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Conformational Preferences of β-Carotene in the Confined Spaces inside Carbon Nanotubes Paul Horn and Miklos Kertesz* Chemistry Department, Georgetown UniVersity, 37th and O Streets, NW, Washington, D.C. 20057-1227 ReceiVed: April 30, 2010; ReVised Manuscript ReceiVed: June 8, 2010
Conformations of β-carotene molecules encapsulated in single walled carbon nanotubes (Car@SWNT) are modeled and evaluated based on experimental Raman spectra from the literature, most notably the emergence of a new band. A combination of molecular mechanics and density functional theory calculations is used in order to shed light on the complex interplay between distortions from equilibrium geometry and increased intermolecular interactions that takes place inside of carbon nanotubes. By optimizing the calculated Raman spectrum relative to the frequency shifts and emerging band observed experimentally, we extract conformational information from the experimental Raman spectrum and generate two structures that together inform about the encapsulated carotene structure. Introduction Independent of potential applications, the behavior of encapsulated molecules provides fascinating challenges for understanding molecular processes via spectroscopy. The conformation that β-carotene adopts inside of a nanotube is of interest in understanding the optical properties of nanotube encapsulated β-carotene both because the mechanism for the energy transfer between the β-carotene and the nanotube depends on the conformation and because the ability of the β-carotene itself to absorb and emit photons depends on its conformation.1,2 Additionally, the subject of conformational changes in encapsulated molecules is of high academic interest.3 High-resolution transmission electron microscopy (HRTEM) has been used to investigate encapsulated molecules; however, the β-carotene molecules are damaged in the process, making a complete determination of the structure by HRTEM difficult.4 Other techniques such as polarization-resolved optical absorption spectroscopy and X-ray diffraction have revealed that the β-carotene is aligned along the nanotube axis and is located off-center, indicating that it interacts with the wall; however, in order to better understand the photon absorption and energy transfer, further information on the conformation of the encapsulated carotene is needed.1 Conversely, vibrational Raman spectroscopy provides information from inside carbon nanotubes.5 The change in the Raman spectrum upon encapsulation of β-carotene informs about the interaction (Figure 1).4,5 The goal of this work is to use accurately calculated vibrational Raman spectra to shed light on the details of the conformational change that the carotene experiences upon encapsulation. We make use of the insight gained from force field molecular mechanics (MM) calculations concerning how the carotene might distort, balancing an increased number of van der Waals (vdW) contacts with the energy cost to distort β-carotene. Instead of calculating the structure directly, we optimize the uniformly scaled calculated Raman spectrum based on the scaled calculated spectra of many carotene structures produced in constrained geometry optimizations. * To whom correspondence should be addressed. E-mail: kertesz@ georgetown.edu.
Figure 1. Experimental Raman spectra of β-carotene, the SWNT, and the encapsulated β-carotene with spectra and spectral assignments adapted from ref 4.
Computational Method In our stepwise method, we first perform MM geometry optimizations on the encapsulated carotene system in order to gain an understanding of likely distortions and to create geometric parameters that mimic these distortions. Next, we explore the parameter space in terms of calculated frequencies using density functional theory (DFT) constrained geometry optimization and frequency calculations. The methodology used in this step is similar to that used by Liu et al. in their investigation of the effects of β-ring rotation on the calculated vibrational frequencies of β-carotene.6 We then use this information to determine the set of parameters that best reproduces the experimental results and calculate this structure in DFT constrained geometry optimizations. Finally, the vibrational frequencies and encapsulated energetics of this structure are analyzed by DFT and MM methods, respectively. Dreiding force field7 (MM) geometry optimizations were performed using Materials Studio on a system consisting of a hydrogen terminated (13, 9) single walled nanotube segment (diameter, 1.504 nm; length, 8.161 nm) as generated by Materials Studio (all internal coordinates constrained to their initial values in the optimization) and a β-carotene molecule located inside the tube in centered and off-center positions.8 The Gaussian ‘03 program was used to perform B3LYP/6-31G* geometry optimization and frequency calculations on the s-cis
10.1021/jp103959s 2010 American Chemical Society Published on Web 06/28/2010
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Figure 2. Two major conformers of unencapsulated β-carotene. Graphic generated using ref 14.
Figure 3. Typical MM optimized geometry of the (13, 9) SWNT encapsulated β-carotene system shown from two different perspectives illustrating the conformation of the β-carotene and the close van der Waals contacts.
and all-trans (Figure 2) conformers of β-carotene without a SWNT.9–12 All frequencies calculated in this work were scaled using a uniform scaling factor of 0.9613, and all calculated spectra were broadened by 10 cm-1 for the determination of band positions.13 Constrained geometry optimization and frequency calculations at the B3LYP/6-31G* level were performed on distortions of the s-cis structure without a SWNT. A fitting of uniformly scaled band frequencies to constrained geometric parameters as well as an optimization of the calculated Raman spectrum were performed using the “Solver” tool in Microsoft Office Excel 2003.15 Additional B3LYP/6-31G* geometry optimization and frequency calculations were performed on isolated carotene structures with geometric parameters constrained as determined in the Raman spectrum optimization. The resulting structures, the s-cis structure, and a Materials Studio generated, hydrogen terminated (13, 9) SWNT segment (same diameter and length as before) were each used in an MM single point energy calculation. The internal coordinates of these structures were then constrained, and an MM geometry optimization was performed on each encapsulated carotene system. Geometric Parameters from MM Calculations In the initial MM calculations, the hydrogen terminated (13, 9) SWNT segment dimensions were chosen such that it was longer than the carotene molecule and had a diameter (1.504 nm) close to that observed in experiment (1.5 nm based on the RBM frequency of 165 cm-1).4,16 All of the internal coordinates of the tube were constrained to their initial values. Regardless of whether the initial geometry was centered or off-center, the carotene always adopted an off-center geometry, conforming
Figure 4. Numbering scheme used for the carbon atoms in β-carotene based on ref 12.
to the contour of the wall of the tube. A typical result for the distortions of the carotene in these optimizations of the Car@SWNT system appears in Figure 3. One observation is that the molecule takes an overall bent shape such that the central conjugated segment is able to make contact with the side of the tube. The molecule also twists such that, in Figure 3, both central methyl groups are above the long conjugated chain and are located toward the center of the tube such that the Ci symmetry is destroyed. Rotation of the end β-rings was also observed. Based on the above results, we have introduced three geometric parameters that are able to mimic the distortions observed in the MM calculations. The first parameter investigated describes an overall bending of the chain and was termed the bending parameter (BP). A specific value of BP was defined by setting the torsions along the conjugated chain that involved, as the second or third atom, carbons bound to a methyl group: 7,8,9,10; 8,9,10,11; 11,12,13,14; 12,13,14,15; and their primed counterparts (Figure 4). Torsions about the same methyl were set to have the same magnitude (e.g., 7,8,9,10 and 8,9,10,11). Additionally, the bend torsions were constrained symmetrically about the predistortion center of inversion such that primed and unprimed torsions were the same and such that the overall
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Figure 5. Image of β-carotene showing the eight torsions (outer torsions yellow and blue, inner torsions blue and red, overlaps in green and purple) about the four methyl groups that are constrained in the definition of the bending parameter (BP).
bending of the molecule broke the Ci symmetry. Of the eight torsions involved, there are only two unique Φ values, with one of them defining the torsions about the two central methyls (Φcentral = 180° + 11,12,13,14 = 180° - 12,13,14,15 = 180° + 11′,12′,13′,14′ = 180° - 12′,13′,14′,15′) and one defining the torsions about the two outer methyls (Φouter = 180° + 7,8,9,10 = 180° - 8,9,10,11 ) 180° + 7′,8′,9′,10′ ) 180° 8′,9′,10′,11′) (Figure 5). The average of these two Φ values is the bending parameter
BP ≡
Figure 6. B3LYP-6-31G* calculated uniformly scaled s-cis and alltrans spectra along with the experimental unencapsulated β-carotene spectrum. Insets show unobserved calculated all-trans carotene peaks at 1337 cm-1 (a) and 1568 cm-1 (b).
Φouter + Φcentral 2
Additionally, the Φ value for the torsions about the outer methyls (Φouter) is always constrained such that
Φouter ≡ Φcentral + 2° Φouter ≡ BP + 1°;
Figure 7. Encapsulated carotene spectrum with band designations adapted from ref 4.
Φcentral ≡ BP - 1°
This condition is imposed to account for the greater ease with which the conjugated molecule can bend near the ends. Another parameter that was investigated is the twisting parameter, TP. The two H-C-C-H torsions adjacent to the center H-C-C-H torsion were both constrained to the same value, TP - 180°, where TP is the twisting parameter (TP 180° ) 15H,15,14,14H ) 15H′,15′,14′,14H′). The third parameter was the ring rotation parameter, RRP. The β-rings were rotated symmetrically with respect to the inversion center by setting the torsion that contains the CdC double bond in the ring and the main carbon chain of the molecule to a given value equal to RRP (RRP ) 8,7,6,5 ) -(8′,7′,6′,5′)). It should be noted that while for the other parameters, BP and TP, the values for an isolated, unconstrained, B3LYP/6-31G* optimized s-cis β-carotene molecule are essentially 0°, the value for RRP in the same calculation is 46.3°. A lower RRP value corresponds to a more planar structure, and a distortion to a structure with more coplanar rings is what was observed in the forcefield calculations. DFT Geometry Optimization and Frequency Calculations The uniformly scaled calculated frequencies for the s-cis and all-trans conformers of β-carotene show the accuracy of the method in computing vibrational frequencies (Figure 6). The scaled frequencies also show that the encapsulated spectrum is not simply a superposition of the stable conformers. The calculated all-trans spectrum contains bands at 1337 cm-1 (Figure 6a) and 1568 cm-1 (Figure 6b) that do not appear in the experimental Raman spectrum of carotene in acetone.
Because of this observation and the fact that the s-cis geometry is 3.0 kcal/mol more stable than the all-trans at the B3LYP/631G* level, the s-cis structure is used as the unencapsulated structure for this work from which frequency shifts and distortion energy costs are measured. Table S1 in the Supporting Information shows the sets of parameters used for each of the 27 constrained geometry optimizations and the results of the subsequent frequency calculations. The frequencies given correspond to eight distinctive vibrational bands from the experimental spectrum (Figure 7). Corresponding frequencies from the experimental encapsulated spectrum and the calculated s-cis structure are also given. It should be noted that the constrained optimized geometries do not correspond to isolated β-carotene potential energy surface minima. It is at these minima that the harmonic approximation used in the frequency calculations is most valid; however, the use of such constrained calculations is necessary in order to bypass expensive DFT calculations of the very large Car@SWNT system.
Fitting and Parameter Optimization The data in Table S1 in the Supporting Information were used in a fitting procedure to produce equations that express the frequency of each of the bands as a function of the three geometric parameters by minimizing the rms error with respect to the scaled calculated frequencies. The function used in the
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TABLE 1: Tabulated Results of the Optimization of Parameters Based on Frequency Shift Matching on Both of the Equation Sets Obtained from Model I and Model II all data 10 best literature frequencies and (model I) (model II) shifts for comparison shift optimized parametersa
TP
23.42
21.12
RRP BP
11.58 9.88 3.3 -18 -13 2 -3 2 7 -5 -2
31.18 6.33 4.8 -10 -13 1 -2 8 5 -3 6
rms error of fitb predicted shiftsb,c ν4 ν4′ ν3 ν3′ ν2 ν2′ ν2′′ ν1
available for β-carotene in CS2 at ambient pressure (960 cm-1).17 This agrees well with the calculated value. The optimizations resulted in two sets of distortion parameters corresponding to models I and II with predicted shifts and rms fitting errors given in Table 1. DFT Calculations Based on Spectrum Optimization
-17d -17d 0 -5 4 -1 -5 -2
a In degrees. b In cm-1. c Frequency shift values projected by the fitted functions for the given parameter values obtained by minimizing the error in projected frequency shifts with respect to literature shift values. d There are no ν4 or ν4′ bands in the experimental spectrum of unencapsulated β-carotene in acetone. Shift values of -17 cm-1 were used. See text for discussion.
fitting was a second order polynomial with form given by the following equation
νi ) Ai + Bi · TP + Ci · TP2 + Di · RRP + Ei · RRP2 + Fi · BP + Gi · BP2
Horn and Kertesz
(i ) 1,2,2′,2′′,3...4′) (1)
The uniformly scaled, calculated vibrational frequencies for both structures determined by constrained geometry optimizations at the B3LYP/6-31G* level using the parameters for models I and II appear in Table 2. Structures produced using both model I and model II (structures I and II) have a ν4 band below 950 cm-1, which reproduces the experimental Raman result well. The 946 cm-1 ν4 band of I is well within the error of the method to the experimental value of 944 cm-1. The rms errors with respect to frequency shifts are only 4.9 and 5.0 cm-1 for structures I and II, respectively. One of the greatest contributions to this error is the shift of the ν1 band, a consequence of increasing the twisting parameter. One concern with structure I when compared to structure II is the significantly greater energy cost required for the distortion. The cost to distort at the B3LYP/6-31G* level is the difference in energy between that of the unconstrained s-cis structure (-977598.5 kcal/mol) and that of each of the constrained structures. High distortion costs are acceptable provided that the energy loss is compensated by increased vdW contact. MM Analysis of Optimized Structures
The data for the fitting were chosen in two ways: (i) In model I, the data for all 27 constrained geometry optimizations (Table S1 (Supporting Information)) were used for completeness. (ii) In model II, only the data for the 10 structures that produced calculated Raman spectra with the smallest rms difference in band position compared to the experimentally observed encapsulated spectrum (first 10 rows of Table S1 (Supporting Information)) were used because these were thought to best represent the most relevant portion of the parameter space. The constants determined in each of the fitting procedures and related rms fitting error can be found in the Supporting Information (Table S2). Both of these sets of eight functions were then used in an optimization procedure which minimized the total rms error for all eight frequency shifts relative to the experimentally observed frequency shift. Working with frequency shifts instead of absolute frequencies is advantageous because it accounts for the shortcomings of the method in reproducing the unencapsulated carotene spectrum. The calculated frequency shift is the difference in band frequency between that projected by the fitted functions and that of the unconstrained B3LYP/6-31G* optimized s-cis structure when scaled. The experimentally observed frequency shift is the difference in frequency between corresponding bands in the encapsulated and unencapsulated experimental Raman spectra. It should be noted that the experimental shift for the ν4 and ν4′ bands cannot be determined from the experimental spectrum of unencapsulated β-carotene in acetone because the associated modes are either symmetry forbidden or very weak.4 A value of -17 cm-1 is used for these shifts, as this is the frequency shift necessary to move the associated modes in the scaled calculated s-cis spectrum (961 cm-1) to the 944 cm-1 frequency observed in the experimental encapsulated spectrum. An experimental value for the ν4 band frequency is
The hydrogen terminated (13,9) tube was used again for the reasons discussed previously. The most important quantity in the evaluation of the structures by force field methods is the net energy change (Enet) upon distortion from the s-cis structure. MM interaction energies for the structures (IEstructure) were computed as the difference in the total energies between the tube and isolated carotene structures (Etube and Eisolated) and the encapsulated system (Eencap).
IEstructure ) Eencap - Etube - Eisolated
(2)
Non-vdW contributions to the total energy do not change in the encapsulated structure calculations because the internal coordinates of each species are frozen. The change in total MM energy is thus the change in vdW energy. Even the unconstrained carotene structure has a favorable interaction energy when placed in the tube; however, the interaction energies for the optimized carotenes are more faVorable. It is the difference between the interaction energy of the s-cis carotene structure (IEs-cis) and the interaction energy of each of the constrained structures (IEstructure) that gives the interaction energy gain (IEgain), the energy that is available to pay the cost of the geometric distortions of the carotene.
IEgain ) IEs-cis - IEstructure
(3)
Force field methods are quite useful for describing vdW interactions; however, force fields cannot accurately evaluate the energy costs for the distortions of the B3LYP/6-31G* optimized carotenes both because the extensive π-conjugation is not accurately represented and because, even accounting for constraints, the structures are not local minima on the
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TABLE 2: B3LYP/6-31G* Frequencies (in cm-1) and Energies for Structures Produced in Constrained Geometry Optimizations with Model I and Model II Parameters
parameters constrained
a
TP RRP BP ν4 ν4′ ν3 ν3′ ν2 ν2′ ν2′′ ν1 rms ν4 ν4 ′ ν3 ν3′ ν2 ν2′ ν2′′ ν1 rms hartrees kcal/mol
scaled calculated frequencyb
frequency rms errorb scaled calculated shiftsb
shift rms errorb energy distortion costd
model I all data
model II 10 best
23.42 11.58 9.88 946 962 998 1007 1163 1192 1215 1522 6.2 -15 -10 3 1 5 5 0 3 4.9 -1557.8878 9.3
21.12 31.18 6.33 949 961 995 1006 1164 1189 1214 1526 7.0 -12 -11 0 0 6 3 0 6 5.0 -1557.8955 4.4
literature frequencies and shifts
944 955 1006 1018 1158 1188 1210 1523 -17c -17c 0 -5 4 -1 -5 -2
a In degrees. b In cm-1. c There are no ν4 or ν4′ bands in the experimental spectrum of unencapsulated β-carotene in acetone. Shift values of -17 cm-1 were used. See text for discussion. d The energy of the given structure (in kcal/mol) relative to that of the B3LYP/6-31G* s-cis structure (-1557.9026 hartrees).
TABLE 3: Results of the MM Analysis of the Spectrum Optimized Carotene Structuresa
s-cisb structure I structure II
Dreiding interaction energy (IEstructure)
Dreiding interaction energy gain (IEgain)
B3LYP/6-31G* distortion cost (Edistort)
net energy change (Enet)
-69.0 -89.5 -85.8
N/A 20.4 16.8
N/A 9.3 4.4
N/A -11.2 -12.3
a Energy values are in kcal/mol. b When encapsulated, the all-trans is 8 kcal/mol more stable than the s-cis; however, both I and II are even more stable when encapsulated.
MM potential energy surface. We thus use the interaction energy gains (IEgain) obtained in the force field calculations and the distortion costs (Edistort) as determined by the B3LYP/
6-31G* energies. These energy values and the net energy change (Enet) upon distortion (eq 4) for each species appear in Table 3.
Figure 8. Images of the MM calculated encapsulated geometries for structures I (a) and II (b).
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Enet ) Edistort - IEgain
Horn and Kertesz
(4)
other spectroscopic problems in supramolecular chemistry that are too large for full DFT treatment.
Structure I has the higher distortion cost; however, it also has a very favorable interaction energy gain due to its increased vdW contact. Structure I (Figure 8a) curves around the inside of the tube, while structure II (Figure 8b) distorts minimally for improved contact with a smaller surface. Based on the small encapsulated energy difference of 1.3 kcal/mol between I and II (more stable), it is likely that both contribute to the description of the carotene conformation inside the nanotube. Figure 9 shows the scaled B3LYP/6-31G* calculated Raman spectra for I and II along with the experimental encapsulated carotene spectrum. The calculated spectra for the two structures are both effective at producing the observed encapsulated spectrum even for bands in the 1250 to 1450 cm-1 range not explicitly fitted in the procedure. The similarity of the two calculated spectra despite the noticeably different structures reveals the limitation of the method especially when one considers the broad bands of the experimental spectrum.
Acknowledgment. Research is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-FG0207ER46472. Support by GridChem is acknowledged for computer time. We are indebted to Prof. Y. Gogotsi and Dr. Vadym Mochalin for useful discussions on problems of encapsulation. Supporting Information Available: Complete ref 9, examples of DFT constrained geometry optimizations (Figures S1-S4), plots of ν1 and ν4 band frequencies as functions of distortion parameters (Figures S5,S6), MM structure of encapsulated carotene without β-rings (Figure S7), vibrational data used in fitting procedure (Table S1), and parameters for fitted functions (Table S2). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes
Conclusion Using a combination of MM and DFT methods, we were able to generate structures for an encapsulated carotene molecule that are energetically favorable and produce calculated Raman spectra that replicate experimental results within the limits of accuracy of the B3LYP/6-31G* method and experimental data. Because of the similarities in encapsulated system energies and calculated band frequencies of the two structures, it is likely that a combination of the two is necessary to describe the conformations of carotene molecules when encapsulated in SWNTs. Methods similar to this one could be used to elucidate
Figure 9. Uniformly scaled B3LYP/6-31G* calculated spectra for structures I and II plotted with the experimental Raman spectrum of encapsulated carotene.
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