Article pubs.acs.org/JPCC
Reliability of Approximate Methods to Study Tip-Functionalized Single-Wall Carbon Nanotubes Tapas Kar,*,† Steve Scheiner,† and Ajit K. Roy‡ †
Department of Chemistry and Biochemistry, Utah State University, Logan, Utah 84322-0300, United States Materials and Manufacturing Directorate, Air Force Research Laboratory, Dayton, Ohio 45433, United States
‡
S Supporting Information *
ABSTRACT: Calculations of the full structure and spectra of large nanotubes can be very demanding of computer resources. The advantages and limitations of the cost-effective same level different basis (SLDB) and selected normal modes (SNM) protocols are elucidated for carboxylated (4,4) armchair and (8,0) zigzag single-wall carbon nanotubes (SWNTs) with varying numbers of COOH groups on the tips of the tubes. While armchair-COOH tubes exhibit CO stretching frequencies in the standard range of 1720−1760 cm−1, zigzag-COOH tubes display a surprising CO band around 1660 cm−1 with much higher intensity, in addition to the higher-frequency CO mode. This low-frequency CO peak is very unusual for a standalone COOH group and is a fingerprint of zigzag tubes.
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INTRODUCTION Regardless of method of synthesis (e.g., arc discharge, laser ablation, and chemical vapor deposition), carbon nanotubes are typically purified by acid treatments. These treatments remove synthetic byproducts such as catalyst particles (Fe, Mn, etc.) and amorphous carbon from raw carbon nanotubes. Liu et al.1 reported the first use of a 3:1 concentrated H2SO4/HNO3 mixture to cut the tangled long single-wall nanotubes into short, open-ended tubes. The oxidation of nanotubes with several other oxidants (such as HNO3, O3, KMnO4, OsO4, H2O2, etc.) was also reported soon afterward.2−6 As a result of chemical oxidation, the tips and the surface or wall of nanotubes are populated with oxygen-containing groups,2,6−13 such as carboxylic acid, ether, ester, quinone, and hydroxyl, to which further functionality may be subsequently attached. Carboxylic acid groups are the most extensively used anchors for further functionalization of single-wall carbon nanotubes (SWNTs) and believed to be the gateway to the arena of functionalized nanotubes. The introduction and removal of specific functional groups at the surface or tips of single-wall carbon nanotubes (SWNTs), following each chemical treatment, are commonly monitored by IR spectroscopy. The carbonyl stretching vibration is strongly absorbing in the IR region and is particularly useful in structural assignments. However, the interpretation of IR spectra to assign the vibrational modes of the attached organic moieties can be difficult and contradictory, due to factors like carboxylated carbonaceous fragments and broken carbon structures, in addition to different diameters and chirality of nanotubes, interaction among functional groups via H-bonding, coupling with other vibrational modes, and so on. To help © 2012 American Chemical Society
overcome these complexities, a complementary theoretical treatment of vibrational modes has been used for several years with great success for a wide range of molecules.14−17 For all their usefulness, calculations of very large systems like nanotubes and their vibrational frequencies are highly computationally demanding. Means of facilitating such calculations, and circumventing some of the inordinate demands, would be extremely useful to incorporate such computations as a workhorse into the analysis of functionalized nanotubes. To address this issue, the present work assesses the reliability of various approximate methods that are highly computationally efficient in the context of carboxylated zigzag and armchair SWNTs. Specifically, Kar and co-workers have18−21 shown the advantages of the same level dif ferent basis (SLDB) protocol over other approximate procedures, such as the widely used ONIOM22,23 for studying chemical modifications of SWNTs at their side-walls (covalent and noncovalent cases) and tips. In this mixed basis function approach, atoms in defined active sites are provided with large sets of basis functions, while smaller sets are applied to the remaining atoms. Previous studies19−21 confirmed that this approach is sufficient to reproduce structure and energetics for sidewall and end-functionalized nanotubes obtained from calculations where large basis functions are used for all atoms. However, those investigations were limited to SWNTs containing only a few functional groups, so little could be said about their accuracy when applied to the more general Received: September 10, 2012 Revised: October 27, 2012 Published: November 10, 2012 25401
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The fully optimized geometries of o-(4,4) and o-(8,0) rings are summarized in Figures 1 and 2, including two key bond
case. In the present investigation, carboxylated (4,4) and (8,0) tubes (representing armchair and zigzag tubes, respectively) are considered that contain as many as eight separate functional groups, both close to and further removed from one another. One issue of particular importance, having to do with the C O band of the COOH groups that has represented a puzzle in these systems, is studied in detail.
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METHOD OF CALCULATIONS Since most commonly applied means of incorporating electron correlation, e.g. MP2, are out of reach for routine use in systems of this size, the more computationally efficient variant of DFT, known as B3LYP,24−26 is considered here. Extensive previous testing has indicated that this approach is quite well suited to evaluate vibrational frequencies and intensities. With regard to basis set, the cost-to-benefit ratio is optimal for 631G*.14−17 The SLDB approximation was incorporated as follows. 6-31G* was used for all atoms of the functional (−COOH) groups and carbons at the first functionalized layer of the tube; diffuse sp functions were added to this split-valence double-ζ quality basis with 5d-polarization functions (631+G*) for the O atom. This hybrid basis is denoted 631G* (O+). The remaining carbon atoms were treated with a smaller split-valence double-ζ 3-21G basis. The 6-31G* set was used for terminal hydrogens at the functionalized end and 321G for hydrogens at the other end of SWNTs. This particular combination of basis functions has been assessed in previous studies and found to yield accurate data. Calculated harmonic vibrational frequencies are generally overestimated (even for more accurate methods, such as MP2, CCSD, etc., as well as for larger basis functions), and a scaling factor is commonly used to better correspond with experimental spectra. For example, a value of 0.965 is recommended for B3LYP/3-21G and 0.960 for B3LYP/631G*.27 The latter value was applied here. All calculations were performed using G0928 code. Theoretical vibrational modes were analyzed using the Molden29 program, and IR plots were generated by the Gabedit30 program. Pristine (no functional groups) nanotube models were obtained using TubeVBS softwate,31 and in the oxidized SWNT models COOH groups and terminal hydrogen atoms were initially positioned using Chemcraft32 software, which is also used to generate SWNT figures and geometry analyses. The specific oxidized-SWNT (o-SWNT) models used in this study are (4,4)-(COOH)x and (8,0)-(COOH)x (x = 8 and 5). The larger value was chosen to engender the maximum effects on the structure and vibrational properties of o-SWNTs. As an intermediate pair of systems, tubes were also considered that included five COOH groups.
Figure 1. Fully optimized geometries of (4,4)-(COOH)8 and (4,4)(COOH)5. Models on left and right sides were obtained using B3LYP/6-31G*(O+) and SLDB, respectively. Two distances (in Å) refer, respectively, to CO of COOH and C−C that links each COOH to the tube.
lengths. The upper value refers to the CO bond of the indicated COOH groups, and the C−C bond linking the COOH to the tube is directly below. Inspection of the figures demonstrates a very close correspondence between SLDB and
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RESULTS AND DISCUSSIONS To assess the reliability of the B3LYP-SLDB method in predicting structures and geometries, data obtained by this approach were compared with those from B3LYP/6-31G*(O +) where all atoms were treated with the same large basis set. Each tube was constructed from 64 carbon atoms with the appropriate number of terminal hydrogens. Results from the full-basis calculations are denoted B3LYP below. In the SLDB approach, all COOH moieties at the functionalized end were treated with 6-31G*(O+), as were 16 carbons of the functionalized layer that connects COOH groups with the tubes; the remaining carbon atoms were represented by 3-21G.
Figure 2. Fully optimized geometries of (8,0)-(COOH)8 and (8,0)(COOH)5. Models on the left and right sides were obtained using B3LYP/6-31G*(O+) and SLDB, respectively. Two distances (in Å) refer, respectively, to CO of COOH and C−C that links each COOH to the tube. Arrows indicate those groups with longer r(CO). 25402
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Table 2. B3LYP ν(CO) (cm−1) and Intensities (km/mol, in Parentheses) of o-(4,4) and o-(8,0) SWNTs and Differencesa in These Quantities Caused by SLDB and SLDB(SNM) Approximations
B3LYP for all tubes, with a maximal variance of only 0.001 Å. The orientations of each COOH unit around the tube are essentially identical for both methods. Most of the CO distances are found to be less than 1.22 Å, i.e., within the standard CO distances (1.20−1.22 Å) of aromatic carboxylic acids. Important exceptions are observed for several of the carbonyl bond lengths in the o-(8,0) tubes of Figure 2, marked by arrows. In these cases, the C−C bond is shorter than a single bond, and CO is rather long, >1.22 Å. Note also that these key bond lengths are reproduced quite faithfully by the SLDB approximation. The efficiency of the latter prescription can be gleaned from the CPU times summarized in Table 1.
(4,4)-(COOH)5
(4,4)-(COOH)8
Table 1. CPU Time (in s) Required for SCF and Frequency Calculations Using Eight Shared Processors and 8 GB Memory number of basis functions
single SCF cyclea B3LYP (4,4)-(COOH)5 (4,4)-(COOH)8 (8,0)-(COOH)5 (8,0)-(COOH)8
(4,4)-(COOH)5 (4,4)-(COOH)8 (8,0)-(COOH)5 (8,0)-(COOH)8
1499 2026 1460 2061
SLDB
B3LYP
635 1178 989 1328 612 1178 983 1328 frequency stepsb
SLDB
(8,0)-(COOH)5
928 1078 928 1078 (8,0)-(COOH)8
B3LYP
SLDB
SLDB (SNM)
644573 982476 628208 1060062
272327 454644 261683 471259
164394 286173 164589 296814
a CPU time from Link 502 of Gaussian09. bCollective CPU times from Links 1110, L1002, and L703. Actual run time is close to 1/8 the CPU times given here plus elapsed time.
B3LYPb
SLDB
SLDB(SNM)
1704 (510) 1711 (255) 1713 (221) 1716 (879) 1735 (349) 1710 (341) 1721 (203) 1723 (175) 1724 (323) 1725 (789) 1744 (488) 1745 (368) 1754 (524) 1655 (535) 1662 (846) 1670 (1026) 1724 (266) 1740 (593) 1649 (1103) 1655 (521) 1668 (968) 1723 (43) 1728 (59) 1740 (829) 1745 (533) 1751 (287)
+3 (−11) +2 (−15) +2 (+6) +2 (−32) +2 (−8) +2 (−14) +2 (−12) +2 (−84) +2 (+112) +3 (−78) +2 (+21) +2 (−21) 0 (+1) +2 (−17) +2 (−17) +2 (−19) +1 (−3) +2 (−7) +2 (+2) +1 (−4) +2 (−5) 0 (−6) +1 (+4) 0 (−70) +1 (+23) +1 (+29)
+3 (−15) +2 (−14) +2 (+5) +2 (−26) +2 (−6) +2 (−15) +2 (−16) +2 (−84) +2 (+118) +3 (−77) +2 (+23) +2 (−20) 0 (+1) +1 (−1) +1 (+7) +2 (+7) 0 (−3) +2 (−4) +1 (+6) +1 (+13) +1 (+2) 0 (−6) +1 (+5) 0 (−75) +1 (+24) +1 (+35)
a Negative − underestimate; positive overestimate. bMost intense peaks are shown in bold.
Compared to B3LYP, each self-consistent field (SCF) step was reduced by more than 50% without loss of accuracy. This efficiency can be traced to the reduction of basis function total by 240 for isoelectronic pairs of o-(8,0) and o-(4,4) tubes. SLDB might thus be even more economical as the tube size increases further and can therefore be recommended in particular for geometry optimizations. It is equally important to examine the reliability and efficiency of SLDB in predicting vibrational stretching frequencies and intensities. The ν(CO) values of the multiple COOH groups obtained from full vibrational analyses using B3LYP/6-31G*(O+) are reported in the first column of Table 2. Changes in ν(CO) and intensity caused by applying the SLDB approximation are listed in the following column. For all cases, SLDB reproduces frequency values very well with a maximum difference of only 3 cm−1. With a few exceptions, SLDB intensities are also fairly close to the full B3LYP values. For a couple of CO modes for x = 8, the intensity discrepancy is more than 70 km/mol. However, the order of intensity of CO modes remains unchanged (see Supporting Information, S1 and S2). For example, the most intense CO peaks from B3LYP are preserved by the SLDB method. Computational times of IR spectra calculations displayed in the lower part of Table 1 clearly indicate the strong advantage of using the SLDB method, i.e., saving about 60% in CPU hours while maintaining a high level of accuracy. The most intense peak of the armchair (4,4)-(COOH)5 and (4,4)-(COOH)8 (Table 2 and Supporting Information, S1) occurs at 1716 and 1725 cm−1, respectively. These values are
within the range of observed frequencies in purified SWNTs, as are all of the CO frequencies in these tubes (1704−1754 cm−1). In contrast, there are two separate ranges of CO frequency for the zigzag tubes. In each case, there are several CO absorbances in the normal range of CO modes, i.e., 1723−1740 cm−1. But again, in each case, there are three lines in the much lower-frequency range of 1649−1670 cm−1. Also of importance, these low-frequency peaks are associated only with the COOH groups that have atypically short C−C and long CO bond lengths. As this frequency range is not a common feature of COOH, but is typical of quinones or their derivatives, most of the previous experimental studies2,5,6,33−37 that observed a peak in this range assigned it to a quinone. However, the results described here indicate that such a peak may be due instead to COOH groups of zigzag tubes. As another issue with implications for making this distinction, the intensity of each such low-frequency band is greater than the regular CO modes of COOH. Such differences in carbonyl stretching modes were first reported20,38 for related zigzag (10,0)(COOH)x tubes with x = 1 and 4. Accuracy of the Selected Normal Mode (SNM) Approach. Analyses of the vibrational data indicate that the CO modes do not couple with any other modes, nor is there H-bonding between COOH groups. Such a scenario is favorable for application of the very efficient and affordable selected normal mode (SNM) approach. In the SNM procedure,39 only selected Cartesian nuclear coordinates, instead of all, are considered in the mass-weighted Cartesian 25403
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Figure 3. Optimized geometries and normalized IR spectra (scaled by 0.960) of (a) tetracarboxylate phenanthrene (C14H6-(COOH)4) and (b) tricarboxylated anthracene (C14H7-(COOH)3), fragments of o-(4,4) and o-(8,0) SWNTs, respectively. Side-views are also shown. CO and C− COOH distances in Å, ν(CO) in cm−1, and intensity in km/mol in parentheses. Lorentzian broadening with fwhm of 20 cm−1 was applied. The same molecules are shown in c and d when the curvature of the full nanotubes is introduced into the structures.
carbon bond length. Slightly longer CO distances (1.215− 1.219 Å) are found in the zigzag model of C14H7-(COOH)3, and the C−COOH bonds are slightly shorter. Despite certain similarities in geometric parameters, the IR spectrum of C14H7(COOH)3 does not exhibit the two CO bands that are observed in the full o-(8,0) tubes. The most intense single C O peak of C14H7-(COOH)3 is 1718 cm−1, which falls in the range of normal CO bands. This value is only slightly higher (1744 cm−1) in the C14H6-(COOH)4 model of the armchair tube. It may be concluded that these small organic molecule models are inadequate for purposes of understanding the vibrational spectra of these tubes. It was thought that perhaps a major weakness of these small molecules as models might be their failure to include the curvature of the tubes. This idea was tested by introducing this curvature into the same systems. The geometries were altered to superimpose each molecule onto the framework of the respective tubes. As displayed in the lower half of Figure 3, the introduction of this curvature vastly improves the ability of these small model molecules to reproduce the geometries and spectra of the full tubes. The fragment of the armchair reproduces the exact 1728 cm−1 band of the full o-(4,4) tube. Importantly, the zigzag fragment now exhibits two separate CO bands, one at 1758 and the other at the low-frequency value of 1658 cm−1, only a few wavenumbers higher than the bands in the full tubes. The biggest error in these curved small molecule models occurs within the intensities, which are underestimated. Comparison of Different DFT Methods and Reliability of B3LYP. B3LYP has demonstrated itself over the years to be one of the most reliable and extensively used DFT methods for studying a variety of simple to complex molecules and in particular organic systems. Its reliability in predicting vibrational frequencies is well established. Nonetheless, given the unusual nature of the low-frequency CO modes, it was thought judicious to check to see whether other methods would
Hessian matrix of second derivatives of the total electronic energy. Diagonalization of this matrix leads to the eigenvectors and eigenvalues that correspond to the selected motions. This approach reduces computation time significantly as one need not compute the entire Hessian. To test the reliability of the SNM strategy, the COOH groups and attached carbons (4n and 2m atoms of o-armchair and o-zigzag, respectively) of each tube were selected for vibrational analyses. In comparison to full B3LYP vibrational calculations, SNM analyses cut the required CPU time to some 25−30% of that used in the full analysis. Nonetheless, the SLDB/SNM CO frequencies and intensities (Table 2, column 3) are very little different than those observed in full SLDB vibrational calculations. This near coincidence augurs well for application of SLDB/SNM for a wide range of endfunctionalized nanotubes, which might make feasible calculations that would otherwise be impractical. Effect of SWNT Curvature and Reliability of Fragment Models. Since calculations of very large tubes are computationally expensive (in particular the frequency calculations), it is common to examine instead a small section of the tubes of interest. Such an approach has been taken in some theoretical studies,40−43 where simple organic molecules, representing the tubes, were used. This approach was tested here wherein armchair and zigzag tubes were modeled by the much smaller tetracarboxylate phenanthrene (C14H6-(COOH)4) and tricarboxylated anthracene (C14H7-(COOH)3), respectively. Geometries were fully optimized, and full vibrational analyses were carried out using B3LYP/6-31G*(O+) for these two systems. The results presented in Figure 3 show first that the steric repulsions between COOH groups cause significant deviations from planarity (see side views in Figure 3) of each parent hydrocarbon. The four CO distances in the (4,4) structural arrangement of C14H6-(COOH)4 are similar to one another, in the narrow range of 1.212−1.215 Å. All four linking C−COOH distances are also quite similar, close to the single carbon− 25404
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This study has also revealed fundamental differences between carboxylated armchair (4,4) and (8,0) zigzag tubes. The latter exhibits a new CO frequency of COOH in the 1650−1670 cm−1 region. Such a peak for a CO mode is unusual for a standalone COOH group. This low-frequency vibration may be a common feature of zigzag tubes.
reproduce this phenomenon. The o-(4,4) and o-(8,0) nanotubes were fully optimized using mPW1PW91, 4 4 B3PW91,24,45,46 PBEPBE,47,48 and BLYP,25,26,49 and the ν(CO) frequencies of the most intense CO peaks are summarized in Table 3. All methods depict similar structural
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Table 3. ν(CO) Frequencies (cm−1) Calculated by Various DFT Methods
a
method (scale factor)a
(4,4)(COOH)8
mPw1PW91 (0.948) B3PW91 (0.957) B3LYP (0.960) PBEPBE (0.986) BLYP (0.992)
1765 1744 1728 1716 1692
S Supporting Information *
(8,0)(COOH)8 1755 1753 1740 1722 1704
ASSOCIATED CONTENT
Expanded normalized IR spectra of o-SWNTs are shown in Figures S1 and S2. This material is available free of charge via the Internet at http://pubs.acs.org.
1671 1670 1651 1645 1619
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Fax: 1-435-797-3390.
Scale factor is taken from ref 27.
Notes
arrangements with only slight differences in CO and C−C bond distances (not shown). All DFT methods agree with B3LYP in terms of predicting a single CO band for o-(4,4) and a dual band for o-(8,0). Although there are small differences in the actual frequencies from method to method, the difference between the two CO stretching bands of o(8,0) remains quite uniform. Thus, it may be concluded that the observation of two separate CO bands is indeed a characteristic of purified zigzag tubes. The low-frequency band around 1660 cm−1 represents a new finding for a COOH group that is not engaged in H-bonding or other intergroup interactions. Range of Application and Limitation of SLDB and SNM Approximations. Both SLDB and SNM methods are highly efficient and reliable for o-SWNTs where acid groups are located only at the tip of the tube. This suggests that such methods might be used for any tip or end-functionalized SWNTs. The reader is cautioned, however, that had there been any coupling between terminal COOH and other modes, such as the CC vibrations of the tube wall, SNM results could not be recommended. Similarly for any side-wall functionalization, comparison with a full vibrational analysis is advisible before using SNM. Moreover, for end-functionalization, if the tubes are significantly distorted beyond the first functionalized layer, one should test the SLDB approach by applying a larger basis set to more carbons. The SNM method is also not recommended for zero-point and thermal vibrational corrections or any thermochemical study as a large number of frequencies are omitted.
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the US DoD High Performance Computing Modernization Program (HPCMP) and NSF (Grant CHE-1026826). We thank the AFRL/DSRC personnel for their support in using the AFRL DoD Supercomputing resources.
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REFERENCES
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CONCLUSIONS Although theoretical IR vibrational analyses for functionalized carbon nanotubes can be a costly proposition, such analyses are crucial for a better understanding of complicated experimental spectra. The same level different basis (SLDB) protocol has proven itself accurate in calculating the characteristic CO modes of COOH groups in SWNTs. This method reduces computation time by more than 60% for IR spectra calculations. Since CO stretching vibrations are not coupled with any other modes, the selected normal mode (SNM) approach was applicable and found to reduce the computation time by another 10−15% while reproducing frequencies quite accurately, to within 3 cm−1. The combination of SLDB with SNM may be very useful in the study of end-functionalized tubes, as well as other related systems, requiring only limited computational resources. 25405
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The Journal of Physical Chemistry C
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