Electron density contour

R F W Bader s theory of atoms in molecules [Bader 1985] provides an alternative way to partition the electrons between the atoms in a molecule. Bader s theory has been applied to many different problems, but for the purposes of our present discussion we will concentrate on its use in partitioning electron density. The Bader approach is based upon the concept of a gradient vector path, which is a cuiwe around the molecule such that it is always perpendicular to the electron density contours. A set of gradient paths is drawn in Figure 2.14 for formamide. As can be seen, some of the gradient paths terminate at the atomic nuclei. Other gradient paths are attracted to points (called critical points) that are... 

Fig. 1.32. (a) Molecular graphs and electron density contours for pentane and hexane. Dots on bond paths represent critical points, (b) Comparison of molecular graphs for bicycloalkanes and corresponding propellanes. (Reproduced from Chem. Rev. 91 893 (1991) with permission of the American Chemical Society.)... 

Figure 13-5. Electron density contour plots (2F0-FC) for 1,6-DHN (left) and flaviolin (right). The contour values are 1.5a (high), 1.0a (medium), and 0.5a (low) from the top to the bottom...

Figure 13-11. The snapshots of the S2 (green) and S3 (cyan) binding states predicted from the calculated potential of mean force superimposed with 1ZB6 (grey) and 1ZDW (purple). The electron density contour of 1ZB6 at the level of 0.5cr is shown as a yellow mesh...

Fig. 31. Approximation of van der Waals cross-sections of inclusion channels in 1 alcohol clathrates21 (dimensions are in A hatched regions represent O atoms of the host matrix continous solid lines indicate surfaces of apolar attribute) (a) 1 MeOH (1 2) (approximately parallel to the 0(I -Cul vectors, cf. Fig. 17a) (b) 1 2-PrOH (1 2) (orientation as before) (c) 1 2-BuOH (1 1) (through a center of symmetry at 1,1/2,1/2, cf. Fig. 30c non-zero electron density contours) (d) 1 ethylene glycol (1 1) (in the plane of the C—C single bonds of a guest molecule, indicated by projected stick models non-zero electron density contours)...

Fig. 32. Packing relations and steric fit of the 26 acetic acid (1 1) clathrate (isomorphous with the corresponding propionic acid clathrate of 26)1U- (a) Stereoscopic packing illustration acetic acid (shown in stick style) forms dimers in the tunnel running along the c crystal axis of the 26 host matrix (space filling representation, O atoms shaded), (b) Electron density contours in the plane of the acetic acid dimer sa First contour (solid line) is at 0.4 eA" while subsequent ones are with arbitrary spacings of either 0.5 and 1 eA 3. Density of the enclosing walls comes from C and H atoms of host molecules.

Figure 5.2 (a) Electron density contour map of the CI2 molecule (see Chapter 6) showing that the chlorine atoms in a CI2 molecule are not portions of spheres rather, the atoms are slightly flattened at the ends of the molecule. So the molecule has two van der Waals radii a smaller van der Waals radius, r2 = 190 pm, in the direction of the bond axis and a larger radius, r =215 pm, in the perpendicular direction, (b) Portion of the crystal structure of solid chlorine showing the packing of CI2 molecules in the (100) plane. In the solid the two contact distances ry + ry and ry + r2 have the values 342 pm and 328 pm, so the two radii are r 1 = 171 pm and r2 = 157, pm which are appreciably smaller than the radii for the free CI2 molecule showing that the molecule is compressed by the intermolecular forces in the solid state. 

Given that X-rays are scattered from the electrons of the molecule and that the electron density contours of the water molecule deviate very little from spherical symmetry, it is not surprising that the X-ray form factors x-ray and 2x-ray differ negligibly ( 1%)Thus, the water molecule appears spherically symmetric to X-rays, and orientational correlation between water... 

Fig. 3. Electron density contours of sperm whale myoglobin at 6 A resolution.

It is also possible to determine accurate electron density maps for the ionic crystal structures using X-ray crystallography. Such a map is shown for NaCl and LiF in Figure 1.45. The electron density contours fall to a minimum—although not to zero—in between the nuclei and it is suggested that this minimum position should be taken as the radius position for each ion. These experimentally determined ionic radii are often called crystal radii the values are somewhat different from the older sets and tend to make the anions smaller and the cations bigger than previously. The most comprehensive set of radii has been compiled by... 

Fig. 22. Electron density contour maps of the three frontier orbitals of electron-deficient M(it-C5Hs)2 fragments. The plots represent a section through the yz plane. [Reproduced from Lauher and Hoffmann (134), by permission of the American Chemical Society.]...

Figure 2.8. Electron density contours for atomic chemisorption on jellium with electron density that corresponds to A1 metal. Upper row Contours of constant electron density in the plane normal to the surface. Center row Difference in charge density between isolated adatom and metal surface, full line gain and dashed line loss of charge density. Bottom row Bare metal electron density profile. Reproduced from [30].

If the absolute intensities of the X-ray reflections are not available— but only relative intensities—the value of the constant term (the equivalent of 000 in the equations given previously) in relation to the other terms of the Fourier series (which are in this case in arbitrary units) is not known the figures for the electron density obtained by calculation, omitting the constant term, will all be wrong by this amount but for the purpose of locating atomic centres, this is of no consequence the image formed by the electron density contours is of precisely the same form. 

Fig. 2.7 (a) Pictorial representation or the electron density in a hydrogen-like 2p orbital compared with lb) the electron density contours Tor the hydrogen-like 2pr orbital of carbon. Contour values are relative to the electron density maximum The xy plane is a nodal surface. The signs (+ and —) refer to those of the original wave function. [The contour diagram is from Ogryzlo, E. A4 Porter G. B J. Client. Ethic. 1963.40. 258. Reproduced with permission. ... 

Fig. 5.6 Electron density contours for the Ht ion bonding (a) and antibonding (b) orbitals.

Fig. 5.14 Electron density contours for (a) H, (b) Li, aXt core (c) Li of, core (d) Li2 cr. (e) Li2. total electron density. IFrom Wahl. A. C. Science 1966. IS/, 961. Reproduced with permission.]...

Consider Fig. 6.32 which is an electron density contour map of the sodium cyanide crystal Interpret this diagram in terms of everything that you know about the structure of solid sodium cyanide. 

Fig. 11.4 Electron density contours for (a) t/. and (b> d.t orbitals. The triangles represent points of maximum electron density. [From Perlmutter-Hayman. B. 1. Clwin. Educ. 1969. 46. 428-430. Reproduced with permission.]...

Fig. 5.18 (a) Total electron density contours for the carbon monoxide molecule. The carbon atom is on the left (b) Total electron density contours for the dinitrogen molecule. [From Bader, R. F. W. Bandrauk, A. D. J. Chem. Phys. 1968,49, 1653. Reproduced with permission.]... 

Wyckoff et al. (62) have provided a preliminary coordinate list of all nonhydrogen atoms in RNase-S. Along with the list is a series of notations on the quality of the map and the fit of the atomic model to the electron density contours. The following comments concerning group accessibilities are based on this coordinate list, but detailed interpretations must be made with caution in view of the uncertainties in many parts of the structure. 

Alanine. Residues 4, 5, 56, 96, 102, and 109 CB clear in positive, but perhaps weak, electron density. Residues 6, 52, 64, 122 CB outside of lowest electron density contour or poorly defined. Residue 5 carbonyl direction apparently pointing somewhat away from helix axis with no obvious explanation. Residues 19 and 20 uncertain region of very poorly defined chain. 

In a later paper, the authors218 analysed the charge re-distribution in more detail, using electron density contour maps. An energy-level correlation diagram shows no correlations between bonding levels and antibonding levels in the product. 

Figure 5 Electron density contours in the plane containing the three nuclei of Pb3. Plus (+) signs indicate maxima and those correspond roughly to 0.04 e/au3. Minus (—) signs indicate minima. The most external contour has a value 0.003 e/au3 and the interval between contours is An = 0.003 e/au3.

Figure 6 Electron density contours in the middle plane containing five atoms in Pbr. Other details are like in Fig. 5.

One interesting species, unknown until recently, is HOF, which was studied by Peslak et al.,611 and electron-density contour maps were described. More recently, it was studied by Kim and Sabin.512 The computed bond lengths with a near-minimal GTO basis were i (H—O) = 1.080 A, J (0—F) = 1.450 A, and 0 = 100.8°, in fair agreement with the results of a microwave investigation. Force constants were computed and the population analysis was reported. A more extensive basis set was used in a similar calculation by Ha, who computed a dipole moment of 2.72 D.513... 

An electron density contour surface piece with a larger Betti number usually (but not necessarily) has the larger surface area. If this is the case, then the sequence of Betti numbers in the above ordering is the same as the decreasing sequence of the Betti numbers. 

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