halogen bonding has emerged in recent 
 years as an effective alternative 
 to hydrogen bonding in directing the formation of self - assembling 
 architectures , with fruitful applications in many chemical and biological systems .
halogen bonding is an electrostatically 
 driven noncovalent interaction between a halogen atom in one molecule 
 and a lone - pair ( n ) or -electron donor in another .
the electron - withdrawing effect of groups covalently 
 bound to a halogen atom depletes the electron density in the n orbital to yield an electropositive  -hole  .
halogen bonding serves as an ideal cohesive force 
 in self - assembling systems due to its linear directionality , tunable bonding strength , and stability in hydrophobic 
 environments .
in addition to attractive 
 intermolecular forces , self - assembling processes may be aided via 
 cooperative effects , i.e. , nonadditive enhancement 
 from polarization and charge transfer .
the existence of cooperativity 
 in halogen bonding has been identified in previous crystallographic and theoretical studies covering molecular complexes 
 with varied structures and strengths .
the present computational 
 study expands upon previous efforts to 
 quantitatively examine cooperative effects and optimal motifs for 
 linear and multiply halogen bonded systems and to consider their potential 
 in the construction of self - assembling architectures . by performing 
 density functional theory ( dft ) calculations , enhanced understanding 

is sought on the nature of cooperativity through geometrical , energetic , 
 and natural bonding orbital ( nbo ) analyses .
factors are considered that could influence interaction strength 
 such as polarization , secondary interactions , and spacing between 
 halogen bonds .
recently , modeling of halogen bonding interactions 
 in supramolecular systems has been facilitated in force fields by 
 representing the -hole as a partial positive point charge attached 
 to the halogen atom .
carter et al . also redesigned a force field for halogen bonds by including 
 angular dependency into the standard lennard - jones potential . in the 
 present work ,
model systems are considered and then extended to construct 
 cylindrical complexes analogous to carbon nanotubes ( cnts ) .
the initial focus was 
 on 4-bromopyridine and 1-bromo-1h - imidazole as prototypical 
 building blocks for construction of larger , self - assembling systems . 

the first issue was to evaluate the structures , interaction strengths , 
 and cooperativity for linear oligomers , namely , ( 4-bromopyridine)n ( 1 ) and ( 1-bromo-1h - imidazole)n ( 2 ) with n = 16 , as shown in figure 1 for n = 4 .
as described in computational 
 methods , dft calculations were carried at the m06 - 2x/6 - 31+g(d , p)-lanl2dzdp - pp 
 level with counterpoise corrections using geometries optimized with 
 the b97x functional and the same basis set .
as reported 
 in table 1 , the calculated average halogen - bond 
 lengths decline and binding energies strengthen steadily with growing 
 oligomer size .
the average halogen - bond strength increases for the 
 4-bromopyridine case from 2.94 to 3.08 kcal / mol for n = 26 .
the interactions and cooperative effects for the imidazole 
 case are notably stronger , advancing from 8.31 kcal / mol for the dimer 
 to 10.30 kcal / mol for the hexamer .
the halogen bonds are also significantly 
 shorter with nbr separations near 2.5  for 2 versus 3.0  for 1 .
the much stronger 
 halogen bonds for 2 are due to the nitrogen atom being 
 covalently bound to bromine , which simultaneously enhances electron 
 withdrawal from bromine by induction and electron donation to the 
 lewis basic nitrogen by resonance .
most of the linear chains of both 
 compounds exhibit nonuniform sequential arrangements , with shorter 
 halogen - bond distances in the middle and longer ones toward the ends 
 of the chains .
this geometric characteristic has also been found in 
 hydrogen - bonded linear chains . as also shown in table 1 ( column 3 ) , including zero - point energy ( zpe ) corrections 

optimized geometries of linear chains for tetramers of 4-bromopyridine 
 ( 1 ) and 1-bromo-1h - imidazole ( 2 ) .
percentage change compared to the 
 dimer . the energy difference 
 between the 
 optimized complex and separated monomers including zero - point energy 
 corrections .
cooperative 
 effects are apparent in the increasing average strengths 
 of the halogen bonds .
further quantification comes from evaluating 
 the average cooperative energy , ecoop , 
 according to eq 1 for oligomers of sizes greater 
 than 2 .
as shown in table 1 , the negative values for ecoop confirm the existence of cooperative effects in all cases , and they 
 increase in magnitude as the chains grow.1 a many - body 
 analysis was also carried out using the standard partition 
 scheme of hankins et al . for the trimers 
 and tetramers .
this requires calculations for each constituent n - mer to yield the total interaction energy as a sum of 
 contributions from two- , three- , and four - body interactions .
the results 
 in table 2 indicate that while the two - body 
 interactions make the dominant contributions to the binding energies , 
 the three - body terms are substantial and increase significantly in 
 going from the trimers to tetramers .
however , the four - body terms 
 are negligible , as found in a previous many - body analysis of hydrogen - bonded 
 hcn chains . to further characterize 
 the cooperative effects
, the investigations 
 considered the changes in dipole moment and intermolecular orbital 
 interactions via nbo analysis .
the average dipole moment per molecule 
 (  ) and the cooperative dipole moment ( coop , defined analogously to the cooperative energy , 
 increase for addition of each monomer ( table 3 ) .
the calculated average intermolecular charge transfer ( qct ) and delocalization energies ( e ) associated with the lone pair on 
 the nitrogen atoms ( nn ) and antibonding orbitals of the 
 c  br bonds ( *c  br ) are also found 
 to increase with the size of the oligomers in table 3 .
moreover , figure 2 illustrates that 
 the average binding energy correlates linearly with average dipole 
 moment ( correlation coefficient = 0.998 and 0.994 ) and transferred 
 charge ( correlation coefficient = 0.954 and 0.981 ) .
overall , the present 
 results support the consensus view that polarization is a major source 
 of the cooperativity in linear halogen - bonded systems .
correlations of the average 
 halogen bonding energy ( exb , 
 kcal / mol ) with the average dipole 
 moment ( debyes ) and average amount of charge transferred ( e ) for linear oligomers of ( 4-bromopyridine)26 on the left and ( 1-bromo-1h - imidazole)26 on the right .
dipole moments in debyes ; transferred 
 charges in e ; energies in kilocalories per mole . while 
 previous studies have focused on the cooperativity in linear systems , 
 the cooperativity in multiply halogen bonded systems was also explored 
 here for a variety of haloazines .
the goals are to identify the factors 
 that influence halogen - bond strengths and to determine optimal halogen - bonding 
 motifs for potential use in self - assembly of large systems .
as shown 
 in figure 3 and table 4 , results were obtained for the singly , doubly , and triply halogen 
 bonded complexes 3a3c .
these cases 
 illustrate alternation of the donor and acceptor sites at the interface . 

the average length of the halogen bonds increases slightly along this 
 series , which likely reflects the unfavorable hh 
 repulsions across the interfaces for 3b and 3c
. nevertheless , the average strength increases from 2.94 to 3.39 
 kcal / mol .
. 10 kcal / mol , which is stronger by 1 kcal / mol than for the tetramer 
 of 1 , also featuring three halogen bonds .
thus , the cooperative 
 effect is somewhat greater for extension of the halogen bonding system 
 laterally rather than in a linear , head - to - tail manner .
halogen bond lengths in the crystal 
 structures from ref ( 32 ) . including zero - point 
 energy corrections .
related systems that incorporate extensive two - dimensional networks 
 of nitrogen  chlorine halogen bonds have been reported in crystallographic 
 studies . in the present work ,
two of the azaaromatic chlorides were revisited : 
 cyanuric trichloride ( 3d ) and 2,4,6,8-tetrachloropyrimido[5,4-d]pyrimidine ( 3e ) .
the dimer and heptamer of 3d and the dimer and tetramer for 3e were optimized . 

as shown in table 4 , the dft results for the 
 halogen - bond lengths for the oligomers of 3d and 3e are in excellent agreement with the findings from the x - ray 
 crystallography .
however , compared to 
 the bromoazines , the dft results reveal that the halogen bonds for 3d and 3e are significantly weaker with average 
 strengths of 1.51.9 kcal / mol .
similarly , in our prior study 
 of the complexes of bromobenzene and chlorobenzene with pyridine , 
 binding energies of 2.7 and 1.9 kcal / mol were obtained .
though the halogen bonds for 3d and 3e are weak , they clearly contribute to the packing 
 and self - assembly reflected in the crystal structures .
stronger halogen bonding interactions can be 
 expected to lead to more robust self - assembling materials .
continuing 
 in figure 3 , 3f and 3 g remove or add a benzene spacer to 3b .
the 
 binding energy for 3f is significantly weaker at 3.37 
 kcal / mol owing to the repulsion between the central bromine atoms ; 
 the monomers shift laterally such that the c  brn 
 angles are 164 rather than the optimal 180 for halogen 
 bonds .
however , the additional spacer in 3 g has little 
 impact yielding an interaction energy nearly the same as that for 3b .
complexes 3h and 3i are isomeric 
 with 3f and 3 g ; however , both donors and 
 both acceptors are now on the same side of the interface .
for hydrogen - bonded 
 systems , this arrangement can lead to stronger binding owing to the 
 favorable secondary electrostatic interactions .
however , in 3h the repulsion between the bromines 
 causes them to bend away from each other with c  c  br 
 angles of 125 and c 
brn angles of 162 , 
 which leads to weak binding of only 3.55 kcal / mol .
the problem is 
 relieved for 3i , though its binding energy of 5.60 kcal / mol 
 is about 0.7 kcal / mol weaker than for 3 g .
there are small 
 structural differences in this case ; the halogen bonds are more linear 
 for 3 g than 3i with c 
finally , 3j and 3k explore replacement 
 of the benzene rings in 3b and 3c with cyclohexyl 
 spacers .
the nbr halogen bond lengths are constant 
 at 3.093.10  ; however , the partially saturated systems 
 exhibit stronger attraction with binding energies of 7.16 and 11.34 
 kcal / mol for 3j and 3k .
the increased attraction 
 can be attributed to reduced hh repulsion across 
 the interfaces . for the unsaturated cases such as 3b , 3c , and 3 g , the shortest interannular hh 
 contacts are 2.82.9  , while they are 3.43.5 
  for the partially saturated 3j and 3k .
thus , for the motifs in figure 3 , 
 the strongest 
 average halogen bond strengths occur when the donor and acceptor sites 
 alternate on each side of the interface and when there is a spacer 
 ring between the 4-bromopyridine subunits as in 3b and 3j .
significant cooperative effects are apparent with the 
 average nbr halogen bond strength climbing from 
 2.94 kcal / mol in the reference dimer 3a to 3.39 kcal / mol 
 in 3c and to 3.78 kcal / mol in 3k .
this analysis 
 assigns all of the net interaction to the halogen bonds , which is 
 an oversimplification since there are at least varying steric effects 
 associated with the hh interactions at the interface . 

however , the results establish that multiply halogen bonded interfaces 
 can be constructed where the net interaction is more attractive than 
 from the sum of the individual halogen bonds . in view of the possible 

influence of secondary electrostatic interactions 
 on complexation , six additional dimers 
 related to 3b were considered ( figure 4 and table 5 ) .
each complex formally 
 has two nbr halogen bonds ; however , the interface 
 has been modified by the arrangement of the donors and acceptors or 
 by modification of the central ring as benzene , pyridine , or pyrazine . 

compound 4a is the isomer of 3b with both 
 donors on one side of the interface and both acceptors on the other . 
 as with 3 g and 3i
, the preference is for 
 the alternating arrangement in 3b over the parallel one 
 in 4a by nearly 1 kcal / mol .
the introduction 
 of the interfacial nn repulsion does weaken the 
 binding to 5.28 kcal / mol .
this replaces the nn 
 repulsion with an nh attraction , and the binding 
 is enhanced to 6.40 kcal / mol .
six alternative doubly halogen bonded complexes 
 with their computed 
 binding energies ebind ( kcal / mol ) 
 from m06 - 2x/6 - 31+g(d , p)-lanl2dzdp - pp calculations .
primary interactions 
 are indicated with solid lines and secondary ones are shown with dashed 
 lines .
distances in ngstroms ; energies 
 in kilocalories per mole . including zero - point energy corrections .
complexes 4d and 4e are 
 the analogues 
 of 3b with the pyrazine and pyridine spacers ; they are 
 also isomers of 4b and 4c with the donor 
 and acceptor arrangement returned to alternating .
these complexes 
 both show some enhancement of the binding energy to 6.79 and 7.03 
 kcal / mol . for 4d the interannular nn 
 repulsion is diminished by a lateral shift such that the nbr 
 halogen bonds become somewhat bifurcated .
the outer c  brn angle is 173 in 4d , while the inner one is 148 , and the central nn 
 distance is 5.0  . in 4e ,
compound 4f is the final possibility in this series ; 
 it is the isomer of 4c with the three nitrogen atoms 
 on one edge .
the binding weakens to 6.06 kcal / mol , which suggests 
 that the basicity of the nitrogens in 4f is lessened 
 by their proximity .
n bonds induces slight concave curvature to the 
 nitrogen - containing edges in 4f that splays out the opposite 
 bromine - containing edges . in turn , this adversely affects the linearity 
 of the halogen bonds in 4f with c  brn 
 angles of 171 vs 178 in 4c .
these results 
 establish that 3b , 3j , 4d , 
 and 4e represent the motifs with the most 
 attractive interactions .
they feature 4-bromopyridine units with edges 
 of alternating donor and acceptor sites separated by a monocyclic 
 spacer .
they can all be extended to potentially yield sheetlike structures 
 in the solid state and a foundation for construction of other self - assembling 
 materials .
the possible 
 utility of the 3b motif for constructing tubelike structures 
 that could self - assemble was considered .
cylindrical belts were generated 
 with alternating 4-bromopyridine and benzene rings and alternating 
 up / down arrangement of the bromopyridines .
dimers of the belts with 
 six ( 5a ) , eight ( 5b ) , and ten ( 5c ) repeats can form six , eight , and ten halogen bonds at each interface 
 ( figure 5 ) .
the dimers were modeled with dft 
 and molecular mechanics calculations . for the latter ,
the fixed - charge 
 force field , opls / cm1ax , was used as well as the version that includes 
 polarization via induced dipoles on non - hydrogen atoms , opls / cm1axp .
both force fields incorporate 
 partial positive charges ( x - sites ) to represent the -holes 
 on chlorine , bromine , and iodine atoms .
the energetic results for 5a , 5b , and 5c are summarized in table 6 . in view 
 of the increasing system sizes , two alternatives for the dft method 
 and basis sets were needed .
the geometry 
 optimizations for 5b and 5c used the smaller 
 6 - 31g(d)-lanl2dzdp - pp basis set with blyp , and the isolated monomer 
 geometry was taken to be the same as in the optimized dimer .
supramolecular 
 tubes can be assembled from cylindrical belts using 
 six ( 5a ) or ten ( 5c ) halogen bonds at each 
 interface . using m06 - 2x//b97x with the 
 6 - 31+g(d , p)-lanl2dzdp - pp basis set . using m06 - 2x//b97 - 1 with the 6 - 31g(d , p)-lanl2dzdp - pp 
 basis set .
using m06l/6 - 31g(d , p)-lanl2dzdp - pp//blyp/6 - 31g(d)-lanl2dzdp - pp . for 3b and 3c as references ,
the binding 
 energies from the force field calculations are found to be in near 
 perfect agreement with the m06 - 2x results .
this is not unexpected 
 since opls / cm1ax was parametrized to reproduce halogen - bonding results 
 from high - end ab initio calculations , specifically , 
 mp2(full)/aug - cc - pvdz(-pp ) .
5c where the best 
 accord is found between the nonpolarizable opls / cm1ax and dft results . 

as noted previously , the cm1a charges include the intramolecular electronic 
 polarization for the monomers , and the intermolecular polarization 
 is generally only essential for modeling interactions with small ions .
notably , along the 5a5c series 
 the binding intensifies from 20 to 27 to 35 kcal / mol .
there is also 
 a gradual increase in the average halogen bond strength from 3.24 
 to 3.46 kcal / mol according to the dft calculations .
this cooperative 
 effect is not mirrored in the force field results which yield the 
 same average halogen bond strength for the two larger systems .
the 
 strong binding for 5a5c indicates 
 that the halogen bonding motifs in 3b , 3j , 4d , and 4e could be used as the cohesive 
 elements to build a wide range of interesting supramolecular systems 
 and nanomaterials .
furthermore , the excellent accord between the dft 
 and opls / cm1ax results supports the use of opls / cm1ax as a computationally 
 efficient means for modeling such systems .
the 
 present study addressed halogen binding strengths and cooperativity 
 in linear and multiply halogen bonded systems . for the linear chains 1 and 2 ,
the cooperativity 
 is largely a polarization effect , as supported by the close correlations 
 of the cooperative dipole moments and amounts of charge transferred 
 with the interaction energies . for multiply halogen bonded complexes 
 such as 3a3c
, the average halogen 
 bond strength was also found to increase with increasing numbers of 
 halogen bonds .
four motifs , 3b , 3j , 4d , and 4e , emerged as having particularly strong 
 halogen bonding and symmetries that allow their elaboration to yield 
 large periodic systems .
this notion was further explored with the 
 demonstration that molecular belts based on 3b could 
 be constructed that should self - assemble into nanotubes .
furthermore , 
 the opls / cm1ax force field was confirmed to accurately predict the 
 interaction energies for halogen - bonded systems as large as the belt 
 dimers 5a5c .
the 6 - 31+g(d , p ) basis set was employed for all atoms with the exception 
 of bromine , for which the lanl2dzdp - pp basis set with pseudopotentials 
 was used .
vibrational frequency calculations 
 were carried out at the same level to confirm that the optimized structures 
 were true minima .
binding energies were obtained by single - point energy 
 calculations with the m06 - 2x or m06l functional using the same basis 
 set as the geometry optimization , and with basis 
 set superposition error ( bsse ) removed via the boys  bernardi 
 counterpoise ( cp ) method .
the accuracy 
 of the results from such computations has been supported by numerous 
 benchmark studies .
natural bond 
 orbital ( nbo ) analysis was performed with gaussian 09 following standard 
 procedures .
geometry optimizations 
 were also performed using molecular mechanics 
 as implemented in the boss 4.9 software .
the opls / cm1ax force field , which features 1.14*cm1a partial atomic 
 charges with extra point charges ( x - sites ) 
 representing the -hole for halogen atoms , was used .
the effects 
 of inclusion of intermolecular polarization with inducible dipoles 
 were considered with the opls / cm1axp force field .
the polarizabilities ,  , for carbon , nitrogen , and bromine 
 were assigned as 1.0 , 1.5 , and 2.0  .
 halogen bonding , due to its directionality 
 and tunable strength , 
 is being increasingly utilized in self - assembling materials and crystal 
 engineering . using density functional theory ( dft ) and molecular mechanics 
 ( opls / cm1ax ) calculations , 
 multiply halogen bonded complexes of brominated 
 imidazole and pyridine are investigated along with their potential 
 in construction of self - assembling architectures . 
 dimers with 110 
 halogen bonds are considered and reveal maximal binding energies of 
 336 kcal / mol . 
 cooperative ( nonadditive ) effects are found 
 in complexes that extend both along and perpendicular to the halogen 
 bonding axes , with interaction energies depending on polarization , 
 secondary interactions , and ring spacers . 
 four structural motifs were 
 identified to yield optimal halogen bonding . for the largest systems , 
 
 the excellent agreement found between the dft and opls / cm1ax results 
 supports the utility of the latter approach for analysis and design 
 of self - assembling supramolecular structures . 
Introduction
Results
Conclusion
Computational Methods