Selection rules for perturbations

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... 

The spin-spin interaction is zero for E states with S < The other selection rules for the Hss operator are g g or u u, but the selection rule E1 1 E is opposite to that for the spin-orbit operator, which is E1 1 E. Note, however, that the spin-spin interaction is zero between triplet and singlet states if both of them are E states (for example, a 3Eq state has only / levels and the universal selection rule for perturbations is e / thus 1E 3E Hss perturbations are e/ / forbidden see the end of Section 3.4.5). 

When a diatomic molecule (guest) is present as a dilute impurity in an inert matrix (host), the selection rules for perturbations and electric dipole allowed transitions can be altered by guest-host interactions. For example, the inevitable absence of cylindrical symmetry (Coot or D h) at a matrix site destroys the distinction between 7r and S orbitals thus A A = 2 transitions [S2 and SO A 3 A— X3E and E-— a1 A (Lee and Pimentel, 1978, 1979) NO 2 — X2II (Chergui, et al., 1988) N2 w1 A — X E+ (Kunsch and Boursey, 1979)] and perturbations (Goodman and Brus, 1977) are quite common. 

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 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. 

Spectroscopy is concerned with the observation of transitions between stationary states of a system, with the accompanying absorption or emission of electromagnetic radiation. In this section we consider the theory of transition probabilities, using time-dependent perturbation theory, and the selection rules for transitions, particularly those relevant for rotational spectroscopy. 

In most of the examples described in this book, the rotational angular momentum is coupled to other angular momenta within the molecule, and the selection rules for transitions are more complicated than for the simplest example described above. Spherical tensor methods, however, offer a powerftd way of determining selection rules and transition intensities. Let us consider, as an example, rotational transitions in a good case (a) molecule. The perturbation due to the oscillating electric component of the electromagnetic radiation, interacting with the permanent electric dipole moment of the molecule, is represented by the operator... 

Dipole selection rules apply for excitation by single photons in the perturbative regime. Selection rules for multiphoton excitation are different (see chapter 9). For excitation by collisions with charged particles or by light beams of high intensity, turned on so fast that the normal conditions of perturbation theory do not apply, then there are no strict selection rules, although various propensity rules may still apply. 

The total parity of a given class of levels (F fine structure component for E-states, upper versus lower A-doublet component for II-states) is found to alternate with 7. The second type of label, often loosely called the e// symmetry, factors out this (—l) 7 or (—l)-7-1/2 7-dependence (Brown et al., 1975) and becomes a rotation-independent label. (Note that e/f is not the parity of the symmetrized nonrotating molecule ASE) basis function. In fact, for half-integer S, it is not possible to construct eigenfunctions of crv in the form [ A, S, E) —A, S, — E)], because, for half-integer S, vice versa.) The third type of parity label arises when crv is allowed to operate only on the spatial coordinates of all electrons, resulting in a classification of A = 0 states according to their intrinsic E+ or E- symmetry. Only A = 0) basis functions have an intrinsic parity of this last type because, unlike A > 0) functions, they cannot be put into [ A) — A)] symmetrized form. The peculiarity of this E symmetry is underlined by the fact that the selection rule for spin-orbit perturbations (see Section 3.4.1) is E+ <-> E, whereas for all types of electronic states and all... 

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 Perturbation selection rules for unsymmetrized 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,... 

Radford (1961, 1962) and Radford and Broida (1962) presented a complete theory of the Zeeman effect for diatomic molecules that included perturbation effects. This led to a series of detailed investigations of the CN B2E+ (v — 0) A2II (v = 10) perturbation in which many of the techniques of modern high-resolution molecular spectroscopy and analysis were first demonstrated anticrossing spectroscopy (Radford and Broida, 1962, 1963), microwave optical double resonance (Evenson, et at, 1964), excited-state hyperfine structure with perturbations (Radford, 1964), effect of perturbations on radiative lifetimes and on inter-electronic-state collisional energy transfer (Radford and Broida, 1963). A similarly complete treatment of the effect of a magnetic field on the CO a,3E+ A1 perturbation complex is reported by Sykora and Vidal (1998). The AS = 0 selection rule for the Zeeman Hamiltonian leads to important differences between the CN B2E+ A2II and CO a/3E+ A1 perturbation plus Zeeman examples, primarily in the absence in the latter case of interference effects between the Zeeman and intramolecular perturbation terms. 

The propensity rules for collision-induced transitions between electronic states and among the fine-structure components of non-1E+ states depend on the identity of the leading term in the multipole expansion of the molecule/collision-partner interaction potential. Alexander (1982a) has considered the dipole-dipole term, which included both permanent and transition dipole contributions. In the limit that first-order perturbation theory applies (not the usual circumstance for thermal molecular collisions), the following collisional propensity rules for the permanent dipole term can be enumerated from the selection rules for both perturbations and pure rotational transitions... 

The electronic selection rule for nonzero electrostatic or nuclear kinetic energy coupling matrix elements is quite simple for diatomic molecules only states with identical electronic quantum numbers can perturb each other. For polyatomic molecules, the situation is not always so simple. It is possible that two electronic states will have different electronic quantum numbers in a high-... 

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... 

Heuristic selection rules for one-photon transitions may be obtained by using Eq. 1.9 or 1.10 for the perturbation W in the expression for c (t), Eq. 1.97. This procedure yields the matrix element... 

In this chapter, we review electronic structure in hydrogenlike atoms and develop the pertinent selection rules for spectroscopic transitions. The theory of spin-orbit coupling is introduced, and the electronic structure and spectroscopy of many-electron atoms is greated. These discussions enable us to explain details of the spectra in Fig. 2.2. Finally, we deal with atomic perturbations in static external magnetic fields, which lead to the normal and anomalous Zeeman effects. The latter furnishes a useful tool for the assignment of atomic spectral lines. 

Kramers and Heisenberg [2], who predicted the phenomenon of Raman scattering several years before Raman discovered it experimentally, advanced a semiclas-sical theory in which they treated the scattering molecule quantum mechanically and the radiation field classically. Dirac [3] soon extended the theory to include quantization of the radiatiOTi field, and Placzec, Albrecht and others explored the selection rules for molecules with various symmetries [4, 5]. A theory of the resonance Raman effect based on vibratiOTial wavepackets was developed by Heller, Mathies, Meyers and their colleagues [6-11]. Mukamel [1, 12] presented a comprehensive theory that considered the nonlinear response functions for pathways in LiouvUle space. Having briefly described the pertinent pathways in Liouville space above, we will first develop the Kramers-Heisenberg-Dirac theory by a second-order perturbation approach, and then turn to the wavepacket picture. 

Although the most fundamental selection rule for rotational spectroscopy is that the molecule should have a nonvanishing permanent dipole moment, we note that molecules without a permanent dipole moment can have perturbation-allowed rotational spectrum [22]. For spherical tops, for example, centrifugal distortion effects can produce a small permanent dipole moment that allows the observation of the rotational spectrum [1, 37]. 

Selection rules for spectroscopic experiments are derived from time-dependent perturbation theory. Transitions are allowed if the integral of the perturbing Hamiltonian and initial and final stationary states is nonzero. For the harmonic oscillator, the allowed transitions are those for which the n quantum number changes by 1, provided that there exists a dipole that varies linearly with the separation distance. (N2 has a zero dipole because of its symmetry, and there is no linear variation with distance.) For the rigid rotator, the selection rules are that the allowed transitions are a change of 1 in the / quantum number, provided that there is a nonzero permanent dipole, po-... 

Using firsf-order time-dependent perturbation theory, find selection rules for elecfro-magnetic radiation inferacting with a particle in a box, taking the particle to possess a net charge. 

VIII. Time-Dependent Perturbations Radiation Theory Time-Dependent Perturbations, 107. The Wave Equation for a System of Charged Particles under the Influence of an External Electric or Magnetic Field, 108. Induced Emission and Absorption of Radiation, 110. The Einstein Transition Probabilities, 114. Selection Rules for the Hydrogen Atom, 116. Selection Rules for the Harmonic Oscillator, 117. Polarizability Rayleigh and Raman Scattering, 118. 

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