Extensions to the MEM

A flat prior density with

ρ^prior_k = N_e

V_cell (2.8)

does not incorporate other information than the number of electrons in the unit cell, which are uniformly distributed over the volume of the unit cell. Whereas a non-uniform prior contains information about atoms and their positions in the unit cell.

The use of a flat prior would result in MEM densities that contain noise and artifacts, of which the effect on the density is larger than effects due to chemical bonding.⁵⁰ The preference of a non-uniform prior above a flat prior as reference electron density has shown to be favorable, because it enhances the quality of the MEM density.¹⁰

2.2. ACCURATE CHARGE DENSITIES BY THE MEM 19

The non-uniform prior is obtained by employment of the independent spherical atom model (ISAM), which provides coordinates and atomic displacement parameters for the computation of the non-uniform prior electron density [see example in Figure 2.1(a)].

Spurious maxima of the MEM densities are eliminated by the method of prior-derived F-constraint (PDC)⁵¹ with

where N_all = N_F +N_{P DC}. The iterations are performed with the summation of Equation 2.9, which includes an extra term (compare to Equation 2.3) incorporating reflections that were not measured. F_prior(H_l) up to a high resolution, e.g. up to sin(θ)/λ = 2.5 Å⁻¹, are obtained from {ρ^prior_k } by discrete Fourier transform. The smallest standard uncertainty of the experimental reflections is selected for σ(H_l).

Due to decreasing scattered intensities with increasing scattering angle, structure factors may be measured as weak or unobserved. By the method of PDC, they are obtained by Fourier transform of the prior density. These calculated structure factors are good estimates for structure factors of high-angle reflections, because mainly core electrons, which are considered to be well described by the ISAM, contribute to high-order reflections. The employment of the PDC enhances the quality of the densities produced by the MEM. However, the method of PDC can only be successful if a certain minimum of resolution of the experimental data is available, e.g. up to sin(θ)/λ >0.9Å⁻¹,⁵¹ which is a requirement on the data that is generally necessary for the purpose of charge density studies.

Within the F-constraint (Equation 2.3) and the PDC (Equation 2.9), respec-tively, static weights have been chosen according to de Vries et al.,⁵² where H_i is the scattering vector of the Bragg reflection and n a small positive integer. The standard MEM employs w_i = 1,^{43, 21, 44} which would lead to a non-Gaussian distribution of the residuals

∆F(H_i)/σ_i^{46, 52} with

∆F(H_i)/σ_i = 1

σ_i [F_obs(H_i)−F_{M EM}(H_i)] . (2.11) The studies of trialanine (Chapter 3)⁴² and α-glycine (Chapter 4)⁵³ have shown, that weighting of H_iⁿ with n = 3, 4 and 5 produce Gaussian distribution of the residuals∆F(H_i)/σ_i (Equation 2.11). Large residuals for low-order reflections, that would occur with w_i = 1, are suppressed and a larger weight is given to reflections with short scattering vectors. In accordance with de Vrieset al.,⁵² a weighting with n = 4 has been chosen for all MEM calculations of the present work. The Gaussian distribution of ∆F(H_i)/σ_i of the studied compounds is displayed in Figure H.1 of Appendix H.

The choice of an optimal value of χ²_aim as stopping criterion for the MEM cal-culation is of high importance, since it determines the point of convergence of the MEM iterations through the criterion C_F² = 0 (Equation 2.3). For employment of C_F^{P DC}2 (Equation 2.9) by the method of PDC instead ofC_F², the convergence of the iterations is still tested by C_F² (Equation 2.3).

The stopping criterion for the standard versions of the MEM corresponds to χ²_aim = 1 (Equation 2.3),^{43, 21, 44} denoted as historical MEM.⁴⁹ χ²_aim equals one when the difference|F_obs(H_i)−F_{M EM}(H_i)|is consistent with the experimental errorσ(H_i).

Theoretically, convergence would be reached with values of χ²_aim < 1, because the average difference |F_obs(H_i) − F_{M EM}(H_i)|/σ(H_i) is expected to be smaller than one.^{47, 46} The behavior, that χ²_aim at the point of convergence is smaller than one, is taken into account by the so-called classical MEM.^{47, 46} If standard uncertainties of observed reflections have been estimated smaller than their true values, values of χ²_aim >1can be determined.

One important tool for the evaluation of the quality of the MEM densities is the difference Fourier map, which visualizes the amount of unfitted density [see exam-ple in Figure 2.1(d)]. By imexam-plementation of that tool into the computer program BayMEM,⁴⁵ it is possible to produce hard evidence for the determination of the optimal value of χ²_aim.

Inspection of dynamic difference maps [ρ^{M EM} −ρ^prior] [see example in Figure 2.1(c)], also denoted as dynamic deformation maps, and difference Fourier maps obtained by the MEM, is employed to determine the optimal value of χ²_aim. It has shown that, if χ²_aim is too small, it would result in overfitted data, leading to

2.2. ACCURATE CHARGE DENSITIES BY THE MEM 21

the incorporation of experimental errors and noise into the densities. The dynamic deformation map in the case of a too lowχ²_aimwould show ripples of the contour lines, while the difference Fourier map would be flat and featureless. Whereas, a too large value of χ²_aim results in a large amount of unfitted data not taken account into the density map because the MEM calculation did not converge yet. Thus, remaining density would be visible as structure larger than the noise in the difference Fourier map, while the [ρ^{M EM} −ρ^prior] map would exhibit smooth contour lines.

Extensive calculations on trialanine and α-glycine (Chapters 3 and 4)^{42, 53} have yielded two different approaches for the purpose of the determination of the optimal value of χ²_aim. One approach employs the classical MEM^{47, 46} initially to determine the optimal value of χ²_aim. The classical MEM cannot be combined with the method of PDC,⁵¹ because it is properly defined only for an F-constraint based on experi-mental data. Thus, the classical MEM is performed without the PDC.⁵¹ The value of χ² at the point of convergence of the classical MEM divided by N_F yields the effective value of χ²_aim, which is subsequently set as χ²_aim for the historical MEM⁴⁹ (Chapter 4).⁵³ Once the value χ²_aim is determined, the historical MEM, employing the PDC⁵¹ and static weighting of H_iⁿ with n = 4 according to de Vries et al.,⁵² is performed to reconstruct the optimized MEM density.

It has been shown, that the χ²_aim obtained in that way, may lead in some cases (Appendices F and G) to overfitted data, which would be visible as ripples in the difference maps. In such cases the value of χ²_aim from the classical MEM is too small and thus used as benchmark for a manual search for the optimal value. For that procedure, the value ofχ²_aim from the classical MEM is gradually increased and used subsequently in the historical MEM. By inspection of the deformation maps and the difference Fourier maps, the optimalχ²_aim is pinpointed (Appendices F and G). Eventually, by this procedure that MEM calculation with the optimal χ²_aim is chosen.

The other method for the determination of the optimal χ²_aim starts with the historical MEM without preceding classical MEM. For that approach, series of his-torical MEM calculations, employing the PDC⁵¹ and static weighting of H_iⁿ with n = 4 according to de Vries et al.,⁵² with arbitrarily chosen values of χ²_aim around one, are performed (Chapter 3).⁴² By inspection of the resulting [ρ^{M EM} −ρ^prior] maps and difference Fourier maps, the point of convergence is estimated by a small amount of unfitted density, visible as structure in the difference Fourier maps, and

incorporated noise, visible as ripples in the difference maps. With respect to the inspection of these maps, that MEM calculation with the optimal χ²_aim is chosen.