The residual density

The difference Ap(r) between the total electron density p(r) and a reference density pref(r), is a measure for the adequacy of the reference density in representing the system. Difference densities Ap(r) are obtained by Fourier summation in which the coefficients AF are equal to the difference between the observed and calculated structure factors. If k is the scale factor, as defined in chapter 4, the difference structure factor AF is given by 

We have introduced the boldface notation to underline that AF is a vector in the complex plane (see Fig. 5.9), because both Fobs and Fcalc are, in general, complex quantities, as is evident from Eq. (5.6). The phase angles p of the two vectors are not necessarily the same, as is further discussed in section 5.2.5. In a different notation we may write, like Eq. (5.5), 

When the model used for Fcalc is that obtained by least-squares refinement of the observed structure factors, and the phases of Fca,c are assigned to the observations, the map obtained with Eq. (5.9) is referred to as a residual density map. The residual density is a much-used tool in structure analysis. Its features are a measure for the shortcomings of the least-squares minimization, and the functions which constitute the least-squares model for the scattering density. 

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Final residual indices of the refinement strategies are given in Table 2. On the residual density maps shown in Figure 1, the maxima and minima do not exceed 0.2 e A-3. 

If we are interested only in the determination of a molecular structure, as most chemists have been, it suffices to approximate the true molecular electron density by the sum of the spherically averaged densities of the atoms, as discussed in Section 6.4. A least-squares procedure fits the model reference density preKr)t0 the observed density pobs(r) by minimizing the residual density Ap(r), defined as follows ... 

The relation between the least-squares minimization and the residual density follows from the Fourier convolution theorem (Arfken 1970). It states that the Fourier transform of a convolution is the product of the Fourier transforms of the individual functions F(f g) = F(f)F(g). If G(y) is the Fourier transform of 9(x)-... 

Since Ap is the Fourier transform of AF, Eq. (5.12) implies that minimization of J (Fobs - Pcaic )2 dr and of J (Fobs - Fcalc)2 dS are equivalent. Thus, the structure factor least-squares method also minimizes the features in the residual density. Since the least-squares method minimizes the sum of the squares of the discrepancies in reciprocal space, it also minimizes the features in the difference density. The flatness of residual maps, which in the past was erroneously interpreted as the insensitivity of X-ray scattering to bonding effects, is an intrinsic result of the least-squares technique. If an inadequate model is used, the resulting parameters will be biased such as to produce a flat Ap(r). 

The deformation density is defined as the difference between the total density and the density calculated with a reference model based on unbiased positional and thermal parameters. The deformation density is obtained by Fourier transform, like the residual density [Eq. (5.9)], but with Fca c from the reference state with which the experimental density is to be compared. 

Two disadvantages of multipole partitioning should be mentioned. The first is that any density not fitted by the model is discarded in the partitioning process. Examination of the residual density is required to ensure the completeness of the set of modeling functions. The second is that very diffuse functions of the model, if included, violate the requirement of locality discussed above, and may lead to counterintuitive results. 

A series of fitting trials using different sets of phosphorus radial functions, Rn r) = NP e r, with the goal of zeroing the residual density (Eq. 9) led to n- 6,6,7,8 for l = 1,2,3 and 4 compared to 4,4,4,4 in Table 1. This result shows that it is fundamental to adjust the radial functions of atoms to small molecule theoretical calculations when no information concerning the radial function is available. 

Figure 36. Effect of bleach potential on bleaching efficacy. The lower the residual density the more efficacious the bleach.

When the application of Eq. (11) to a least squares analysis of x-ray structure factors has been completed, it is usual to calculate a Fourier synthesis of the difference between observed and calculated structure factors. The map is constructed by computation of Eq. (9), but now IFhid I is replaced by Fhki - F/f /, where the phase of the calculated structure factor is assumed in the observed structure factor. In this case the series termination error is virtually too small to be observed. If the experimental errors are small and atomic parameters are accurate, the residual density map is a molecular bond density convoluted onto the motion of the nuclear frame. A molecular bond density is the difference between the true electron density and that of the isolated Hartree-Fock atoms placed at the mean nuclear positions. An extensive study of such residual density maps was reported in 1966.7 From published crystallographic data of that period, the authors showed that peaking of electron density in the aromatic C-C bonds of five organic molecular crystals was systematic. The random error in the electron density maps was reduced by averaging over chemically equivalent bonds. The atomic parameters from the model Eq. (11), however, will refine by least squares to minimize residual densities in the unit cell. 

The distribution of the residual density of states has theoretically been calculated by Onishi et al.54 In Fig. 17 the behaviour of the T T= constant plot for Bi221237 and T1222336 are attributed to crystal imperfections. Non-annealed as-grown T12223 has a 7, of 115 K where the linear part in 1/7 is... 

Figure 18 shows the relations between Tc and the residual density of states Ares55 that are obtained from the expressions,... 

Fig. 18. Tc plotted against the residual density of states obtained from 1 / T ( ) and K(Q) in YBa2(Cui. vZnx)07, from 1/77 in T12223, Bi2212, UPd2Al3, Lai.ssSro.isCuC + Znl% and Sr2Ru04.

A similar sorption isotherm has been published for 2500-year-old waterlogged alder 21). EMC values were consistently about twice as great as comparable values for recent wood over the entire hygroscopic range. Additional data collected in Table III include EMC values at or near 100% relative humidity for fir, alder, and a series of oak finds. In many cases the residual density values are greater than 100%, probably because of enhanced shrinkage affecting the determination of air-dry density rather than a com-... 

Fig. 4.3 Final model of the tetrameric InCls, with coordinating Et2NH and N-H-Cl hydrogen bonds in the same orientation as in Figure 4.2. All hydrogen atoms bound to carbon have been omitted for clarity the hydrogen atoms bound to nitrogen are at the positions of the residual density maxima shown in Figure 4.2.

There are also many high residual electron density peaks with values of more than 1 e /A. Omission of the most disagreeable reflections with AF /s.u. > 6 (files nonml-03. ) lowers the / -values and also the residual density, but the result is not yet satisfactory. Disorder or solvent molecules are not detectable. 

Estimates of the Copper residual state can be made by assuming the material is at the same state as if it had achieved its final velocity by a single shock and then had rarefied to one atmosphere. The residual temperature of the Copper initially shocked to 830 kbar and then rarefied to a free-surface velocity of 0.3 cm/fisec is 768 K the residual density is 8.688 g/cc, comparable to the initial density of 8.903. Calculations were performed with Copper initial conditions of 8.903 g/cc and 300 K and of 8.688 g/cc and 768 K. Since the changed equation of state results in only slightly changed explosive shock pressure, the calculated results were insensitive to the Copper jet initial conditions. 

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