a relationship was found between the ammonia adsorption energy (E ads ) and the Al-O a distance

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1 Suorting Information 1. Analysis of relationshi between E ads and local geometric arameters θ and φ. 2. Numerical analysis of relationshi between the Al-O (a) distance (a) and structural arameters (b and ω). 3. Sulementary figures. 4. CIF files of otimized clusters. 1. Analysis of relationshi between E ads and a local geometric arameters θ and φ. In the section of 2. E ads and structural arameters within Si(OH)Al unit in Discussion, we found a relationshi between E ads and a (AlO a distance), as shown in Figure 6. We then tried to correlate E ads with other local geometric arameters (θ and φ). No correlation was found between a and θ, as shown in Figure S1, and no ositive relation between θ and E ads was seen in Figure 7. No deendence of E ads on φ was also found, as shown in Figure S2. 2. umerical analysis of relationshi between the Al-O (a) distance (a) and structural arameters (b and ω). As stated in the section of 2. E ads and structural arameters within Si(OH)Al unit in Discussion, a relationshi was found between the ammonia adsortion energy (E ads ) and the Al-O a distance (a). Subsequently, it was attemted to show the deendence of the Al-O a distance a on the crystal structure through the arameters b and ω. Definitions of these symbols are shown in Figures 3 and 4; b and ω are related to distance and relative orientation, resectively, between the two tetrahedra at both ends of the Si(OH)Al unit (see Figure 3). The distance a had a weak correlation with b, as shown in Figure S3, but some deviations 1

2 were observed. The distance a showed also a weak correlation with ω, as shown in Figure S4. Then, it was attemted to show the distance a by a simle function of b and ω as follows. a = Xb + Yω + Z (S1) The best fitting was obtained where X, Y and Z were 0.29, and 0.56, resectively, as shown in Figure S5. The ositive X and Y suorts that the relationshi is urely geometric, as a should be strained when b and/or ω were increased; the longer b should give longer lengths for all bonds in the Si(OH)Al unit, when ω is fixed; the larger ω should move O a to the uer direction in Figure 3, elongating a, when b is fixed. Thus, E ads could be related with a, and a could be related with b and ω. By combining these relationshis, a simle relationshi was found for the grou A (bidentates in relatively oen saces), as shown in Figure 9. E ads (67b ω) (2) Figures S1, S2 and 7 show no correlations. This demonstrates that not every arameter can be related to acid strength (E ads ) but only those containing a hysical insight. Our results recisely indicate that only two simle geometrical arameters can be related to acid strength (E ads ): one of the arameters, a (AlO a bond length), relates to the local structure of the Si(OH)Al unit, and the other relationshi is a combination of two arameters (b and ω), both relate to the distance and relative orientations of the tetrahedra located at both ends of the Si(OH)Al Brønsted site. 2

3 3. Sulementary figures. 150 θ / degree Figure S1: Plots of θ (SiO a Al angle) against a (Al-O a distance). The crystal tye is shown by the following colors; MFI (red), FER (black), FAU (blue), MOR (green), BEA (urle) and MWW (brown). The coordination environment of NH 4 + is shown by the following symbols; the coordination environment was A (,, and : H=H, H=H *, H=H=* and H H, resectively), B ( and : H H * and H H=*, resectively) or C ( and : H H=H=** and H H H, resectively) secies shown in Figure 2. shows the results based on the eriodic method, while the others were based on the cluster method. 3

4 E ads / kj mol ϕ / degree Figure S2: Plots of E ads (calculated value) against φ (rotation angle, showing the locations of oxygens surrounding the Si(OH)Al unit as dislayed in Figure 4). The crystal tye is shown by the following colors; MFI (red), FER (black), FAU (blue), MOR (green), BEA (urle) and MWW (brown). The coordination environment of NH 4 + is shown by the following symbols; the coordination environment was A (,, and : H=H, H=H *, H=H=* and H H, resectively), B ( and : H H * and H H=*, resectively) or C ( and : H H=H=** and H H H, resectively) secies shown in Figure 2. shows the results based on the eriodic method, while the others were based on the cluster method. 4

5 b / A Figure S3: Relationshi between a (AlO a distance) and b (distance between triangles at both ends of O 3 Si(OH)AlO 3 unit) in cases where NH 4 + formed A (,, and : H=H, H=H *, H=H=* and H H, resectively) secies shown in Figure 2 on MFI (red), FER (black), FAU (blue), MOR (green), BEA (urle) and MWW (brown). shows the results based on the eriodic method, while the others were based on the cluster method. 5

6 ω / degree Figrue S4: Relationshi between a and ω (lanar angle between triangles at both ends of O 3 Si(OH)AlO 3 unit) in cases where NH 4 + formed A (,, and : H=H, H=H *, H=H=* and H H, resectively) secies shown in Figure 2 on MFI (red), FER (black), FAU (blue), MOR (green), BEA (urle) and MWW (brown). shows the results based on the eriodic method, while the others were based on the cluster method. 6

7 (b/A ) (ω/degree) Figure S5: Deendence of a on 0.29b ω in cases where NH 4 + formed A (,, and : H=H, H=H *, H=H=* and H H, resectively) secies shown in Figure 2 on MFI (red), FER (black), FAU (blue), MOR (green), BEA (urle) and MWW (brown). shows the results based on the eriodic method, while the others were based on the cluster method. 7