Proton vibrational wave

An electronically adiabatic proton transfer reaction may be either vibrationally adiabatic or vibrationally non-adiabatic. Vibrationally adiabatic refers to the situation in which the proton responds instantaneously to the solvent, while vibrationally non-adiabatic refers to the opposite limit. The adiabatic proton vibrational wave functions are calculated if the Schrodinger equation is solved for fixed values of Zp. 

The two-dimensional electron transfer diabatic free energy surfaces in Figure 7 have been analyzed with the Golden Rule rate expression given in Eq. 46. This analysis suggests that FT and EPT are possible for both systems, but FT is the dominant path due to significant overlap between the proton vibrational wave... 

The distance between the proton donor and acceptor also affects the rates and mechanisms of PCET reactions. As this distance decreases, the barrier along the proton coordinate rp decreases and eventually disappears. As illustrated in Figures 3-5, the height of this barrier determines the number of localized proton vibrational states. In particular, if the barrier along the proton coordinate is very low or nonexistent, the proton vibrational wave functions are mixtures of a and b, so the distinction between ET and EPT is unclear. For systems in which the potential is a double well along the proton coordinate, however, the rate of EPT decreases as the barrier along the proton coordinate increases due to the decrease of the overlap of the proton vibrational wave functions for the a and b states. 

Drukker, K., Hammes-Schiffer, S. An analytical derivation of MC-SCF vibrational wave functions for the quantum dynamical simulation of multiple proton transfer reactions Initial application to protonated water chains. J. Chem. Phys. 107 (1997) 363-374. 

Once the gas phase Hamiltonian is parametrized as a function of the inner-sphere reaetion coordinate(s), the free energy is calculated as a function of the proton coordinate(s), the scalar solvent coordinates, and the inner-sphere reaction coordinate(s). Note that this approaeh assumes that the optimized geometries of the VB states are not significantly affected by the solvent. For proton transfer reactions, the proton donor-acceptor distance may be treated as an additional solute reaction coordinate that ean be incorporated into the molecular mechanical terms describing the diagonal matrix elements hf- and, in some cases, the off-diagonal matrix elements (/io)y. If the inner-sphere reaction coordinate represents a slow mode, it is treated in the same way as the solvent coordinates. As discussed throughout the literature, however, often the inner-sphere reaction coordinate must be treated quantum mechanically [27, 28]. In this case, the inner-sphere reaction coordinate is treated in the same way as the proton coordinate(s), and the vibrational wave functions depend explicitly on both the proton coordinate(s) and the inner-sphere reaction coordinate(s). 

A very peculiar dynamics has been revealed in the Ca(OH)2 crystal by means of inelastic neutron scattering technique [26]. It has been found that anharmonic terms must be included, which mix the vibrational states of the OH and lattice modes. In particularly, the lattice modes have successfully been represented as the superposition of oxygen and proton synchronous oscillators, and it appears that the proton bending mode Eu is strongly coupled to the lattice modes. The contribution of the proton harmonic wave functions has been taken as the zero-order approximation. 

Another approach has been proposed in Ref. 191. The approach is based on the model of small polaron and makes it possible to extend the range of the model. In particular, it includes the influence of vibration wave functions on the tunnel integral and provides a way of the estimation of diagonal and off-diagonal phonon transfers on the proton polaron mobility. It turns out that the one phonon approximation is able significantly to contribute to the proton mobility. Therefore, we will further deal with matrix elements constructed on... 

The 2D model of the linear B- -H-A fragment, assuming a strong coupling between the proton (AH stretch) and low-frequency (B- -A stretch) coordinates, was introduced by Stepanov [9, 10]. It seems to be the simplest model enabling one to interpret the different specific features of H-bonded systems [11-16]. In terms of this model, the vibrational wave function of the H-bonded system is written as... 

In this formalism the mobility is determined partly by the value of AW (which occurs as AW ) and partly by the scattering of ion-state waves by lattice vibrations in the form of phonons. Because the ion-state band is so narrow and the difference between proton vibrational levels is so large compared with typical phonon energies, it is not possible for (k) to be scattered to a new state (k ) by emission or absorption of a single phonon. Instead, so that energy and momentum can be conserved, scattering must occur by the simultaneous emission and absorption of a pair of phonons of nearly equal energy. The analysis is therefore rather complicated but, if we assume that orientational defects are present in sufficient concentration that polarization effects do not block ion paths, the... 

In vibrational Raman scattering, which is the primary technique of interest in studies of proton conductors, the Born approximation is invoked to write the state i > as the product of an electronic wave function, e>, and a vibrational wave function, d >, i.e. i> = k> o >. The subscript m on the vibrational wave function designates the mth normal vibrational mode. Usually the transition j k occurs between the vibrational level v in the ground electronic state g and the vibrational level v also in the ground electronic state, so the transition jy - k> may be written Gy m + f>- Here the harmonic oscillator selection rule =... 

The free-energy profile is calculated by the FEP/US method (see section 16.3.3.3). However, at each step of the molecular dynamics simulation, the vibrational energy and the wave function of the transferred proton are determined from a three-dimensional Schrodinger equation and are included in the FEP/US procedure. In addition, dynamical effects due to transitions among proton vibrational states are calculated with a molecular dynamics with quantum transition (MDQT) procedure in which the proton wave function evolution is determined by a time-dependent Schrodinger equation. This procedure is combined with a reactive flux approach to calculate the transmission... 

The infrared spectra of some monosubstituted and the unsubstituted oxadiazoles show the vibration bands of the protons directly attached to the heterocycle in position 3 or 5- The wave length is 3.19—3.22 p, i.e. almost the same as for an ethylenic CH bond. 

Ultrafast proton transfer. The diffusion-controlled limit for second-order rate constants (Section A3) is 1010 M 1 s 1. In 1956, Eigen, who had developed new methods for studying very fast reactions, discovered that protons and hydroxide ions react much more rapidly when present in a lattice of ice than when in solution.138 He observed second-order rate constants of 1013 to 1014 M 1 s These represent rates almost as great as those of molecular vibration. For example, the frequency of vibration of the OH bond in water is about 1014 s . The latter can be deduced directly from the frequency of infrared light absorbed in exciting this vibration Frequency v equals wave number (3710 cm-1 for -OH stretching) times c, the velocity of light (3 x 1010 cm s ). 

Films of the complexes are stable in water at a pH of 7 while they dissolve at pH 5. This can be explained by the pKa value of retinoic acid, which is, for example, 6.05 in 150 mM NaCl and 6.49 in 5 mM NaCl [163]. Therefore, the anionic retinoic moieties within the complexes will be protonated at pH values lower than the pKa which lead to the cleavage of the ionic bonds in the complexes. The first experiments to evaluate the release properties of retinoic acid from thin films of the complexes were performed by using FTIR and surface tension measurements. Films were immersed in solutions of 0.15 m sodium chloride at pH 5 for both methods. The increase of the absorbance at 1255 cm-1 (C-0 stretch vibration) [186] in the FTIR spectra was used as a qualitative measure for the release of retinoic acid from the PEI-retinoate complexes. For comparison, the spectra of the complex and of the non-com-plexed retinoic acid are shown at wave numbers around 1255 cm 1 (Fig. 26, insert curves a and b). The time-dependency of the absorbance, which is a relative measure of the amount of released retinoic acid, is shown in Fig. 26. It can be seen that the increase of the absorbance, and therefore the release... 

The only way in which such molecules can be demonstrated to occur as linear vibrating pairs of atoms, is by confinement as guest species in crystals. Even this situation is contingent on directed interaction with the host lattice, in the absence of which the guest appears structureless, or disordered. The general conclusion must be that protons, like electrons, appear as point particles only in close confinement. Protons and neutrons must, like electrons, logically be considered as distortions of the aether as compressible and flexible fluids. Despite differences in mass and topological structure these different particles must therefore all have quantum-mechanical properties. In observation they would display the type of behaviour that seems to imply a dual wave-particle structure. 

---