Proton vibrational states

As noted in Section 34.2, the proton coordinate (such as that of a quantum particle) should be eliminated from the free-energy surfaces used for calculation of the activation free energy. The characteristics of the proton are reflected in the energies at the points of minimum of these free-energy surfaces, which involve the energies of the initial, E°, and final, E°f, ground proton vibrational states, respectively. This is denoted by the superscript 0 in the free-energy surfaces Uf (P) and U° (P). 

Two papers have attempted to compare the observed infrared lines of the base pairs in the region of the hydrogen-bond absorption with the calculated data. Rein and Svetina have calculated the proton vibrational states and relative transition probabilities for two guanine-cytosine hydrogen bonds. Their preliminary results seemed to be consistent with the absorption peak at 3489 cm (0.436 eV) reported by Pitha et for the hydrogen bond N-H stretching mode of... 

The description of PCET reactions is particularly challenging due to the quantum mechanical behavior of the ET electrons, the PT electrons, and the transferring protons. The adiabatic mixed electronic/proton vibrational states are calculated when the following Schrbdinger equation is solved for fixed solvent coordinates... 

If all four of the new basis states are included, the adiabatic mixed electronic/proton vibrational states are exactly the same as those obtained with the original four VB states. Moreover, the diabatic mixed electronic/proton vibrational states for the new basis states are exactly the same as the adiabatic states obtained with the settings Ho)ia,2a = ( o)ia,26 = ( o)i6,2a = ( )i6,26 = (as described at the end of the previous section). 

The proton vibrational states can be calculated for each of the two electronic states by solution of the Schrodinger equation... 

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. 

Since the dielectric continuum representation of the solvent has significant limitations, the molecular dynamics simulation of PCET with explicit solvent molecules is also an important direction. One approach is to utilize a multistate VB model with explicit solvent interactions [34-36] and to incorporate transitions among the adiabatic mixed electronic/proton vibrational states with the Molecular Dynamics with Quantum Transitions (MDQT) surface hopping method [39, 40]. The MDQT method has already been applied to a one-dimensional model PCET system [39]. The advantage of this approach for PCET reactions is that it is valid in the adiabatic and non-adiatic limits as well as in the intermediate regime. Furthermore, this approach is applicable to PCET in proteins as well as in solution. 

Statistical averaging over all reactants , Sp, and summation over products proton vibrational states, Sp are included. This is important for kinetic isotope effects but often reduces to complete dominance of the ground vibrational states v = w = 0. 

Figure 10.14 Proton diabatic free energy curves versus the solvent reaction coordinate for individual reactant (n ) and product (rip) proton vibrational states. Several transitions are qualitatively indicated.

In a general formulation, excited proton vibrational states are included in the PT rate as a sum over all state-to-state PT rates from a proton reactant state... 

Figure 10.16 Log k versus AG n = 300 K) for H including excited proton vibrational states (solid lines). Dotted lines indicate individual contributions from 0-0, 0—1,1—0, and 0-2 transitions. Rate constants were calculated with Eqs. (10.36) and (10.37).

As for the ion AH itself, the calculation for the entirely nonadiabatic reaction does not require any serious limitations in the choice of the model. The calculation may be performed, taking account of all the intramolecular vibrations. However, for simplicity, we shall consider at first a simple model in which the reaction leads to a change of the characteristics of only the proton vibrational states and thus to be able to treat also partly adiabatic reactions (Figure 1). 

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... 

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