Maximum entropy method (MEM)

The MEM considers electron densities on the same grid over the unit cell as have been used for computation of the corresponding prior densities. The informational

Table 5.2: Details of MEM calculations. The initial RF is the RF value for the prior, the final RF is the RF value for the MEM-optimized electron density map (MEM electron density). Δρ(min/max) refers to the minimum and maximum values in the difference Fourier map calculated for the MEM electron density.

α-Glycine D, L-Serine L-Alanine Ala-Tyr-Ala_Etoh

Grid size (˚A) 0.04 0.04 0.04 0.04

No. of pixels 128×288×144 256×216×128 144×324×144 216×216×324 IAM PRIOR

χ²_aim 0.3131 0.55 0.70 1.35

Initial R_F 0.0253 0.0356 0.0327 0.0406

Final R_F 0.0105 0.0180 0.0193 0.0255

Δρ(min/max) (e/˚A³) -0.15/0.13 -0.18/0.20 -0.18/0.18 -0.29/0.26 IAM-HO PRIOR

χ²_aim 0.3131 0.55 0.70 1.40

Initial R_F 0.0259 0.0404 0.0325 0.0409

Final R_F 0.0108 0.0183 0.0193 0.0258

Δρ(min/max) (e/˚A³) -0.20/0.14 -0.19/0.19 -0.19/0.18 -0.30/0.24 INV PRIOR

χ²_aim 0.90 0.80 1.05 2.80

Initial R_F 0.0143 0.0196 0.0217 0.0252

Final R_F 0.0088 0.0129 0.0157 0.0196

Δρ(min/max) (e/˚A³) -0.12/0.11 -0.13/0.15 -0.16/0.16 -0.23/0.19 MP PRIOR

χ²_aim 0.85 0.90 0.90 2.50

Initial R_F 0.0125 0.0176 0.0184 0.0224

Final R_F 0.0086 0.0144 0.0142 0.0186

Δρ(min/max) (e/˚A³) -0.11/0.11 -0.13/0.17 -0.16/0.16 -0.20/0.20

5.3. COMPUTATIONAL DETAILS 79 coordinates x_k; ρ^prior_k = ρ^prior(x_k) is the corresponding value of the prior density.

The maximum of S is searched for variation of {ρk} subject to the F-constraint CF² = 0, with (Sakata and Sato, 1990; Hofmann et al., 2007a). where Fobs(H_i) is the phased observed structure factor of Bragg reflection H_i with standard uncertaintyσ(H_i) and static weightwi. FM EM(H_i) is obtained by discrete Fourier transform of the electron density{ρk}. The summation in Eq. (5.2) extends over all measured reflections NF.

The MEM is an iterative procedure, where the value of χ²aim defines the point of convergence through CF² = 0. Phases of Fobs(H_i) are the phases of the calculated structure factors of the structure model. The model thus enters into the procedure in two ways: as values of the model density{ρ^prior_k }in the expression of S [Eq. (5.1)], and as reflection phases in the constraint on the data [Eq. (5.2)].

MEM calculations have been performed with the computer program BayMEM (van Smaalen et al., 2003). Four MEM-electron densities—denoted by ρ^MEMIAM (x), ρ^MEMIAM−HO(x),ρ^MEMINV (x) andρ^MEMMP (x)—have been generated for each compound, with a prior given by the dynamic model density of the IAM, IAM-HO, INV and MP models, respectively. Following procedures given in Hofmann et al. (2007a), we have determined optimal values of χ²_aim for each of the sixteen MEM calculations (Table 5.2). Previous values of χ²_aim for the IAM priors of α-glycine, L-alanine and Ala-Tyr-Ala_Etoh are basically confirmed (Netzel and van Smaalen, 2009). Almost the same values are presently found for the IAM-HO priors. Input data for BayMEM for the INV and MP priors have been generated on the basis of the final refinements with XD2006, instead of JANA2006 that has been used for IAM and IAM-HO priors. XD2006 and JANA2006 employ different weighting schemes, i.e. different sets of standard uncertainties of the reflections (the instability factor cannot be used in XD2006). This corresponds to smaller standard uncertainties in XD2006.

Accordingly, following the procedure by Hofmann et al. (2007a), we find larger

Figure 5.1: C1–C2–N plane of density maps ofα-glycine. (a, b) residual density (difference Fourier map) with contours at 0.05 e/˚A³; (c, d) dynamic deformation density [Eq. (5.4)]

with contours at 0.05 e/˚A³; and (e, f) MEM density with contours at 0.2 e/˚A³ up to 2.5 e/˚A³. For (a, c, e) the IAM prior, and for (b, d, f) the IAM-HO prior has been used.

The numbers on the axes indicate the distance in ˚A with respect to an arbitrarily selected origin. Solid lines denote positive values, dotted lines denote negative values and dashed lines are the zero contour.

5.3. COMPUTATIONAL DETAILS 81

Figure 5.2: C1–C2–N plane of density maps ofα-glycine. (a, b) residual density (difference Fourier map) with contours at 0.05 e/˚A³; (c, d) dynamic deformation density [Eq. (5.4)]

with contours at 0.05 e/˚A³; and (e, f) MEM density with contours at 0.2 e/˚A³ up to 2.5 e/˚A³. For (a, c, e) the INV prior, and for (b, d, f) the MP prior has been used. The numbers on the axes indicate the distance in ˚A with respect to an arbitrarily selected origin. Solid lines denote positive values, dotted lines denote negative values and dashed lines are the zero contour.

Figure 5.3: C1–C2–N plane of difference density maps [Eq. (5.3)] ofα-glycine for (a) INV prior and (b) MP prior. The numbers on the axes indicate the distance in ˚A with respect to an arbitrarily selected origin. Contours are at 0.05 e/˚A³; solid lines denote positive values, dotted lines denote negative values and dashed lines are the zero contour.

optimal values of χ²_aim for INV and MP priors than in the case of the IAM priors (Table 5.2).

Bindzus and Iversen (2012) have recently employed the residual density analysis of Meindl and Henn (2008) as a criterion for finding the optimal value ofχ²_aim. We like to stress that the method of Hofmann et al. (2007a)—by its very principle—

leads to smooth MEM electron density maps with zero or very few spurious maxima, thus facilitating a meaningful topological analysis of the resulting electron density maps.

MEM densities and dynamic model densities have been visualised by four types of maps. Contour maps of sections of the density itself show atomic maxima as well as BCPs [Figs. 5.1(e),(f) and 5.2(e),(f)].² Difference Fourier maps provide the residual density ΔρM EM of remaining misfit between model and data [Figs.

5.1(a),(b) and 5.2(a),(b)]. The difference between the MEM density ρ^MEMPRIOR(x) and the prior density ρ^PRIOR(x) is defined asρ^diff(x) with [Fig. 5.3]

ρ^diff(x) =ρ^MEMPRIOR(x)−ρ^PRIOR(x) (5.3) where PRIOR stands for any of the four types of structure model. Finally, in analogy with static deformation densities, the dynamic deformation density is [Figs.

2Sections of maps similar to Figs. 5.1–5.3 are given for the other three compounds in the supplementary material B.

5.3. COMPUTATIONAL DETAILS 83

Table 5.3: Topological properties of covalent bonds of α-Glycine: ρBCP (e/˚A³; first line) and ∇²ρBCP (e/˚A⁵; second line) for eight different density maps.

Dynamic Model density MEM density

Bond IAM IAM-HO INV MP IAM IAM-HO INV MP

C1–O1 2.042 2.044 2.636 2.701 2.482 2.526 2.749 2.735 12.76 12.30 -17.18 -19.44 8.03 -10.66 -15.93 -15.25 C1–O2 2.016 2.020 2.598 2.648 2.341 2.376 2.611 2.601 7.15 6.96 -21.83 -23.61 7.18 -3.68 -15.53 -14.48 C1–C2 1.184 1.183 1.696 1.698 1.552 1.566 1.681 1.694 0.24 0.28 -14.28 -13.28 -12.29 -13.75 -14.00 -15.16 C2–N 1.400 1.401 1.749 1.657 1.500 1.518 1.656 1.649 1.88 1.86 -11.65 -10.21 -7.62 -10.01 -6.89 -7.67

5.1(c),(d) and 5.2(c),(d)]

ρ^def(x) =ρ^MEMPRIOR(x)−ρ^IAM∗(x). (5.4) ρ^IAM∗(x) is the dynamic model density constructed from the IAM^∗, which is defined as an IAM obtained by removing any MP parameters from the model. So, for IAM and IAM-HO priors, IAM^∗ is equal to the respective model and ρ^def(x) = ρ^diff(x), while for INV and MP priors, IAM^∗ borrows atomic positions and ADPs from the respective models and it differs from the IAM.