Density Functional Theory Calculations of the Non-Resonant and Resonant X-ray Emission Spectroscopy of Carbon Fullerenes and Nanotubes

The non-resonant X-ray emission spectroscopy of fullerenes and carbon nanotubes is studied with density functional theory in conjunction with short-range corrected functionals. For C60 and C70 the X-ray emission spectra are insensitive to modest structural changes, and absorption onto the fullerene cage has the greatest e↵ect with a broader less structured band observed with the high energy ⇡ band reduced in intensity. For carbon nanotubes the X-ray emission spectra are shown to be weakly dependent on the length and chirality of the nanotube. However, some variation with the diameter of the tube is observed in both resonant and non-resonant spectra.


Introduction
The increasing availability of synchrotron sources and free-electron lasers has led to spectroscopic techniques in the X-ray region being applied in a wide range of research areas [1][2][3][4]. One advantage of these techniques compared with other spectroscopic methods is that they are element specific and can provide 5 a local probe of structure. More recently, these techniques have been used to 1 study ultrafast chemical processes through time-resolved measurements [5][6][7].
There are several commonly used spectroscopic techniques in the X-ray region which can provide complementary information. For example, X-ray emission spectroscopy (XES) probes the occupied orbitals while X-ray absorption spectroscopy (XAS) probes the unoccupied orbitals. Resonant inelastic X-ray scattering (RIXS) represents an extension of XES wherein the intermediate state is a core-excited state rather than the core-ionised state in XES. Simulations of Xray spectroscopy can play an important role in interpreting and understanding spectra measured by experiment. Within the framework of density functional 15 theory (DFT), X-ray absorption spectra are commonly simulated using either transition-potential or time-dependent density functional theory (TDDFT) approaches [8,9]. Accurate core-excitation energies can be obtained from TDDFT calculations with the use of short-range corrected (SRC) functionals [10] and it has been shown how these calculations can be applied to study large systems 20 [11].
There has been less focus on the calculation of XES, one approach is to apply TDDFT or equation of motion coupled cluster theory to a reference determinant describing the core-ionised state [12], and these approaches have been applied to 25 study the XES of organic [13] and inorganic [14,15] molecules. Alternatively, X-ray emission spectra can be determined directly from a Kohn-Sham DFT calculation. In this approach the transition energy is evaluated from the orbital energies of the valence orbital (✏ v ) and core orbital (✏ c ) The associated intensity can be determined from the following transition matrix where a valence orbital ( v ) is taken to be the initial state and the final state is a core orbital ( c ). Recently, it has been shown that this approach can provide accurate X-ray emission spectra when the DFT calculation is performed with a SRC exchange-correlation functional [16]. These functionals were introduced 35 for the calculation of X-ray absorption spectra and incorporate an increased mode is known to depend on the diameter of the CNT. A further band is the disorder-induced D band, which is associated with sp 3 defects in CNTs [20].
CNTs can also be studied using fluorescence [21] and nuclear magnetic resonance spectroscopies [22,23], and there has been a considerable e↵ort focused on accurate simulations of the spectroscopy of these systems, for examples see 60 references [24][25][26][27][28][29][30]. Spectroscopic studies have also shown that hydrogenation of fullerenes has a significant e↵ect on its electronic structure. The ability to tune the photophysical properties of C 60 H x has potential applications in a range of devices where carbon based materials have a number of advantages compared to commonly used transition metal based quantum dot chromophores [31].
Carbon nanomaterials have also been studied with X-ray spectroscopy. Kawai and Motoyama reported X-ray emission spectra for solid C 60 and C 70 [32]. The experimental spectra were compared with spectra simulated from the 2p density of states from a Hartree-Fock theory calculation. The X-ray emission spectrum 70 of C 60 has also been measured and calculated by Heggie and co-workers [33].
The core-hole was treated as a nitrogen impurity in the Z+1 approximation and the spectrum simulated from the p-projected density of occupied states. The calculations reproduced the bands observed in experiment. Other fullerenes have also been studied, for example the X-ray emission spectra of di↵erent iso-75 mers of C 82 have been calculated [34] and the X-ray emission spectrum of bulk K 3 C 60 measured [35]. The XES of hydrofullerene (C 60 H 36 ) [36] and fullerene fluoride (C 60 F 24 ) [37] has been studied. For both of these molecules, the distinct bands present in the X-ray emission spectrum of C 60 are not evident and only a broad peak is observed. RIXS spectra have been reported for C 60 and C 70 80 [38,39]. For C 60 there is a significant variation between the spectra excited at di↵erent photon energies.
X-ray emission spectra for nanotubes have also been reported [40][41][42]. In one study [40] four peaks were identified and spectra simulated based upon 85 semi-empirical PM3 calculations using finite nanotube structures with the end carbons capped with hydrogen atoms. Only small di↵erences between the spectra for zigzag and armchair nanotubes were observed. X-ray emission spectra of multi-walled nanotubes showed a small contraction in the high energy region of the spectrum with decreasing diameter [41]. Resonant X-ray emission spectra 90 showed a strong dependence on the diameter of the tubes at an excitation energy of 285.5 eV, corresponding to a core! ⇡ ⇤ excitation [43]. It was suggested that this might provide a technique to determine the mean tube diameter. In this paper DFT calculations are used to study the X-ray emission spectra of carbon fullerenes and nanotubes and study the sensitivity of the spectra to modifica-95 tions of the structure. Subsequently, the extension of the calculations to study the RIXS spectra of these materials is explored.

Computational Details
The structure of the fullerenes and nanotubes were optimised at the B3LYP/6-31G* level of theory [44,45]. The nanotube calculations used finite length 100 nanotubes with enclosed (capped) ends, similar to those used in previous work [25]. X-ray emission spectra were computed according to equations 1 and 2 from a Kohn-Sham DFT calculation using the short-range corrected SRC1r1 exchange-correlation functional [10]. Details of the methodology underlying the calculations have been described fully elsewhere [16], and a more brief account 105 is given here. In the SRC1r1 functional, the electron repulsion operator is partitioned according to The first and third terms of equation 3 are treated with HF exchange and DFT exchange used for the remaining terms. The four parameters in the functional, C SHF , C LHF , µ SR and µ LR , determine the amount of Hartree-Fock exchange in 110 the short and long range, and these parameters were optimised to reproduce a set of core-excitation energies [10]. Two sets of parameters were optimised, the first for excitations at the K-edge of first row nuclei and the second for excitations from second row nuclei, and these functionals are denoted SRC1r1 and SRC1r2, respectively. These functionals correct the orbital energy of the 115 core orbitals and result in more accurate transitions energies for X-ray emission when evaluated from the di↵erent in the orbital energies. In the evaluation of the transition matrix elements to determine the intensity, the final and initial states are approximated by the valence and core molecular orbitals associated with the transition (see equation 2).

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The 6-31G* basis set was also used for the calculations of the X-ray emission spectra. For C 60 , there is little di↵erence between the spectrum computed with this basis set and spectrum computed using the larger 6-311G* basis set except that the spectrum for the 6-31G* is about 0.4 eV higher in energy. A 125 comparison between these two calculated spectra is shown in the Supplementary Content. However, the small size of this basis set allows larger systems to be studied more readily. Spectra were generated by convoluting the transitions with Lorentzian functions with a width of 1 eV. All calculations were performed with the Q-CHEM software package [46].

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A RIXS spectrum for C 60 was computed using a similar approach but with a reference determinant corresponding to a core to lowest unoccupied molecular orbital (LUMO) excited state. C 60 has three degenerate LUMOs, however, the resulting spectrum does not depend on which of these are occupied. The 135 core excited state is maintained during the self-consistent field calculation by using the maximum overlap method [47]. The B3LYP functional in conjunction with the 6-31G* basis set was used for these calculations as suggested by previous work [16]. A spectrum is also computed by averaging over 100 structures taken from an ab initio molecular dynamics simulation for the excited state.

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These structures were sampled at equal time intervals from a 24 femtosecond simulation at 300 K. and 283.0 eV in C 60 , and 280.8 eV and 282.7 eV in C 70 arise from ⇡ orbitals. In general the two molecules have similar spectra. One noticeable di↵erence is the relative intensity of the two high energy bands associated with the ⇡ orbitals.

Results and Discussion
The calculations show that the lower energy band is more intense, while for C 70 the intensities are similar. This feature is also evident in the experimental 155 spectra.
In order to explore the sensitivity of the X-ray emission spectra to structural modifications of the fullerene cage, Figure 2 shows computed spectra for C 60 with an encapsulated Li + ion, a vacancy defect and with adsorbed hydro-160 gen atoms (hydrofullerene C 60 H 36 ). For each system the structure has been re-optimised and for C 59 the ground state is a triplet state. Encapsulation of Li + results in no observable a↵ect on the computed X-ray emission spectrum.
The vacancy defect leads to some minor changes in the spectrum, particularly would be expected to be reduced further. The band corresponding to C in the spectrum does arise from an orbital that has ⇡ character, but this orbital also has C-H character. These findings are consistent with the experimental spectrum for C 60 H 36 [36]. Based upon the systems studied here, the X-ray emission spectra of fullerenes are insensitive to modest structural changes. Absorption  approach [48,49], however, such an approach is currently beyond our current capabilities for a system the size of C 60 .
The accuracy of the calculated non-resonant X-ray emission spectra for C 60 195 and C 70 provides some confidence that the methodology can be applied to carbon nanotubes. The calculated spectrum for a (9,0) nanotube of length 30Å is shown in Figure 4 with an experimental spectrum available in the literature [41].
The computed spectrum has been shifted by the value of -1.4 eV determined for in the intensity at about 280 eV. Experimental studies have shown a dependence of the intensity on the diameter of the nanotube in RIXS measurements [43]. Computing non-resonant X-ray emission spectra for these nanotube models is challenging, and the method used for C 60 where the core-excited state is considered with the Kohn-Sham calculation proved problematic to converge.

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Consequently, we have adopted a more crude approach to estimate the RIXS spectra of these systems. In this approach one carbon atom is replaced by a nitrogen atom and a calculation performed on the resulting doublet state. The rationale for this approach is that the additional electron occupied the LUMO and the nitrogen nuclei represents a carbon nuclei with a core-hole. This is 240 similar to the scheme used elsewhere to calculate non-resonant X-ray emission of C 60 [33]. The predicted excitation energies are too high, however, in the case of C 60 the resulting spectrum has a similar shape to the spectrum computed with a core-hole. The resulting spectra for nanotubes using this approach are also shown in Figure 6. These spectra have been shifted by -102.05 eV so that 245 the maximum aligns with the non-resonant spectra. The calculated spectra do show a reduction in the intensity of the peaks either side of the most intense band as the diameter of the tube increases. This trend does match observations from experiment [43] but the magnitude of the changes in the calculations is much smaller.

Conclusions
The calculation of the non-resonant and resonant X-ray emission spectra of carbon fullerenes and nanotubes using DFT has been investigated. Nonresonant X-ray emission spectra can be computed accurately with short-range corrected exchange-correlation functionals and modest sized basis sets. The 255 spectrum for C 60 did not show significant changes on encapsulation of a Li + ion or the presence of a vacancy defect. Absorption of hydrogen (hydrofullerene) gave a broad, less structured band with a reduction in intensity of the high energy bands associated with ⇡ orbitals. Similarly, for nanotubes the nonresonant spectra were insensitive to changes in length and chirality but did show 260 some dependence on the diameter of the nanotubes. Simulating the resonant X-ray emission spectra (RIXS) is more challenging and the agreement with experiment is poorer. Two approaches have been used, one involving explicit consideration of the core-excited state and a simpler scheme where a carbon atom is substituted with nitrogen. The calculations show that for C 60 the most 265 intense bands arise from transitions from orbitals, with higher nervy bands corresponding to ⇡ orbitals. Inclusion of vibrational coupling through sampling structures from an ab initio molecular dynamics simulation had little e↵ect on the resulting spectrum compared with the spectrum for the optimised ground state structure. For nanotubes. the calculations indicate some dependence on 270 the diameter of the nanotube in non-resonant spectra.