Vibrational wave function, proton

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 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 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 =... 

Fig. 2 (a) Two inde[>endent single-well potential curves for the vibrating OH groups, (b) The double-well potential curve for the hydrogen-bonded OH group (-OH 0= vs. =0- HO-). The solid curve indicates a symmetric combination of the two individual proton wave-functions and the broken curve is for an antisymmetric combination. The tunnelling splitting (Ao) is also shown. 

The vibrationally adiabatic proton wave functions provide the most useful description for PCET reactions. For typical single proton transfer reactions, the lowest adiabatic vibrational state is a double well along Zp, as shown in Figure 2a. In gen-... 

A small proton polaron is different in some aspects from the electron polaron that is, the hydrogen atom is able to participate in the lattice vibrations in principle (in any case it is allowable for excited states see Section II.F), but the electron cannot. This means that one more mechanism of phonon influence on the proton polaron is quite feasible. That is, phonon fluctuations would directly influence wave functions of the protons and thereby contribute to the overlapping of their wave functions. In other words, phonons can directly increase the overlap integral in concept. Such an approach allows one to describe the proton transfer without using the concept of transfer from site to site through an intermediate state. 

The presence of the polar mode oriented in the direction of maximum conductivity gives grounds to postulate the feasibility of the polaronic mechanism of motion of the protons in this crystal. As was shown in the previous subsection (see also Refs. 57,58, and 191) the proton can traverse a comparatively large distance between the nearest sites of the protonic sublattice (0.4 to 0.8 nm) with the participation of vibrational quanta, that is, phonons the virtual absorption of such a quantum can appreciably increase the resonance integral of overlapping of the wave functions of the proton on the nearest sites see expression (318). 

The one-dimensional quadratic potential V = kx2 has been used for the description of covalent binding. The ground-state wave functions for a simple harmonic oscillator, /t and iR, have been used to describe the proton in the left and right wells. The force constant k has been determined from the stretch-mode vibrational transitions for water occurring at 3700 cm-1. The ground-state energy for the proton is 0.368 x 10-19 J. The tunneling barrier is AE = 4 x 10-19 J. 

Fitting procedures give information on wave functions via mean-square displacements (ufj for each vibration and effective oscillator masses. It transpires that proton dynamics for bending modes correspond very closely to isolated harmonic oscillators with a mass of 1 amu [Ikeda 2002], They are largely de-... 

The presence of two minima always leads to a modification of the harmonic vibrational structure. If the two minima are equivalent, the lowest level will have one symmetric and one antisymmetric proton wave function. The larger the barrier, the closer the two lowest vibrational levels, until they become almost degenerate. The wave function of the proton is then a superposition of the symmetric state and the antisymmetric state, localized in either of the two minima. 

---