1

Ab initio potential energy surfaces of propane dimer Jukka-Pekka Jalkanen, Riina Mahlanen, Tapani A. Pakkanen Department of Chemistry, University of Joensuu, FIN-80101 Joensuu, Finland Richard L. Rowley Department of Chemical Engineering, Brigham Young University, Provo, Utah 84602

(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 –1.625 kcal

mol-1. The complete ab initio energy surface was fitted to a simple model consisting of pair-

wise-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.

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 dioxide1,

water2-4, methane,5 hydroxylamine6 and methanol7 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,8-11

3

ethylene,12,13 acetylene14) and their combinations with small alkanes15,16 have been published

previously. Dimer systems of small polar carbon compounds (carbon dioxide,17-19 methanol20,

acetone21) have also been studied, as well as noble gas molecules combined with carbon

monoxide22 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 systems27,28 and their

combinations15 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

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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. Methane12 and benzene10

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 Gaussian94.32 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.

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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,

))exp1(1()( 2)( *rrArU −−−−−= ε (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

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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 ccl-

edges.

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 (-1.625 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 C2h 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 -1.122 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

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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 ethane28 and methane5 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

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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 methane5 and

ethane28 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

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

HHCHCC EEEE 64489Point ++= (2)

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

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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

−−

= TFF ii

p1

exp (3)

where Fi and Fi-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

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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 14.588 (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 0.8835 (kcal mol-1)2, while the best (route 23) is less than

0.003 (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

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the ethane-ethane set, both produce nearly equivalent curves. Figure 6 shows a comparison of

the interatomic potentials obtained for methane-methane,5 ethane-ethane28 and propane-

propane 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 monomers5,28. 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. Cross-

interactions 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.

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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 paper5 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

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 CH2 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 ethane-

ethane ab initio data. The reported best fit for ethane28 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

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

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, pair-

wise 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 CH2-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 CH2-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.

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Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A. Petersson, J. A. Montgomery, K.

Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, J.

Cioslowski, B. B. Stefanov, A. Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W.

Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S.

Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C. Gonzalez, J. A. Pople,

Gaussian, Inc., Pittsburgh PA, 1995.

33 S. F. Boys, F. Bernardi, Mol. Phys., 19 (1970) 553

34 E. Fraschini, A. J. Stone, J. Comp. Chem., 19 (1998) 847

35 J.-R. Hill, J. Comput.Chem., 18 (1997) 211

36 J. L. Banks, G. A. Kaminski, R. Zhou, D. T. Mainz, B. J. Berne, R. A. Friesner, J. Chem.

Phys., 110 (1999) 741

37 W. L. Goffe, G. D. Ferrier, J. Rogers, J. of Econometrics, 60 (1994) 65

38 J. Haataja, Optimointitehtävien ratkaiseminen, CSC Tieteellinen laskenta Oy, Helsinki

Finland 1995

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 C2V

Table 2. Parameters for propane-propane interaction energy curves fitted with equation (1). Route ε

kcal mol-

1

A Å-1

r* Å

Route ε kcal mol-

1

A Å-1

r* Å

Route ε kcal mol-

1

A Å-1

r* Å

1) a-a 0,2660 1,3863 5,8954 36) c-abc 0,7942 1,4996 5,2270 71) bc-cc l 1,1682 1,3384 4,5598 2) a-b 0,3377 1,5056 5,4334 37) c-acc 0,5879 1,4616 4,8516 72) bc-cc s 1,2494 1,3677 4,7766 3) a-c 0,3794 1,4090 5,7789 38) c-bcc 1,1797 1,4027 4,7922 73) bc-abb 0,9328 1,4643 4,2016 4) a-ab 0,6192 1,5183 5,1989 39) c-cccc 0,8830 1,3684 5,6490 74) bc-abc 1,0254 1,4520 4,1561

5) a-ac 0,4207 1,3795 5,7815 40) ab-ab 135 0,6979 1,5870 4,5733 75) bc-acc 0,8663 1,4571 5,4549

6) a-bb 0,4769 1,5252 5,0455 41) ab-ac 0,7621 1,5249 5,2046 76) bc-bcc 1,3758 1,4580 4,0107 7) a-bc 0,8249 1,3801 4,7174 42) ab-bb 0,8851 1,4426 3,9666 77) bc-cccc 1,0495 1,4642 5,0282 8) a-cc l 0,5101 1,3058 5,5198 43) ab-bc 1,1509 1,5417 4,2879 78) cc l-cc l 0,5392 1,4703 5,7535 9) a-cc s 0,4743 1,5884 6,3545 44) ab-cc l 1,0479 1,3175 4,4871 79) cc l-cc s 0,6761 1,3577 5,7504 10) a-abb 0,6011 1,4029 4,8250 45) ab-cc s 0,9250 1,4803 5,1942 80) cc l-abb 1,0616 1,3180 4,4569 11) a-abc 0,6484 1,4034 4,8120 46) ab-abb 0,7147 1,4641 4,2891 81) cc l-abc 1,2000 1,3317 4,3956 12) a-acc 0,4648 1,4868 6,3698 47) ab-abc 0,7869 1,4748 4,2557 82) cc l-acc 0,6798 1,4289 6,1244 13) a-bcc 0,9124 1,3635 4,5763 48) ab-acc 0,6863 1,4891 5,5112 83) cc l-bcc 1,0848 1,4588 4,6942 14) a-cccc 0,7249 1,3574 5,6135 49) ab-bcc 1,0091 1,4326 4,1063 84) cc l-cccc 0,9632 1,2636 5,4216

15) b-b 0,4114 1,4152 4,6332 50) ab-cccc 0,8788 1,4624 5,0584 85) cc s-cc s 180 0,5500 1,5491 6,4070

16) b-c 0,4793 1,5006 5,3341 51) ac-ac 0,3819 1,4326 6,0361 86) cc s-abb 0,9472 1,3876 4,8105 17) b-ab 0,7627 1,3907 4,2292 52) ac-bb 0,4602 1,5190 5,1860 87) cc s-abc 1,0004 1,3649 4,8336 18) b-ac 0,4816 1,4840 5,3548 53) ac-bc 0,9491 1,4186 4,8041 88) cc s-acc 0,5987 1,4858 6,4183 19) b-bb 0,5648 1,4375 4,2344 54) ac-cc l 0,5304 1,4207 5,8073 89) cc s-bcc 1,2162 1,4005 4,7958 20) b-bc 0,9573 1,3763 4,1379 55) ac-cc s 0,4684 1,5807 6,4435 90) cc s-cccc 0,8895 1,3873 5,7642 21) b-cc l 0,6832 1,3002 4,8449 56) ac-abb 0,7654 1,4342 4,8328 91) abb-abb 0,6002 1,5114 4,3902 22) b-cc s 0,6033 1,4542 5,2937 57) ac-abc 0,8008 1,4173 4,8348 92) abb-abc 0,9609 1,4942 4,0928 23) b-abb 0,7497 1,3973 4,2195 58) ac-acc 0,4822 1,5316 6,4642 93) abb-acc 0,6467 1,5040 5,5341 24) b-abc 0,7565 1,3877 4,2346 59) ac-bcc 1,0046 1,4544 4,7775 94) abb-bcc 1,1898 1,4229 3,9268 25) b-acc 0,5459 1,4188 5,6070 60) ac-cccc 0,7323 1,3871 5,7426 95) abb-cccc 0,8137 1,4203 5,0840 26) b-bcc 1,1097 1,4192 4,0061 61) bb-bb 0,5374 1,4614 4,0690 96) abc-abc 0,6799 1,4957 4,3781 27) b-cccc 0,9181 1,3080 4,7769 62) bb-bc 1,1285 1,4333 3,8799 97) abc-acc 0,7548 1,4622 5,4510 28) c-c 0,4157 1,5586 6,4199 63) bb-cc l 0,6136 1,3922 4,8722 98) abc-bcc 0,9436 1,4715 4,2045 290) c-ab 0,7561 1,3838 4,8976 64) bb-cc s 0,6028 1,4819 5,1157 99) abc-cccc 0,8742 1,4649 5,0595 30) c-ac 0,5346 1,4307 5,7482 65) bb-abb 0,9038 1,4302 3,9149 100) acc-acc 0,4722 1,5270 6,8290 31) c-bb 0,6174 1,4902 4,9773 66) bb-abc 0,9424 1,4185 3,9113 101) acc-bcc 0,9804 1,4349 5,3294 32) c-bc 0,9252 1,4737 5,1950 67) bb-acc 0,5588 1,4864 5,3813 102) acc-cccc 0,7070 1,4424 6,1968 33) c-cc l 0,6990 1,3327 5,6343 68) bb-bcc 1,1223 1,4580 3,9091 103) bcc-bcc 1,0506 1,4647 4,2089 34) c-cc s 0,7742 1,4175 5,7042 69) bb-cccc 0,9183 1,3326 4,6064 104) bcc-cccc 1,1127 1,4375 4,9674 35) c-abb 0,7436 1,4971 5,2356 70) bc-bc 45 1,0657 1,4665 4,2509 105) cccc-cccc 0,7765 1,4576 5,8757

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 Ethane22 E/kcal mol-

1 Methane17 E/kcal mol-

1* 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 ε (kcal mol-1) A (Å-1) r* (Å) C-C 0,16105 1,2655 4,1844 C-H 0,55162 2,2744 2,544 H-H 0,45284*10-4 1,2550 6,1543

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 0,1954 36 c-abc 0,0391 71 bc-cc l 0,0579 2 a-b 0,1666 37 c-acc 0,2371 72 bc-cc s 0,0360 3 a-c 0,2329 38 c-bcc 0,0394 73 bc-abb 0,0547 4 a-ab 0,0538 39 c-cccc 0,5106 74 bc-abc 0,0399 5 a-ac 0,2117 40 ab-ab 135 0,1162 75 bc-acc 0,1884 6 a-bb 0,0784 41 ab-ac 0,0101 76 bc-bcc 0,2654 7 a-bc 0,0289 42 ab-bb 0,0370 77 bc-cccc 0,3458 8 a-cc l 0,2226 43 ab-bc 0,1760 78 cc l-cc l 0,6321 9 a-cc s 0,1011 44 ab-cc l 0,0603 79 cc l-cc s 0,0407 10 a-abb 0,0614 45 ab-cc s 0,0315 80 cc l-abb 0,0607 11 a-abc 0,0164 46 ab-abb 0,0470 81 cc l-abc 0,0894 12 a-acc 0,2763 47 ab-abc 0,0168 82 cc l-acc 0,1354 13 a-bcc 0,0798 48 ab-acc 0,0878 83 cc l-bcc 0,0448 14 a-cccc 0,3153 49 ab-bcc 0,0688 84 cc l-cccc 0,0749 15 b-b 0,0626 50 ab-cccc 0,2841 85 cc s-cc s 180 0,4360 16 b-c 0,1067 51 ac-ac 0,0198 86 cc s-abb 0,0120 17 b-ab 0,0104 52 ac-bb 0,2580 87 cc s-abc 0,1255 18 b-ac 0,1341 53 ac-bc 0,1287 88 cc s-acc 0,0710 19 b-bb 0,2772 54 ac-cc l 0,1960 89 cc s-bcc 0,0217 20 b-bc 0,0165 55 ac-cc s 0,2712 90 cc s-cccc 0,8100 21 b-cc l 0,1767 56 ac-abb 0,0943 91 abb-abb 0,0717 22 b-cc s 0,2394 57 ac-abc 0,0074 92 abb-abc 0,0786 23 b-abb 0,0027 58 ac-acc 0,0172 93 abb-acc 0,1350 24 b-abc 0,0314 59 ac-bcc 0,2218 94 abb-bcc 0,2873 25 b-acc 0,0962 60 ac-cccc 0,2396 95 abb-cccc 0,0269 26 b-bcc 0,3965 61 bb-bb 0,4640 96 abc-abc 0,0699 27 b-cccc 0,5343 62 bb-bc 0,1929 97 abc-acc 0,0873 28 c-c 0,2737 63 bb-cc l 0,0498 98 abc-bcc 0,0321 29 c-ab 0,0375 64 bb-cc s 0,4470 99 abc-cccc 0,5660 30 c-ac 0,0937 65 bb-abb 0,0152 100 acc-acc 0,0509 31 c-bb 0,1138 66 bb-abc 0,0349 101 acc-bcc 0,1852 32 c-bc 0,0762 67 bb-acc 0,0378 102 acc-cccc 0,1121 33 c-cc l 0,2599 68 bb-bcc 0,0301 103 bcc-bcc 0,0612 34 c-cc s 0,1381 69 bb-cccc 0,1670 104 bcc-cccc 0,1047 35 c-abb 0,0593 70 bc-bc 45 0,1186 105 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 ΣError 7,9250

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 2,5143 1.5586 0.9132 2,8 9,9420 7.5303 5.8163 5,12 4,7471 2.8702 3.5224 4,2 0,4953 -0.0011 0.2464 3,2 1,4626 0.5177 0.8929 5,31 1,6455 0.6272 1.0370 4,4 -0,4031 -0.6544 0.0632 3,4 0,1283 -0.4014 0.2806 5,50 0,1518 -0.3567 0.2586 4,8 -0,7880 -0.8846 0.0093 3,6 -0,4326 -0.7354 0.0917 5,69 -0,4962 -0.7381 0.0585

5 -0,7273 -0.8076 0.0064 3,8 -0,6121 -0.8011 0.0357 5,88 -0,7170 -0.8342 0.0137 5,2 -0,6268 -0.6984 0.0051 4 -0,6165 -0.7502 0.0179 6,07 -0,7353 -0.7999 0.0042 5,4 -0,5221 -0.5848 0.0039 4,2 -0,5508 -0.6553 0.0109 6,27 -0,6683 -0.7121 0.0019 5,6 -0,4277 -0.4797 0.0027 4,4 -0,4656 -0.5508 0.0073 6,46 -0,5740 -0.6084 0.0012 5,8 -0,3479 -0.3880 0.0016 4,6 -0,3833 -0.4522 0.0047 6,65 -0,4792 -0.5067 0.0008

6 -0,2825 -0.3107 0.0008 4,8 -0,3119 -0.3655 0.0029 6,85 -0,3946 -0.4149 0.0004 6,2 -0,2297 -0.2471 0.0003 5 -0,2529 -0.2923 0.0016 7,04 -0,3231 -0.3358 0.0002 6,4 -0,1875 -0.1955 0.0001 5,2 -0,2053 -0.2321 0.0007 7,23 -0,2642 -0.2695 0.0000 6,6 -0,1537 -0.1541 0.0000 5,4 -0,1673 -0.1832 0.0003 7,62 -0,1779 -0.1707 0.0001 6,8 -0,1267 -0.1212 0.0000 6 -0,0933 -0.0885 0.0000 8,01 -0,1218 -0.1066 0.0002 7,2 -0,0875 -0.0744 0.0002 6,8 -0,0464 -0.0328 0.0002 8,80 -0,0607 -0.0408 0.0004 7,6 -0,0618 -0.0455 0.0003 7,6 -0,0250 -0.0121 0.0002 9,58 -0,0326 -0.0154 0.0003

8 -0,0446 -0.0278 0.0003 10,37 -0,0186 -0.0058 0.0002 8,4 -0,0327 -0.0169 0.0002

ΣError* 1,2541 ΣError* 1,3474 ΣError* 1,3775 ΣError** 7,1637 ΣError** 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

23

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

Ha

Hc

Hb

C3

C2

24

a b c ab ac bb bc cc l cc s abb abc acc bcc cccc a 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

b 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

c 0,416 0,756 0,535 0,617 0,925 0,699 0,771 0,744 0,794 0,588 1,180 0,883

ab 1,170 0,762 0,885 1,151 1,048 0,925 0,715 0,787 0,686 1,009 0,732

ac 0,448 0,460 0,949 0,530 0,468 0,765 0,801 0,482 1,005 0,732

bb 0,643 1,128 0,614 0,603 0,904 0,942 0,559 1,122 0,918

bc 1,394 1,168 1,249 0,933 1,025 0,866 1,376 1,049

cc l 1,112 0,676 1,062 1,200 0,680 1,085 0,963

cc s 0,767 0,947 1,000 0,599 1,216 0,890

abb 1,038 0,961 0,647 1,190 0,814

abc 1,216 0,755 0,944 0,874

acc 0,506 0,980 0,707

bcc 1,625 1,113

cccc 0,981

Energy kcal mol-1

Color

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 Figure 4. Potential well depth of propane-propane interactions (in kcal mol-1)

25

-1

-0.5

0

0.5

1

1.5

2

2.5

3

2 3 4 5 6 7 8 9 10

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

Figure 5. Potential energy surfaces of monomer A methyl hydrogen. Same point is scanned with different monomer B orientations.

r/Å

E/(k

cal m

ol-1

)

26

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

a-ccsa-accb-abb

b-abcb-abab-abc

ac-abbac-abcc-ac

ac-cclbb-bccbb-bc

ccl-abbab-cclccs-bccc-bcc

ab-ccccabc-cccc

Figure 6. Similarity of some ab initio propane dimer potential energy surfaces. Similar routes have same color.

r/Å

E/(k

cal m

ol-1

)

27

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.

-2

0

2

4

6

8

10

12

2 3 4 5 6 7 8 9 10

bb-cccc 90, MP2bb-cccc 90, fitbb-bb 90, MP2bb-bb 90, fitccs-ccs 90, MP2ccs-ccs 90, fitcccc-cccc 90, MP2cccc-cccc 90, fitccl-ccl 180, MP2ccl-ccl 180, fit

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.

-1

-0.5

0

0.5

1

1.5

2

1 2 3 4 5 6 7

PCC

ECC

MCC

PCH

ECH

MCH

PHH

EHH

MHH

r/Å

E/(k

cal m

ol-1

)

r/Å

E/(k

cal m

ol-1

)