Determination of parameters

MEM calculations according to the procedures described in section 4.2 result in MEM-optimized electron densities, ρ^{M EM}(x), that still depend on two parameters:

the value of χ²_aim [eqn (4.2)] and the value of n [eqn (4.3)]. The historical MEM employs χ²_aim = 1 and n = 0, reflecting the expectation value of χ² being equal to N_F for weights proportional to the inverse square of the s.u.’s [eqn (4.2)]. It has been shown that this estimate ofχ² is too pessimistic and leads to underfitted data, while n = 0usually leads to a distribution of values of

∆F(H_i)/σ_i = 1 σi

[F_obs(H_i)−F_{M EM}(H_i)] (4.4) that is far from the required Gaussian distribution.^{52, 46} The classical MEM is based on an alternative stopping criterion for the iterative procedure that is based on both the constraint and the entropy.⁴⁶ It is properly defined only for the case of n = 0 and an F-constraint based on the data, i.e. it cannot be combined with PDC.

In a first approach we have performed several runs of the MEM based on the F-constraint without PDC and with values of n equal to three (H3 weights), four

4.3. RESULTS 57

(H4) and five (H5), respectively. Iterations were stopped according to the criterion of the classical MEM approach.⁴⁶ The value of χ² at convergence can be computed for each run. Divided byN_F it provides an effective value for χ²_aim that would have led to the same density as the classical MEM in a MEM procedure withχ²_aimand eqn (4.2) as stopping criterion. Effective values of 0.1762, 0.3131 and 0.5504 for χ²_aim have been obtained by the classical MEM with weights H3, H4 and H5, respectively.

These values of χ²_aim were then used in MEM calculations withχ²_aim and eqn (4.2) as stopping criterion, now with an F-constraint including PDC. The quality of the resulting maps and the fit to the data were analysed by inspection of the difference maps ρ^{M EM}(x)−ρ^prior(x) and difference Fourier maps of ∆F(H_i) (Fig. 4.2).

It appears that the calculation with H5 and χ²_aim = 0.5504 has stopped too early, because the difference Fourier map is structured in the region of the atoms (Fig. 4.2e). The calculation with H3 and χ²_aim = 0.1762 results in a difference Fourier map without any structure, but the contours in the difference density are not perfectly smooth anymore, thus suggesting that some noise of the data has been fitted. Finally, the calculation with H4 and χ²_aim = 0.3131 provides both a featureless difference Fourier map and a difference density that is smooth. For the present compound and data it is thus found that the combination of weights H4 and a stopping criterion defined byχ²_aim = 0.3131 represents the optimal fit to the data.

This result does not allow the conclusion that weights other than H4 are inap-propriate, because other weights might lead to the correct ρ^{M EM}(x) for values of χ²_aim different from the values determined by the classical MEM. Therefore, we have performed a series of MEM calculations for all combinations of χ²_aim equal to 0.2, 0.3, 0.4, 0.5 and 1.0 and weights H0, H3, H4 and H5.

A first evaluation of the quality of the fits to the data is provided by the his-tograms of the residuals [eqn (4.4)]. For χ²_aim = 1 (represents the historical MEM) and weights H0 an unfavorable distribution of residuals is obtained with a few out-liers of large values of |∆F/σ|, in accordance with the discussion in the literature (Fig. 4.3a).⁵² MEM runs with weights H3, H4 or H5 result in distributions of resi-duals much closer to the Gaussian distribution than MEM runs with H0 do, but notable differences are still present. The Gaussian distribution is much better ap-proximated for MEM calculations withχ²_aim =0.3 (Fig. 4.3b). Weights H0 still have outliers but they are less severe than for the calculation with χ²_aim = 1. Outliers are not present in MEM calculations with weights H3, H4 and H5, while only minor

Figure 4.2: Sections of area 6 × 6 Ų through the C(1)–O(1)–O(2) plane of dif-ference Fourier maps and difdif-ference maps ρ^{M EM} − ρ^prior for selected MEM calcula-tions. (a) Difference Fourier map for weights H3 and χ²_aim = 0.1762. (b) Difference map with ∆ρ(min/max) = −0.37/0.56 electrons/ų. (c) Difference Fourier map for weights H4 and χ²_aim = 0.3131. (d) Difference map with ∆ρ(min/max) = −0.36/0.56 electrons/ų. (e) Difference Fourier map for weights H5 andχ²_aim = 0.5504. (f) Difference map with ∆ρ(min/max) = −0.33/0.54 electrons/ų. Contour lines are at intervals of 0.05 electrons/ų. Solid lines are contours of positive value, dotted lines are negative contours, and dashed lines represent the contour of zero value.

4.3. RESULTS 59

Figure 4.3: Distribution of residuals∆F(H_i)/σ_i [eqn (4.4)] for weights H3, H4, H5 and H0. (a) χ²_aim = 1. (b)χ²_aim =0.3. The insets show magnifications of the outer regions of the curvatures.

differences are found between histograms obtained for MEM calculations with the latter three weights. We have chosen H4 as weights for the final calculations.

Difference Fourier maps and ρ^{M EM}(x) have been analyzed for MEM calcula-tions with weights H4 and stopping criteria provided by χ²_aim = 0.2, 0.3131, 0.5 and 1.0 (Figs. 4.2c and 4.4). Maps of similar appearances have been obtained for H3 and H5. The maps for different values of χ²_aim confirm the observations made in Fig. 4.2. Too high values of χ²_aim lead to underfitted data as represented by structured difference Fourier maps. Too low values of χ²_aim lead to fitting of the noise in the data as represented by very flat difference Fourier maps and a noisy appearance of the contour lines in ρ^{M EM}(x) and ∆ρ^{M EM}(x). The optimal value of χ²_aim is approximately 0.3. We have chosen the MEM electron density obtained with χ²_aim = 0.3131 and weights H4 for a more detailed analysis by the AIM theory.

A good fit to the data thus requires weights Hn different from 1 (n > 0), while the precise value of n is not of large influence. H4 seems to be a good choice. The optimal value of χ²_aim can be obtained by inspection of difference Fourier maps and ρ^{M EM}(x) or ∆ρ^{M EM}(x). These results are in accordance with the results obtained for trialanine.⁴² Weights H4 are also the weights recommended by de Vries et al.⁵²