Electron density increment

A determination of the molecular weight by small-angle x-ray scattering requires, therefore, knowledge of the scattering intensity as a function of protein concentration and angle of observation, as well as of the electron density increment, ( — ), which replaces the refractive index... 

The constant depends on the radiation used, as does (dp Jdc ), which is the refractive index increment for light, the electron density increment for X-rays, and the neutron scattering density increment for neutrons. 

The atom and bond concepts dominate chemistry. Dalton postulated that atoms retained their identities even when in chemical combinations with other atoms. We know that their properties are sometimes transferable from one molecule to another for example, the incremental increase in the standard enthalpy of formation of a normal hydrocarbon per CHj group is —20.6 1.3 kJmol . We also know that more often there are subtle modifications to the electron density. 

In this plot, we can see that if we increase the pressure, the energy also will be increased but the rate of this increment will be different for each state. The results discussed for the PIAB model are particular situations of generalizations reported for systems confined with Dirichlet boundary conditions [2]. We must remember these results for further discussion through this chapter. Let us conclude this section with the remark that the state dependence of the effective pressure at the given value of Rc can be analogously understood in terms of the different electron densities and their derivatives at the boundaries. In most general case of atoms and molecules, scaled densities may have to be employed in order to include the excited states. In the next section, we present some basic results on such connections between wave function and electron density. 

Bearing in mind that chemical shifts are usually not dependent on the solvent or concentration, but may be partly dependent on the anion37, quaternization of alkyl-phosphines causes an upfield shift (1-7 ppm) for the near carbons of the alkyl chain, the effect becoming only slightly detectable on C(y) carbons37,94. Where triarylphosphines are quaternized, the chemical shift of the phenyl ipso carbon Cf moves upfield whereas that of the Cpara carbon moves downfield, as expected from the polarization of the 71-electron density in the phenyl ring36,37. For the aromatic carbon atoms, the increments... 

Nolo that (he mathematical function goes to zero because ihe volume of the incremental shell, iIV, goes 10 zero at r = 0. As we have seen, however, there is electron density at the nucleus for s orbitals... 

Figure 3-23 (A) Stereoscopic a-carbon plot of the cystolic aspartate aminotransferase dimer viewed down its dyad symmetry axis. Bold lines are used for one subunit (subunit 1) and dashed lines for subunit 2. The coenzyme pyridoxal 5 -phosphate (Fig. 3-24) is seen most clearly in subunit 2 (center left). (B) Thirteen sections, spaced 0.1 nm apart, of the 2-methylaspartate difference electron density map superimposed on the a-carbon plot shown in (A). The map is contoured in increments of 2a (the zero level omitted), where a = root mean square density of the entire difference map. Positive difference density is shown as solid contours and negative difference density as dashed contours. The alternating series of negative and positive difference density features in the small domain of subunit 1 (lower right) show that the binding of L-2-methylaspartate between the two domains of this subunit induces a right-to-left movement of the small domain. (Continues)...

Carbon-13 shifts of enamines [342] follow the behavior described for other donor substituted alkenes (Sections 4.4.3 and 4.6.2). Electron release by the dialkylamino group has two consequences The inductive electron withdrawal at the a alkene carbon is reduced (Za 10-15 ppm) compared with the a increments of aliphatic amines (Table 4.43). Further, electron density at the fi olefinic carbon increases, as indicated by considerable shieldings in pyrrolidino- and morpholinoalkenes. 

To utilize Eq. 7.11, Kohn and Sham introduced the idea of a fictitious reference system of noninteracting electrons which give exactly the same electron density distribution as the real system has. Addressing electronic kinetic energy, let us define the quantity A(T[p ] ) (don t confuse Greek delta A, an increment, with the differential operator del V) as the deviation of the real electronic kinetic energy from that of the reference system ... 

The variations of the one-electron densities 6Pria with a = a, ,v,( and the polarity (/ — P[l) of the bond with m = 1 deserve some discussion. As it is seen from eqs. (3.86), (3.105) each bond incident to an atom contributes an increment to the quasitorque and to the pseudotorque acting upon its hybridization tetrahedron. In the equilibrium these increments separately sum up to zero. We can think that the equilibrium shape and orientation of the hybridization tetrahedron is obtained within a TATO DMM model applied to the entire system. Then, within such a model, there exists an atom corresponding to the left end of the bond with m = 1 having number Li according to our previous notation. The HOs obtained in this approximation provide an initial guess for HOs in the system including those of the atom Pi, which... 

If, on the other hand, the electron density map calculated at some resolution r is somehow improved in quality in real space, then if it is transformed to produce structure factors, phases at somewhat higher resolution, r + Ar, can be computed that do have some measure of validity. Improvement of an electron density map thus allows gradual extension of phases in reciprocal space to higher resolution, and ultimately to an electron density map of sufficient quality and detail that a model can be constructed. This is another example of those bootstrap, incremental procedures so common to X-ray crystallography. 

FIGURE 10.2 A small area of a raw electron density map of a protein, directly from the Fourier calculation as it comes off the computer. The location of each number on the plane corresponds to a specific x, y, z fractional coordinate in the unit cell. In general, one of the three coordinates will be constant for the entire plane, and rows and columns will correspond to the other two coordinates. The value of the number at each position is p(x, y, z), the electron density at that point. Contours are incrementally drawn around areas having p(x, y, z) greater than certain values. This yields a topological map of the electron density on each plane of the unit cell. 

FIGURE 10.5 Several consecutive electron density sections, separated by a constant increment along the z axis, are stacked and displayed on a light box. The continuity of the protein, here the protease from streptomyces, is clear in many places. The dots are used in the early interpretation of the map to mark points, such as putative a carbon positions, along the chain. The first objective in interpreting an electron density map is generally to deduce the overall fold of the polypeptide. Only later are side chains identified and oriented. 

Frequently, with smaller and well-diffracting structures (W , < 700 and dp > 10), all atoms of the structure can be written out as the initial model by the program and they just have to be named correctly (as in Fig. 9.11) and refined. The refinement process (see Refs. 110 —1121) uses incremental movement of the atom coordinates and atomic displacement parameters (commonly called as thermal parameters ) of the structure solution model using the so-called least-squares method. The model (the calculated structure factors) is fitted against the measured data (the observed structure factors) and the R-factor (see above. Section 9.2.1) is calculated. With larger structures or if the unit cell contains light atom solvent molecules (C, H, O, N atoms only), some atoms, sometimes even 50% of all atoms, cannot be located from the first very crude electron density map (calculated from the ab initio phase set). However, those atoms which are chemically feasible (based on the proposed molecular structure) can be fed into the calculation of the calculated structure factors Peak ( cafc will approach Fobs when a more accurate model is... 

Starting from these disparate systems, we construct a whole range of intermediate systems, all with identical electron density but with incrementally increasing strengths for the electron-electron repulsion term ... 

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