Perturbation selection rule

For homonuclear molecules, the g or u symmetry is almost always conserved. Only external electric fields, hyperfine effects (Pique, et al., 1984), and collisions can induce perturbations between g and u states. See Reinhold, et al., (1998) who discuss how several terms that are neglected in the Born-Oppenheimer approximation can give rise to interactions between g and u states in hetero-isotopomers, as in the HD molecule. An additional symmetry will be discussed in Section 3.2.2 parity or, more usefully, the e and / symmetry character of the rotational levels remains well defined for both hetero- and homonuclear diatomic molecules. The matrix elements of Table 3.2 describe direct interactions between basis states. Indirect interactions can also occur and are discussed in Sections 4.2, 4.4.2 and 4.5.1. Even for indirect interactions the A J = 0 and e / perturbation selection rules remain valid (see Section 3.2.2). 

The and e/f labels are really two different bookkeeping devices for the same physical property, but the e/f labels are more convenient, mainly for optical transition and perturbation selection rules. 

This e/f degeneracy between the two same-TV components will be lifted by interaction with a 2II state. If the potential curves of the 2 + and 2II states are identical and the configurations of the 2 + and 2II states axe basis functions require that the following interactions be considered. The 2n1//2 state experiences two types of Afl = 0 interactions with 2E]f/,2 spin-orbit,... 

Several other kinds of information are available from the information in Fig. 5.7. Two consecutive vibrational levels of A1 are crossed by the same vibrational level of e3 . The Q branch (F2) crossing occurs in va = 1 and 0 at J = 48.9 and 61.5. Since the deperturbed A1 term energies at these two J-values are known by interpolation, accurate values for B(e3 ) and F(e3 ) can respectively be determined from the slope and intercept of the straight line drawn through these va = 1, J = 48.9 and va == 0, J = 61.5 term values. Alternatively, the J-values of all three 1II 3 (Fi,F2,F3) crossings are determined accurately at many perturbations, for example, (56.5, 61.5, 66.0) in Si160 A1 v = 0. Each 3X TV-level consists of three near-degenerate J components. The perturbation selection rule is A J = 0, thus the F3 and F3... 

Callis PR, Scott TW and Albrecht AC (1983) Perturbation selection rules for multiphoton electronic spectroscopy of neutral alternant hydrocarbons. J Chem Phys 78 16-22... 

They are caused by interactions between states, usually between two different electronic states. One hard and fast selection rule for perturbations is that, because angidar momentum must be conserved, the two interacting states must have the same /. The interaction between two states may be treated by second-order perturbation theory which says that the displacement of a state is given by... 

Two states /a and /b that are eigenfunctions of a Hamiltonian Hq in the absence of some external perturbation (e.g., electromagnetic field or static electric field or potential due to surrounding ligands) can be "coupled" by the perturbation V only if the symmetries of V and of the two wavefunctions obey a so-called selection rule. In particular, only if the coupling integral (see Appendix D which deals with time independent perturbation theory)... 

For forbidden transitions in atoms and molecules this phenomenon may be experimentally observed in spectra induced by collisions. As is known, the selection rules on some transitions may be cancelled during collision. The perturbers are able to induce a dipole moment of transition having the opposite direction in successive collisions due to intercollisional correlation. Owing to this, the induced spectra do involve the gap (Fig. 1.7), the width of the latter being proportional to the gas density [46, 47], Theorists consider intercollisional correlation to be responsible for the above phenomenon [48, 49, 50]. 

In collaboration with E.L. Sibert, we have learned to interpret these Franck-Con-don forbidden, pure torsional band intensities in S,-S0 absorption spectra quantitatively and thus place the key ml+ assignment on firm ground.27 The forbidden bands follow the selection rule Am = 3, so we need a perturbation of the form Vel cos 3a. Working in an adiabatic representation with the S0 and S, electronic states denoted by y0(g a) and /,( a) and the torsional states by m" and m, the electric dipole transition moment is,... 

The zeroth-order Hamiltonian and the spin-orbit part of the perturbation are diagonal with respect to the quantum numbers K, S, P, Ur, It, t>c, and lc-The terms of H involving the parameters aj, ac, and bo are diagonal with respect to both the lT and lc quantum numbers, while the hi term connects with one another the basis functions with l T = lT 2, l c = Zc T 2. The c terms couple with each other the electronic species —A and A. The selection rules for the vibrational quantum numbers are v Tjc = vT/c, t)j/c 2, vT/c 4. 

The predominant term in the perturbing potential V is of the form er, equal to the electric dipole moment operator. This is the origin of the selection rule that if ( 0, er i) = 0, the perturbed secular equation will not mix the states xpo and t/ i) so that the transition tpo ip i will not occur. 

A further variation on the theme of emission is circularly polarized emission, where chiral interactions, for example between a lanthanide complex and a chiral ligand in solution, can be studied. Selection rules have been given619 based on S, L and / values for 4/states perturbed by spin-orbit coupling and 4/ electron-crystal field interactions, and four types of transition were predicted to be highly active chiroptically. These are given in Table 12. 

The selection rule (4.138) differs from previously discussed selection rules in that it holds well for nonradiative transitions, as well as for radiative transitions. In deriving (4.138), we made no reference to the operator d, beyond the statement that it did not involve the nuclear spin coordinates. For any time-dependent perturbation that does not involve nuclear spin, the selection rule (4.138) will hold. Thus molecular collisions will not cause nonradiative transitions between symmetric and antisymmetric rotational levels of a homonuclear diatomic molecule. If we somehow start with all the molecules in symmetric levels, the collisions will not populate the antisymmetric levels. 

The transition operator J t) is determined by the perturbed Hamiltonian H and, in particular, transforms in the same manner as T does under the symmetry group of H0. The manner in which J(t) and Y transform determines selection rules, through the use of the Wigner-Eckart theorem. 

Selection rules also arise on considering the point-group symmetry of tfo(Qeq). In the case of electric dipole radiation the perturbation Y, which describes the interaction with the radiation field, may be expressed in terms of the x, y, z components of the dipole moment operator r. The operators (t) transform as the x, y, or z components of r. 

The remark made previously about the applicability of the selection rules for predissociation reactions now becomes clearer, since these selection rules merely describe properties of the matrix element v2. That is, although no assumption about the decay process has been directly introduced, the lifetime against decay will take a form similar to that obtained from first-order, time-dependent perturbation theory, and therefore be proportional to p2v2. 

The well-known selection rules proposed by Woodward and Hoffman25 to predict the stereochemical course of electrocyclic reactions can be viewed as emphasizing the symmetry requirements for electronic coupling of final and initial states. The rules are expressed in terms of rotatory motions required to convert one electronic state into another, so the matrix element is really vibronic rather than pure electronic. In terms of this paper, it appears that Woodward and Hoffman have identified necessary rotation properties of the perturbation operator. 

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