Quantum resonances

The reactant R2 can also be considered to be a solvent molecule. The global kinetics become pseudo first order in Rl. For a SNl mechanism, the bond breaking in R1 can be solvent assisted in the sense that the ionic fluctuation state is stabilized by solvent polarization effects and the probability of having an interconversion via heterolytic decomposition is facilitated by the solvent. This is actually found when external and/or reaction field effects are introduced in the quantum chemical calculation of the energy of such species [2]. The kinetics, however, may depend on the process moving the system from the contact ionic-pair to a solvent-separated ionic pair, but the interconversion step takes place inside the contact ion-pair following the quantum mechanical mechanism described in section 4.1. Solvation then should ensure quantum resonance conditions. 

Quantum Resonance Spectrum of the H3+ Molecular Ions for J=0. An Accurate Calculation Using Filter Diagonalization. 

At those values of the thickness, 7 = 1, as if the barrier does not exist. Such a quantum resonance phenomenon is currently the subject of intense theoretical and experimental research (see for example, Escapa and Garcia, 1990). 

S. Zhdanovich, C. Bloomquist, J. Floss, I. Sh. Averbukh, J. W. Hepburn, and V. Milner. Quantum resonances in selective rotational excitation of molecules with a sequence of ultrashort laser pulses. Phys. Rev. Lett., 109(4) 043003 (2012). 

Fig. 9.23. Square-well model for electronic mixing between two discrete states. The displacement toward resonance is derived from modulation of the energy levels by the coupling of the electronic levels to the nuclear motion of the surrounding medium. In configuration A, the electron is localized at the donor site B corresponds to the condition of quantum resonance between the two states C corresponds to the nuclear configuration in which the electron becomes localized on the acceptor site (Reprinted from R. J. D. Miller, G. McLendon, A. J. Nozik, W. Schmickler, and F. Willig, Surface Electron Transfer Processes, p. 4, copyright 1995 VCH-Wiley. Reprinted by permission of John Wiley Sons, Inc.)...

In particular, irregular vibrational spectra with Wignerian level spacing statistics have been observed this last decade for a number of highly excited molecules [3-7]. On the other hand, many recent works have characterized the reactive dynamics in terms of quantum resonances, which allows a rigorous definition of metastable states with finite lifetimes and hence of dissociation rates [4, 8-10]. 

Moreover, new semiclassical methods have been developed that are based on the Gutzwiller and Berry-Tabor trace formulas [12, 13]. These methods allow the calculation of energy levels or quantum resonances in systems with many interfering periodic orbits, as is the case for chaotic dynamics. 

Figure 19. Comparison of the quantum (filled circles, long dashes) and the classical (solid lines) rotational product distributions of C>2(n = 0) following the dissociation of HO2 for four energies as indicated the precise energies of the corresponding quantum resonances are 0.1513, 0.2517, 0.3507, and 0.4471 eV, respectively. Also shown are the distributions obtained from PST (short dashes). All distributions are normalized so that the areas under the curves are equal. The arrows on the 7 axis indicate the highest accessible state at the respective energy and the vertical bars on the classical curves indicate7sACM, the highest populated state according to the SACM. (Reprinted, with permission of the American Institute of Physics, from Ref. 37.)...

At this point, we have a comprehensive picture of quantum resonance states. The discussion above shows that the non-Hermitian stationary resonance solutions of Section 3 are real flesh and blood beings even in the Hermitian world dictated by the TDSE. As a wave packet evolves with time,... 

M.J. Redmon, R.E. Wyatt, Quantum resonance structure in the three-dimensional F+H2 reaction, Chem. Phys. Lett. 63 (1979) 209. 

Figure 4.4. Bonino s 1934 formula for benzene. The internal arrows show the negative carbon atoms and the corresponding quantum resonance between forms I and II. The horizontal arrows indicate that the configurations of the carbon atoms have the same probability.

Another unexpected feature of the quantum kicked rotor is the existence of resonances that lead to quadratic energy gain. Quantum resonances in the kicked rotor were first analysed by Izrailev and Shepelyan-skii (1979). They showed that whenever the quantum control parameter T is a rational multiple of tt, the quantum energy grows quadrati-... 

The EPR evidence indicates that 4 is a triplet diradical. Especially important is the obsowation of a half-field (double-quantum) resonance for 4 at 1450 G. The room temperature EPR spectrum of the solid reaction mixture is also consistent with a triplet diradical. The central peak with 5.8 G width is accompanied by side peaks characteristic of a triplet state species [8], with iD = 28 G, = 2G(Fig. 2). 

The pulse sequence INADEQUATE (Incredible Natural Abundance DoublE QVAntum Transfer Experiment) was developed by Freeman to suppress the usual (single-quantum) resonances and exhibit only the satellite (double-quantum) resonances. The pulse sequence is 90°-T-180y-T-90°-A-90. The homonuclear 180° pulse refocuses field inhomogeneities, but allows the vectors from different coupling arrangements to contin-... 

Fig. 3.2.1 [Giin2] Energy level diagram and powder spectra of the deutron. The wideline spectrum consists of two superimposed powder spectra. The double-quantum resonance at frequency a)2Q appears independent of molecular orientation.

Vibrational Relaxation. Stochastic processes, including vibrational relaxation in condensed media, have been considered from a theoretical standpoint in an extensive review,502 and a further review has considered measurement of such processes also.503 Models have been presented for vibrational relaxation in diatomic liquids 504 and in condensed media,505 using a master-equation approach. An extensive development of quantum ergodic theory for relaxation processes has been published,506 and quantum resonance effects in electronic to vibrational energy transfer have been considered.507 A paper has also considered the coupling between vibrational relaxation and molecular electronic transitions.508 A theory has also been outlined for the time-resolved electronic absorption spectrum of a molecule undergoing collisional vibrational relaxation.509... 

The probability of W exchange decreases with hAco. However, when the defect of resonance is big (hAco tico), multi-quantum resonant W -exchange processes become important (Nikitin, 1970) ... 

The probabiUty of the multi-quantum resonant exchange can be estimated as... 

Mandelshtam, V.A., Taylor, H.S. The quantum resonance spectrum of the H3+ molecular ion for / = 0. An accurate calculation using filter diagonatization, J. Chem. Soc. Faraday Trans. 1997, 93, 847-60. 

To follow the scale of complexity, the review is divided into three parts. The first two parts deal with the key concept of effective Hamiltonians which describe the dynamical and spectroscopic properties of interfering resonances (Section 2) and resonant scattering (Section 3). The third part. Section 4, is devoted to the resolution of the Liouville equation and to the introduction of the concept of effective Liouvillian which generalizes the concept of effective Hamiltonian. The link between the theory of quantum resonances and statistical physics and thermodynamics is thus established. Throughout this work we have tried to keep a balance between the theory and the examples based on simple solvable models. 

The two previous secfions were devoted to modeling quantum resonances by means of effective Hamiltonians. From the mathematical point of view we have used two principal tools projection operators that permit to focus on a few states of interest and analytic continuation that allows to uncover the complex energies. Because the time-dependent Schrodinger equation is formally equivalent to the Liouville equation, it is attractive to try to solve the Liouville equation using the same tools and thus establishing a link between the dynamics and the nonequilibrium thermodynamics. For that purpose we will briefly recall the definition of the correlation functions which are similar to the survival and transition amplitudes of quantum mechanics. Then two models of regression of a fluctuation and of a chemical kinetic equation including a transition state will be presented. 

We outline briefly in this section how to link the theory of quantum resonances to statistical physics and thermodynamics by extending the concept of effective Hamiltonian as recently discussed in Ref. [60]. The quantum Liouville-von Neumann equation is written in the form... 

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