Contour difference maps

As a rule with few exceptions, OH moieties are hydrogen bonded to an acceptor. Potential H-atom positions lie on a cone. Finding the correct position on this cone can be tricky when the difference electron-density map does not show a single suitable maximum. SHELXL (Sheldrick, 1997b) provides an option to find the optimal position by way of an electron-density calculation around a circle (see Chapter 3 for details). Inspection of contoured difference maps for hydrogen atom positions should be attempted in less obvious settings. 

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). 

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)...

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...

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 . 

Field Effects. Once the frame of reference has been established, other properties of molecules, such as the electrostatic field, can be compared as well. Because the electrostatic properties can be sampled on a grid, differences between the values of two moleculescan be calculated and a difference map contoured. Such difference maps (428) highlight more clearly the similarities and differences between molecules. Hopfinger (429) integrated the difference between potential fields and showed this parameter to be useful in QSAR studies. 

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).

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 3 Valence Fourier difference map in the least squares plane of uradL Contours in units of 0.2 e A. Dotted line is 0.2 e A 1 (after Stewart ). 

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.

Fig. 26.10 Fq - Fc difference maps, (a) Detection of errors in the model. The side-chain of Aspi A in human thrombin was deiiberateiy moved to a wrong position and an F — Fc difference map was calcuiated. The modei used to caiculate Fc is indicated in thick solid iines, the correct position of the side-chain is indicated in thick broken lines. Negative contours (4a beiow mean) are drawn in thin broken lines and positive contours (4a above mean) are drawn in thin solid iines. Strong negative difference density is present around the wrongiy piaced side-chain while strong positive density is present at the position where the side-chain shouid be according to the experimental data. These maps are extremely useful to spot errors in the model, (b) Fo - Fc omit map (see text) contoured at 3.5a of an inhibitor bound to porcine pancreatic eiastase. Protein atoms, used for calcuiation of Fc are indicated in thin iines, the inhibitor which has been removed from the modei is drawn in thick lines. This Fo - Fc density map has been caiculated with a model which contains no information whatsoever about the inhibitor. The difference density shown is therefore entireiy due to the experimental data and can be used to verify the correctness of the placement of the inhibitor.

Fig. 3. Electron density difference map of tetrafluoroterephthalodinitrile in the molecular plane (from [27]). Contour lines are drawn at intervals of 0.075 electrons per A, positive contours fuU Unes, negative contours dashed, zero contour dotted. Note the weak density in the...

Most often the results of a CoMFA study are presented in graphical form, with contours for favorable and unfavorable regions of the different fields e.g. Figure 60). Difference maps were proposed as tools to analyze and identify areas of interest with respect to activity and selectivity, if two different types of biological activities are compared [1011]. 

Fig. 1.18 Electron density difference maps for the (CH3)2TeCl2 molecule. On the left (a), in the equatorial plane defined by the atoms Te, S1, and C2, showing the Te-C bond and Te(I V) lone-pair densities. Contours are at 0.03e/A3 with negative contours broken. On the right (b), in the plane perpendicular to the plane of the left, showing the Te-Q (chlorine) bond and the Te(IV) lone-pair densities (Ref. [53], 1983 American Chemical Society)...

Electron density difference map for the [Co(N02)6] ion in the C0N4 plane. Within a solid contour the electron density has been depleted and within a dotted contour it has increased. Adapted and reproduced with permission from S. Ohba, K. Toriumi, S. Sato and Y. Saito, Acta Ctyst, (1978) B34, 3535. 

Figure 7.3 An example experimental electron density difference map for (Z)-N-methyl-C-phenylnitrone. The solid (dashed) contours are for an increased (decreased) electron density compared with a simple sum of noninteracting atomic densities. The inset in the top left shows a chemical structure of the same molecule in roughly the orientation in the contour plot, (Source Hibbs DE, Hanrahan JR, Hursthouse MB, Knight DW, Overgaard J, Turner P, Piltz RO, Waller MP (2003) Organic Blomolecular Chemistry 1 1034-1040.)...

Figure 3 Pepstatin difference map (3.7 A resolution) superimposed on the relevant sections of the 2.5 A native map. The thick black lines represent the pepstatin contours. 

Sketch the probability of finding an electron in the 2s orbital of hydrogen at distance r from a hydrogen nucleus as a function of r as a contour map with heavy lines at high probability and light lines at low probability. How does this distribution differ from the Is orbital ... 

Comparison of Alignment Charts and Cartesian Graphs. There are typically fewer lines on an alignment chart as compared to Cartesian plots. This reduces error introduced by interpolation and inconsistency between scales. For example, to find a point (x,j) on a Cartesian graph one draws two lines, one perpendicular to each axis, and these reference lines intersect at the point x,j). This point (x,j) may correspond to some finite value found by rea ding a contour map represented by a family of curves corresponding to different values of the function. 

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.

We need ways to visualize electrons as particle-waves delocalized in three-dimensional space. Orbital pictures provide maps of how an electron wave Is distributed In space. There are several ways to represent these three-dimensional maps. Each one shows some important orbital features, but none shows all of them. We use three different representations plots of electron density, pictures of electron density, and pictures of electron contour surfaces. 

It Is usual to visualize the spatial distribution of p(x) within an area A by contouring measured or estimated values p (29c) nodes 29c grid. The resulting estimated map necessarily differs... 

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