Residual density

A main source of model bias lies in the choice of exponents in the single-exponential-type functions r exp (-ar) that are commonly used as the radial parts of the deformation functions this choice is often more of an art than a science [4]. Very little is known about the optimal values to be used for elements other than those of the first two rows. Selection of the best value for the exponents n is usually carried out by systematically varying exponents and monitoring the effects on the R indices and/or residual densities [8, 9]. The procedure can in some cases be unsatisfactory, as is the case when very diffuse functions centred on one atom are used to model most of the density in the bond, and even some of the density on neighbouring atoms [10]. 

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. 

Figure 1. Residual density in Si,—O - Al and Sii-Oi0-Si3 planes of scolecite after the multipole refinement 4 in Table 2. Contour interval 0.1 e A 3 negative contours are dashed, zero contour omitted.

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 ... 

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. 

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). 

As discussed in the previous section, a residual density calculated after least-squares refinement will have minimal features. This is confirmed by experience (Dawson 1964, O Connell et al. 1966, Ruysink and Vos 1974). Least-biased structural parameters are needed if the adequacy of a charge density model is to be investigated. Such parameters can be obtained by neutron diffraction, from high-order X-ray data, or by using the modified scattering models discussed in chapter 3. 

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. 

Palladium Monosilicide, PdSi, is obtained as brilliant bluish grey fragments on treating any Pd-Si alloy, containing above 60 per cent, of silicon, with dilute potash. The free silicon dissolves, leaving the silicide as residue. Density 7-31 at 15° C. 

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 Fourier syntheses of various residual density maps based on x-ray and neutron diffraction measurements seem to indicate that present-day diffraction data have sufficient information to pursue quantitative charge density analysis. One route is by a least squares analysis of x-ray data with generalized x-ray-scattering factors. However, published applications of the method do not lend themselves to a critical evaluation of... 

The d-wave model has also been supported strongly by the impurity effect. Ishida et al. found a linear temperature dependence of Cu on 1 /T at low temperatures, when Zn is added as an impurity to YBCO7, as seen in Fig. 15, and proposed a residual density of states to appear for E—>0 as in the inset of... 

Fig. 16.48 This residual density of states was pointed out theoretically to appear in the unitarity limit scattering by non-magnetic impurities in p- or d-wave superconductors in a heavy fermion study.49 From this result it became possible to explain the BCS-like temperature dependence of the penetration depth, A,50 which supported strongly the. 9-wave pairing model in high-7 , superconductors at an early stage, in terms of the d-wave + impurity model.51 53...

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.

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