Probability proton density

For a given neutron, the probability of taking part in a nuclear combination is proportional to the proton density. High densities favour marriage, whilst low densities lead... 

It has already been pointed out that nitrosation is probably the first step in diazotization. Ingold (1952) describes the reaction as N-nitrosation and classifies it as an electrophilic substitution, together with related processes such as the formation of 4-nitrosophenol, an example of a C-nitrosation. It was probably Adamson and Kenner (1934) who first applied these ideas to diazotization and realized that in aniline itself the electron density at the nitrogen atom is greater than in the anilinium ion, so that the base is more reactive. On the other hand, the nitrosoacidium ion (3.1), the addition product of nitrous acid and a proton, is a more powerful electrophilic reagent than the HN02 molecule. They therefore represented the first step of diazotization as in Scheme 3-5. 

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].

At the platinum electrode the individual steps of the four-electron reaction cannot be studied separately. Slope b has its usual value of about 0.12 V, but in contrast to what is seen at the mercury electrode, the polarization is practically independent of solution pH (i.e., the potential at a given current density shifts by 0.06 V in the negative direction when the pH is raised by a unit). It follows that the reaction rate depends on hydrogen ion concentration. The step in which an electron and a proton are transferred while the 0-0 bond is broken is probably the ratedetermining step. 

Figure 2.9.3 shows typical maps [31] recorded with proton spin density diffusometry in a model object fabricated based on a computer generated percolation cluster (for descriptions of the so-called percolation theory see Refs. [6, 32, 33]).The pore space model is a two-dimensional site percolation cluster sites on a square lattice were occupied with a probability p (also called porosity ). Neighboring occupied sites are thought to be connected by a pore. With increasing p, clusters of neighboring occupied sites, that is pore networks, begin to form. At a critical probability pc, the so-called percolation threshold, an infinite cluster appears. On a finite system, the infinite cluster connects opposite sides of the lattice, so that transport across the pore network becomes possible. For two-dimensional site percolation clusters on a square lattice, pc was numerically found to be 0.592746 [6]. 

The two wave functions are now interpreted as (ip1)2 — probability density of the proton... 

Solutions to the Schrodinger equation Hcj) = E(f> are the molecular wave functions 0, that describe the entangled motion of the three particles such that (j) 4> represents the density of protons and electron as a joint probability without any suggestion of structure. Any other molecular problem, irrespective of complexity can also be developed to this point. No further progress is possible unless electronic and nuclear variables are separated via the adiabatic simplification. In the case of Hj that means clamping the nuclei at a distance R apart to generate a Schrodinger equation for electronic motion only, in atomic units,... 

According to Scheme 11, the isomeric ortho para) product ratios are established during the collapse of the radical pair in (64) (most probably at the positions of AN+- with the highest electron density). Furthermore, the absence of a measurable kinetic isotope effect in the decay of the deuterated analogue (C6D5OCH.v) in Table 3 is predicted from Scheme 11 since the proton loss occurs in a subsequent, rapid step (65). The absence of a deuterium kinetic isotope effect also indicates that the presence of pyridine in the triad in (63) does not lead to the nitroanisoles by an alternative... 

If the fraction of surface that is covered with protons is smaller than 1 (%h 1), the surface density of singly, doubly, and triply protonated surface sites (B, C, and D, respectively (see Fig. 5.6)) can be described as probability functions of the surface protonation Cf. ... 

A weakening of the critical metal-oxygen bonds occurs as a consequence of the protonation of the oxide ions neighboring a surface metal center and imparting charge to the surface of the mineral lattice. The concentration (activity) of D should reflect that three of such oxide or hydroxide ions have to be protonated. If there is a certain numer of surface-adsorbed (bound) protons whose concentration (mol nr2) is much lower than the density of surface sites, S (mol 2), the probability of finding a metal center surrounded with three protonated oxide or hydroxide ions is proportional to (CJ/S)3. Thus, as has been derived from lattice statistics by Wieland et al. (1988), the activity of D is related to (C )3, and the rate of proton-promoted dissolution, Rh (mol nrr2 lr1), is proportional to the third power of the surface protonation ... 

Fig. S-S8. Electron levels of dehydrated redox particles, H ld + bh /h = H,d, adsorbed on an interface of metal electrodes D = state density (electron level density) 6 = adsorption coverage shVi - most probable vacant electron level of adsorbed protons (oxidants) eH(d = most probable occupied electron level of adsorbed hydrogen atoms (reductants) RO.d = adsorbed redox particles.

Fig. 6-S8. Probability density for the energy level of interfadal redox electrons in adsorbed redox particles of proton-hydrogen and hydroxyl-hydroxide on the electrode interface of semiconductor ADS = adsorption > ost probable...

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