Electron density average

Crystals of pronase-released heads of the N2 human strains of A/Tokyo/3/67 [44] and A/RI/5+/57 were used for an x-ray structure determination. The x-ray 3-dimensional molecular structure of neuraminidase heads was determined [45] for these two N2 subtypes by a novel technique of molecular electron density averaging from two different crystal systems, using a combination of multiple isomorphous replacement and noncrystallographic symmetry averaging. The structure of A/Tokyo/3/67 N2 has been refined [46] to 2.2 A as has the structures of two avian N9 subtypes [47-49]. Three influenza type structures [50] have also been determined and found to have an identical fold with 60 residues (including 16 conserved cysteine residues) being invariant. Bacterial sialidases from salmonella [51] and cholera [52] have homologous structures to influenza neuraminidase, but few of the residues are structurally invariant. 

The high solvent content of macromolecular crystals leads to another way to modify the electron density. The electron density map is the space average of all the unit cells in the crystal, so atoms that are in random positions (as in liquid water) in different unit cells will not show up as peaks. Why They do not obey the periodicity of the crystal (remember that the FTs concerned periodic functions) and so are called disordered. The crystal then consists of ordered molecules, where the electron density is the same in each unit cell, and so visible, and disordered solvent, where the electron density averages to zero. In order to make physical sense, the electron density also has to be positive a property not imposed by the FT. Consequently, electron density peaks outside the macromolecule are noise and can be got rid of, as can negative electron density inside the macromolecule. We can therefore apply these conditions modify the initial electron density so that it is zero outside the molecules and positive within them. This, as for noncrystallographic averaging, alters the electron density map to conform to what must be true, and the map is thus a better representation than the initial map. 

Figure 8 (a) A guanine quartet with pseudo-electron density, averaged water occupancy... 

Dispersion forces, (a) The distribution of electron density averaged over time in a neon atom is s)rmmetrical, and there is no net polarity, (b) Temporary polarization of one neon atom induces temporary polarization in adjacent atoms. Electrostatic attractions between temporary dipoles are called dispersion forces. 

In any case, it is clear that proper treatment of the solvent effect, both static and dynamical, is fundamental for reliable evaluation of the CT transition s stability in the condensed phase. When using continuum solvation models, a state-specific approach combined with an accurate description of the excited-state electron density (averaging procedures of the excited-state density such as those usually employed in CASPT2 should be avoided) is mandatory, since LR-PCM/TD-DFT strongly underestimates the stability of transition with even partial CT character. 

Ratio of neutrals to ions Electron density Average electron energy Average neutral or ion energy... 

Electron density averaged over the entire film. 

If we consider the scattering from a general two-phase system (figure B 1.9.10) distinguished by indices 1 and 2) containing constant electron density in each phase, we can define an average electron density and a mean square density fluctuation as ... 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

You should remember the basic physical idea behind the HF model each electron experiences an average potential due to the other electrons (and of course the nuclei), so that the HF Hamiltonian operator contains within itself the averaged electron density due to the other electrons. In the LCAO version, we seek to expand the HF orbitals i/ in terms of a set of fixed basis functions X X2 > and write... 

In this equation, (/ (/ a)) is the spherical average of the electron density at distance/ A from nucleus A. The symbol. .. )r=o means that we have to evaluate the mean value of the quantity and then set R to zero in the result. The symbol (... (Ra = 0)) means that we have to evaluate the mean value of the quantity when it is measured at Ra = 0. 

The and operators determine the isotropic and anisotropic parts of the hyperfine coupling constant (eq. (10.11)), respectively. The latter contribution averages out for rapidly tumbling molecules (solution or gas phase), and the (isotropic) hyperfine coupling constant is therefore determined by the Fermi-Contact contribution, i.e. the electron density at the nucleus. 

In the HF molecule, the distribution of the bonding electrons is somewhat different from that found in H2 or F2. Here the density of the electron doud is greater about the fluorine atom. The bonding electrons, on the average, are shifted toward fluorine and away from the hydrogen (atom Y in Figure 7.9). Bonds in which the electron density is unsymmetrical are referred to as polar bonds. 

Figure 3-5. Disorder-averaged density of stales />(< ) in the FGM for three values ol Hie dimensionless disorder strength g=0, i.c., no disorder (eurve a), g=0.25 (curve b), g=4 (curve e). The (tec electron density of stales was set to unity.

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

Fig. 2.—Electron distribution for hydrogen-like states the ordinates are values of D. Z-1. 10 8, in which D = 4mxp, with p the electron density. The vertical lines correspond to r, the average value of r.

Averages of properties require integrals over CSFs which can readily be written for one- and two-electron operators, insofar the Slater determinants and the MSOs are orthonormal by construction, in terms of one- and two-electron density matrices. 

---