three different types of carbon nanotubes are experimentally observed : armchair , zigzag and chiral . these carbon structures are finished with semispheres containing pentagons and hexagons , being formally parts of fullerenes . due to their structure , the cnts are hydrophobic , strongly interact with light and possess interesting electrical and physical properties [ 13 ] . modification of hydrophobic carbon nanotubes by allowing stronger intermolecular interactions , leading to solubility is expected upon addition of selected small molecules covalently bonded to the ( a ) end , ( b ) surface , or ( c ) both the end and surface ( mixed ) of swcnts [ 2 , 413 ] . the rim structure of swcnt in case of zigzag or armchair open ended cnt shows a different pattern . zigzag carbon nanotubes show metallic or semiconductor properties and their ends shows  saw - tooth  like shape . for example , oh , cooh or nh2 functionalized cnt are easily transformed and could bear longer chains connecting antibodies or drugs . end - substituted swcnt are by - products of mild oxidation and carboxylic , carbonyl and hydroxyl groups are frequently formed . most previous works [ 46 ] concentrated on single functionalized swcnts and the impact of substituent on physical properties of modified versus pristine structure ( for example , density of states , homo - lumo gap ) . the pristine cnts are insoluble in water and organic solvents and this is a serious hindrance in their industrial applications , for example , as efficient nanocomposites [ 1 , 14 , 15 ] . earlier works demonstrated a possibility of transforming inert and hydrophobic cnts , into soluble forms [ 1417 ] . this was accomplished by subsequent chemical modification of water soluble cnts , containing cooh groups [ 1417 ] . unfortunately , little is known about the systematic changes of energy of zigzag and armchair swcnts upon consecutive replacement of rim hydrogen atoms by cooh groups . density functional theory ( dft ) and , in particular , the exchange - correlation b3lyp hybrid density functional is widely used in molecular modeling studies to predict structure , spectroscopic parameters and energy changes of small , middle and large size molecules [ 7 , 8 , 1820 ] . due to the large size of cnts , the dft calculations with relatively small basis sets ( 3 - 21 g or 6 - 31 g * ) , and also am1 and pm3 semiempirical methods , have been used for theoretical description of molecular structure and other parameters of finite models of cnts . in this study , as an extension of our previous works on hydroxyl substituted swcnts [ 19 , 21 , 22 ] , we would like to get a more detailed information at the atomistic level on the open - end cnt modification with cooh , up to full substitution with nine ( zigzag ) or ten ( armchair ) carboxylic groups . reliable exchange - correlational b3lyp hybrid density functional and basis sets of relatively small size ( 3 - 21 g and 6 - 31 g * ) , enabling completing fairly large scale calculations were selected . full structure optimization of unsubstituted open - ended ( with dangling bonds on carbon saturated with hydrogen atoms ) , and cooh - modified swcnts were performed . several models of swcnt were selected , including ( 9,0 ) zigzag and ( 5,5 ) armchair structures with one and three layers ( strings ) of hexagon units . ir and raman harmonic frequencies were calculated in case of one layer with one to 9/10 cooh substituents . all positive frequencies ensured ground state structure of the optimized system . for comparison purposes only , the calculations with small model molecules including methane , benzene , anthracene and phenanthrene before , and after replacing one hydrogen atom with the carboxyl group were performed at the same level of theory . in addition , to verify the basis set quality impact , these calculations were performed with a large basis set ( 6 - 311++g(3df,2pd ) . energy of one cooh group formation at nanotube terminated initially with h atoms was calculated by considering a hypothetical reaction:1\documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \text{swcnt } } - { \text{h } } + { \text { c}}{{\text{h}}_{{3}}}{\text{cooh } } \to { \text{swcn } } - { \text{cooh } } + { \text { c}}{{\text{h}}_{{4}}}. $ $ \end{document } energies of adding subsequent groups ( e in kcal  mol ) were assumed as follows:2\documentclass[12pt]{minimal } 
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				 \begin{document}$$ { \text{d}}{{\text{e}}_{\text{n } } } = { } \left [ { { \text{e}}\left ( { { \text{swcnt}}{{\left ( { \text{cooh } } \right)}_{\text{n } } } } \right ) + { \text{e}}\left ( { { \text{c}}{{\text{h}}_{{4 } } } } \right ) } \right]-\left [ { { \text{e}}\left ( { { \text{swcnt}}{{\left ( { \text{cooh } } \right)}_{{{\text{n } } - { 1 } } } } } \right ) + { \text{e}}\left ( { { \text{c}}{{\text{h}}_{{3}}}{\text{cooh } } } \right ) } \right ] , $ $ \end{document}where n  =  1,2,  9 ( 10 ) . initially , the calculations were conducted at b3lyp/3 - 21 g level of theory . qualitatively , the changes in energies obtained with smaller basis set ( 3 - 21 g ) were similar to those , obtained at b3lyp/6 - 31g*. thus , the final results , obtained with the larger basis set will be only discussed . the carboxylic group can be considered as an asymmetric substituent with two different ends ( o atom vs. oh group ) and their position in respect to the rim of the cnt be positioned in a way which minimizes the interactions with the neighboring h - atoms and/or forms h - bonds with other cooh substituents . this was tested on model systems ( scheme  1 ) by rotating the cooh substituent relative to the aromatic ring plane ( changing the dihedral angle cring  cring c  =  o).scheme 1small model molecules ( benzoic , anthracene-9-carboxylic and phenathrene-4-carboxylic acids small model molecules ( benzoic , anthracene-9-carboxylic and phenathrene-4-carboxylic acids the energy landscape of cooh rotation in case of monosubstituted benzene , phenanthrene and anthracene are shown in fig .  1 . in case of benzene , the energy minimum , corresponding to favorable carboxylic orientation , coplanar with the ring , is observed and the perpendicular position , e.g. , at both sides of the ring , are about 8  kcal  mol higher . in case of anthracene , the energy minimum corresponds to about 45 degree deviation of cooh plane from rings plane and there are also two maxima of the same height ( at about 3  kcal  mol ) for the perpendicular orientation . the rotation of cooh in phenanthrene at position 4 ( see scheme  1 ) leads to an asymmetric shape of energy curve . the basis set effect on the position of energy maxima upon cooh rotation in c6h5cooh is also shown in fig . thus , upon improving the basis set quality from 6 - 31  g * to 6 - 311++g * * and 6 - 311++g(3df,2pd ) the barrier height slightly decreases ( from 7.85 to 6.73 and 6.52  kcal  mol ) . it is apparent that the barrier height decreases by about 1.3  kcal  mol upon significantly improving the basis set quality.fig . 1energy change upon rotation of cooh substituent relative to ring plane in ( a ) benzoic acid and ( b ) anthracene-9-carboxylic acid and phenathrene-4-carboxylic acid energy change upon rotation of cooh substituent relative to ring plane in ( a ) benzoic acid and ( b ) anthracene-9-carboxylic acid and phenathrene-4-carboxylic acid in fig . 2a and b are shown optimized structures of zigzag and armchair cnt consisting of three ring layers with a single cooh substituent at the rim.fig . 2optimized structures of model ( a ) zigzag and ( b ) armchair swcnts with a single cooh substituent at the rim ( dimensions in  ) optimized structures of model ( a ) zigzag and ( b ) armchair swcnts with a single cooh substituent at the rim ( dimensions in  ) in fig . 3 are shown energy landscapes of single cooh group rotation attached to zigzag and armchair swcnts . in this case , the preferred geometry is observed for both -oh and =  o ends of carboxylic group outside the tube ( on the circumference ) . two energy maxima are observed for cooh group oriented along the tube radius and the slightly lower one corresponds to oh being outside the tube . in case of zigzag cnt , the energy minimum corresponds to c  c  c  =  o angle of about 5 degrees , ( substituent on the circumference ) and the highest maximum corresponds to about 90 degrees ( c  =  o outside the tube ) . the other maximum ( with c  =  o inside , or oriented toward the tube center ) is slightly lower . similarly to fig .  1a , the improvement of basis set quality from 3 - 21 g to 6 - 31 g * leads to energy barrier lowering by about 4  kcal  mol . in the case of carboxylic group rotation at the armchair rim , the situation is similar and the corresponding barrier heights are 9 and 7  kcal  mol and the energy minimum is observed at about 10 degrees.fig . 3energy change upon rotation of cooh substituent at the rim of model ( a ) zigzag and ( b ) armchair swcnts energy change upon rotation of cooh substituent at the rim of model ( a ) zigzag and ( b ) armchair swcnts up to nine carboxylic substituents were placed consecutively at the zigzag rim ( see fig . , a kind of threefold symmetry was observed . nevertheless , some funnel shape deformation and increase of the tube - end diameter was observed.fig . 4optimized structures of model ( a ) zigzag and ( b ) armchair swcnts fully functionalized with cooh substituents at the rim . threefold symmetry is indicated for zigzag nanotube optimized structures of model ( a ) zigzag and ( b ) armchair swcnts fully functionalized with cooh substituents at the rim . threefold symmetry is indicated for zigzag nanotube the armchair model consisting of three hexagon layers with up to nine cooh groups at one rim was also stable . in addition , upon complete functionalization of one hexagon layer ( the shortest armchair nanotube model ) a stable system was also observed ( fig . however , all attempts to obtain fully functionalized one end of a longer tube , containing three layers of hexagons , failed . this was probably due to steric crowding at the relatively rigid tube skeleton end . in the next step , starting from model systems of methane and benzene , we examined the energetics of substitution process calculated according to eq .  2 ( see table  1 ) . addition of zpv correction changes the substitution energy slightly while hydroxylation is more favorable in the case of benzene.table 1comparison of carboxylation and hydroxylation energy ( kcal  mol ) calculated at b3lyp/6 - 311++g(3df,2pd ) level for two model compounds according to eq . 2ch4c6h6-coohe3.8(e  +  zpv)4.9-oh e29.439.9(e  +  zpv)26.538.4 in agreement with formula ( 2 ) an opposite sign to that in ref . is given comparison of carboxylation and hydroxylation energy ( kcal  mol ) calculated at b3lyp/6 - 311++g(3df,2pd ) level for two model compounds according to eq . 2   in agreement with formula ( 2 ) an opposite sign to that in ref . 2vs . the number of cooh substituents for zigzag and armchair are displaced in fig . 2 for model zigzag and ( b ) armchair swcnts functionalization at two basis set sizes . for better visualization the data points .  2 for model zigzag and ( b ) armchair swcnts functionalization at two basis set sizes . for better visualization that the carboxylation energy for the first hydrogen atom at the rim of zigzag nanotube formed from three layers is about 13  kcal  mol . this differs from the calculated previously   hydroxylation energy of about 35  kcal  mol . significantly smaller energy increments are needed for adding three to nine carboxylic groups . in fig . 5b is shown a similar carboxylation energy dependence on replacement of consecutive hydrogen atoms at the rim of armchair cnt model . the first carboxylation energy is higher than for zigzag model ( about 8.5 vs. 13  kcal  mol ) whereas for the second group this energy is about 1 vs. 3  kcal  mol , for armchair and zigzag models , respectively . however , some oscillation of energy is observed , with minima located at odd numbers of cooh . this resembles the results for hydroxylation energy pattern observed previously for armchair cnt [ 21 , 22 ] . the reason of this behavior was explained earlier as a result of different h - bond ring pattern formation at the rim . the present density theory studies using b3lyp hybrid functional indicate a possibility of cooh - functionalization of one end of zigzag cnt with one to nine substituents . however , the end of armchair nanotube can not be fully functionalized with cooh groups ( one hydrogen atom remains unsubstituted ) . on the basis of the performed b3lyp/6 - 31 g * calculations it appears that the replacement of one hydrogen atom at the rim of the zigzag cnt model is a more exothermic process than for armchair model ( 13 vs 8.5  kcal  mol ) . a gradual and nearly linear increase of energy is observed for subsequent carboxylation , starting from two to nine groups on a zigzag nanotube end .