Mixed parity states

Hund, one of the pioneers in quantum mechanics, had a fundamental question of relation between the molecular chirality and optical activity [78]. He proposed that all chiral molecules in a double well potential are energetically inequivalent due to a mixed parity state between symmetric and antisymmetric forms. If the quantum tunnelling barrier is sufficiently small, such chiral molecules oscillate between one enantiomer and the other enantiomer with time through spatial inversion and exist in a superposed structure, as exemplified in Figs. 19 and 24. Hund s theory may be responsible for dynamic helicity, dynamic racemization, and epimerization. 

In the "nonrigid symmetric-top rotors" (such as NH ), the second-order Stark effect is observed under normal circumstances. Indeed, field strengths of the order of 1 600 000 [V/m] are required to bring the interaction into the first-order regime in this case [18]. In contrast, very weak interactions suffice to make the mixed-parity states and appropriate for the description of optically active systems. Parity-violating neutral currents have been proposed as the interaction missing from the molecular Hamiltonian [Eq.(1)] that is responsible for the existence of enantiomers [14,19]. At present, this hypothesis is still awaiting experimental verification. 

The fact that states with definite parities are almost never observed experimentally cannot be explained alone by the persistence of mixed-parity states. Obviously, symmetry-breaking phenomena are in operation whether or not external perturbations are present. For species in condensed media, symmetry breaking is brought about by intermolecular interactions. In the extreme case of solids, these interactions are so strong that it is proper... 

Intermolecular interactions are capable of stabilizing broken-symmetry states in other ways. For example, contrary to intuitive expectations, random molecular collisions tend to preserve mixed-parity states [23]. 

These relations are the same as the parity rules obeyed by the second derivative of the second entropy, Eqs. (94) and (95). This effectively is the nonlinear version of Casimir s [24] generalization to the case of mixed parity of Onsager s reciprocal relation [10] for the linear transport coefficients, Eq. (55). The nonlinear result was also asserted by Grabert et al., (Eq. (2.5) of Ref. 25), following the assertion of Onsager s regression hypothesis with a state-dependent transport matrix. 

If the rare-earth ion is immersed in a crystal field, the perfect symmetry of the free ion is destroyed, leaving parity in some cases not quite a good quantum number. Under this circumstance, electric-dipole transitions become quite possible. It was Van Vleck (25), in his classic 1937 paper The Puzzle of Rare Earth Spectra, who first pointed out that the weak electric dipole emission was due to this mixing of states of opposite parity by the crystal field. 

Figure 3. In (a) the potential curve is unsymmetric with respect to the equilibrium position 0 of the nucleus. The crystal field in this case causes mixing of the even and odd parity states. In (b) there is symmetry with respect to the nucleus when it is at 0, but vibration carries the ion to the unsymmetric point P [from Ref. (25)].

In Fig. 5. The structure of the negative parity states is as follows. The low lying 3 and 5 states appear to be rather complicated as their population does not follow the (2J+1) population expected for a simple multiplet. In contrast, the 7 , 9 and 10 appear to have equivalent f7/2 il3/2 strength and thus have rather simple structure. On the %other hand, no evidence is found for the 4 or 8 states and only weak evidence that the state at 3.3 MeV is a 6 state. Configuration mixing is a possible, though not a certain, reason for the absence of these states. In any case, there does not appear to be a simple vf7/2 v13/2 multiplet. 

Using (8.432) and (8.433) the Stark energies for J = 2, S2 = 2 can be readily calculated and the results are presented in figure 8.50 the initial splitting of the /1-doublets was determined from the electric resonance study to be 7.351 MHz for the v = 0 level. In small electric fields the parities of the states are essentially preserved, and transitions between the /1-doublets have their full electric dipole intensities. At higher electric fields, however, the opposite parity states are mixed and the electric dipole intensity decreases. It follows that so far as the intensities of the electric resonance transitions are concerned, low electric fields are desirable. On the other hand, Stern, Gammon,... 

Considering the crystal electric field as a first-order perturbation, the mixing of states with higher energy and opposite parity, nla"[S"l"]J"M") may be represented by iff") with A) and Z ) defined as... 

The nonlinear susceptibility is evaluated using third-order perturbation theory, and resonant enhancement is readily demonstrated to occur. Four-wave mixing is a useful experimental technique to extend the energy range available to tunable dye lasers [468]. It is also of interest that processes involving excitation by three photons allow transitions between even and odd parity states to be excited, as do single-photon transitions. 

In mixed-parity MBPT, we modify Hq by adding the weak-interaction h v to the HF potential. This approach leads to a generalization of the single-particle states in which each state acquires an opposite-parity admixture,... 

The Judd-Ofelt theory explains how the observation of strictly parity forbidden transitions results from non-centro-symmetric interactions that lead to a mixing of states of opposite parity. One of the most obvious mechanisms is simply the coupling of states of opposite parity by the odd terms of the crystal field expansion. The transition, giving a... 

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. 

If there is no inversion symmetry at the site of the rare-earth ion, the uneven cry.stal field components can mix opposite-parity states into the 4/"-configurational levels (Sect. 2.3.3). The electric-dipole transitions are now no longer strictly forbidden and appear as (weak) lines in the spectra, the so-called forced electric-dipole transitions. Some transitions, viz. those with AJ = 0, 2, are hypersensitive to this effect. Even for small deviations from inversion symmetry, they appear dominantly in the spectrum. 

---