Density difference maps

Typical examples of density- and density difference maps for ion-molecule complexes are shown in Chapter V. For our purpose we do not need the detailed spatial information which is rather confusing for a discussion of charge transfer. We proposed therefore 113> to choose an axis of the complex (2-axis), which is characteristic for the type of interaction that we are interested in. In Li+... OH2 or Li+... OCH2 this is evidently the twofold symmetry axis C2. In other cases like e.g. F-... HOH or (H20)2, the connection line of the three atoms forming the hydrogen bond will be appropriate ). After defining a 2-axis in this way we can calculate a density difference curve Aq(z) by simple integration (35) ... 

Fig. 18. The active site region of the electron density difference map between N-carbobenzoxy-L-alanine-elastase at —SS C and native elastase at the same temperature. The resolution is 3.5 A. The bilobed feature is consistent with the binding of the alanyi portion of the substrate to the oxygen of the catalytic serine, with weak interaction of the carbobenzoxy group to the surface of the enzyme.

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). 

After the crystal structure of the compound has been solved, or deduced, from the X-ray data, the initial parameters (atomic positions, bond lengths, and bond angles) are only approximate and have to be improved. The usual method employed is that of least-squares refinement, although electron-density difference-maps and trial-and-error procedures are also used. Electron-density difference-maps give the approximate difference between the actual structure and the trial structure. 

The development of synchrotron radiation as an X-ray source404 416 418 has permitted accumulation of data for electron density difference maps in less than 1 s and it is expected that such data can eventually be acquired in 1 ps.419-421 If a suitable photochemical reaction can be initiated by a picosecond laser flash, a substrate within a crystalline enzyme can be watched as it goes through its catalytic cycle. An example is the release of inorganic phosphate ions from a "caged phosphate" (Eq. 3-49) and study of the reaction of the released phosphate with glycogen phosphorylase (Chapter 12).422/423... 

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... 

Fig. 23. Stereodrawings of the dinucleotide phosphonate UpcA bound to RNase-S. Histidine 119 is in position IV where it is forced to be by the adenosine. Lysine 41 is in the position found in DNP-Lys 41 derivative. In the native protein it is lower and closer to the phosphate position, but it is not long enough to contact the phosphate. The UpcA is an interpretation of the 2-A resolution electron density difference map and the CH, which is bound to the phosphorus is crowding His 119. Some further adjustments may be necessary.

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...

The charge density difference maps from the Gunnarsson et al." calculations are compared with experiment107 and with Hartree-Fock results108 in Figure 12. There are differences between theory and experiment which are not presently resolved to the author s knowledge. 

Figure 12 Charge density difference maps 0/N2 from ref. 99, compared with HF results of ref. 108...

Charge-density difference maps from the band calculation (crystal minus superimposed free atoms) clearly show charge transfer, but the limited basis set precludes detailed comparison with experiment. 

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 . 

All theoretical studies agree with this second picture [13-17,39,99]. This is shown by the plots of the electron density maps and in particular by density difference maps which show very clearly the electron localization at the center of the vacancy [39,55]. This is true not only for the bulk but also for the surface of MgO, Fig. 5. The localization of the electrons in the center of the vacancy is an indirect proof of the highly ionic nature of MgO. In fact, the electrons are trapped in the cavity by the crystalline Madelung potential. Calculations performed on cluster models have shown that in absence of the external field the electrons tend to distribute more over the 3s levels of the Mg ions around the vacancy [38]. The localization of the electron in the center of the vacancy is... 

It would be interesting to be able to assign electron populations to the atoms in a molecule, or at least to have a way in which to see how electron densities change as a result of the introduction of substituents. In the latter case, an electron density difference map can be quite revealing. However, it does not provide a ready means for giving quantitative expressions for population changes, and is not useful for giving absolute values for populations. 

In the preceding discussion it was assumed that the central bond of bicyclobutane is normal , In other words, the bonding is due to some electron density between the two bridgehead carbons. However, it is worth pointing out that in one case this assumption has been challenged. Thus, electron-density difference maps obtained for 4 show that although the distance between C(l) and C(6) is not unusual for a C-C bond (1.574 A), no residual electron density was observed between these two carbon atoms. However, whatever its physical meaning is, this seems to be a unique case since none of the many theoretical... 

The development of synchrofron radiafion as an X-ray source ° " has permitted accumulafion of data for elecfron density difference maps in less fhan 1 s and if is expected that such data can eventually be acquired in 1 If a suifable photochemical... 

However, no correction for decomposition was applied to the data since recollection of a small number of intensity data well-dispersed in reciprocal space indicated that decomposition was not isotropic, and hence a simple correction could not be made (14 ). Further, even more complex forms of decomposition corrections could not explain the observed intensity changes. It thus seemed best to allow this systematic error to remain in the data, rather than to introduce additional errors. In this crystal the maximum decay was small but nevertheless significant however, there were no unusual features in either the final electron density difference map or the thermal parameters (see Tables I and III). During exposure to x-rays it was noted that the crystal became gradually darker, changing from an initially colorless prism to yellow, brown, and finally a deep brown-black. While this particular crystal was not subjected to further analysis, other crystals of this material behaved similarly under x-, 7- or UV irradiation. Product isolation was carried out as described below. Details of the structure analysis, in outline form, are given in Table I. Atomic coordinates appear in Table II, while temperature factors appear in Table III. 

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... 

Mg , and Ca (n =2, 3) clusters at the Mpller-Plesset electron correlation level (MP4) and the SCF level, using a reasonably large basis set [6-311 + G(3df)]. The 2-and 3-body decompositions of the interaction energy at the MP4 and SCF levels, the NBO population analysis and the electron density difference maps allow to elucidate the nature of bonding in alkaline-earth clusters 2001 by Academic Press. 

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. 3. Substrate binding site on Taka-amylase A deduced from electron density difference maps with the enzyme-maltose complex and model building. The seven saccharide binding sites are numbered. Presumed catalytic amino acids Asp-206 and Asp-297 surround the sessile glycoside bond. Glu-230 is considered as a possible catalytic amino acid as well because of its proximity to the reaction center. Adapted from Matsuura et al. (262) with permission from J. Biochem (Tokyo).

C4. Using the quantities IF bJ - IF i l as coefficients and the calculated phases, a density difference map is constructed. This map indicates where adjustments of the current approximation Kj of the theoretical nuclear configuration is required. One potential problem involves the effects of cut-off errors related to limitations of with regard to scattering angles. Another potential problem is that the phases may need adjustments. Nevertheless, the difference density map so obtained is expected to indicate the likely components of the nuclear geometry correction AK- for step D. 

C5. Another density difference map that provides input for estimating the nuclear geometry correction AKj is obtained by using the quantities CF s caic coefficients and the calculated phases, where the cut-off errors are no longer relevant. This map is expected to indicate the dominant components in the nuclear geometry correction AK for step D. 

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...

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