The algorithms in EDMA have been validated by a series of calculations on model electron densities for which we have been able to compute the topological properties analytically. Unfortunately, electron densities corresponding to realistic structure models are not accessible to simple analytical computation. On the other hand, computer programs are available which compute topological properties analytically from wave functions or multipole parameters, but here we were unable to calculate exactly the corresponding gridded electron densities. Therefore, we have constructed a simple structure-like model, for which we can accurately compute both the gridded densities and, analytically, the topological properties. The model consist of two Gaussian peaks placed in monoclinic unit cells:
ρG(r) = Ne where Ne is the number of electrons of the ”atom” and σ is the width of the peak.
Both ”atoms” are equal in model 1, while in the second model the ”atoms” have different width and different numbers of electrons (Table 3.1). Atom A has been placed on the origin; atom B has been put between grid points.
3.5. VALIDATION OF EDMA 31
Table 3.2: Properties of model 1 at the BCP obtained by EDMA and obtained analytically (last row). The BCP is located at (x, x, x).
Grid Laplacian
Resolution Location Electron Value Absolute Relative Points (˚A) xBCP density (e/˚A3) (e/˚A5) error (e/˚A5) error (e/˚A5)
643 0.055 0.124106 0.4769 148.8759 0.0188 0.126
723 0.049 0.124109 0.4769 148.8688 0.0117 0.079
843 0.042 0.124139 0.4773 148.9519 0.0948 0.637
963 0.036 0.124113 0.4769 148.8627 0.0056 0.038
1083 0.032 0.124126 0.4771 148.9089 0.0518 0.348 1283 0.027 0.124115 0.4769 148.8618 0.0047 0.032 1323 0.027 0.124121 0.4770 148.8892 0.0321 0.216 1443 0.024 0.124115 0.4769 148.8616 0.0045 0.030 1923 0.018 0.124116 0.4769 148.8615 0.0044 0.030 2163 0.016 0.124116 0.4769 148.8614 0.0043 0.029
Analytical 0.124117 0.4769 148.8571 – –
Table 3.3: Properties of model 2 at the BCP obtained by EDMA and obtained analytically (last row). The BCP is located at (x, x, x).
Grid Laplacian
Resolution Location Electron Value Absolute Relative points (˚A) xBCP density (e/˚A3) (e/˚A5) error (e/˚A5) error (10−3)
643 0.055 0.122933 0.5326 172.2412 0.0326 0.190
723 0.049 0.122939 0.5326 172.2343 0.0257 0.149
843 0.042 0.122975 0.5329 172.3021 0.0935 0.543
963 0.036 0.122947 0.5326 172.2297 0.0211 0.123
1083 0.032 0.122963 0.5327 172.2506 0.0420 0.244 1283 0.027 0.122951 0.5326 172.2288 0.0202 0.117 1323 0.027 0.122959 0.5326 172.2285 0.0199 0.116 1443 0.024 0.122963 0.5326 172.2284 0.0198 0.115 1923 0.018 0.122954 0.5326 172.2259 0.0173 0.101 2163 0.016 0.122954 0.5326 172.2251 0.0165 0.096
Analytical 0.122956 0.5325 172.2086 – –
Electron density maps have been generated with the computer program PRIOR for both models and several grid sizes (Tables 3.2 and 3.3). The most important result of the calculations is that all topological quantities at the BCP are determined to a very good accuracy regardless of the grid spacing. The error in thex-coordinate of the BCP is always below 8×10−5 ˚A; the error in the density at the BCP is never larger than 0.0004 e/A3; and the relative accuracy of the Laplacian is always better than 10−3. Finer grid spacings give more accurate values. However, the accuracy does not increase monotonically with increasing grid size. The best results are obtained for grid spacings where the BCP is located near a grid point, while grids where the BCP is between the grid points (843, 1083, 1323) result in somewhat larger discrepancies. In any case the errors introduced by interpolation are much smaller than errors from other sources, like experimental errors and methodological artifacts. The integrated charges of the atoms in Model 1 do not differ from the expected value of 37.5 electrons by more than 10−4 electrons, regardless of the resolution. In Model 2 the integrated charges of atoms 1 and 2 are 37.49971 and 56.50029 electrons, respectively, for the finest grid of 2163 points. The results for coarser grids do not deviate by more than 10−4 electrons, and in most cases only by 10−5 electrons. These results show that the integration procedure works well, and that neglecting the partitioning of the grid points at the borders between atomic basins introduces negligible errors.
Models 1 and 2 comprise two atoms lying on the diagonal of the unit cell. We have performed another series of calculations for models where the orientation of the interatomic vector as well as the distance between the atoms A and B vary in a wide range (other parameters are those of model 2). Computations for a grid size of 1283 points show large variations of the densities and Laplacians at the BCPs for the different models (Table 3.4), but error values are consistent with those of model 2 (Table 3.3), regardless of the positions of the atoms and the distance between them.
It can be concluded that EDMA is able to calculate the properties of critical points and atomic basins with very good accuracy. The original version of EDMA (Palatinus, 2003) has been used to determine the atomic positions (atomic modula-tion funcmodula-tions) of aperiodic [LaS]1.14[NbS2] and Rb2ZnCl4 (van Smaalen et al., 2003;
Li et al., 2011). Properties at BCPs and ionic charges (number of atoms in atomic basins) have been determined for several amino acids and tripeptides (Netzel et al., 2008; Netzel and van Smaalen, 2009). A major difference between the original and current versions of EDMA is the algorithm for determination of the atomic basins (Section 3.3.2). The original algorithm (Palatinus, 2003; Henkelman et al., 2006)
3.5. VALIDATION OF EDMA 33
Table 3.4: Properties at the BCPs of models with different distances between the atoms.
The grid resolution is 1283 points.
Electron density at BCP Laplacian
Distance EDMA Analytical EDMA Analytical Absolute Relative (˚A) (e/˚A3) (e/˚A3) (e/˚A5) (e/˚A5) error (e/˚A5) error (10−3) 0.9905 28.6163 28.6139 2960.8328 2960.9189 -0.0861 -0.029 1.1171 11.4283 11.4270 1762.2181 1762.1835 0.0346 0.020 1.1325 10.1361 10.1351 1630.1352 1630.1174 0.0178 0.011 1.1767 7.1353 7.1337 1285.7641 1285.6844 0.0797 0.062 1.3949 1.0344 1.0339 296.6280 296.5378 0.0902 0.304 1.4335 0.7099 0.7097 218.3109 218.2778 0.0331 0.152 1.4623 0.5326 0.5325 172.2370 172.2153 0.0217 0.126 1.4639 0.5242 0.5241 169.9820 169.9503 0.0317 0.187 1.4725 0.4803 0.4803 158.0710 158.0726 -0.0016 -0.010 1.4932 0.3888 0.3886 132.4896 132.4274 0.0622 0.470 1.4993 0.3651 0.3649 125.6588 125.6279 0.0309 0.246
1.5367 0.2471 0.2469 90.3665 90.3293 0.0372 0.412
1.5880 0.1421 0.1421 56.3160 56.3006 0.0154 0.274
and the new procedure (Section 3.3.2; (Tang et al., 2009)) are both available in EDMA. A comparison of the two methods for analysis of the electron density cor-responding to the independent atom model of α-glycine (Netzel et al., 2008) shows a substantial difference in the volumes and integrated charges of the atomic basins (Table 3.5), illustrating the need for advanced methods of determination of zero-flux boundaries. EDMA is also used frequently to locate atoms and assign element types in electron densities solvedab initio by charge flipping. To highlight just one of the numerous references, EDMA was used in the analysis of structures in the Al-Cu-Ta system, some of them ranking among the largest known inorganic structures (Weber et al., 2009).
Table 3.5: Number of electrons and Volumes of the atomic basins for the electron density of the independent atom model of α-glycine, obtained with EDMA at a grid of 0.04 ˚A resolution.
Compare with Netzel et al. (2008). Method ’on grid’ corresponds to the algorithm de-scribed by Palatinus (2003) and Henkelman et al. (2006). Method ’near grid’ corresponds to the algorithm by Tang et al. (2009), improved according to this work (Section 3.3.2)
Method ’on grid’ Method ’near grid’
Atom Electrons Volume (˚A3) Electrons Volume (˚A3)