Ab initio potential energy surfaces of propane dimer

Transcription

1 Ab initio potential energy surfaces of propane dimer Jukka-Pekka Jalkanen, Riina Mahlanen, Tapani A. Pakkanen Department of Chemistry, University of Joensuu, FIN Joensuu, Finland Richard L. Rowley Department of Chemical Engineering, Brigham Young University, Provo, Utah (Received ABSTRACT The potential energy surface of a model propane dimer was systematically mapped with quantum chemical calculations. The calculations included approximately 12 separation distances between the monomers for each of 121 different relative geometries, or 1487 different configurations. The generated potential energy map reveals that the most attractive interactions are those having a maximum number of close contacts between carbon and hydrogen. The potential well depth of the most attractive orientation found was kcal mol -1. The complete ab initio energy surface was fitted to a simple model consisting of pairwise-additive interatomic potentials, each modeled with a modified Morse function of interatomic distance. The resultant model accurately represents the entire propane dimer ab initio energy surface. The efficacy of the generated parameter set was tested with previously published ethane dimer energies and propane routes not included in fitting. The new parameter set is consistent with these results indicating a high level of transferability for the interatomic C-H, C-C, and H-H potentials obtained. 1

2 INTRODUCTION Potential energy surfaces of non-bonded dimers have been of great interest in the past few years. Understanding intermolecular forces is important when trying to apply computational methods to large systems containing hundreds of molecules. However, quantum chemical calculations for systems of this size cannot yet be carried out in a reasonable amount of time. Systematic mapping of ab initio potential energy surfaces for N-mers is difficult because the number of combinations of different orientations in space increases rapidly with N. A simplification commonly used in molecular dynamics (MD) simulations is to treat the total energy as a sum of pair interactions plus multi-body corrections. In practice, these multi-body corrections are often ignored or assumed small. The effect of neglecting multi-body interactions is often compensated for by using pair interaction parameters adjusted to match limited experimental data. These so-called effective pair potentials, which include contributions of the remaining N-2 molecules outside the interacting pair, may not be particularly transferable to other properties or simulation conditions. Fitting the model parameters so as to provide the best agreement with macroscopic properties may also hide model inadequacies that limit predictive capabilities. Quantum chemical calculations of interactions between two molecules have been a popular way of studying potential energy surfaces of various molecules. True pair potentials may be more transferable and the contributions for multi-body effects, while not easily calculated, are clearly and rigorously defined. Although ab initio studies of multi-body effects for molecules of interest are scarce, they have been studied recently for carbon dioxide 1, water 2-4, methane, 5 hydroxylamine 6 and methanol 7 by comparing ab initio results to fitted potentials. In this work we extend previous work on the calculation of actual pair potentials for dimers. Ab initio dimer interaction calculations of unsaturated hydrocarbons (benzene,

3 ethylene, 12,13 acetylene 14 ) and their combinations with small alkanes 15,16 have been published previously. Dimer systems of small polar carbon compounds (carbon dioxide, methanol 20, acetone 21 ) have also been studied, as well as noble gas molecules combined with carbon monoxide 22 or carbon dioxide. 23 Most of these studies included an attempt to fit ab initio data to an analytical functional form. Quantum chemical studies of saturated hydrocarbons have received a surprisingly small amount of attention so far, despite their importance to the petroleum industry. Studies for methane, 5,12,20,24-26 and ethane dimer systems 27,28 and their combinations 15 have been conducted previously, but systematic studies using larger alkane dimers are scarce. Tsuzuki et al. have reported a study concerning basis set effects on a propane dimer, 9 and computations by Gupta et al. compared MD-simulations and ab initio calculations for propane. 29 In this paper, we extend the work started with methane and ethane by Rowley et al, to include studies of propane dimer. The potential energy surface for the propane dimer is probed in considerable detail, using 1487 points and 121 different relative orientations. We present an analysis of the energy surface and a simple and accurate method of representing the ab initio data that can be used in MD simulations. COMPUTATIONAL DETAILS Accurate ab initio studies of interactions between hydrocarbon molecules require rigorous methods. With no permanent charges, the interactions are dominated by weak induced dipole-induced dipole attractions. Electron correlation methods are required to capture this behavior. In this work, second-order many-body perturbation theory (MP2) was used because it offers a good description of non-bonding interactions between hydrocarbon molecules at moderate computational cost. Most of the correlation effects are captured at MP2 level using a 3

4 basis set of adequate size and interaction energies are comparable to higher order MP4(SDTQ) calculations. 27 However, use of electron correlation makes the choice of basis set crucial, because interaction energy is strongly dependent on basis set size. Methane 12 and benzene 10 dimer attraction was considerably increased by introducing diffuse functions to both carbon and hydrogen. The effect of basis set size on the interaction energy of various dimer systems has been studied by several authors. 9,13,15,16,19,21,25,30,31 In this work, the propane monomer structure was optimized at the MP2/6-311+G(2df,2pd) level using Gaussian This equilibrium structure, basis set and electron correlation method were used without relaxation for all dimer calculations. This facilitates use of the resultant model in MD simulations of rigid molecules. Structural details and atom labeling of the unique nuclei for the optimized monomer are given in Table I and Figure 1. The interaction energy of a propane dimer system was studied using the supermolecular approach. Energies of the dimer were calculated at varying separation distances along routes of constant relative orientation between the two dimers. Basis set superposition error was eliminated with the counterpoise correction. 33 Relative dimer orientations were chosen to represent all combinations of vertices, edges and faces. These are illustrated in Figure 2. There are three unique vertices, five faces, and six edges. Counting each combination only once leads to 105 different routes, all of which were calculated in this work. In addition, a more complete mapping of the energy surface would require additional rotations of a monomer about the intermolecular approach axis (see Figure 3), and is of secondary interest because of the small impact that the more distant nuclei have on the dimer energy. Therefore, only 16 rotated orientations were included. 4

5 RESULTS A total of 1239 data points were calculated for the 105 main routes; 248 data points were also calculated for the 16 rotated routes. The voluminous counterpoise-corrected pair energies are available from the authors. We provide here simplified representations of the potential energy surface. Data points along individual routes were fitted independently to a single Morse potential, * A( r r ) 2 U ( r) = ε (1 (1 exp ) ) (1) using least squares. Parameters describing these energy curves are collected to Table II. Due to the large amount of numerical data, the calculated data points are not presented. Equation (1) with the parameters in Table II represents the ab initio energy curves for energies between 0.1 and 3.0 kcal mol -1. In Eq. (1), the parameter r* represents the location of the potential well minimum along the route, e shows the well depth, and A describes the slope of repulsion. The intermolecular distance is measured as a separation (in Å) between C2 carbon atoms unless otherwise stated. A topographical plot of well depths is shown in Figure 4 in which cells are colored according to the depth of the attractive well. Darker colors represent strong attractive interactions between monomers, lighter colors represent less attractive interactions, and numbers show well depths. For some routes, calculations were repeated at different intermolecular rotation angles. Figure 4 depicts the well depth of the most attractive interaction. As is to be expected, all routes show an attractive region due to electron correlation. It is noteworthy that routes with a bcc-face orientation appear to be the most attractive. Vertex-vertex routes have the least attractive minima of those studied, which is understandable due to the short distances between closely interacting hydrogens. In these 5

6 routes hydrogen atoms approach head-on. In general, routes that avoid hydrogen-hydrogen interactions and optimize carbon hydrogen interactions at closer distances are the most attractive. These include routes containing bcc-, cccc-, and abc-faces and ab-, bc- and ccledges. Most of the deep minima are concentrated inside the attractive triangle, formed by routes bc-bc, a-bcc and bcc-bcc in Figure 4. As can be seen from this figure, the bcc-bcc180 route was found to be the most attractive ( kcal mol -1. It contains many interlocking atoms and unscreened attractive C-H interactions, referred to here as cross interactions. It is also one of the structures that Tsuzuki et al. reported in their paper. 9 The main focus of their paper was to study the effect of basis set on interaction energy, but Table I and Figure 1 of their paper reveal that the geometry studied by them is very close to our bcc-bcc 180 route. Both have C 2h symmetry, but Tsuzuki et al. report a C2-C2 separation of 3.8 Å and an interaction energy of 1.85 kcal mol -1. Tsuzuki et al. used a smaller, optimized basis set (aug(df,pd)-6-311g** and MP2 method) in their calculations. Optimization of orbital exponents was not considered in this work. Additionally, the previously reported geometry was obtained from MP2/6-31G* calculations, while our monomer was optimized with the same basis set used in our dimer calculations. Both the differences in monomer geometry and basis set affect the calculated interaction energy. Results of a previous study of propane potential energy surfaces suggested that the deepest minimum occurs when two propane monomers form a T-shaped structure. 29 This would correspond to a bb-bcc route in our naming system. In our calculations, an energy minimum of kcal mol -1 was encountered on this route. Direct comparison of energies to that previous study is not viable because of the smaller basis set used in that study. Our calculations show that the stacked route (bcc-bcc) is not as attractive as the bb-bcc route, if intermolecular rotation is completely neglected. However, if the bcc-bcc orientation is 6

7 rotated 180 degrees so that all closely interacting atoms are interlocked, it becomes very favorable and has a deep attractive minimum. Our results suggest, that there are large variations in the potential energy with relative orientation. For example, Figure 5 shows energy curves when monomer A in fixed orientation is scanned with monomer B in variable orientations. Probing a propane monomer with a single atom is unlikely to yield the diversity seen here. 34,35 Furthermore, rotation of monomer B about the approach axis will further complicate the picture and limit the ability of a test atom approach to obtaining intermolecular potentials. Figure 5 also shows that some routes have very similar energy curves. Table III lists routes of similar or nearly equivalent potential energy. Steep repulsion occurs when like vertices, edges or faces come in close contact with each other. In these cases hydrogen and carbon atoms of the monomers begin to overlap when viewed along the approach axis. Routes 55, 28, 85 and 40 illustrate this. The softest repulsions occur when monomers approach in such a manner that hydrogens and carbons are interlocked, as in routes 84, 21, 8, 44 and 80. This can be seen by examining the A parameters in Table II, where large values of A denote steep repulsion and vice versa. Recently published papers on ethane 28 and methane 5 dimer interactions are in good agreement with these qualitative interpretations and with the magnitudes of similar kinds of routes. A comparison of corresponding routes in methane, ethane and propane is presented in Table IV. Although quantitative comparisons for different molecules are problematic, since intermolecular rotation angles are not necessarily the same, the trends illustrated in Table IV are consistent. For all three dimers the least attractive route is a vertex-vertex route. Likewise, the potential well depth of a vertex-vertex route increases as the number of atoms in the dimer increases due to more interatomic attractions. This trend can be seen in all of the routes in Table IV, although the incremental difference in going from ethane to propane seems to be larger than from methane to ethane. Methane-methane calculations revealed that the face-face 7

8 route, rotated by 60 degrees so that hydrogens were interlocked, was the most stable. Face-face interactions seem to be the most stable for propane dimers as well. In the case of ethane two interlocking edges produced a deep minimum, but the face-face case was found to be nearly as energetically favorable. FITTING OF THE POTENTIAL ENERGY SURFACE WITH ATOMIC PAIR POTENTIALS A convenient representation of the potential energy surface for use in MD simulations is a summation of pair potentials between the atomic sites. An important issue to be resolved is whether the complex and diverse potential energy surface illustrated in Figures 4 and 5 can be represented adequately by the sum of atomic pair interactions. Previous work on methane 5 and ethane 28 showed that pair-wise-additive interatomic potentials for several simple functional forms (Lennard-Jones, exp-6, etc.) were incapable of reproducing the complex nature of the full dimer potential surface. However, the modified Morse function, Eq. (1), was able to effectively model the surface under the assumption of pair-wise additivity. 5,28,30 Equation (1) does not include separate charge terms. This was a deliberate choice because all electron distribution and correlation effects included in the ab initio calculations can be effectively included in the parameters. For example, electrostatic potential calculations at various dimer separations show that partial charges on atoms change as intermolecular distance changes. 36 This effect is already included in our parameter set, obviating the need to introduce fixed partial charges on atoms from artificial assignment methods and the need to include Coulombic terms for nonpolar molecules. Furthermore, this decision helps to keep the potential model as simple as possible without losing accuracy. In this work fitting was done using the potential energies for the 105 main routes. Each route was included only once, even if 8

9 some routes were studied at different intermolecular rotation angles. The data used in regression consisted of 1239 calculated points. The potential energy at each point is a sum of all 121 atomic pair interactions, which we represent as E = 64E (2) Point 9ECC + 48ECH + HH In this equation, all carbon atoms are assumed equivalent regardless of their neighbors; likewise, all hydrogen atoms are equivalent. There are therefore nine carbon-carbon interactions, 48 cross interactions between carbon and hydrogen and 64 hydrogen-hydrogen interactions. This leads to total of nine parameters for the C-H, H-H and C-C Morse potentials. Points with a repulsion larger than 3 kcal mol -1 were not included in fitting. This was done to ensure that the attractive part of the potential well is adequately fitted and not ignored by larger residuals that can occur with the much larger repulsions at shorter distances. Nevertheless, shorter distances and larger repulsions are also adequately represented by the parameters obtained. Several different fitting algorithms were tested to find a robust method for searching parameter space. Best suited for our work seemed to be the simulated annealing method (SA). Goffe et al. showed that this method is applicable to a variety of optimization problems, 37 and their regression program was modified by us to find a global minimum for our problem. According to Goffe et al., the simulated annealing algorithm is a very robust method capable of dealing with large combinatorial problems comprised of exponential and non-continuous functions. These are difficult for traditional fitting algorithms. 38 Simulated annealing is a stochastic global optimization algorithm, which covers only a part of parameter space. The basic idea of the method is to accept some moves away from an apparent minimum toward 9

10 which the solution is moving in order to avoid becoming stuck in a local minimum. A sequence of three steps is used repeatedly: 1) calculate the function s value using current parameter values, 2) change the parameters by a variable amount, determined by the ratio of uphill to downhill moves, then 3) recalculate the function and apply Monte Carlo moves. If the new recalculated value is lower than the current best solution, store the parameters and the function s value as the new best solution, or if the new value is higher than the previous value, use the Metropolis criterion with the transition probability given by Fi 1 Fi T p = exp (3) where F i and F i-1 correspond to function values at trials i and i-1. As in all Monte Carlo methods, if p is greater than a generated uniform random number, the point is accepted; else the new point is rejected. After a certain number of iterations, step lengths and the temperature are adjusted. In our application of the annealing algorithm, temperature is viewed simply as a variable that controls the allowed step size on the energy surface. Temperature control is implemented with a reduction multiplier. This allows the fitting run to start with a very rough scan of parameter surface and large uphill moves. Step length is gradually decreased during the SA run. Parameter changes are adjusted to accept half of the moves. As temperature decreases, smaller and smaller uphill moves are accepted. The algorithm starts from a very rough picture of the potential energy surface surrounding the initial point. Ideally, SA concentrates on the most promising area and converges to global minimum. The algorithm terminates when either a maximum number of function evaluations is 10

11 reached or it cannot minimize the function further. Conventional optimization algorithms converged relatively quickly to a nearby minimum, but an abundance of local minima on potential energy surface made regression difficult. SA regressions for the propane dimer data took several days on a Compaq ES40 Alpha server, but produced considerably better results than all other methods that we have tried. Propane data were fitted using five randomly generated initial guesses. All of these converged to the same minimum. Although there were some small variations observed in the parameter sets, they were within statistical uncertainty and the squared residual value changed only by about 0.05 (kcal mol -1 ) 2 in the converged solutions. Table V shows the best parameter set for propane dimer data using the modified simulated annealing algorithm. Using these parameters with equation (1), the sum of all squared residuals for the 105 main routes was (kcal mol -1 ) 2, the average error per route was less than 0.14 (kcal mol - 1 ) 2, and the average error per data point is <0.012 (kcal mol -1 ) 2. These numbers are a bit misleading, because the errors are not distributed equally amongst the routes. The error for the most poorly fitted route (route 105) is (kcal mol -1 ) 2, while the best (route 23) is less than (kcal mol -1 ) 2. The combined error of the five least accurately fit routes is 3.38 (kcal mol - 1 ) 2 constituting almost one fourth of the total residual. Curiously, those routes with the highest residuals all included c-type hydrogen atoms. The cccc-cccc route had a particularly large residual (see Figure 2). The largest fitting errors were concentrated on the repulsive side of the potential due to the nature of least squares method. Table VI presents a summary of errors for all routes when the parameters in Table V and equation (1) were used. The interatomic parameters for propane are somewhat different than those obtained in earlier work on ethane. In the present work, epsilon for the H-H interaction was limited to values greater than zero. This was done mainly to prevent the turnover feature of the modified Morse potential at short distances. Despite the difference in H-H parameters when compared to 11

12 the ethane-ethane set, both produce nearly equivalent curves. Figure 6 shows a comparison of the interatomic potentials obtained for methane-methane, 5 ethane-ethane 28 and propanepropane dimers. The similarity in the potentials is evident even though different fitting methods were employed, and several general conclusions can be drawn from the comparison concerning the important issue of transferability. The C-C attraction is strongest in the propane-propane dimer. The C-C potential in the methane dimer is somewhat softer and shallower than in either propane or ethane. The C-C attraction in ethane is slightly less attractive than in propane. It is evident, that the cross interactions dominate the attractions between the monomers 5,28. While there is a difference in the apparent well-depth of the C-H potential between the three dimers, the minimum occurs at approximately the same distance. The C-H minimum for methane dimers is substantially smaller than in either ethane or propane, suggesting that transferability may not extend to methane where there is no C-C bond. It is also evident that the usual combining rules for interactions between unlike atoms are not valid. Figure 6 shows that these cross-interactions differ greatly from arithmetic and geometric means commonly used to approximate cross-interactions from the like interactions. Fixing cross-interaction parameters to geometric or arithmetic means for H-H and C-C unnecessarily restricts the solution in parameter space producing interactions that are not physically correct. While it is common to apply Lorentz-Berthelot (LB) combining rules in MD simulations, our results show that this is a practice that should be discontinued. Crossinteractions should be directly and independently obtained if accuracy in the simulation is expected. As the LB rule has no theoretical basis except for the case of equal size atoms with equal electronegativity, its use has been a matter of convenience because of the difficulty in the past of determining cross-interactions. 12

13 TESTING THE POTENTIAL MODEL It is important to understand that validation of the model pair-wise-additive potential is best done in three steps: (1) testing convergence of the ab initio molecular dimer potential with respect to level of theory and basis set size, (2) verification that the sum of interatomic pair potentials adequately represents the complex, multi-dimensional molecular dimer interactions surface generated from ab initio calculations, (3) testing the extrapolation capability of the model to new routes not used in the regression, and (4) testing the extrapolation of the interatomic potential parameters to other molecules. Item 1 was investigated in a previous paper^5; the remaining 3 items are discussed in this paper. The efficacy of the atomic pair-wise additive potential must be compared to true molecular pair potentials that are obtained from the ab initio calculations. The question here is not how accurately these potentials can reproduce experimental results in MD simulations, but how accurately they reproduce the molecular ab initio surface. The latter is the exact solution for the molecular pair that we desire to represent. Comparisons of MD simulation results with experimental data, on the other hand, would mix uncertainties of the atomic model developed here with the additional question of multi-body interactions mentioned in the introduction. This latter question is one that we have addressed in a previous paper 5 where we have shown that there are systematic ways to approach the inclusion of three-, four-, and higher-body interactions as needed. A. Propane-propane: Remaining 16 routes To test the predictive ability of the new interatomic potential models, the remaining 16 routes not included in the parameter regression training set were treated as a test set. Errors for these 16 cases are given in Table VII. In general, the predictive ability of the interatomic 13

14 potentials is satisfactory. Only three routes stand out having errors larger than 1.0 (kcal mol -1 ) 2. These three routes are bb-cccc rotated 90, bb-bb rotated 90 and ccs-ccs rotated 90 degrees. The former two involve direct interactions between CH 2 groups. The latter one is for methyl group edges approaching each other at an angle of 90 degrees. Table VIII shows the calculated MP2 energies of these routes and predicted energies based on the parameter set given in Table V and equation (1). In Table VIII, there are two data points with energies over 3 kcal mol -1. They are presented here to show the quality of predicted repulsions. While the absolute error increases when large repulsions are included in the comparison set, the results appear significantly worse. This is because of the steep slope of the repulsive part of the curve where a small difference in the spatial function produces a large absolute error in energy. This larger absolute error can be misleading and should be viewed relative to the magnitude of the repulsion itself. For example, Figure 7 shows the potential energy curve for one poorly predicted repulsive region using the interatomic potentials developed in this work. It also shows one of the average and one of the better predictions. As can be seen, on a relative basis, the repulsions are reasonably accurate and show the appropriate distances at which repulsion occurs even though the absolute deviation in energy at a fixed distance may be large. The predicted curve even in the worst case, is quite similar to the ab initio results. B. Ethane dimer data Another test for transferability was conducted using the previously published ethaneethane ab initio data. The reported best fit for ethane 28 using the modified Morse potential for the interatomic interactions yielded a total error of 4.19 (kcal mol -1 ) 2. The corresponding error obtained using the parameters of Table V regressed solely from propane dimer data was 8.48 (kcal mol -1 ) 2. There were 8 routes for which the potential energy minimum was accurately 14

15 predicted and 14 routes where the energy minimum was too shallow; none were predicted as too attractive. In ethane two routes (19 and 22) show errors larger than 1.0 (kcal mol-1) 2 : 2.53 and 1.07 (kcal mol -1 ) 2, respectively. Route 19 was previously found to be the most attractive for ethane. In this orientation the two monomers were brought together such that all methyl groups were interlocked and the carbon-carbon bonds were perpendicular. Using the propane parameters produced too shallow a minimum for this route. The potential energy curve was slightly too repulsive, but the attractive tail was accurately reproduced. The propane parameter set was also used to predict the ethane repulsion data, which included energies up to several hundreds of kilocalories. The routes for which the propane parameters accurately predicted the attractive wells also described the repulsive potential well. These more accurately predicted routes were for the most part ones involving direct contact of methyl groups. The largest errors were for those routes in which vertices, edges and faces approach the carbon-carbon bond directly. This suggests that more accuracy could be obtained if the C2 and C3 carbon atoms and the H3 and H2 hydrogen atoms are not treated equivalently, but the overall agreement is encouraging that general C-C, C-H, and H-H interactions may be used in n-alkanes. CONCLUSIONS We have done a systematic mapping of the propane dimer potential energy surface obtained from MP2/6-311+G(2df,2pd) calculations. Ab initio results were fitted to a simple Morse function and a set of 9 parameters were obtained that reproduce the 105 energy curves with very good accuracy. The unweighted least squares fitting scheme tends to emphasize the large repulsion energies. For this reason, only energies smaller than 3.0 kcal mol -1 were included in the regression. Even so, the repulsive energies constituted more than half of the 15

16 total residual error even though they constituted only about 1/6 of the total number of points. The efficacy and transferability of the newly regressed parameter set for the interatomic, pairwise additive potentials was tested with unfitted propane dimer orientations and previously published ethane dimer data. Errors were on the same order of magnitude as those in the regression. A slight underprediction of well depth and a systematically stronger repulsion of the ethane dimer potential energy curves was observed, but our parameter set can be applied to ethane data with good accuracy. Furthermore, a more detailed study of ethane and propane parameters reveals that both sets are very similar. Inclusion of the propane CH 2 -group slightly alters all the parameters, but the potential curves are remarkably similar. Though a more accurate description of the energy surface could be obtained by distinguishing between the methyl C and H atoms and the methylene C and H atoms, the good results shown here when all the C atoms and all the H atoms were treated as equivalent is an important simplification that suggests a good deal of transferability of the interatomic potentials obtained. We plan to investigate larger molecules because an accurate description of CH 2 -interactions may be of more importance when the potential model is transferred to larger molecules. The energy map created in this work serves as a useful example to illustrate the complexity of propane dimer potential energy surfaces. REFERENCES 1 S. Tsuzuki, W. Klopper, H. P. Lüthi, J. Chem. Phys., 111 (1999) M. Masella, J. P. Flament, J. Chem. Phys., 107 (1997) M. P. Hodges, A. J. Stone, S. S. Xantheas, J. Phys. Chem. A, 101 (1997) G. C. Groenenboom, E. M. Mas, R. Bukowski, K. Szalewicz, P. E. S. Wormer, A. van der Avoird, Phys. Rev. Lett., 84 (2000) R. L. Rowley, T. A. Pakkanen, J. Chem. Phys., 110 (1999)

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19 Table 1. Propane monomer structural parameters Bond length (Å) Angle ( ) Dihedral angle ( ) Ha-C3 1,08865 Ha-C3-C2 111,8398 Ha-C3-C2-C3 180,000 Hb-C2 1,09095 Hb-C2-C3 109,6011 Hb-C2-C3-Ha 58,175 Hc-C3 1,08991 Hc-C3-C2 110,7576 Hc-C3-C2-C3 59,6173 C3-C2 1,52205 C3-C2-C3 111,9374 Point group C 2V Table 2. Parameters for propane-propane interaction energy curves fitted with equation (1). Route e kcal mol - 1 A r* Å -1 Å Route e kcal mol - 1 A r* Å -1 Å Route e kcal mol - 1 A r* Å -1 Å 1) a-a 0,2660 1,3863 5, ) c-abc 0,7942 1,4996 5, ) bc-cc l 1,1682 1,3384 4,5598 2) a-b 0,3377 1,5056 5, ) c-acc 0,5879 1,4616 4, ) bc-cc s 1,2494 1,3677 4,7766 3) a-c 0,3794 1,4090 5, ) c-bcc 1,1797 1,4027 4, ) bc-abb 0,9328 1,4643 4,2016 4) a-ab 0,6192 1,5183 5, ) c-cccc 0,8830 1,3684 5, ) bc-abc 1,0254 1,4520 4,1561 5) a-ac 0,4207 1,3795 5, ) ab-ab 135 0,6979 1,5870 4, ) bc-acc 0,8663 1,4571 5,4549 6) a-bb 0,4769 1,5252 5, ) ab-ac 0,7621 1,5249 5, ) bc-bcc 1,3758 1,4580 4,0107 7) a-bc 0,8249 1,3801 4, ) ab-bb 0,8851 1,4426 3, ) bc-cccc 1,0495 1,4642 5,0282 8) a-cc l 0,5101 1,3058 5, ) ab-bc 1,1509 1,5417 4, ) cc l-cc l 0,5392 1,4703 5,7535 9) a-cc s 0,4743 1,5884 6, ) ab-cc l 1,0479 1,3175 4, ) cc l-cc s 0,6761 1,3577 5, ) a-abb 0,6011 1,4029 4, ) ab-cc s 0,9250 1,4803 5, ) cc l-abb 1,0616 1,3180 4, ) a-abc 0,6484 1,4034 4, ) ab-abb 0,7147 1,4641 4, ) cc l-abc 1,2000 1,3317 4, ) a-acc 0,4648 1,4868 6, ) ab-abc 0,7869 1,4748 4, ) cc l-acc 0,6798 1,4289 6, ) a-bcc 0,9124 1,3635 4, ) ab-acc 0,6863 1,4891 5, ) cc l-bcc 1,0848 1,4588 4, ) a- cccc 0,7249 1,3574 5, ) ab-bcc 1,0091 1,4326 4, ) cc l-cccc 0,9632 1,2636 5, ) b-b 0,4114 1,4152 4, ) ab-cccc 0,8788 1,4624 5, ) cc s-cc s 180 0,5500 1,5491 6, ) b-c 0,4793 1,5006 5, ) ac-ac 0,3819 1,4326 6, ) cc s-abb 0,9472 1,3876 4, ) b-ab 0,7627 1,3907 4, ) ac-bb 0,4602 1,5190 5, ) cc s-abc 1,0004 1,3649 4, ) b-ac 0,4816 1,4840 5, ) ac-bc 0,9491 1,4186 4, ) cc s-acc 0,5987 1,4858 6, ) b-bb 0,5648 1,4375 4, ) ac-cc l 0,5304 1,4207 5, ) cc s-bcc 1,2162 1,4005 4, ) b-bc 0,9573 1,3763 4, ) ac-cc s 0,4684 1,5807 6, ) cc s-cccc 0,8895 1,3873 5, ) b-cc l 0,6832 1,3002 4, ) ac-abb 0,7654 1,4342 4, ) abb-abb 0,6002 1,5114 4, ) b-cc s 0,6033 1,4542 5, ) ac-abc 0,8008 1,4173 4, ) abb-abc 0,9609 1,4942 4, ) b-abb 0,7497 1,3973 4, ) ac-acc 0,4822 1,5316 6, ) abb-acc 0,6467 1,5040 5, ) b-abc 0,7565 1,3877 4, ) ac-bcc 1,0046 1,4544 4, ) abb-bcc 1,1898 1,4229 3, ) b-acc 0,5459 1,4188 5, ) ac-cccc 0,7323 1,3871 5, ) abb-cccc 0,8137 1,4203 5, ) b-bcc 1,1097 1,4192 4, ) bb-bb 0,5374 1,4614 4, ) abc-abc 0,6799 1,4957 4, ) b- cccc 0,9181 1,3080 4, ) bb-bc 1,1285 1,4333 3, ) abc-acc 0,7548 1,4622 5, ) c-c 0,4157 1,5586 6, ) bb-cc l 0,6136 1,3922 4, ) abc-bcc 0,9436 1,4715 4, ) c-ab 0,7561 1,3838 4, ) bb-cc s 0,6028 1,4819 5, ) abc-cccc 0,8742 1,4649 5, ) c-ac 0,5346 1,4307 5, ) bb-abb 0,9038 1,4302 3, ) acc-acc 0,4722 1,5270 6, ) c-bb 0,6174 1,4902 4, ) bb-abc 0,9424 1,4185 3, ) acc-bcc 0,9804 1,4349 5, ) c-bc 0,9252 1,4737 5, ) bb-acc 0,5588 1,4864 5, ) acc-cccc 0,7070 1,4424 6, ) c-cc l 0,6990 1,3327 5, ) bb-bcc 1,1223 1,4580 3, ) bcc-bcc 1,0506 1,4647 4, ) c-cc s 0,7742 1,4175 5, ) bb-cccc 0,9183 1,3326 4, ) bcc-cccc 1,1127 1,4375 4, ) c-abb 0,7436 1,4971 5, ) bc-bc 45 1,0657 1,4665 4, ) cccc-cccc 0,7765 1,4576 5,

20 Table 3. Similarities on propane dimer interaction energy curves. Numbers correspond to route numbering of Table 2. Similar routes 9, 12 23, 24, 17, 47 56, 57 30, 54 68, 62 80, 44 89, 38 50, 99 Table 4. Potential energy minimum of corresponding routes in methane, ethane and propane. Propane E/kcal mol - 1 Ethane 22 E/kcal mol - 1 Methane 17 E/kcal mol - * acc-acc -0,472 F1-F1(route 2) -0,429 FF Ecl ~-0,3 abb-acc -0,647 F1-F2(route 4) -0,620 a-a -0,266 V-V(route 5) -0,228 VV Ecl -0,107 a-acc/c-acc -0,465 V-F1(route 7) -0,409 VF ~-0,275 a-abb/c-abb -0,601 V-F2(route 8) -0,531 ccs-ccs -0,550 E2-E2(route 10) -0,317 EE ~-0,18 a-ccs -0,474 V-E2(route 12) -0,347 VE ~-0,22 ab-acc -0,686 F1-E1(route 14) -0,602 ccs-acc -0,599 F1-E2(route 16) -0,407 a-ab -0,619 V-E1(route 21) -0,538 Deepest min Deepest min Deepest min bcc-bcc 180-1,625 E1-E1 90 ( route 19) -1,038 FF St -0,30 * Energy values taken from figure 2 in reference 17 Table 5. Propane parameters for modified Morse parameters. Interaction e (kcal mol -1 ) A (Å -1 ) r* (Å) C-C 0, ,2655 4,1844 C-H 0, ,2744 2,544 H-H 0,45284*10-4 1,2550 6,

21 Table 6. Propane fitting errors of each route. Error is in (kcal mol -1 ) 2 Route # Name Error Route # Name Error Route # Name Error 1 a-a 0, c-abc 0, bc-cc l 0, a-b 0, c-acc 0, bc-cc s 0, a-c 0, c-bcc 0, bc-abb 0, a-ab 0, c-cccc 0, bc-abc 0, a-ac 0, ab-ab 135 0, bc-acc 0, a-bb 0, ab-ac 0, bc-bcc 0, a-bc 0, ab-bb 0, bc-cccc 0, a-cc l 0, ab-bc 0, cc l-cc l 0, a-cc s 0, ab-cc l 0, cc l-cc s 0, a-abb 0, ab-cc s 0, cc l-abb 0, a-abc 0, ab-abb 0, cc l-abc 0, a-acc 0, ab-abc 0, cc l-acc 0, a-bcc 0, ab-acc 0, cc l-bcc 0, a-cccc 0, ab-bcc 0, cc l-cccc 0, b-b 0, ab-cccc 0, cc s-cc s 180 0, b-c 0, ac-ac 0, cc s-abb 0, b-ab 0, ac-bb 0, cc s-abc 0, b-ac 0, ac-bc 0, cc s-acc 0, b-bb 0, ac-cc l 0, cc s-bcc 0, b-bc 0, ac-cc s 0, cc s-cccc 0, b-cc l 0, ac-abb 0, abb-abb 0, b-cc s 0, ac-abc 0, abb-abc 0, b-abb 0, ac-acc 0, abb-acc 0, b-abc 0, ac-bcc 0, abb-bcc 0, b-acc 0, ac-cccc 0, abb-cccc 0, b-bcc 0, bb-bb 0, abc-abc 0, b-cccc 0, bb-bc 0, abc-acc 0, c-c 0, bb-cc l 0, abc-bcc 0, c-ab 0, bb-cc s 0, abc-cccc 0, c-ac 0, bb-abb 0, acc-acc 0, c-bb 0, bb-abc 0, acc-bcc 0, c-bc 0, bb-acc 0, acc-cccc 0, c-cc l 0, bb-bcc 0, bcc-bcc 0, c-cc s 0, bb-cccc 0, bcc-cccc 0, c-abb 0, bc-bc 45 0, cccc-cccc 0,8835 SSR 14,588 Table 7 Suitability of regressed parameter set for routes not included in fitting. Errors in (kcal mol -1 ) 2. Route Error Route Error Route Error Route Error bb-cccc 90 1,2541 bc-bc 135 0,3738 abb-abb 90 0,1632 acc-acc 180 0,0444 ab-ab 45 0,0768 ccl-ccl 90 0,2990 abb-abb 180 0,1447 bcc-bcc 180 0,6750 ac-ac 90 0,0271 ccl-ccl 180 0,7232 abc-abc 90 0,7222 cccc-cccc 30 0,6039 bb-bb 90 1,3474 ccs-ccs 90 1,3775 abc-abc 180 0,0353 cccc-cccc 90 0,0574 SError 7,

22 Table 8. Comparison of predicted and calculated propane-propane energies of three worst fitted routes in kcal mol -1. SR stands for squared residuals. bb-cccc 90 bb-bb 90 ccs-ccs 90 r/å MP2 Fit SR r/å MP2 Fit SR r/å MP2 Fit SR 4 2, ,8 9, ,12 4, ,2 0, ,2 1, ,31 1, ,4-0, ,4 0, ,50 0, ,8-0, ,6-0, ,69-0, , ,8-0, ,88-0, ,2-0, , ,07-0, ,4-0, ,2-0, ,27-0, ,6-0, ,4-0, ,46-0, ,8-0, ,6-0, ,65-0, , ,8-0, ,85-0, ,2-0, , ,04-0, ,4-0, ,2-0, ,23-0, ,6-0, ,4-0, ,62-0, ,8-0, , ,01-0, ,2-0, ,8-0, ,80-0, ,6-0, ,6-0, ,58-0, , ,37-0, ,4-0, SError* 1,2541 SError* 1,3474 SError* 1,3775 SError** 7,1637 SError** 4,9000 * Sum of squared residuals in (kcal mol -1 ) 2, only points having energy <3 kcal mol -1 included ** Sum of squared residuals in (kcal mol -1 ) 2, all calculated data points included 22

23 Hb Ha C2 C3 Hc Figure 1. Propane atom labeling Figure 2. Propane monomer faces and edges. Propane faces, edges are named after their vertices. For example cccc-face consists of four c-type hydrogens. Colors correspond to: Green=cccc, red=acc, yellow=abb, cyan=abc, pink=bcc. Propane ab edge is between yellow and cyan faces, ccs between green and red faces, ccl between pink and green faces, bc between cyan and pink faces, bb between CH2 carbons and ac between red and cyan faces. Figure 3. Intermolecular rotation 23

24 a b c ab ac bb bc cc l cc s abb abc acc bcc cccc a b c ab ac bb bc cc l cc s abb abc acc bcc cccc 0,266 0,338 0,379 0,619 0,421 0,477 0,825 0,510 0,474 0,601 0,648 0,465 0,912 0,725 0,411 0,479 0,763 0,482 0,565 0,957 0,683 0,603 0,750 0,757 0,546 1,110 0,918 0,416 0,756 0,535 0,617 0,925 0,699 0,771 0,744 0,794 0,588 1,180 0,883 1,170 0,762 0,885 1,151 1,048 0,925 0,715 0,787 0,686 1,009 0,732 0,448 0,460 0,949 0,530 0,468 0,765 0,801 0,482 1,005 0,732 0,643 1,128 0,614 0,603 0,904 0,942 0,559 1,122 0,918 1,394 1,168 1,249 0,933 1,025 0,866 1,376 1,049 1,112 0,676 1,062 1,200 0,680 1,085 0,963 0,767 0,947 1,000 0,599 1,216 0,890 1,038 0,961 0,647 1,190 0,814 1,216 0,755 0,944 0,874 0,506 0,980 0,707 1,625 1,113 0,981 Energy kcal mol -1 0,2-0,3999 0,4-0,5999 0,6-0,7999 0,8-0,9999 1,0-1,1999 1,2-1,3999 >1,4 Color Figure 4. Potential well depth of propane-propane interactions (in kcal mol -1 ) 24

25 3 2.5 E/(kcal mol -1 ) a-a a-b a-c a-ab a-ac a-bb a-bc a-cc l a-cc s a-abb a-abc a-acc a-bcc a-cccc r/å Figure 5. Potential energy surfaces of monomer A methyl hydrogen. Same point is scanned with different monomer B orientations. 25

26 E/(kcal mol -1 ) a-ccs a-acc b-abb b-abc b-ab ab-abc ac-abb ac-abc c-ac ac-ccl bb-bcc bb-bc ccl-abb ab-ccl ccs-bcc c-bcc ab-cccc abc-cccc r/å Figure 6. Similarity of some ab initio propane dimer potential energy surfaces. Similar routes have same color. 26

27 E/(kcal mol -1 ) 2 PCC ECC 1.5 MCC PCH 1 ECH MCH PHH 0.5 EHH MHH r/å Figure 7. Comparison of parameter sets of methane, ethane and propane. Three letter acronyms PCC=propane C-C, PCH=propane C-H, PHH=propane H-H etc. 12 E/(kcal mol -1 ) bb-cccc 90, MP2 bb-cccc 90, fit bb-bb 90, MP2 bb-bb 90, fit ccs-ccs 90, MP2 ccs-ccs 90, fit cccc-cccc 90, MP2 cccc-cccc 90, fit ccl-ccl 180, MP2 ccl-ccl 180, fit r/å Figure 8. Some propane dimer routes not included in fitting. Squares mark ab initio energies of bb-cccc 90, diamonds bb-bb 90, triangles ccs-ccs 90, circles cccc-cccc 90, crosses ccl-ccl 180 route. Lines represent predicted energy curves. 27