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Chemical Physics Letters 425 (2006) 347–352

Substituent constants for predicting aromatic molecularquadrupole moments

Dianne Tran, Shana Beg, Aimee Clements, Michael Lewis *

Department of Chemistry, Saint Louis University, 3501 Laclede Avenue, Saint Louis, MO 63103, United States

Received 9 April 2006Available online 10 May 2006

Abstract

Work in our group has shown a direct relationship between arene–cation binding enthalpies and the aromatic quadrupole moment(Hzz); the more negative the Hzz value, the greater the arene–cation binding enthalpy. The ability to predict aromatic Hzz values from thearomatic substitution pattern would therefore allow for the simultaneous prediction of the arene–cation binding enthalpy. Substituentconstants were developed for the accurate prediction of Hzz values for mono-, di-, tri- and tetra-substituted aromatics. The Hzz valuespredicted from the developed substituent constants perform very well in determining relative arene–cation binding enthalpies.� 2006 Elsevier B.V. All rights reserved.

1. Introduction

Non-covalent interactions involving aromatic p-electrondensity [1,2] are important in organic reaction development[3,4], protein folding [5–8], enzyme–substrate recognition[9,10], solid-state organization [11], the function of ionchannels [12] and in the design of organic semiconductors[13]. The aromatic molecular quadrapole moment, Hzz, iswidely used to explain the nature of the cation–arene com-plex (cation–p interaction), the anion–arene complex(anion–p interaction) and the arene–arene complex [1].Dougherty showed that the aromatic electrostatic potentialis all that need be considered when predicting the cation-binding energy of substituted aromatics [14], and werecently showed that the cation-binding of substituted aro-matics is directly related to the aromatic quadrapolemoment [15]. This is not surprising since the aromaticHzz value describes the electrostatic potential. Given therelationship between the aromatic Hzz value and arene–cat-ion binding energies, substituent constants that wouldallow for the prediction of the aromatic quadrapole

0009-2614/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2006.05.011

* Corresponding author. Fax: +1 314 977 2521.E-mail address: LewisM5@slu.edu (M. Lewis).

moment would be very valuable since these substituentconstants would also allow for the prediction of aro-matic-cation binding energies. We report here a set of qua-drapole moment substituent constants that allow for theaccurate prediction of the Hzz value of mono-, di-, tri-and tetra-substituted aromatics. We also evaluate the abil-ity of these substituent constants to predict arene–cationbinding enthalpies.

2. Computational methods

Structures were optimized, frequency calculations wereperformed and quadrapole moment tensor componentswere determined at the RHF/6-311G** level of theory.Among numerous theoretical methods, including the corre-lated MP2 and QCISD(T) methods, and more flexible basissets that included diffuse functions and increased polariza-tion functions, this method was shown to give the bestagreement with experiment for calculating aromatic quad-rupole moments [16]. This is illustrated by the Hzz valuesfor benzene and hexafluorobenzene; the experimental val-ues are �8.7 ± 0.5 DA and +9.5 ± 0.5 DA [17], respec-tively, and the RHF/6-311G** calculated values are�8.76 DA and +10.06 DA. Calculations of iodo-substi-tuted aromatics employed the MIDI-X basis set for iodine.

348 D. Tran et al. / Chemical Physics Letters 425 (2006) 347–352

All calculations were performed using the GAUSSIAN03 suiteof programs [18].

3. Theoretical approach and results

The aromatic quadrupole moment, Hzz, is a 3 · 3 tensor(Eq. (1)) and for planar aromatics the off-diagonal elementsare zero and Hzz is determined by Eq. (2). Thus, any effortto determine substituent constants for aromatic Hzz valuesis inherently an effort to determine substituent constantsfor the aromatic quadrupole moment diagonal tensor com-ponents. We approached this endeavor by using benzene asthe standard and determining the affect on the tensor com-ponents Qxx, Qyy and Qzz as a result of replacing an aro-matic hydrogen atom with a substituent. We oriented themolecule such that one of the benzene C–H bonds wasalong the y-axis and the z-axis was orthogonal to the

Hzz ¼Qxx Qyx Qzx

Qxy Qyy Qzy

Qxz Qyz Qzz

264

375 ð1Þ

Hzz ¼ Qzz � 0:5ðQxx þ QyyÞ ð2Þ

hydrocarbon framework (Fig. 1). As illustrated in Fig. 1,one of the hydrogen atoms on the y-axis was denoted asoccupying the ipso-position (Hi) and the other hydrogenatom was denoted as occupying the para-position (Hp).The four hydrogen atoms in the xy-plane were denotedas occupying the ortho- and meta-positions (Ho and Hm).It is important to recognize from Fig. 1 that substitutionat the ipso-position will, for instance, affect Qxx differentlythan substitution at the ortho-position. Therefore, Qxx, Qyy

and Qzz substituent constants must be developed for substi-tution at the ipso-, ortho-, meta- and para-positions.

With all aromatics oriented as described in Fig. 1, wedetermined substituent constants for the diagonal tensorcomponents Qxx, Qyy and Qzz using Eqs. (3)–(6). The sub-stituent constants for substitution at the ipso-position(Qxx,i, Qyy,i and Qzz,i) were simply determined by subtract-ing the quadrupole moment tensor components of benzenefrom the quadrupole moment tensor components of theipso-substituted benzenes, as described by Eq. (3) (Qaa,i

Fig. 1. Orientation of aromatics in xyz-coordinate system. Hydrogenatoms on y-axis denoted as occupying ipso- and para-positions (Hi andHp); hydrogen atoms in the xy-plane denoted as occupying ortho- andmeta-positions (Ho and Hm).

denotes Qxx,i, Qyy,i and Qzz,i), and the results are shownin Table 1. Eqs. (4)–(6) describe how the substituent con-stants were determined for substitution at the ortho-, meta-and para-positions, and this is a different approach thanwas used for the ipso-position. For instance, substituentconstants for substitution at the ortho-position (Qxx,o, Qyy,o

and Qzz,o) were determined by calculating the quadrupolemoment tensor components of the ortho-disubstituted aro-matics and subtracting the quadrupole moment tensorcomponents of the corresponding mono-ipso-substitutedaromatic. By keeping the ortho substituent constant andvarying the ipso substituent we calculated a set of ortho

substituent constants for each substituent; this allowed usto determine an average value and standard deviation forQxx,o, Qyy,o and Qzz,o and these are shown in Table 2.Employing the same approach gave us substituent constantaverage values and standard deviations for substitution atthe meta- and para-positions (Tables 3 and 4). The fact thatthe standard deviations for all ortho, meta and para substi-tuent constants are small validates our approach. Eqs. (3)–(6) thus allowed us to calculate substituent constants thatrepresent the affect on Qxx, Qyy and Qzz as a result ofreplacing an aromatic hydrogen atom with varioussubstituents.

ð3Þ

ð4Þ

ð5Þ

ð6Þ

We did not include the substituent constants obtainedfrom nitro and hydroxyl substitution at the ipso-positionwhen calculating the average values and standard devia-tions of the ortho, meta and para substituent constants.Thus, for the fluoro substituent the average Qxx,o valuewas determined by Eq. (7) and all other Qaa,o, Qaa,m and

Table 1Substituent constants (DA) for substitution at the ipso-position

Substituent Qxx,i Qyy,i Qzz,i

F �2.4839 �1.1559 �3.2500Cl �10.5095 �5.3593 �11.8682Br �16.3572 �9.8039 �17.7254I �27.1484 �19.5982 �28.2905CN �7.6043 �5.4203 �9.3164NO2 �16.0391 �6.8467 �13.0376NH2 �4.6353 �8.3429 �6.4009CH3 �6.8269 �8.2035 �3.3062OH �3.2971 �5.2875 �5.0304

Table 2Substituent constants (DA) for substitution at the ortho-position

Substituent Qxx,o Qyy,o Qzz,o

F �8.5320 0.2780 �3.18730.1152 0.1815 0.1197

Cl �16.5526 �7.2195 �11.75600.7557 0.3786 0.2662

Br �21.3337 �13.1699 �17.64450.8386 0.5031 0.1632

I �29.2076 �24.6594 �28.27110.9211 0.6665 0.0761

CN �26.8400 �0.5788 �9.32110.4648 0.6563 0.1043

NO2 �31.1250 �6.7729 �14.80921.7004 1.2557 1.2537

NH2 �2.7063 �7.1694 �6.53270.5691 0.7218 0.3866

CH3 �6.1790 �7.9640 �6.28120.3156 0.3121 0.0454

OH �0.8285 �1.1625 �5.01960.2008 0.7488 0.0337

Mean values are in normal font and standard deviations are in italics.

Table 3Substituent constants (DA) for substitution at the meta-position

Substituent Qxx,m Qyy,m Qzz,m

F �8.5708 �9.8020 �3.26060.0419 0.6321 0.0418

Cl �16.6305 �18.8675 �11.79500.3547 0.7557 0.1177

Br �21.5070 �24.1451 �17.66380.4285 0.7586 0.0468

I �29.9777 �33.9000 �29.91590.7404 3.1274 4.3701

CN �26.1197 �28.3462 �9.30910.2500 1.5766 0.0396

NO2 �30.1032 �35.8903 �13.03720.2044 1.7230 0.0454

NH2 �2.9556 �0.9768 �6.61310.2475 0.3933 0.1064

CH3 �6.1869 �5.3576 �6.28790.1914 0.3706 0.0185

OH �9.8070 7.1663 �5.04650.0667 0.6758 0.0252

Mean values are in normal font and standard deviations are in italics.

Table 4Substituent constants (DA) for substitution at the para-position

Substituent Qxx,p Qyy,p Qzz,p

F �2.5258 �20.9733 �3.22770.0411 1.0621 0.0323

Cl �10.5896 �31.4074 �11.84800.0535 1.4952 0.0216

Br �16.4441 �35.5746 �17.70490.0557 1.6649 0.0314

I �27.6537 �40.3013 �27.55881.9383 2.2418 1.8781

CN �7.7013 �61.9391 �9.32760.0695 2.9665 0.0786

NO2 �16.1571 �65.7770 �13.05470.0492 3.0770 0.0932

NH2 �4.6492 3.6096 �6.51510.0859 2.3968 0.3189

CH3 �6.8243 �3.7399 �6.28070.0045 0.8620 0.0189

OH �3.6299 �6.6124 �4.99610.0487 0.6666 0.0664

Mean values are in normal font and standard deviations are in italics.

D. Tran et al. / Chemical Physics Letters 425 (2006) 347–352 349

Qaa,p were determined in the same manner. The primaryreason for omitting the Qaa,o, Qaa,m and Qaa,p values deter-mined with nitro and hydroxyl substituents at the ipso-position when determining average Qaa,o, Qaa,m and Qaa,p

values and standard deviations was because of significantdiscrepancies between the ortho substituent constants whena nitro or hydroxyl group occupies the ipso-position andthe average ortho substituent constants when all other sub-stituents in this study occupy the ipso-position. This can beexplained by recognizing Eq. (4) assumes the substituent inthe ipso-position does not significantly vary in geometrywhen a second substituent is introduced at the ortho-posi-tion. While this assumption should generally be fine formeta- and para-substitution (Eqs. (5) and (6)) there aremany scenarios where for ortho-substitution problemsmight exist. For instance, in cases with bulky ipso- orortho-substituents, the position of the ipso-substituentwould be affected by sterics; in cases where two stronglyelectronegative substituents occupy the ipso- and ortho-

positions the position of the ipso-substituent would beaffected by electrostatic repulsion; in cases that allow forhydrogen-bonding between the substituents at the ipso-and ortho-positions the position of the ipso-substituentwould be affected. Inspection of the aromatic structuresshows that the positions of the ipso-nitro and ipso-hydroxyl

350 D. Tran et al. / Chemical Physics Letters 425 (2006) 347–352

substituents are affected by ortho substitution. When anitro substituent is in the ipso-position its position isaffected by electrostatic repulsion from electronegative sub-stituents (F, Cl, NO2) at the ortho-position. When a hydro-xyl substituent is in the ipso-position it is affected byortho-substituents that are able to participate in hydro-gen-bonding (NH2, OH).

ð7Þ

4. Accuracy of the quadrupole moment tensor component

substituent constants in predicting aromatic quadrupole

moments and arene–cation binding energies

The primary goal of this study was to develop a set ofquadrupole moment substituent constants for the accurateprediction of aromatic molecular quadrupole moments,Hzz. As explained above, this requires the prediction ofthe aromatic substituent constants Qxx, Qyy and Qzz andTable 5 illustrates how theses substituent constants weredetermined for 1-bromo-2-chloro-4-cyanobenzene usingthe substituent constants in Tables 1–4. For all three sub-stituent constants the base value is benzene and, for 1-bromo-2-chloro-4-cyanobenzene, the substituent constantsfor ipso-Br, ortho-Cl and para-CN are employed. Table 6compares the Hzz values obtained from the substituent con-stants in Tables 1–4 with the calculated Hzz values for 1-bromo-2-chloro-4-cyanobenzene and 5 other tri- andtetra-substituted aromatics. As was the case for the calcu-lation of the quadrupole moment substituent constants,Hzz values were calculated using the RHF/6-311G** levelof theory. The substituent constants do an excellent jobpredicting the aromatic Hzz values; the predicted Hzz valuesin Table 6 are remarkably close to the calculated values,validating the predictive capability of the substituent con-

Table 5Predicted Hzz value of 1-bromo-2-chloro-4-cyanobenzene using the substituen

Qxx = Qxx,benzene + Qxx,ipso-B

Qxx = �31.6800 � 16.3572 �Qyy = Qyy,benzene + Qyy,ipso-B

Qyy = �31.6800 � 9.8039 �Qzz = Qzz,benzene + Qzz,ipso-Br

Qzz = �40.4300 � 17.7254 �Hzz = Qzz � 0.5(Qxx + Qyy)

a All values in DA.

stants for tri- and tetra-substituted aromatics. The substitu-ent constants do not perform well in predicting the Hzz

values for penta- and hexa-substituted aromatics and thisis probably because there is significant steric crowding inthese molecules, causing substituents to move significantlywith respect to mono-substituted aromatics. Since Eqs. (4)–(6) assume that the position of each substituent is not

affected by other substituents, the movement of substitu-ents in penta- and hexa-substituted aromatics causes theexperienced difficulty in predicting their Hzz values.

Computational studies in our laboratories have shown adirect correlation between the aromatic Hzz value and thecation binding enthalpy [15] and thus one reason for calcu-lating quadrupole moment tensor component substituentconstants was to predict aromatic cation binding enthalpiesbased on the aromatic substitution pattern. Using theequation for the line of best fit that describes the relation-ship between arene–Na+ binding enthalpies (BE) and thearomatic Hzz value (Eq. (8)) [15], we determined the pre-dicted Na+–arene binding enthalpies based on the quadru-pole moments derived from our substituent constants. Thisprediction was done for the three tri-substituted aromaticsand the one tetra-substituted aromatic investigated in ourNa+–arene binding enthalpy study [15] and the resultsare shown in Table 7. The predicted values are in relativelygood agreement with the calculated values, especially whenyou consider the predicted values were simply determinedfrom the aromatic substitution pattern, the quadrupolemoment substituent constants and Eq. (8), while thecalculated binding enthalpies required over 100 h ofcomputational time on the NCSA IBM P690supercomputer.

t constants from Tables 1–4a

r + Qxx, ortho-Cl + Qxx,para-CN =16.5526 � 7.7013 = �72.2911

r + Qyy, ortho-Cl + Qyy,para-CN =7.2195 � 61.9391 = �110.6425+ Qzz, ortho-Cl + Qzz,para-CN =11.7560 � 9.3276 = �79.2390

= �79.24 � 0.5(�72.29 � 110.64) = +12.23

Table 6Comparison of Hzz values predicted from substituent constants withcalculated Hzz, valuesa

Molecule Predicted Hzz Calculated Hzz

+12.23 +11.72

+16.50 +11.01

�6.18 �6.59

�8.79 �8.43

+2.92 +0.61

�1.32 �3.47

+6.08 +5.45

a Predicted Hzz values determined using substituent constants in Tables1–4. Calculated Hzz values determined at the RHF/6-311G** level oftheory.

Table 7Comparison of calculated Na+–arene binding enthalpies and bindingenthalpies predicted from the quadrupole moment substituent constantsa

Aromatic CalculatedHzz (DA)

PredictedHzz (DA)

Calculated Na+

bindingenthalpy(kcal/mol)

PredictedNa+ bindingenthalpy(kcal/mol)

C6H3F3 0.69 1.68 �9.27 �6.85C6H3(CN)3 21.12 24.29 2.78 5.90C6H2F4 4.08 5.54 �4.23 �4.41C6H2(CN)4 27.84 35.87 8.13 14.81

a Calculated Na+ binding enthalpies used the MP2(full)/6-311G** levelof theory because of the excellent agreement with experiment (calculatedbenzene–Na+ binding enthalpy: DH298 = �24.51 kcal/mol; experimentalgas-phase binding enthalpy: DH298 = �22.13 ± 1.39 kcal/mol) [19]. Tri-substituted aromatics have D3h symmetry and tetra-substituted aromaticshave D2h symmetry.

D. Tran et al. / Chemical Physics Letters 425 (2006) 347–352 351

BE ¼ 0:6334ðHzzÞ � 7:9148 ð8Þ

5. Summary

A set of quadrupole moment substituent constants weredeveloped that give excellent agreements with calculatedvalues for tri- and tetra-substituted aromatics. Obviously,since mono- and di-substituted aromatics were employedin the development of the quadrupole moment tensor com-ponent substituent constants, the prediction of Hzz valuesof mono- and di-substituted aromatics are also predictedvery well. Since previous work in our group has shown adirect relationship between arene–cation binding enthalpies

and the aromatic quadrupole moment, the predicted Hzz

values allow for the prediction of relative arene–cationbinding enthalpies; the more negative the Hzz value thegreater the arene–cation binding enthalpy. The calculationof arene–cation binding enthalpies is computationallydemanding, and therefore the developed aromatic quadru-pole moment substituent constants are highly valuable.

Acknowledgements

This work was supported by the Research Corporation(CC6345) and the National Center for SupercomputingAllocations (CHE050039N) through an allocation on theIBM P690 system.

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