Proton wave functions

Thus, overcoming the activation barrier is performed here by fluctuation of the solvent polarization to the transitional configuration P, whereas electron-proton transmission coefficient is determined by the overlap of the electron-proton wave-functions of the initial and final states. 

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-... 

In 1969, Thomas published two papers [11,12] in which a molecular structure theory was developed without invoking the Bom-Oppenheimer approximation. In these publications and two further papers published in 1970 [13,14], Thomas studied methane, ammonia, water and hydrogen fluoride adding the kinetic energy operators of the protons to the electronic hamiltonian and using Slater-type orbitals centered on the heavier nuclei for the protonic wave functions. Over the years, a number of authors [15-23] have attempted the development of a non-Bom-Oppenheimer theory of molecular structure, but problems of accuracy and/or feasibility remain for applications to arbitrary molecular systems. 

Proton Wave Functions in Palladium Studied by Ab Initio Calculations and Inelastic Neutron Scattering Methods... 

Calculations of the proton wave functions and the Inelastic scattering cross sections... 

The beauty of this method is that it yields the actual proton wave function for each excited state. We can thus directly use Fermi s Golden Rule to predict the inelastic neutron scattering cross-section for comparison with experiment. 

CALCULATIONS OF THE PROTON WAVE FUNCTIONS AND THE INELASTIC SCATTERING CROSS SECTIONS... 

It is, of course, not possible to include a statistical weighting function exp — WJkT) directly in the proton wave function, since this must actually be inserted in the density matrix. The total jump rate, however, amounts essentially to a summation of quantum contributions like (9.68) from energy levels below the barrier top together with a classical contribution like (9.59) frorh higher levels. 

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. 

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... 

In this paper, we present a three dimensional (3D) model in order to describe the spatial extension of the proton wave functions in a more realistic way. Furthermore, each potential well associated to the molecules A and B will be assumed to be spherical with a finite constant depth Vq. Qualitatively, the 3D nature of the wells will considerably reduce the overlap between the wave functions of the two individual wells, thus decreasing the resonant proton transfer rate. The constant depth of the wells avoids the unrealistic presence of a sharp peak at... 

Very recently, Singleton has suggested a new way of calculating isotope effects illustrated by the bromonium ion [61]. This method could possibly also be used for tautomeric systems. Another new approach is the multicomponent molecular orbital method for direct treatment of nuclear quantum effects [62]. The basic idea is to incorporate the nuclear wave function and in particular the proton wave function directly into the electronic structure calculation. This approach has great potential but has so far been tested only for secondary isotope effects on chemical shifts [63]. The geometric isotope effect has also been looked into based on Pauling valence-bond orders [20]. 

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