Proton coordinate

Figure 43. Probability density of the wavepacket after 30 fs of field-free propagation of the accelerated wavepacket as a function of (a) proton coordinate, (b) N=C bond length, (c) proton momentum, and (d) N=C bond momentum. Taken from Ref. [41].

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). 

Thus, for a transition between any two vibrational levels of the proton, the fluctuation of the molecular surrounding provides the activation. For each such transition, the motion along the proton coordinate is of quantum (sub-barrier) character. Possible intramolecular activation of the H—O chemical bond is taken into account in the theory by means of the summation of the probabilities of transitions between all the excited vibrational states of the proton with a weighting function corresponding to the thermal distribution.3,36 Incorporation in the theory of the contribution of the excited states enabled us in particular to improve the agreement between the theory and experiment with respect to the independence of the symmetry factor of the potential in a wide region of 8[Pg.135]

If prior to the electron transition the potential energy surface along the proton coordinate r had a minimum corresponding to a stable chemical bond, various situations are possible after the change of the electron state due to the electron transition ... 

Measurement of pH-dependent equilibria can also be used to identify coordination isomerization reactions in addition to stepwise dissociation, such as in the case of the iron(III) complex of exochelin MN (59). Here, a combination of spectrophotometric and potentiometric titration characterized multiple equilibria involving second-sphere protonation, coordination isomerization, and stepwise dechelation, and is illustrated in Fig. 8. 

The formula (5.14) has been applied, for example, by Hutchison and McKay144 to determine the proton coordinates in Nd(III)-doped lanthanum nicotinate dihydrate crystals, and by Balmer et al.101) to localize the charge compensator H+ in Co(II)-doped a-Al203. 

An asymmetric hydrogen bond is common even where a proton coordinates two equivalent anions. The rc-bond repulsive forces between two coordinated anions tend to prohibit a close X-H-X separation, so competition between the two equivalent anions for the shorter X-H bond may set up a double-well potential for the equilibrium proton position between the two coordinated anions. With oxide anions, an O-H-O separation greater than 2.4 A sets up a double-well potential and creates an asymmetric hydrogen bond, which we represent as O-H O. Although displacement toward one anion may be energetically equivalent to a displacement toward the other, one well is made deeper than the other by an amount AH, as a result of the motion of the proton from the centre of the bond. 

We have seen that copper(II) is a slowly relaxing metal ion. Magnetic coupling of copper to a fast relaxing metal ion increases the electron relaxation rate of copper, as clearly shown by the NMRD profiles of tetragonal copper(II) complexes reacting with ferricyanide (105) (Fig. 38). The electron relaxation time, estimated from the relaxation rate of the water protons coordinated to the copper ion, is 3 x 10 ° s, a factor of 10 shorter than in the absence of ferricyanide. 

This section intends to provide the reader with some examples of how the high relaxivity challenge has been tackled for Gd(III) complexes so far. For the sake of clarity, this section has been divided in three sub-sections in relation to the specific relaxation parameter considered for relaxivity enhancement, namely the hydration state of the metal centre, the tumbling rate of the CA, and the exchange rate of the mobile protons coordinated to the paramagnetic center. 

The potential (6.37) corresponds with the previously discussed projection of the three-dimensional PES V(p,p2,p3) onto the proton coordinate plane (pi,p3), shown in Figure 6.20b. As pointed out by Miller [1983], the bifurcation of reaction path and resulting existence of more than one transition state is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of the PES (6.37) was carried out by Benderskii et al. [1991b], The existence of the onedimensional optimum trajectory with q = 0, corresponding to the concerted transfer, is evident. On the other hand, it is clear that in the classical regime, T > Tcl (Tc] is the crossover temperature for stepwise transfer), the transition should be stepwise and occur through one of the saddle points. Therefore, there may exist another characteristic temperature, Tc2, above which there exists two other two-dimensional tunneling paths with smaller action than that of the one-dimensional instanton. It is these trajectories that collapse to the saddle points at T = Tcl. The existence of the second crossover temperature Tc2 for two-proton transfer was noted by Dakhnovskii and Semenov [1989]. 

Bencini, A., Bianchi, A., Garcia-Espana, E., Micheloni, M., Ramirez, J. A., Proton coordination by polyamine compounds in aqueous solution. Coord. Chem. Rev. 1999, 188, 97-156. 

Let us assume that the lifetime of the delocalized state is limited by proton transfer between one of the base pairs such that as soon as one proton coordinate cross the XT/CT intersection, the delocalized state collapses to from a localized state. Assuming the usual Condon separation between the nuclear and electronic dynamics, we can write this within the non-adiabatic Marcus approximation... 

The lack of a deuterium isotope effect [29, 32, 34, 38, 51] may be attributed to the reaction coordinate not being identibed with the proton coordinate. There is precedent for this in other systems [52-54]. 

It must be remembered, furthermore, that the identification of the H-atom translocation mode is not equivalent to the identification of the reaction coordinate. We have attributed the absence of a deuterium isotope effect on the excited-state H-atom transfer (for the 10-ps component in hypericin and hypo-crellin A) to the zero-point energy in the proton coordinate lying above the barrier, with the H-atom being effectively delocalized between the two oxygen atoms. Consequently, the reaction coordinate for the excited-state H-atom transfer cannot be identified with the proton coordinate, and it must be concluded that other intramolecular motions are in fact responsible for the process. Temperature-dependent measurements indicate that these motions are extremely low amplitude, Ea 0.05 kcal/mol for hypericin [37]. Because the nature of this motion is not yet identified, we refer to it as the skeleton coordinate [48, 71, 82]. We propose that it is the time scale for this latter conformational change... 

Figure 4.1. Electron and proton coordinates. X, Y, Z are space-fixed axes with origin at the nucleus.

In both of the above treatments, spherical tensor and cartesian, we have factored the quadrupole interaction into the product of two terms, one of which operates only on functions of proton coordinates within the nucleus and the other only on functions of coordinates of electrons and protons outside the nucleus. We shall see in subsequent chapters that the spherical tensor form is rather more convenient for the calculation of matrix elements of 3Cq. However, we shall find this easier to appreciate once we have considered some of the theory of angular momentum in chapter 5 so we defer discussion until later. 

Bransted acidity or basicity develop when the H2O molecule dissociates. Two different hydroxyl coordinations can develop on the MgO surface. As shown in Fig. 4.57, one hydroxyl becomes end-on coordinated to Mg2+, the proton coordinates to the basic bridging oxygen anion. 

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