden electric dipole transition

For the evaluation of probabilities for spin-forbidden electric dipole transitions, the length form is appropriate. The velocity form can be made equivalent by adding spin-dependent terms to the momentum operator. A sum-over-states expansion is slowly convergent and ought to be avoided, if possible. Variational perturbation theory and the use of spin-orbit Cl expansions are conventional alternatives to elegant and more recent response theory approaches. 

L. L. Lohr Jr., J. Chem. Phys., 45, 1362 (1966). Spin-Forbidden Electric-Dipole Transition... 

Let us consider how independent /i(i ) 2 effects contribute to the v E) for the hydrogen halides, HX (X = I, Br, and Cl). The curves shown on Fig. 7.6 correspond to relativistic adiabatic potential energy curves (respectively 0 dotted, 0+ dashed, 1 and 2 solid) for HI obtained after diagonalization of the electronic plus spin-orbit Hamiltonians (see Section 3.1.2.2). The strong R-dependence of the electronic transition moment reflects the independence of the relative contributions of the case(a) A-S-Q basis states to each relativistic adiabatic II state. The independent experimental photodissociation cross sections are plotted as solid curves in Fig. 7.7 for HI and HBr. Note that, in addition to the independent variations in the A — S characters of each fl-state caused by All = 0 spin-orbit interactions, all transitions from the X1E+ state to states that dissociate to the X(2P) + H(2S) limit are forbidden in the separated atom limit because they are at best (2Pi/2 <— 2P3/2) parity forbidden electric dipole transitions on the X atom. In the case of the continuum region of an attractive potential, the energy dependence of the dissociation cross section exhibits continuity in the Franck-Condon factor density (see Fig. 7.18 Allison and Dalgarno, 1971 Smith, 1971 Allison and Stwalley, 1973). 

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],... 

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. 

Two MO calculations have been performed on the dimer (NO>2. Vladimiroff has concluded that a cyclic structure is most stable. Shancke and Boggs carried out an ab initio calculation on the cw-isomer and expressed reservation about some of the geometrical parameters calculated by Vladimiroff and the strongly contracted basis set he used. In an additional mechanism for the spin-forbidden a II-x II transition of nitric oxide, consideration has been given to the coupling of two NO molecules and the subsequent co-operative optical transition via a one-photon, spin-allowed, electric-dipole mechanism. It was concluded that this co-operative mechanism is as important as the conventional spin-orbit interaction mechanism. 

Electric-dipole transitions between 4f levels of rare earth ions are strictly forbidden, because the parity does not change (Laporte s selection rule). We shall now consider as an example the transitions between the and F levels of the Eu ion. The electric-dipole transition between these levels is forbidden not only because of the above-mentioned Laporte s selection rule, but also because the spin quantum number S of the total angular momentum changes (from 2 to 3). 

E and E. Transitions from the ground A, state to the E state is dipole allowed, but transitions to all the triplet states are spin-forbidden. The Aj Aj, and the Ai A2 transitions are electric dipole-forbidden but become allowed by vibrcnic coupling to the E level. A vibration of t or symmetry makes the former absorption allowed, while the latter is allowed only by a vibration of e symmetry. 

For example, the singlet-triplet transitions in ethylenic compounds generally have tmax <3C 1. The fact that spin-forbidden transitions can be observed at all shows that the transition moment, f electric dipole operator. This operator also contains small terms such as quadrupole operators and spin-orbit operators. The latter is the part of any dynamical operator which couples orbital and spin angular moments this term is responsible for the appearance of weak triplet — singlet absorption spectra. 

Figure 2.14. The coupling scheme by which the spin-forbidden a 3A2(rc,7t ) <— X Ai(n2) transition borrows oscillator strength from the B 1A1(jt,jt ) <— X iAi(n2) electric dipole allowed transition.

The quality of the SOC calculation in O2 can be checked by estimation of the fc Sj" — A3E transition probability. The transition is forbidden by selection rules for electric dipole radiation with account of SOC, and occurs as magnetic dipole spin-current borrowing intensity from microwave transitions between spin-sublevels of the ground state [41]. 

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