Non-empirical model potentials

8.3.1

Dispersion models

In order to use the repulsion model derived from the overlap calculations, a form o f the dispersion attraction between molecules must be formulated. A description o f the physical origins o f this contribution to intermolecular energies is given in Chapter 3. It is not yet possible to derive an accurate model o f dispersion directly from the charge distribution o f a molecule the size o f chlorobenzene, so the dispersion model must either be derived from simple semi-empirical models or taken from previously determined coefficients for organic molecules. As a consequence, the atom- atom dispersion model, unlike the electrostatics and repulsion, must be treated as isotropic. Previous applications o f the overlap model have combined the repulsion model with C& dispersion coefficients from density functional theory calculationst^ ^^’^^l or from the Slater-Kirkwood relationship to atomic polarisabilities.t^-^^'^^l M odels derived from either o f these m ethods have been successful, so there is no a priori reason to choose one over another and, so, several models are tested in the present study, based on either DFT or Slater-Kirkwood derived coefficients and using several different sources for the parameters.

As a first model, we take Cg coefficients derived by loannou and Amost^-^^l through time- dependent density functional theory. In his work, loannou started to build up a database o f atom -atom Cô coefficients by calculating the total Ce dispersion by integrating the calculated polarisabilities over imaginary frequencies;

C e^ = — (ico)ay (fru)dct) (8.1)

^ 0

By partitioning the molecular polarisabilities into atomic contributions within Heusden and Amos' (AP2) distribution s c h e m e , a t o m - a t o m dispersion coefficients were derived for a series o f C, H, N, and O containing small molecules. Though this method is theoretically rigorous and undoubtedly quite accurate, there are several drawbacks. Firstly, the resulting model represents the pure C& dispersion and, so, misses any higher attractive contributions such as the dipole-quadrupole atomic

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Chapter 8. Chlorobenzenes II 181 fluctuations that contribute to Cg dispersion. These higher order contributions can be significant and are often absorbed into the fitting o f the term in model potentials and can be absorbed into param eters in the Slater-Kirkwood relationship. A second problem with using these DFT coefficients in the present study is that the test set o f molecules studied by loannou did not include any aromatic rings, so the dispersion contributions o f the ring carbons must be assumed to be the sam e as a non- arom atic sp' carbon. Additionally, no chlorine-containing molecules were studied, so the Cl coefficients must be derived from another source. Mitchellt^-^^J used these DFT Cg coefficients in a model potential for cyanuric chloride (C3N3CI3), deriving the Cl Cl coefficients from the relationship proposed by Slater and Kirkwood,t^-^^l relating the dispersion interaction between molecules to their polarisabilities. In a distributed atom model o f the molecule, the dispersion coefficient betw een atoms o f type I and k is

r">^' __ 2 ________________________ /o

^ { a , / N f f + { a j N f f '

where a , is the atomic polarisability and N f the effective number o f electrons for an atom o f type i. Literature values for the atomic polarisability were used and the overall model was successful for m odelling the crystal structure o f cyanuric chloride. We take the same approach to derive a for use w ith loannou's carbon and hydrogen parameters, taking the atomic polarisability o f Cl from Miller'st^-2^1 values, which were parameterised to a large set o f experimental m olecular polarisabilities. We set the parameter, (= 7 for Cl) and derived all other Cl X coefficients from the usual geometric combining rules (equation 3.13, Chapter 3).

Four other dispersion models were investigated, taking all o f the Cg coefficients from the Slater- Kirkwood formula. M i l I e r ' s t ^ - 7 7 ] atomic polarisabilities were used in two o f the models, as well as a

set that we have fitted specifically for this series o f molecules. The geom etry o f each m olecule in the series w as optimised at the MP2(fc)/6-31G(c(p) level o f theory and the polarisability tensor was calculated at this structure using the same level o f theory (Table 8-1) by differentiating the electron density with respect to an applied electric field. Atomic polarisabilities were fitted to this series so as to m inim ise the root-mean-squared deviation between the sum o f atomic and mean molecular polarisabilities. The number o f C(-H) and H atoms in a molecule are equal, so cannot be fitted independently and, hence, atomic polarisabilities o f C(-H) and C(-Cl) had to be set equal to each

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Chapter 8. Chlorobenzenes II 182 other. The fit o f to the ab initio calculated mean molecular polarisabilities, a , is very good across the entire series (Table 8-1).

The resulting values differ from those taken from M iller's work (Table 8-2) - they are sm aller in m agnitude, which is a result o f the underestimate o f ab initio calculated values, which are known to system atically underestimate molecular p o l a r i s a b i l i t i e s . O u r values are approxim ately 20% too low com pared to the experimental values for mono- and all three dichlorobenzenes (Table 8-1). Here, we use the calculated atomic polarisabilities without adjustment, but such a systematic error can be corrected with a scaling factor, which we investigate in Section 8.4.3. Relative atomic polarisabilities also differ between the sets, reflecting differences between atom types averaged over the small, specific set o f molecules and the large parameterisation set used by Miller,

Table % -\.A b initio calculated m olecular polarisabilities, sum o f fitted atomic polarisabilities, and experim ental molecular polarisabilities for the series o f chlorobenzenes (Bohr^).

MP2/6-31G(<i,p) calculated

molecule CCyy, _«Z: a = { a , , + a ^ + a ^ ) / 3 fitted expt. [8-29]

mono 25.44 72.93 94.24 64.21 63.87 82.67 o-di 29.55 89.13 105.20 74.62 75.27 95.62 m-di 29.64 109.89 86.33 75.29 75.27 96.02 p-di 29.66 75.97 121.02 75.55 75.27 95.83 1,2,3-tri 34.28 115.18 107.73 85.73 86.68 - 1,3,5-tri 34.48 113.42 113.42 87.11 86.68 - 1,2,3,5-tetra 38.76 118.63 136.97 98.12 98.08 - 1,2,4,5-tetra 38.78 108.77 147.66 98.40 98.08 - penta 43.00 152.96 132.55 109.50 109.48 - hexa 47.21 158.12 158.12 121.15 120.88 -

Polarisabilities are given with x _L to the ring and y and z in the plane o f the molecule, with the z-axis along the C2 axis, where applicable.

Table 8-2. Atomic polarisabilities from Millert^-^^l and our fitting to ab initio calculations (Bohr^).

source _^CL a„

M iller 15.622 9.124 2.612

ab initio fitted 13.553 6.594 2.151

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Chapter 8. Chlorobenzenes II 183 Further complicating the choice o f a dispersion model is the definition o f the num ber o f effective electrons, N f in equation (8.2). The most straightforward choice is to set N f equal to the num ber

o f valence electrons, N,,, o f atom type i {i.e. N f = 7, N f = 4, N f = 1 ), which we use for dispersion models 2 & 3 (M iller and fitted atomic polarisabilities, respectively). Comparison o f accurate C^ coefficients and atomic polarisabilities for the rare gases yields

N f = 1.422, N f = 3.810, N f = 5.404, N f = 6.309, and N f = 7.253 ,[8 ^0] gn below the num ber o f valence electrons, a trend followed by many small polyatomic molecules as well. Recognising the need for a systematic approach to determining N f for developing model potentials,

H algrent^-^'] proposed the following scheme

for H and He, = 0.80 and 1.42 for C -» Ne, = 1.17 + 0.33N ,

for Si -+ Ar, = 3.00 + 0.30A^„ (8.3)

for Ge -> Kr, = 3 .9 0 + 0.30 for Sn -+ Xe, N-^' = 4.85 + 0.30Æ,,

from which reasonable reproduction o f accurate dispersion coefficients was possible. Our final two dispersion models use this algorithm, along with Miller's (model 4) and our fitted (model 5) atomic polarisabilities. These models are denoted as MillerNcu and FitN^jj-, respectively, and the resulting Cg coefficients are given in Table 8-4.

Table 8-3. Dispersion models.

1 - DFT DFT Ce coefficients from loannou, supplemented with MillerNvCl parameters 2 - MillerNy, Slater-Kirkwood; M iller atomic polarisabilities and A/^ =

3 - F i t N , Slater-Kirkwood; fitted atomic polarisabilities and = N„

4 - MillerNcj) Slater-Kirkwood; M iller atomic polarisabilities and N^^ from Halgren 5 - FitNejf Slater-Kirkwood; fitted atomic polarisabilities and A^^ from Halgren

Table 8-4. Atom-atom Ce dispersion coefficients (kJ'AVmol).

atom pair D FT M illerN v FitNv M illerNeff FitNeff

Cl Cl 7063.98 7063.98 5708.00 6029.57 4872.15 Cl C 3199.75 4103.02 1464.30 3363.78 2372.52 Cl H 940.08 1134.52 882.15 991.93 771.05 C C 1449.38 2383.25 1464.30 1880.35 1155.32 C H 425.82 659.11 445.97 553.77 375.46 H H 125.11 182.49 136.43 163.22 122.02

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Chapter 8. Chlorobenzenes 11 184 The five models show a wide variability in the magnitude and ratio o f atom-atom dispersion coefficients, highlighting the need for further work in validating the different approaches. The C C/H H ratio is fairly constant over the five models, while the main differences are in overall m agnitude and the relative magnitude o f Cl X to ‘non-chlorine’ coefficients. The very large Cl Cl/C C ratio in the DFT model indicates that the mixing o f different methods results in an imbalance in the model. The Cl Cl coefficients are relatively stronger in the models derived from our calculated molecular polarisabilities than in the models using M iller’s atomic values.

8.3.2

Deriving the proportionality constant, K, for intermolecular repulsion

The repulsion between molecules is taken from the overlap model, described in C hapter 7. The charge density overlap between molecules is described by equations (7.2), (7.11), and (7.12) and, for the transferable model, the parameters in Table 7-10. With the model describing intermolecular charge density overlap, only the proportionality constant, K, is required to complete the repulsion m odel.

Two approaches have been used in earlier studies to obtain the proportionality constant, K, betw een overlap and repulsion energy in (7.2). Intermolecular perturbation theory (IM PT) can be used to calculate the contributions to intermolecular interaction energies. Details o f the IM PT approach are given in Chapter 3. Price and co-workers^^- ^ 8-20,32] applied the overlap model to a range o f m olecular systems small enough to calculate IM PT interaction energies. Using current software, the limit on IM PT calculations is 255 basis functions.[ An important use o f the overlap m odel is as an alternative to fitting a model potential to an accurate ab initio surface for large m olecules, where the number o f points necessary for such a fit is impractical. By only using IM PT calculations to fit K, orders o f magnitude fewer calculations are required. However, the size o f m olecule can be so computationally restrictive that even a few IMPT calculations are impractical. Hence, a second strategy was suggested by M itchell and co-workers,t^-^^l and was used by Tsui et ût/[8-22,23] developing a model for chlorothalonil, C6Cl4(CN)2. The proportionality constant was em pirically fitted to reproduce the volume o f the known crystal structure. The resulting values o f K w ere in the broad range predicted from IM PT calibrations on a variety o f other molecules and led to success in crystal structure prediction for the molecule.t^-^^l

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C hapter 8. Chlorobenzenes II 185 B oth o f the ab ove strategies {i.e. em pirical fitting and IM PT) have b een tested here. T o calculate IM PT interaction en ergies for the sm allest chlorob en zen e, the basis set u sed for overlap calcu lation s e x c e e d s the lim it on basis functions. H ence, a sm aller 6-2\G {d) basis set w as u sed for IM PT

calcu lation s on ch lorob en zen e dim ers. Such calculations, used to calibrate the proportionality b etw een charge d en sity overlap and repulsive energy are described in S ection 8 .S .2 .2 . For em p irically fitting to know n structures, the proportionality constant w as fitted to a sm all subset o f the crystals and the transferability tested on the rem ainder o f the set. T his approach is described in S ection 8 .3 .2 .1 .

8.3.2.1 F ittin g K to know n stru c tu re s

T he three polym orphs o f p -d ich lorob en zen e sam ple a range o f the im portant atom -atom contacts and form a suitable subset for fitting K to know n structures. T he anisotropy o f the Cl Cl contact is reasonably sam pled - the a polym orph has a c lo se en d -to-sid e interaction, there is a roughly en d -to-

end contact in the (3 form , and the y polym orph has tw o c lo s e Cl Cl contacts sam p lin g sid e-to -sid e and interm ediate ( Z C -C L ---C = 1 3 7 ° ) geom etries. T he carbon atom s are also w ell sam pled w ith o ffse t Ti-Tt stack in g in the a and (3 forms and a tilted ed g e-to -fa ce interaction in the y m od ification .

Several H H, H Cl and tilted H ti contacts are also sam pled in the three m od ification s.

T he structures w ere relaxed on the potential energy surface defined by a D M A m od el o f the electrostatics, the g iv en dispersion m odel, and the repulsion from the overlap m odel w ith K varied over a reasonable range (typ ically K = 4 - 10 a.u.). T he sum o f relative errors in the crystal volu m e for the three polym orphs w as plotted against K to find the value that best reproduced all three sim u ltan eou sly. O ne reason for the d evelopm ent o f the chlorob en zen e m odel potential is to test its ab ility to reproduce lattice dynam ical properties o f the crystals, w hich are sen sitiv e to detail o f m olecu lar p osition s and orientations as w ell as lattice parameters. H ence, K w as actually fitted to the

structural drift factor, F [8-34] (introduced in Chapter 6)

= + ( lO A x f + Z (8-4)

T = a , f ) , c X = < x . p . Y

w here A 0 is the rigid-body rotation (in d egrees) on lattice energy m inim isation. Ax is the rigid-body translation (A ), and Ax and A% are the changes in lattice parameters.

A s w ell as com paring the overlap m odel derived potential w ith em pirical m odels in their ab ility to reproduce the know n crystal structures, the m odel w ill be used in crystal structure prediction studies on on e or m ore o f the ch lorobenzenes. To be consistent with such future studies, all calcu lation s

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Chapter 8. Chlorobenzenes II 186