Transferability of Buckingham Parameters for Short-Range Repulsion between Topological Atoms

The repulsive part of the Buckingham potential, with parameters A and B, can be used to model deformation energies and steric energies. Both are calculated using the interacting quantum atom energy decomposition scheme where the latter is obtained from the former by a charge-transfer-based energy correction. These energies relate to short-range interactions, specifically the deformation of electron density and steric hindrance, respectively, when topological atoms approach each other. In this work, we calculate and fit the energies of carbonyl carbon, carbonyl oxygen, and, where possible, amine nitrogen atoms to the repulsive part of the Buckingham potential for 26 molecules. We find that while the steric energies of all atom pairs studied display exponential behavior with respect to distance, some deformation energy data do not. The obtained parameters are shown to be transferable by calculating root-mean-square errors of fitted potentials with respect to energy data of the same atom in, as far as possible, all other molecules from our data set. We observed that 36% and 10% of these errors were smaller than 4 kJ mol–1 for steric and deformation energy, respectively. Thus, we find that steric energy parameters are more transferable than deformation energy parameters. Finally, we speculate about the physical meaning of the A and B parameters and the implications of the strong exponential and exponential-linear piecewise relationships that we observe between them.


Addi onal Computa onal Details
The scan range is the range of interatomic distances over which the deforma on and steric energies were studied.Wilson and Popelier used a range of 70% to 130% of the sum of van der Waals radii of the two atoms involved 1 .In this work, we used ranges star ng at larger interatomic distances and ending at the same distance as Wilson's range.This was done to reduce the effect of hydrogen atom overlap in the dimers as described in the main text.
Table S1.Scan ranges used for deforma on and steric energy calcula ons.Moreover, we no ced a secondary benefit to using a narrower scan range: scans performed using narrower ranges give lower values for the fi ed average, minimum and maximum RMSE, as shown in Tables S2 and S3.For each deforma on energy plot, data points were removed star ng from the smallest distance because removing data points from the furthest distance would lead to li le or no change in fi ng due to the exponen al shape of the repulsive Buckingham curve.The remaining data points were then fi ed to the same Buckingham poten al, eq S1, giving new values of  and .
Range / Å Although progressively narrowing the range further improves average RMSEs, it also diminishes the exponen al behaviour of the data.Therefore, we chose a scan range star ng from 2.3 Å for oxygen, and from 2.7 Å for carbon, in order to reduce the effect of non-exponen al behaviour at small distances on the fits while preserving the exponen al shape of the data, which would be lost if the trunca on were taken too far.Indeed, narrowing the scan range reduces non-exponen al behaviour (from errors at small distances).Narrowing the scan range by too much eliminates exponen al behaviour (because the data will almost lie on a straight line).
Using eq 14 in the main text (derived from equa ons first formulated by Costales et al. 2 ) we calculated the energe c contribu on from charge transfer,  , by rewri ng eq 14 as where  is the electron count of an atom in a system (here a dimer),  is the electron count of an atom in a reference state (here a monomer), and the ionisa on poten al  is the difference in energy between an atom with a charge of ⌈⌉ (ceiling of noninteger , e. g. 2 if N 1.6), and that same atom with a charge of ⌊⌋ loor of noninteger , e. g. 1 if  1.6 .For example, the ionisa on poten al (more rigorously called energy) of a carbon atom with a charge between +1 and +2 would be calculated as IP E C E C .The energies of the atoms and ions used to find these differences are given in Table S4 and were calculated in GAUSSIAN09 3 at the B3LYP/aug-cc-pVTZ level of theory.

Nitrogen
While the main text focused on carbonyl carbon and oxygen atoms, data regarding nitrogen atoms are presented in this sec on.As with carbon and oxygen, the nitrogen (appearing in only 10 out of 26 molecules) deforma on and steric energies were fi ed to a repulsive Buckingham poten al of the form given in eq S1, to yield the parameters  and .The plots in this sec on were made in the same way as Figures 4 and 5

Fi ed Parameters
Tables S5 to S8 list the parameters obtained by fi ng deforma on ("Def") and steric ("Str") energy data to a repulsive Buckingham poten al with the form of eq S1.Two scan ranges are shown: (i) the original, wider range used by Wilson and Popelier (Table S6 for oxygen and Table S8 for carbon), and (ii) the narrower range used in the main text (Table S5 for oxygen and Table S7 for carbon).
Table S5.Values of the  and  parameters along with the RMSEs of their fits to the deforma on and steric energy data for the oxygen atom.The  and  values presented are from the 2.3-4.0Å scan.Molecular dipole moments,||, are also given.

Molecule
Def  / kJ mol Occasionally, fits using our scan range resulted in extremely high  and  values due to the steric energy becoming nega ve at large interatomic separa ons, which would imply that the steric interac on becomes a rac ve at those distances.Two examples are shown in Table S9, where rounding the charge differently by using a ceiling func on instead of a floor func on, or using the next (meaning ⌊⌋ instead of ⌊⌋ 1 ) ionisa on poten al (actually energy) led to more reasonable parameter values and lower RMSE.While unphysical, the phenomenon of nega ve steric energies has been observed previously 2 with hydrogen atoms at distances smaller than 2.7 Å in relaxed ammonia dimer scans, where it was a ributed to changing geometries during the scan.More similar to the current work, the data in that work 2 for a frozen methane dimer scan seems to show nega ve steric energies for the carbon atom at distances greater than around 3.5 Å although this is not men oned in their text 2 .In the current work, where only frozen scans were performed, the nega ve steric energies occur for 12 of the 26 molecules studied, which are listed in Table S10.Currently, we can only offer a qualita ve descrip on of this phenomenon.In cases where the steric energy does not become significantly nega ve, such as ethanal and formamide, this could be due to numerical instability.More extreme cases such as cyclopropenone tend to occur when there is poorer contact between the carbonyl carbon atoms.However, poor contact does not necessarily lead to nega ve steric energies.Perhaps a way of quan fying the contact between atoms such as the interatomic surface area should be accounted for along with charge transfer in a further correc on for steric energy.Table S10.Distance at which the carbonyl carbon atoms exhibit nega ve steric energy.The most nega ve energy observed across the scan range is also given.To allow for a fair comparison of transferability for carbon deforma on and steric parameters, Figure 5b in the main text only contains molecules for which an exponen al deforma on energy curve could be fi ed; the other molecules' deforma on energy data displayed non-exponen al behaviour.Figure S3b shows the transferability of the steric parameters for the carbon atom in all 26 molecules.

Proper es
Wilson and Popelier performed dimeric scans of noble gases and small molecules, and fi ed deforma on energy to Buckingham poten als 1 .Figure S4 shows a plot of  against  from these fits.The lower  value here compared to those of >0.9 in Figures 6 and 7 in the main text is due to the larger variety of systems studied in the work of Wilson and Popelier.Plots of  and  against the molecular (monomeric) dipole moment and volume of a topological atom were created as a first a empt to link the parameters to a physical property.These are shown in Figure S5 (oxygen) and Figure S6 (carbon) for the dipole moment, and in Figure S7 for the volume.Monomeric molecular dipole moments were calculated using GAUSSIAN09 3 at the B3LYP/aug-cc-pVTZ level of theory.Topological atom volumes were calculated using the gs30 quadrature and Proaim method of the program 4 AIMAll19.
The figures in this sec on demonstrate the possible links between  and  that were examined in our current study.Figure S4 demonstrates that the exponen al rela onship between  and  is not limited to our study, the specific systems we studied, or indeed to homo-atomic interac ons because it includes some hetero-atomic interac ons.Figures S5, S6, and S7 show that there is no straigh orward rela onship between  and  , and the dipole moment or the volume of topological atoms.Nevertheless, there are weak correla ons between oxygen deforma on  and  and molecular dipole moment in Figures S5a and S5c due to the connec on between molecular dipole moment and charge transfer.
in Sec on 4 of the main text.Transferability is shown in FigureS1, while FigureS2plots the fi ed  values against the fi ed  values.Compared to carbon and oxygen, transferability of nitrogen is significantly worse, with no transferability RMSE values smaller than 4 kJ mol -1 in FigureS1.Moreover, the exponen al rela onship between the nitrogen deforma on  and  parameters is weaker than the corresponding rela onships for carbon and oxygen shown in Figures6 and 7in the main text, and absent for nitrogen steric  and .

Figure S1 .
Figure S1.Heatmaps showing the transferability of  and  parameters of the nitrogen atom in 10 molecules for (a) deforma on energy and (b) steric energy.Diagonal cells represent the RMSE between the fi ed Buckingham poten al and the original data it was fi ed to, while off-diagonal cells represent transferability RMSEs of the Buckingham curve plo ed using the  and  values of the molecule in the row compared to the data of the molecule in the column.

Figure S2
Figure S2 Rela onship between the (a) deforma on and (b) steric energy  and  parameters for nitrogen atoms.

Figure S3 .
Figure S3.Heatmaps showing the transferability of  and  parameters of the carbonyl carbon atom for (a) deforma on energy in 20 molecules and (b) steric energy in all 26 molecules.Diagonal cells represent the RMSE of the fi ed Buckingham poten al and the data it was fi ed to, while off-diagonal cells represent transferability RMSEs of the Buckingham curve plo ed using the  and  values of the molecule in the row compared to the data of the molecule in the column.

Figure S4 .
Figure S4.Rela onship of  and  parameters from the work of Wilson and Popelier 1 .

Figure S6 .
Figure S6.Plots of carbon  and  parameters against molecular dipole moment for (a) deforma on , (b) steric , (c) deforma on , and (d) steric .

Figure S7 .
Figure S7.Example plots of (a) parameter  , and (b) parameter  against atomic volume at 2.6 Å intermolecular separa on for carbon steric energy.

Table S3 .
Minimum, maximum and average carbon deforma on RMSE (in kJ mol -1 ) when changing the scan range."In deforma on" refers to the subset of molecules where deforma on energy displayed exponen al behaviour.

Table S4 .
Atomic energies calculated at the B3LYP/aug-cc-pVTZ level of theory taking into account spin mul plici es.

Table S6 .
Values for the  and  parameters along with the RMSEs of their fits to the deforma on and steric energy data for the oxygen atom in the 2.1-4.0Å scans (original Wilson-Popelier range).Only 25 molecules are listed because malonaldehyde is absent due to methylene hydrogens being too close together, leading to an error in GAUSSIAN09 3 .

Table S7 .
Values for the  and  parameters along with the RMSEs of their fits to the deforma on and steric energy data for the carbon atom in the 2.7-4.5 Å scans.Blanks due to errors or non-exponen al behaviour.

Table S8 .
Values for the  and  parameters along with the RMSEs of their fits to the deforma on and steric energy data for the carbon atom in the 2.3-4.5 Å scans (original Wilson-Popelier range).Blanks due to errors or non-exponen al behaviour.

Table S9 .
Effect of using different values of charge  when calcula ng charge transfer for carbonyl carbon of fluoroace c acid and cyclopropenone.