Toward understanding physical origin of 2175Å extinction bump in interstellar medium
Xing-Yu Ma, Yan-Yan Zhu, Qing-Bo Yan, Jing-Yang You, Gang Su
11 Toward understanding physical origin of 2175Å extinction bump ininterstellar medium
Xing-Yu Ma , Yan-Yan Zhu , Qing-Bo Yan , Jing-Yang You , Gang Su
3, 1* School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China. Center of Materials Science and Optoelectronics Engineering, College of Materials Science and OptoelectronicTechnology, University of Chinese Academy of Sciences, Beijing 100049, China. Kavli Institute for Theoretical Sciences, and CAS Center of Excellence in Topological Quantum Computation,University of Chinese Academy of Sciences, Beijing 100190, China * E-mail: [email protected] (GS)
ABSTRACT
The 2175 Å ultraviolet (UV) extinction bump in interstellar medium (ISM) of the Milky Way was discovered in1965. After intensive exploration of more than a half century, however, its exact origin still remains a bigconundrum that is being debated. Here we propose a mixture model by which the extinction bump in ISM is arguedpossibly relevant to the clusters of hydrogenated T-carbon (HTC) molecules (C H ) that have intrinsically a sharpabsorption peak at the wavelength 2175 Å. By linearly combining the calculated absorption spectra of HTC mixtures,graphite, MgSiO and Fe SiO , we show that the UV extinction curves of optional six stars can be nicely fitted. Thispresent work poses an alternative explanation toward understanding the physical origin of the 2175 Å extinctionbump in ISM of the Milky Way. Key words: (ISM:) dust, extinction.
1. INTRODUCTION
The interstellar extinction refers to the phenomenon that interstellar dust between earth and stars absorbs andscatters starlight. In 1965, Stecher discovered unexpectedly with sounding rocket observations that there is a 2175 Åbump feature in interstellar ultraviolet (UV) extinction curves (Stecher 1965). This extinction bump was laterconfirmed in tens stars by means of Orbiting Astronomical Observatory 2 (Bless & Savage 1972). Since then,several satellite observatories have helped to verify this bump feature toward many sightlines (Fitzpatrick & Massa1986; Savage & Sembach 1996; Fitzpatrick & Massa 2007; Zafar et al. 2018). The subsequent observation indifferent galaxies shows that such a 2175 Å extinction bump feature in interstellar medium (ISM) is almostubiquitous and pronounced in the Milky Way (Zafar et al. 2018; Draine 1989). It is commonly believed fromobservations that the 2175 Å extinction bump in ISM is from carbonaceous materials (Draine 1989). Despiteintensive exploration more than a half century, however, its exact physical origin is still unclear. A few possiblecandidates including graphite, nongraphic carbon, diamond and polycyclic aromatic hydrocarbons were proposed,but a consensus is yet far to reach.Earlier studies suggested that graphite grains could be responsible for the bump feature (Stecher & Donn 1965;Borg 1967; Papoular et al. 2009). Fitzpatrick and Massa discovered that the position of 2175 Å bump was ratherstable, with variations in the full width at half maximum (FWHM) of the profile in reddened galaxy stars(Fitzpatrick & Massa 1986). Draine and Malhotra pointed out that different shapes and sizes of graphite mayproduce variations in FWHM, but these were accompanied with position changes of the UV absorption peak (Draine& Malhotra 1993), showing that graphite alone cannot explain the origin of 2175 Å extinction bump. Meanwhile,diamond was also proposed for the interstellar dust (Saslaw & Gaustad 1969). It was later argued that diamondmight be the dust component to fit far ultraviolet (FUV) extinction curve and could give a smooth extinction butwithout distinctive features (Tielens et al. 1987), implying that diamond could not be the physical origin of the 2175Å bump feature.In addition, aromatic carbon or polycyclic aromatic hydrocarbons (PAHs) (Donn 1968; Vdovykin 1970) weresuggested for the bump feature in ISM as well. PAHs are organic molecules composed of C and H atoms in forms ofmultiple aromatic rings, whose typical structures include coronene, pyrene and anthracene. It was experimentallydiscovered that the UV-visible absorption curve of PAH mixtures was compatible with the interstellar extinctioncurve, but the UV absorption spectra of anthracene and pyrene had a remarkable absorption peak at 2380 Å (Joblin,Léger & Martin 1992), which is far different from 2175 Å in the UV extinction curve. A silicate-graphite-PAHmodel was then proposed by adjusting the UV absorption peak of PAH using an empirical grain size distributionfunction to better fit the interstellar extinction curves with several parameters (Weingartner & Draine 2001).However, Mathis argued that the bump peak is sensitive to the PAH shapes (Mathis 1998), and Allamandola et al. reported that the absorption spectra of interstellar dust should include sp hybridization bonds (Allamandola et al.1993). These studies thus pose challenges on PAHs as possible physical origin of the extinction bump, leavingactive debate on this intractable issue.In this circumstance, different candidates other than the aforesaid carbon forms to explain the extinction bumpin ISM of the Milky Way should be sought for. Several years ago, a new carbon allotrope named T-carbon (Figure1(a)) was proposed (Sheng et al. 2011), which is formed by replacing each atom in cubic diamond by a carbontetrahedron. T-carbon possesses the same space group mF as diamond and contains anisotropic sp hybridizedbonds. This novel carbon allotrope was successfully synthesized in different laboratories recently (Zhang et al. 2017;Chen et al. 2020). It is interesting to note that the measured UV-visible optical absorption spectra of T-carbon has aprominent peak at the wavelength of 2250 Å (Zhang et al. 2017), which is not far from 2175 Å. This fact stronglyhints that T-carbon might be correlated with the interstellar extinction bump in some manner. As T-carbon isinclined to form under a negative pressure circumstance according to our calculations, we speculate that it isconducive to form T-carbon and/or its fragments (molecules or clusters) in interstellar space, as the latter isgenerally in a state of negative pressure. Because the interstellar space is full of neutral hydrogen (Kalberla et al.2005, 2009), it is the most possible that T-carbon in interstellar space may exist in the form of molecules or clusterspassivated by hydrogen atoms, forming the hydrogenated T-carbon (HTC) molecules or clusters. Figure 1.
Structures of T-carbon and hydrogenated T-carbon molecule. (a) T-carbon. (b) Hydrogenated T-carbon (HTC) molecule (C H ). The red and blue balls represent C and H atoms, respectively. (c) Calculatedmolecular orbital energy levels of HTC molecules together with the molecular orbital maps of the lowestunoccupied molecular frontier orbitals (LUMOs) and the highest occupied molecular frontier orbitals(HOMOs), with the LUMO-HOMO gap of 5.417 eV. The green and yellow regions represent the positive andnegative phases of the orbital wave function. The calculations show that the 2175 Å absorption peak of HTCmolecule is intrinsically generated from the excitation from the HOMO (with the orbital symmetry E) to the10th level (degenerate in energy with the 11th and 12th levels a nd the orbital symmetry T2 ) above the LUMO (with the orbital symmetry A1) .
2. HTC MOLECULE, GRAPHITE AND SILICATES
The structure of HTC molecule (with point group T d , the same as T-carbon) is comprised of a supercell of T-carbonwith passivation of hydrogen atoms, containing 40 carbon and 16 hydrogen atoms (C H ), as presented in Figure1(b). (The calculation methods can be found in Appendix A1.) There are three distinct bonds and four different bondangles in HTC molecules close to those in T-carbon (Table A1 in Appendix A2). The passivation of hydrogen atomshas little effect on C-C bond length and bond angles. The HTC molecule possesses sp hybridized bonds. Themolecular vibration calculations show that HTC molecules are kinetically stable (Table A2 in Appendix A4 and A5).The UV-visible absorption spectra of HTC molecules are found to have a very sharp absorption peak positioned atthe wavelength of 2175 Å (Figure 2(a)), which is in accordance with the excitations from the highest occupiedmolecular orbitals (HOMO) to the 10th level (degenerate in energy with the 11th and 12th levels) above the lowestunoccupied molecular orbitals (LUMO) (Figure 1c), indicating that the physical origin of the interstellar UVextinction bump might be intrinsically related to HTC molecules or clusters.We also calculated the UV absorption peaks of HTC molecule by using the PBE (Perdew et al. 1996), M06-2X(Zhao et al. 2006) and ωB97 (Chai et al. 2008) methods comparing with B3LYP and found that the main peak of theUV absorption spectra of HTC molecule by different methods is surprisingly stable and consistent (Figure A1 inAppendix A3). As the interstellar dust is usually comprised of molecules and/or clusters with various sizes, webelieve that in ISM the HTC molecules in most cases may exist in the form of HTC mixtures consisting of HTCclusters with distinct sizes, leading to broadening of absorption peaks. Figure 2(a) presents the calculated sharpabsorption peaks (shown by straight lines) of HTC molecule as well as those broadened absorption spectra of HTCmixtures at the absorbed wavelengths as labeled. The calculation details for HTC mixtures can refer to the nextsection. It is clear that the central peak of the overall broadened absorption spectra of HTC mixtures appears at 2175Å (the red curve in Figure 2(a), which is a superposition of all broadened spectra of HTC mixtures with various sizesat all absorbed wavelengths). It should be remarked here that as the absorption peaks of HTC molecule are so smallat wavelengths other than 2175 Å, they are almost covered up when superposing all broadened spectra of HTCmixtures, giving rise to a profile of the total absorption spectra of HTC mixtures centered at the wavelength 2175 Å.In addition, we also found there are eight peaks of IR spectra of HTC molecule at wavelengths 5.77, 6.23, 7.54, 8.38,12.00, 12.65, 16.27 and 18.12 µm (The details can be found in Appendix A6) that are almost accord with theobserved peaks in NGC 2023 and NGC 7023 at 5.70, 6.25, 7.63, 8.59, 11.92, 12.70, 16.42 and 17.75 µm,respectively (Sellgren et al. 2010).Moreover, because silicate (Gilman 1969) and Mg/Fe-containing olivines (Campins & Ryan 1989; Jäger et al.1998) as well as graphite grains might also be the components of interstellar dust, the UV absorption spectra offayalite (Fe SiO ), enstatite (MgSiO ) and graphite should be included (Figures 2b-2d) to generate better fittingresults. The calculations reveal that the UV-visible absorption spectrum of Fe SiO shows a dramatic rising trend at0-9.92 eV (0-8 μm -1 ) (Figure 1(b)), which agrees with the rising linear background proposed by Fitzpatrick(Fitzpatrick 1999). The absorption spectrum of MgSiO possesses an absorption edge at 8.31 eV (6.7 μm -1 ), whichshows a rising trend in the 8.31-9.92 eV (6.7-8.0 μm -1 ) FUV range, being consistent with the upward trend of theFUV curve (Figure 2(c)). The UV spectra of MgSiO and Fe SiO are consistent with the previous results ( Jiang &
Guo 2004;
Tokár et al. 2010). The UV absorption spectrum of graphite has a distinct absorption peak of 4.30 eV(3.47 μm -1 ) in the range of 0-9.92 eV (0-8 μm -1 ) (Figure 2(d)).The red lines in Figure 2 represent the smoothed spectral curves. Since the observed spectra in experimentsare usually the averaged results owing to the particles or molecules randomly distributed and oriented in interstellarspace, the observed data should be the averaged ones of particles or molecules of various sizes. Note that such asmoothing does not alter the position of the central peak and the shape of spectra. Thus, it is reasonable to make thesmoothing when fitting to the realistic observed curves in interstellar medium. The calculation details can be foundin next section and Appendix. Figure 2.
Calculated UV absorption spectra of HTC molecule, HTC mixtures, Fe SiO , MgSiO and graphite.(a) HTC molecule, where a strikingly sharp peak appears at the wavelength of 2175 Å. The red curve representsthe overall superposed broadened absorption spectra of HTC mixtures, where the peak is positioned at 2175 Å.The straight solid lines are for HTC molecule and dashed curves are for HTC mixtures at the absorbedwavelengths labeled in the legend of (a), which are obtained by Eq. (2). (b) Fe SiO , (c) MgSiO , and (d)Graphite, where red solid lines represent the smoothed spectral curves by Savitzky-Golay polynomialregression.
3. MODEL AND FITTING EQUATIONS
To fit the UV extinction bump curves around 3-8 µm -1 in ISM, we take a mixture model in which four kind ofprimary ingredients including HTC mixtures, graphite, MgSiO and Fe SiO grains are included. We chooseoptionally six stars in the Milky Way such as BD582292, HD239689, HD239742, HD239722, HD283800 andHD147889 as examples, because they have well-documented observatory data in literature. The four starsBD582292, HD239689, HD239742 and HD239722 locate in Trumpler 37 and the extinction curves were obtainedby International Ultraviolet Explorer (IUE) (Valencic et al. 2003). HD283800 is the Taurus star through the Taurusdark cloud at a distance of about 150 parsec, where its extinction curve was obtained through IUE (Whittet et al.2004). The extinction curve of HD147899 star was found in a publication (Jenniskens & Greenberg 1993).The fitting equation, which was frequently employed to fit 2175 Å bump in previous works (Fitzpatrick &Massa 2007; Li et al. 2001; Blasberger et al. 2017), is given by Fitzpatrick and Massa (Fitzpatrick & Massa 1988,1990) with the absorption cross-section of Li and Draine (Li & Draine 2001) in the form of I Fit = α I HTC + α I Graphite + α I MgSiO + α I Fe SiO where α -α are linear fitting parameters, and I HTC , I
Graphite , I
MgSiO3 and I
Fe2SiO4 stand for the absorption intensities ofHTC mixtures, graphite, MgSiO and Fe SiO , respectively, as displayed in Figure 2.The intensities I Graphite , I
MgSiO3 and I
Fe2SiO4 can be obtained directly by means of first-principles calculationsbecause graphite, MgSiO and Fe SiO are all assumed in solid grains, as given in Figs. 2(b)-2(d). The calculationmethods can be found in Appendix A1.To calculate I HTC , we employ the equation of absorption intensity for mixtures given by Weingartner andDraine (Weingartner & Draine 2001) in the form of I HTC = (2.5πlg (e)) dn ada a Q ext a dlna dn a = Cn H a −3.5 da where n(a) is the number density of HTC mixtures, a is the size of HTC mixture, Q ext is the extinction efficiencyfactor, which can be evaluated with an assumption of spherical grains by using a Mie theory derived from BHMIE(Bohren & Huffman 1983), n H is the number density of H nuclei, and C is a constant.The extinction efficiency factor (Q ext ) can be obtained by the following equations (Bohren & Huffman 1983) Q ext = n=−1∞ (2n + 1)Re(a n + b n ) a n = mψ n mk ψ n' k −ψ n k ψ n' mkmψ n mk ξ n' k −ξ n k ψ n' mk b n = ψ n mk ψ n' k −mψ n k ψ n' mkψ n mk ξ n' k −mξ n k ψ n' mk ψ n x = xj n (x) , ξ n x = xh n(1) (x)k = 2πa/λ , m = ( ε real2 + ε imag2 + ε real )/2 + i ( ε real2 + ε imag2 + ε imag )/2 (1)(2)(3)(4) (7) (6)(5)where k is the size parameter, λ is the wavelength, j n is the spherical Bessel function, h n(1) is th e first class ofspherical H ankel functions, m is the refractive indices of clusters, ε real and ε imag are the real and imaginary part ofthe dielectric function of mixtures , respectively. The dielectric function of HTC mixtures can be calculated by the expression (Bohren & Huffman 1983) ε = ε real + iε imag = 1 + A μ −μ (μ −μ ) +μ γ + iA μγ(μ −μ ) +μ γ where μ is the photon frequency, μ is the photon frequency at the peak of dielectric function, A is the amplitude ofthe peak of dielectric function and γ is the broadening width of the peak of dielectric functions.The mixtures with different sizes usually have distinct parameters (μ , A and γ) in dielectric function. Tocalculate these three parameters for HTC mixtures of various sizes, we suppose that the photon frequency at thepeak position of dielectric function for HTC mixtures with various sizes distributes randomly near the correspondingpeak position of an HTC molecule, obeying a relation μ a = μ p + r × a−a molecule a max β + ∆ where μ p is the photon frequency at the peak of dielectric function for an HTC molecule, r is a random frequency inthe range of -0.5 μm -1 4. RESULTS AND DISCUSSION The fitting results according to Eq. (1) are presented in Figure 3. One may see that the extinction curves of the sixstars are nicely fitted by the UV absorption spectra of HTC mixtures, graphite, MgSiO and Fe SiO grains, wherethe adjusted coefficient of determination (goodness of fit) for fittings is about 97% with the mean squared error ofabout 0.016.To show the contributing proportion of the four ingredients in fitting the UV extinction feature bump in ISM ofthe Milky Way, we take the column density ratio as an indicator. The column density ratio between the fouringredients can be obtained according to the formula given by Weingartner and Draine (Weingartner & Draine 2001)in the form of n HTC : n Graphite : n MgSiO : n Fe SiO = α ρ HTC : α ρ Graphite : α ρ MgSiO : α ρ Fe SiO ρ HTC = Cn H m molecule a a molecule3 a −3.5 da where α -α are linear fitting parameters in Eq. (1), Cn H = 100 cm , and m molecule is the mass of HTC molecule(8.3×10 -22 g). The column density of HTC mixtures (ρ HTC ) can be acquired by Eq. (13), which yields 1.121 g/cm (Adetail calculation is given in Appendix A7). The column densities of ρ Graphite =2.259 g/cm for graphite, ρ MgSiO3 =3.965g/cm for MgSiO , and ρ Fe2SiO4 =4.671 g/cm for Fe SiO are adopted. We choose BD582292 star as an example. Thefitting results give that α , α , α , and α are 1.14×10 -6 , 1.46×10 -6 , 0.72×10 -6 and 3.50×10 -6 , respectively. The columndensity ratio of the four substances (HTC mixtures, graphite, MgSiO and Fe SiO ) in fitting to this star can beobtained to be 5.37%:13.87%:12.01%:68.75% by Eq. (12). It is interesting to mention that the column density ratioof carbonaceous dust (HTC mixtures and graphite) and silicate dust (MgSiO and Fe SiO ) in our fittings is about0.24, which is not far from that of carbonaceous dust and silicate dust 0.38 in a previous work (Weingartner &Draine 2001). The column density ratios in other stars are listed in Table 1, which shows the results similar toBD582292 star.The present fitting results reveal that, as shown in Figure 3, the 2175 Å UV extinction bump in ISM may bepossibly contributed by HTC mixtures (~5.72% in column density ratio), graphite (~12.42%), MgSiO (~19.83%)and Fe SiO (~62.03%), while the central peak at 2175 Å in the UV extinction bump feature may be primarilyattributed to HTC clusters. (12)(13)α α α α R (%) n HTC : n graphite : n MgSiO : n Fe SiO (%)BD582292 1.14 1.46 0.72 3.50 96.09 HD239689 1.20 1.40 1.35 3.40 96.71 HD239742 1.43 1.35 0.90 3.86 97.36 HD239722 0.90 1.40 0.92 3.60 96.71 HD283800 1.20 1.40 1.74 3.45 98.04 HD147889 1.40 1.00 1.50 1.80 97.15 Mean 1.21 1.34 1.19 3.27 97.01 Figure 3. Fitting results for the normalized extinction curves of the six stars of (a) BD582292, (b) HD239689, (c)HD239742, (d) HD239742, (e) HD283800 and (f) HD147889. Diamonds denote the observed data adapted from thereferences as indicated. The red solid lines are fitting curves of the observed data in accordance with Eq. (1). Theother lines are marked in the legend of (a). The expression in each figure shows the linear fittings with differentcolumn density ratios (in percentage) of four substances (HTC mixtures, graphite, MgSiO and Fe SiO ). The 2175Å bump feature in extinction curves is nicely reproduced for the six stars. Table 1. The fitting parameters, adjusted coefficients of determination, column density ratio of the four substances(HTC mixtures, graphite, MgSiO and Fe SiO ) in fitting the UV extinction curves of six stars BD582292, HD239680,HD239742, HD239722, HD283800 and HD147889. The corresponding mean values are included. Note. Observation data for BD582292, HD239689, HD239742 and HD239722 are from Valencic et al. 2003. Datafor HD283800 and HD147899 are from Whittet et al. 2004 and Jenniskens & Greenberg 1993, respectively.0 5. SUMMARY AND CONCLUSIONS In this work, we proposed a mixture model including HTC mixtures as well as graphite, MgSiO and Fe SiO grainsto explain the 2175 Å UV extinction bump feature of the interstellar medium in the Milky Way by fitting to theextinction curves of optional six stars. Our calculations are in good agreement with the observational results,indicating that the extinction bump feature may be caused by HTC mixtures (~5.2% in column density ratio),graphite (~12.5%), MgSiO (~19.9%) and Fe SiO (~62.4%), which also expounds the possible physical origin ofthis mysterious bump feature. The present work gives an addition to possible explanations for this phenomenon,which shows that the mixtures of hydrogenated T-carbon molecules with an intrinsic sharp absorption peak at thewavelength of 2175 Å may play essential roles in explaining the UV extinction curves of interstellar dust in theMilky Way. With the present proposals we anticipate that more observations and explorations on this mixture modelbased on HTC clusters in ISM of the Milky Way would be conducted in future. ACKNOWLEDGMENTS This work is supported in part by the National Key R&D Program of China (Grant No. 2018YFA0305800), theStrategic Priority Research Program of CAS (Grant No. XDB28000000), the NSFC (Grant No. 11834014), andBeijing Municipal Science and Technology Commission (Grant No. Z191100007219013). The calculations wereperformed on Era at the Supercomputing Center of Chinese Academy of Sciences. APPENDIXA1. Calculation Methods The calculations of hydrogenated T-carbon (HTC) molecules are completed by Gaussian 09 (Frisch et al. 2016). Inparticular, the geometric optimization and harmonic vibrational frequencies were performed within the Becke three-parameter Lee-Yang-Parr (B3LYP) exchange-correlation functional (Becke 1993; Lee et al. 1988) and 6-311G**Gaussian basis set. Nuclear magnetic shielding tenors were calculated at GIAO-B3LYP/6-311G** level. Based onthe time-dependent density-functional theory (Uchino et al. 2000; Raghavachari 2002), the ultraviolet (UV)absorption spectrum of HTC was obtained using B3LYP (13.9% exact exchange energy) and TZVP basis set. TheB3LYP (13.9% exact exchange energy)/TZVP calculations also give the HOMO-LUMO gap. The calculated dataare extracted by Multiwfn program (Lu & Chen 2012).All geometric optimization of crystals was performed within the density functional theory (DFT) asimplemented within the Vienna ab initio simulation package (VASP) (Kresse & Furthmuller 1996) with projectedaugmented wave (PAW) method (Blöchl 1994; Kresse et al. 1999) and the exchange-correlation interactions weretreated within spin-polarized gradient approximation and Perdew-Burke-Ernzerhof generalized gradientapproximation (PBE-GGA) (Perdew 1996). The plane-wave cutoff energy is taken as 500 eV and 8×8×8 k -pointssampling are used for the ground state calculation. The lattice geometries and atomic positions were fully relaxeduntil the energy and force were converged to 10 -6 eV and 0.01 eV/Å.1In practical calculations, the on-site Coulomb correlated potential for Fe ions was chosen as U=4.5 eV, asderived for fayalite (Cococcioni et al. 2003), and Hund’s exchange J=0.9 eV was obtained using the Racahparameters for Fe ions (Zaanen 1990).The GW approximation (Hedin 1965, 1999) was applied to obtain the quasiparticle (QP) energies via theperturbative solution to the Dyson equation [ − ℏ e ∇ + V ion + V H + E nkQP ]ψ nkQP = E nkQP ψ nkQP where m e is the mass of electron, ℏ is the reduced Planck constant, V ion is the electrostatic potential contributed byions, V H is the Hartree potential, and E nkQP is the quasi-particle energy.For MgSiO crystal, the optical excitation energies and exciton wave functions are determined through Bethe-Salpeter equation (BSE) (Salpeter & Bethe 1951; Strinati 1984; Rohlfing & Louie 2000) E ckQP − E vkQP A vckS + v ' c ' k ' vck K eh v ' c ' k ' A v ' c ' k ' S = Ω S A vckS where E ckQP and E vkQP are the quasi-particle energies for conduction and valence bands, K eh and Ω S are the electron-hole interaction kernel and excitation energy, respectively.The imaginary part of MgSiO crystal is evaluated from the excitation energies and exciton wave functionswith the following expression (Rohlfing & Louie 2000) ε ω = ω S λ S δ(ω − Ω S )where λ and is the polarization vector of the incident light and velocity operator, respectively, and is thetransition matrix element.The imaginary part of Fe SiO and graphite are given by ε μν ω = π V lim q→0 1q c,v,k k δ(ε ck − ε vk − ω) × μ ck+ e μ q μ vk μ ck+ e μ q μ vk ∗ where c and v denote the conduction and valence states, respectively, V is the cell volume, e μ are unit vectors forthree Cartesian directions, w k are k-points weight, and μ ck is the cell periodic part of the wave function at the point k.The absorption coefficient α ω is calculated as follows (Saha et al. 2000) α ω = 2ω ε ω + ε ω − ε ω where ε ω and ε ω are the real and imaginary parts of the dielectric function, respectively.The kinetic energy cutoff for GW and BSE is 400 eV, 10 occupied and 10 unoccupied orbitals are used tobuild the electron-hole interaction kernel.The adjusted coefficient of determination ( R adjusted2 ), employed to evaluate the goodness of fit, is defined as R = 1 − i (y iExp −y iFit ) i (y iExp −y iExp ) R adjusted2 = 1 − (1−R )(N−1)N−1−p (14)(15)(16)(17)(18)(19)(20)2where y iExp is the experimental observation value, y iFit is the fitting value, N is the number of data and p is thenumber of fitting parameters. The closer to 1 the value of R adjusted2 , the better the fitting degree on experimentalvalues.The mean squared error (MSE) represents the expected value of the square of the difference between thefitting values and experimental data, which is defined as MSE = 1N iN (y iExp − y iFit ) A2. Structural parameters of HTC molecule In Figure 1(b) of the text, the chemical environments of C1 and C4 atoms in each carbon tetrahedron, C2 and C3atoms, and H1 and H2 atoms in an HTC molecule are the same (Figure 1(b)), respectively. Electrons of LUMOs andHOMOs (Figure 1(c)) are mainly concentrated on the interior of the HTC molecule and C-C bonds in tetrahedrons,respectively. Table A1 gives the structural parameters of an HTC molecule, including the bond lengths of C-C andC-H atoms, and the bond angles between C atoms at different positions.Bond length (Å)C-H1 1.702C-H2 1.701C1-C2 1.485C1-C4 1.490C2-C3 1.482C4-C5 1.424Bond angle ( ° )C1-C2-C3 60.07C1-C3-C2 60.20C1-C3-C4 59.90C2-C1-C3 59.87C3-C4-C5 144.68 Table A1. The structural parameters of HTC molecule (21)3 A3. Comparison between different DFT methods We calculated the UV absorption spectra of HTC molecule by using the PBE (Perdew et al. 1996), M06-2X(Zhao et al. 2006) and ωB97 (Chai et al. 2008) methods comparing with B3LYP, and found that the main peakof the UV absorption spectra of HTC molecule by different methods is surprisingly stable (PBE: 216.6 nm,M06-2X: 218.0 nm, ωB97: 217.9 nm and B3LYP: 217.5 nm), as presented in Figure A1. Therefore, theseresults show that our calculations are very reliable. Figure A1. Calculated UV absorption spectra of HTC molecule by different methodsB3LYP, PBE, M06-2X and ωB97.4 A4. Excited States, molecule orbital transition, orbital symmetry of main transition andcontributions of transition for HTC molecule In Figure 2(a), we have listed five main peaks with the intensity and corresponding excitation energies,oscillator strengths, excited states, molecular orbitals transition and contributions of transition. As shown inFigure 2(a) and Table A2, the 2175 Å absorption peak corresponds to the excitation energy 5.6956 and 5.7052eV, where the latter has the greatest contribution, and the three excited states 27, 28 and 29th are the origin ofexcitation energy 5.7052 eV that is mainly generated from the excitation from HOMO (127, 128th) (withorbital symmetry E) to the 10th level (degenerate in energy with the 11th and 12th levels with orbitalsymmetry T2) above LUMO (with orbital symmetry A1). Excitationenergy(eV) Oscillatorstrengths Excited states Molecular orbital transitionand orbital symmetry of maintransition Contribution (%)5.6199 0.0225 18 126 -> 129 12.2473127 -> 133 33.6725127 -> 140 11.1685128 -> 133 15.4846128 -> 140 24.287719 124 -> 129 12.1584127 -> 141 35.0167128 -> 134 48.548720 125 -> 129 12.1712127 -> 135 39.5605127 -> 139 6.7029128 -> 135 9.2924128 -> 139 28.5345 125 -> 129 3.9649126 -> 129 13.8664127 -> 133 2.0779128 -> 134 13.122625 124 -> 129 11.6905125 -> 129 2.5910 127 -> 133 9.5677128 -> 133 4.399826 128 -> 134 2.5439124 -> 129 6.1411 127 -> 135 12.6796128 -> 135 2.9783126 -> 129 5.9430127 -> 133 19.5163 Table A2. Excitation energy, oscillator strengths, excited states, molecular orbitals transition, orbital symmetryof main transition and contributions of transition for an HTC molecule5 27 127 -> 137 2.4589127 -> 140 17.9592128 -> 133 8.9752 28 124 -> 129 5.8256 128 -> 134 27.7855128 -> 138 3.500729 125 -> 129 5.8011127 -> 135 22.5228127 -> 136 2.8379127 -> 139 10.5864128 -> 135 5.2904 Note . The 127 th and 128 th molecular orbitals are the highest occupied molecular frontier orbitals (HOMOs) that aredegenerate in energy. The 129 th molecular orbital is the lowest unoccupied molecular frontier orbital (LUMO). Thegreatest contribution for molecular orbital transition and orbital symmetry are shown in boldface font.7 A5. Vibrational frequencies of HTC molecule Table A3 presents the vibrational frequencies of the HTC molecule, showing that it is kinetically stable, as noimaginary mode is observed.Mode Frequencies (cm -1 ) Mode Frequencies (cm -1 )1 50.2855 82 814.69552 50.2855 83 818.83673 50.2855 84 820.07894 63.5687 85 820.07895 63.5687 86 820.07896 71.1983 87 834.98547 71.1983 88 834.98548 71.1983 89 837.98449 115.9677 90 837.984410 115.9677 91 837.984411 115.9677 92 843.593612 184.9283 93 843.593613 184.9283 94 843.593614 184.9283 95 848.793615 197.4205 96 848.793616 218.7256 97 850.361917 218.7256 98 850.361918 218.7256 99 850.361919 270.2911 100 852.003220 299.6154 101 859.588221 299.6154 102 859.588222 299.6154 103 859.588223 314.4171 104 860.069924 314.4171 105 860.069925 317.1261 106 860.069926 317.1261 107 1113.84727 317.1261 108 1113.84728 329.7647 109 1113.84729 329.7647 110 1116.037330 329.7647 111 1116.037331 332.2067 112 1116.037332 332.2068 113 1116.124633 339.6876 114 1116.124634 339.6876 115 1116.124635 339.6876 116 1120.806936 345.3286 117 1140.962637 543.431 118 1140.962638 543.431 119 1145.088639 543.431 120 1145.088640 553.6009 121 1145.088641 553.6009 122 1163.341142 553.6009 123 1163.341143 567.6182 124 1163.341144 567.6182 125 1179.841445 569.0809 126 1228.2396 Table A3. The vibrational frequencies of the HTC molecule846 569.0809 127 1228.239647 569.0809 128 1228.239648 573.9239 129 1297.019149 588.8335 130 1297.019150 588.8335 131 1361.752751 588.8335 132 1361.752752 590.3399 133 1361.752753 590.3399 134 1474.23754 590.3399 135 1548.978655 622.0563 136 1548.978656 622.0563 137 1548.978657 633.7698 138 1651.601358 633.7698 139 1651.601359 633.7698 140 1651.601360 635.2303 141 1701.704861 671.6014 142 1701.704862 671.6014 143 1776.135863 671.6014 144 1776.135864 675.7037 145 1776.135865 675.7037 146 1804.992666 696.5178 147 3316.764767 704.7426 148 3316.764768 704.7426 149 3316.764769 704.7426 150 3316.829670 720.6237 151 3316.829671 720.6237 152 3316.829672 720.6237 153 3321.197173 723.899 154 3321.197174 723.899 155 3321.197175 811.2774 156 3321.237476 811.2774 157 3336.83477 813.1699 158 3336.83478 813.1699 159 3336.884679 813.1699 160 3336.884680 814.6955 161 3336.884681 814.6955 162 3337.06269 A6. Infrared, Raman and NMR spectra of the HTC molecule The all calculations are performed by Gaussian 09 (Frisch et al. 2016). In particular, the infrared (IR) and Ramanspectra were calculated within the Becke three-parameter Lee-Yang-Parr (B3LYP) exchange-correlation functional(Becke 1993; Lee et al. 1988) and 6-311G** Gaussian basis set. Nuclear magnetic resonance (NMR) spectra wascalculated at GIAO-B3LYP/6-311G** level.Figure A2 presents the infrared (IR), Raman and NMR spectra of an HTC molecule. The IR and Raman spectraspread up to more than 3 μm. The region of a sharp peak at 3.1 μm corresponds to the C-H stretching modes as thesame mode in Raman spectra, and the region more than 5 μm corresponds to the C-C stretching, C-C-C bending,and C-C-H bending modes in which active modes are different for IR and Raman spectra, as shown in Figs. A2 (a)and (b). Figure A2. The infrared (IR), Raman and NMR spectra of HTC molecule. (a) IR, which is scaled by afactor of 0.9679. (b) Raman. (c) NMR, where the inset shows the chemical shifts of H atoms.0Figure A2 (c) shows the NMR spectra of the HTC molecule. There are also two regions. Two peaks appear inthe left region whose relative degeneracy is 1:3, which could be ascribed to two unique types of H atoms [total 16 Hatoms (1×4, 1×12)]. The lower peak is ascribed to H1 atoms, and another peak is ascribed to H2 atoms. Four peaksappear in the right region, three of which have the same degeneracy, and the other has a third of the degeneracy,which is ascribed to the four unique types of C atoms [40 sp C atoms (3×12, 1×4)], where a×b represents “a”unique types, and in one type, there are “b” symmetrical equivalent atoms.As shown in Fig. A2 (a), we found eight peaks of IR spectra of HTC molecule at wavelength 5.77, 6.23, 7.54,8.38, 12.00, 12.65, 16.27 and 18.12 µm that are almost consistent with the observed peaks in NGC 2023 and NGC7023 at 5.70, 6.25, 7.63, 8.59, 11.92, 12.70, 16.42 and 17.75 µm, respectively (Sellgren et al. 2010). The other twopeaks of HTC molecule at 9.00 and 9.27 µm may be covered by the IR spectra of other stuffs (e.g. PAH molecules),which is like the case for the peak at 7.04 μm of C60 molecule (Sellgren et al. 2010). A7. Calculation of column density of HTC mixtures The column density of HTC mixtures (ρ HTC ) can be calculated according to the formula given by Weingartnerand Draine (Weingartner & Draine 2001) ρ HTC = Cn H m molecule a a molecule3 a −3.5 da = Cn H × m molecule a molecule3 × 2( a 춰ᘴ − a molecule ) where we take Cn H = 100 cm , m molecule is the mass of HTC molecule (8.3×10 -22 g), a molecule is the size of HTCmolecule (1 nm), and a max is the maximum size of HTC mixtures (around 500 nm). By Eq. (14), we can obtainthe column density of HTC mixtures being about 1.121 g/cm . If we take Cn H = 3.23×10 -4 cm , ρ HTC will be3.6×10 -6 g/cm . However, the fitting results give that α , α , α , and α are 0.3426, 1.46×10 -6 , 0.72×10 -6 and3.50×10 -6 , respectively. The column density ratio of the four substances (HTC mixtures, graphite, MgSiO andFe SiO ) in fitting to this star can be obtained to be 5.37%:13.87%:12.01%:68.75% by Eq. (12), which isconsistent with the result of Cn H = 100 cm . Thus, the value of Cn H does not affect the column density ratio. DATA AVAILABILITY The data underlying this article will be shared on reasonable request to the corresponding author. (22)(23)1