Electron-equivalent gradients

We can calculate the vertical (e.g., cross isopycnal) molar- and electron-equivalent gradients as a function of depth and use them as a constraint for proposed reactions. The approach used was first to calculate the vertical molar depth gradients (moles per liter per meter equals moles per meter to the fourth power) and then to multiply those gradients by the number of electrons required for the appropriate redox reactions see Table I). These gradients were calculated against depth rather than density because depth is the traditional unit for gradients. If the gradients are divided by the density, their... 

Figure 7. Electron-equivalent gradients (in moles of electrons per meter to the fourth power) into the oxic-anoxic interface from above and below.

Another difficulty with the metal oxide hypothesis is that the observed sum of the vertical electron equivalent gradients of Mn(II) and Fe(II) is much less than that of sulfide (Figure 7). In a simple vertical, steady-state system where the upward flux of Mn(II) and Fe(II) results in oxidized particulate metal oxides, which in turn settle to oxidize sulfide, the electron gradients of Mn(II) + Fe(II) would equal that for sulfide. The fact that they do not equal it suggests that the vertical flux of Mn(II) + Fe(II) would not produce sufficient particulate metal oxides. This problem would be solved if the particulate oxides were produced primarily at the boundaries and transported into the interior (40). 

Their electron equivalent gradients match within a factor of 2. 

We have used transferable atom equivalent (TAE) descriptors [116,117] that encode the distributions of electron density based molecular properties, such as kinetic energy densities, local average ionization potentials, Fukui functions, electron density gradients, and second derivatives as well as the density itself. In addition autocorrelation descriptors (RAD) were used and represent the molecular geometry characteristics of the molecules, while they are also canonical and independent of 3D coordinates. The 2D descriptors alone or in combination with the latter 3D descriptors were calculated for 26 data sets collated by us from numerous publications. These data sets encompass various ADME/TOX-related enzymes, transporters, and ion channels as... 

Bader s atoms-in-molecules theory [69, 70] is based on the topology of the electron density p(r) and enables to partition the three-dimensional real space into non-overlapping domains called basins . To achieve such separation, one looks at the so-called gradient paths (GPs), which are the equivalent of field lines in classical electromagnetism. A GP is defined as a curve C such that the electron density gradient pif) (which can stem from theoretical calculations as well as from X-ray diffraction experiments) is tangent to C at every C point. In general (non-nuclear... 

In an equivalent manner we utilize J, to eliminate the x-component of the electron concentration gradient from 7 y... 

I he function/(r) is usually dependent upon other well-defined functions. A simple example 1)1 j functional would be the area under a curve, which takes a function/(r) defining the curve between two points and returns a number (the area, in this case). In the case of ni l the function depends upon the electron density, which would make Q a functional of p(r) in the simplest case/(r) would be equivalent to the density (i.e./(r) = p(r)). If the function /(r) were to depend in some way upon the gradients (or higher derivatives) of p(r) then the functional is referred to as being non-local, or gradient-corrected. By lonlrast, a local functional would only have a simple dependence upon p(r). In DFT the eiK igy functional is written as a sum of two terms ... 

Forces of Adsorption. Adsorption may be classified as chemisorption or physical adsorption, depending on the nature of the surface forces. In physical adsorption the forces are relatively weak, involving mainly van der Waals (induced dipole—induced dipole) interactions, supplemented in many cases by electrostatic contributions from field gradient—dipole or —quadmpole interactions. By contrast, in chemisorption there is significant electron transfer, equivalent to the formation of a chemical bond between the sorbate and the soHd surface. Such interactions are both stronger and more specific than the forces of physical adsorption and are obviously limited to monolayer coverage. The differences in the general features of physical and chemisorption systems (Table 1) can be understood on the basis of this difference in the nature of the surface forces. 

In Chapter 16, we studied the so-called primary properties, which I defined as those that could be obtained by direct calculation from the electron density (or equivalently the wavefunction). We also touched on derivative or gradient properties. It is now time to mm our attention to those properties that measure the response of a system to an external field. In the language of Boys and Cook, these are the induced properties. 

Recent work has shown that bacteria, in common with chloroplasts and mitochondria, are able, through the membrane-bound electron transport chain aerobically, or the membrane-bound adenosine triphosphate (ATP) anerobically, to maintain a gradient of electrical potential and pH such that the interior of the bacterial cell is negahve and alkaline. This potential gradient and the electrical equivalent of the pH difference (1 pH unit = 58 mV at 37°C) give a potential difference across the membrane of 100-180 mV, with the inside negative. The membrane is impermeable to protons, whose extmsion creates the potential described. 

Superoxide ions, 02, are readily formed by the transfer of electrons from Fs centers on MgO or from Mo(V) on Mo/Si02 to molecular oxygen (7, 9). The value of g3 for 02 is particularly sensitive to the crystal field gradient at the surface and thus varies from one metal oxide to another (10). In fact, the spectrum of 01 on MgO indicates that the ions are held at four distinctly different sites (11,12). The oxygen-17 hyperfine splitting (Table I) for 170170- on MgO confirms that both oxygen atoms are equivalent, on supported molybdenum the atoms are nonequivalent, suggesting a peroxy-type bond to the metal (7,13). 

Oxidizible substrates from glycolysis, fatty acid or protein catabolism enter the mitochondrion in the form of acetyl-CoA, or as other intermediaries of the Krebs cycle, which resides within the mitochondrial matrix. Reducing equivalents in the form of NADH and FADH pass electrons to complex I (NADH-ubiquinone oxidore-ductase) or complex II (succinate dehydrogenase) of the electron transport chain, respectively. Electrons pass from complex I and II to complex III (ubiquinol-cyto-chrome c oxidoreductase) and then to complex IV (cytochrome c oxidase) which accumulates four electrons and then tetravalently reduces O2 to water. Protons are pumped into the inner membrane space at complexes I, II and IV and then diffuse down their concentration gradient through complex V (FoFi-ATPase), where their potential energy is captured in the form of ATP. In this way, ATP formation is coupled to electron transport and the formation of water, a process termed oxidative phosphorylation (OXPHOS). 

In (22.37) ip is the wave function of an atom with motionless nucleus. The one-electronic submatrix element of the gradient operator (n/ V ni/i) is non-zero only for h = / 1. Therefore, the matrix element of operator (22.38) inside a shell of equivalent electrons vanishes and one has to account for this interaction only between shells. For the configuration, consisting of j closed and two open shells, it is defined by the following formula [156] ... 

As a final example, it should be noted that in the presence of valinomycin, K+ is taken up by mitochondria to compensate for the H+ lost in forming the proton gradient. This work confirms the ratio of four K+ taken up per pair of electrons passing the energy-conserving site and so is equivalent to the H+/site ratio.85 The protein responsible for K+/H+ antiport has been identified.86 Other potassium transport processes have been described.87... 

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