Density difference contour maps

Figure 1.17. Electron charge density difference contour map for CO on Ni(100) and CO on Ni(100)/H in atop sites, derived from DFT calculations.

Fig. 5. Density difference contour maps Ap for the H2 system at several internuclear distances. The maps for R = 8.0 and 6.0 are drawn one-half scale relative to the others. (Reproduced from Bader and Chandra, 1968.)...

Fig. 7. Maps of the electronic charge density in the (110) planes In the ordered twin with (111) APB type displacement. The hatched areas correspond to the charge density higher than 0.03 electrons per cubic Bohr. The charge density differences between two successive contours of the constant charge density are 0.005 electrons per cubic Bohr. Atoms in the two successive (1 10) planes are denoted as Til, All, and T12, A12, respectively, (a) Structure calculated using the Finnis-Sinclair type potential, (b) Structure calculated using the full-potential LMTO method.

Figure 6.3 Constant electron density envelope maps for SCI2 for three different contour values (a) p = 0.001 au, (b) p = 0.200 au and (c) p = 0.133 au. (a) This constant density envelope shows the practical outer boundary of the molecule broadly corresponding to the van der Waals envelope, (b) This constant density envelope demonstrates that for higher p values the envelope becomes disconnected into three surfaces each encompassing a nucleus, (c) This constant density envlope is plotted at the highest p value for which the molecular envelope is still connected or encompasses the whole molecule.

FIG. 11.11 Electron-density difference maps on Li2BeF4 calculated with all reflections < sin 6/1 = 0.9 A"1 (81 K). (a) Based on the neutral atom procrystal model, (b) based on the ionic model. Contour levels are drawn at intervals of 0.045 eA"3.1 Full lines for positive density, dashed lines for negative and zero density. The standard deviation, estimated from [2Lff2(F0)]1/2N, is 0.015 eA-3. Source Seiler and Dunitz (1986). 

Fig. 9a and b. Difference Fourier maps calculated from Laue diffraction data showing maltoheptose bound in phosphorylase b. The Laue map shown in a is calculated with a subset of 9029 unique data at 2.5 A resolution. A positive contour at half maximal peak height is shown, b is an enlargement of a and shows 4 of the 7 sugar units, the 3 central units have the highest occupancies. Side chain movements produce the two extra lobes of density. (Figures courtesy of J. Hajdu)... 

FIGURE 24. Electron-density difference maps of207. (a) Section in the plane of atoms C1, C3 and C5, approximately perpendicular to the C2—C4 axis of the propellane unit. Note the density peaks in the centers of the three-membered rings, around the projection of C2 and C4. Contour lines are at 0.05 e A 3 intervals, (b) Section in the plane of the three-membered ring C2, C3, C4. Contour lines are at 0.025 e A 3 intervals. Full lines mark positive, dashed lines negative regions. Reproduced by permission of Verlag Helvetica Chimica Acta from Reference 319... 

To compare apo- and holo-forms of proteins after both structures have been determined independently, crystallographers often compute difference Fourier syntheses (Chapter 7, Section IV.B), in which each Fourier term contains the structure-factor difference FAc>/c(-—F 0. A contour map of this Fourier series is called a difference map, and it shows only the differences between the holo-and apo- forms. Like the FQ — Fc map, the FAoio—F map contains both positive and negative density. Positive density occurs where the electron density of the holo-form is greater than that of the apo-form, so the ligand shows up clearly in positive density. In addition, conformational differences between holo- and apo-forms result in positive density where holo-protein atoms occupy regions that are unoccupied in the apo-form, and negative... 

Figure SCF-MO density difference map for LiF in uniform electric field along internuclear axis minus electron density of field-free molecule.11 Contours are drawn corresponding to values of the electron density as follows A= — 0.800, B= -0.400, C= -0.200, D= -0.080,..., J=0,... S=0.800 electrons...

Fig. 10 a Contour map of the valence electron densities of the tetragonal C60 polymer on the (001) plane (leftpanel) and (010) plane (right panel). The difference between each neighbor contour is 0.021 atomic units. The projected C-C network on the (001) plane is also shown, b Its electronic energy band. The material is a semiconductor with a smaller band gap than the pristine fee C60 [37]... 

Fig. 9. Polymorphism in p-nitrophenol static deformation density in the plane of the phenyl rings for the a- and the (3-forms (contours at 0.1 eA-3). Intramolecular and lone-pair regions exhibit many differences. Relief maps of the Laplacians in the inteimolecular hydrogen bond region are also shown (range -250 to 250 eA-5). In the a-fonn, H(l) bonds not only with 0(3) but also with 0(2) and N(l) of the neighboring nitro group (reproduced with permission from Kulkami et al. [61]).

Fig. 2.2. Contour maps of the total molecular charge density (p ,) and the difference densities with respect to the superposition of the directionally prepared free atoms (Apatoms) and free ions (Apio s) for LiF (after Smith, 1977, and Bader, 1964 reproduced with the publisher s permission).

There are a number of ways of monitoring the distribution of electron density in any molecular entity. The total density can be computed at a number of points in space and presented as a contour map or some three-dimensional representation. Shifts are easily examined by density difference maps which plot the difference in density between two different configurations. For example, the density shifts caused by H-bond formation can be taken as the difference between the complex on one hand, and the sum of the densities of the two non interacting subunits on the other, with the two species placed in identical positions in either case. Comparisons with x-ray diffraction data have verified the validity of this ap-proach. Also, the total density of the complex itself can be examined for the presence of critical points that indicate H-bonding interactions . 

Fig.7 Contour maps of charge density p and difference density Ap...

Fig. 5 Contour map of difference charge density, p (p (Si04Na )) — p (Si04 )), around Si(5) and 0(3) ions. Solid, dotted and dashed lines indicate positive, negative and zero contour lines, respectively.

Fig. 6.13 Representations of the valence m.o.s of HF (only one of the two tt orbitals is shown) by contour plots and a three-dimensional grid in the case of the vacant (anti-bonding) m.o. Plot of DD (density difference) illustrates the detailed structure of the difference map between the total electron density and the atomic contributions if no bond was formed (full lines for increase of electron density and dotted lines for decrease).

FIGURE 10.11 Here is a two-dimensional representation of what one would likely see in an Fo — Fc difference Fourier map, where the Fc were calculated from a model that includes an arginine side chain, shown here, which was misplaced. A region of negative difference electron density (dashed contour lines) would superimpose upon the position incorrectly occupied by the side chain, while the positive density (solid contour lines) would indicate the location to which it should be shifted. 

This function is similar to the electron density function given earlier. Here, P(uvw) is the value of the Patterson function at Patterson coordinates u, v, w these are the traditional coordinate symbols (instead of x, y, z) used for squared ( F jt,p) space. All other symbols have their usual meaning. The Patterson function is a Fourier summation using the intensities as coefficients and setting all equal to 0. The resulting contoured map will have peaks corresponding to vector differences between all atoms in the structure. A vector between an atom and itself is a zero vector therefore, the Patterson functions always have a very large peak at u,v,w = 0, 0, 0. 

Bader et al. first clarified the molecular topography of density distributions. For example, they defined the width and length of a molecule in terms of the density contours of 0.002 a.u. The density difference maps... 

Figure 1 The electronic density difference maps for the Be3 trimer partitioned for 2-body (a) and 3-body (b) contributions and the total difference density distribution (c). The plot is done in the plane of Be3. The spacing between the contours is 0.001 electron/bohr. The contour with no density charge axe labeled with zeros while solid lines indicate the enhancement of electronic density.

Figure 5 reports a comparison between the contour maps of the density difference, Ap(r), the Kullback-Leibler integrand, Ah(r), and the entropy displacement function [equation (94)], A (r), for the planes of sections shown in Fig. 4. The corresponding central bond profiles of the density and entropy difference functions are compared in Fig. 6. The optimized geometries of propellanes have been determined from the UHF calculations (GAMESS program) using the 3-21 G basis set. The contour maps have been obtained from the DFT calculations (deMon program) in DZVP basis set.

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