Effects of exchange-correlation potentials on the density functional description of C_60 versus C_240 photoionization
Jinwoo Choi, EonHo Chang, Dylan M. Anstine, Mohamed El-Amine Madjet, Himadri S. Chakraborty
aa r X i v : . [ phy s i c s . a t m - c l u s ] O c t Effects of exchange-correlation potentials on the density functional description of C versus C photoionization Jinwoo Choi, EonHo Chang, Dylan M. Anstine, Mohamed El-Amine Madjet, and Himadri S. Chakraborty ∗ Department of Natural Sciences, D.L. Hubbard Center for Innovation and Entrepreneurship,Northwest Missouri State University, Maryville, Missouri 64468, USA Qatar Environment and Energy Research Institute,Hamad Bin Khalifa University, Qatar Foundation, P.O Box 5825, Doha, Qatar (Dated: July 13, 2018)We study the photoionization properties of the C versus C molecule in a spherical jelliumframe of density functional method. Two different approximations to the exchange-correlation (xc)functional are used: (i) The Gunnerson-Lundqvist parametrization [Phys. Rev. B , 4274 (1976)]with an explicit correction for the electron self-interaction (SIC) and (ii) a gradient-dependentaugmentation of (i) by using the van Leeuwen and Baerends model potential [Phys. Rev. A , 2421(1994)], in lieu of SIC, to implicitly restore electrons’ asymptotic properties. Ground state resultsfrom the two schemes for both molecules show differences in the shapes of mean-field potentialsand bound-level properties. The choice of a xc scheme also significantly alters the dipole single-photoionization cross sections obtained by an ab initio method that incorporates linear-responsedynamical correlations. Differences in the structures and ionization responses between C andC uncover the effect of molecular size on the underlying physics. Analysis indicates that thecollective plasmon resonances with the gradient-based xc-option produce results noticeably closerto the experimental data available for C . PACS numbers: 61.48.-c, 33.80.Eh, 36.40.Cg
I. INTRODUCTION
Fullerene molecules are a highly stable form ofnanoscopic carbon allotrope that can exist at room tem-perature. Therefore, they are routinely attractive candi-dates for spectroscopic studies in understanding aspectsof fundamental physics both in their vapor and con-densed matter phases. Technologically also, fullereneshold the prospect of exciting applications in solid statequantum computations [1, 2], improving the supercon-ducting ability of materials [3], biomedical fields [4],contrast-enhancement research for magnetic resonanceimaging (MRI), and improving organic photovoltaic de-vices [5]. Therefore, investigations of the response ofthese materials to radiations are valuable. One directionof these studies is to understand the collective responseof fullerene electrons to relatively low-energy photons. Inan infinite system like graphite, the incoming oscillatoryelectric field induces plane-wave type plasma oscillationsin the electron cloud within the system’s translationalsymmetry. This can only quantize a surface plasmonquasi-particle, but not the longitudinal (compressional)volume plasmon, since light is a transverse wave. Butwhen the medium has a boundary, the broken transla-tional symmetry enables the plasma wave to reflect andinduce other eigen modes of oscillations, including thevolume quantization. In particular, for finite systemswith boundaries in all directions, such as fullerenes andmetallic nanoclusters, photospectroscopy reveals multi- ∗ [email protected] ple plasmons that were measured [6, 7]. The photoelec-tron angular distribution asymmetry [8] and the emissiontime delay [9] at the surface plasmon of C also predictedinteresting behaviors. The other direction of fullerenestudies involves the response to photons whose energyis higher than the plasmon excitation energies. Thesephotons with their shorter wavelengths begin to resolvethe fullerene molecular geometry, entering the spectralregion of photoelectron diffraction. This effect resultsinto the occurrence of a series of cavity minima observedin the ionization spectra as the integer multiples of thephotoelectron half wavelength fit the molecular redii atcertain energies [10]. The effect also accompanies a beat-ing modulation in the ionization spectra as a signature ofC molecular width [11]. Emission delay spectroscopypredicted structures at these minima [12].Since the first observation of C giant plasmon reso-nance [13], theoretical studies with various levels of ap-proximation and success formed a large body of publishedresearch, an account of which up until 2008 can be foundin Ref. [14]. After 2008, there have been mainly two linesof theoretical calculations that attempted to account forthe atomistic details of the fullerene carbon-core on atruncated icosahedral geometry. One involves the geo-metric optimization of the C structure by the commer-cially available DMol3 software followed by the calcula-tion of Kohn-Sham ground state and then its linear re-sponse to the incoming radiation [15]. The other uses thegeneral access OCTOPUS software to directly solve thetime-dependent density functional equations for excitedC to subsequently Fourier transform the density fluc-tuation to obtain the dynamical structure factor utilizedto derive the electron energy-loss signal [16]. However,in spite of these important new developments, the jel-lium approximation to C ion-core, a model based onwhich we have developed a linear-response density func-tional methodology known as the time-dependent localdensity approximation (TDLDA), has seen a significantrange of success over last several years and continues toremain relevant [17]. This is because of the ease andtransparency of this model to capture the primary, ro-bust observable effects and to access the key physics thatunderpins the photo-dynamics and related spectroscopy.Let us cite two sets of results from our methods that di-rectly connected the experiments: (i) Our calculationshave predicted the photoionization of a second plasmonat a higher energy whose first observation was reportedin our joint publications [6, 18] with the experimentalgroup for gas phase C anions; a subsequent experimentaccessed this new plasmon even for the neutral C [19].(ii) Another experiment-theory joint study of ours re-vealed oscillations in C valence photoemissions provid-ing trains of diffraction minima mentioned above [10].Besides these pivotal results, our jellium-based study alsoextended to the photoionization of several atomic end-ofullerene molecules [20–24] and the C @C bucky-onion [25]. For some of these fullerene systems, TDLDAinvestigations of the photoemission time-delay [9, 12, 26]and multitudes of resonant inter-Coulombic decay pro-cesses [27–29] were also carried out with reasonable suc-cess.One limitation of the Kohn-Sham density functionalmethod is its approximate treatment of the electron ex-change. This is because the exchange interaction can onlybe fully treated in a non-local theory such as Hartree-Fock (HF) that exactly cancels out all self-interactions,restoring correct 1 /r behavior at r → ∞ . In most ofour previous calculations involving C and its deriva-tive endo-C compounds we used a widely utilized ap-proximate scheme [30] of exchange-correlation (xc) func-tional augmented by an orbit-by-orbit elimination ofself-interaction [31] originally proposed by Perdew andZunger [32]. A different scheme is to use the gradient-corrected xc potential of van Leeuwen and Baerends [33]that intrinsically approximates the correct long distanceproperties. While we adopted the latter in some of ourmost recent works, no detailed study on the comparativeabilities between the two schemes has yet been made.This is the primary objective of the current work thatconsiders the photoionization of the fullerene molecule.Along with C , a larger spherical fullerene, C , hasalso been considered to further broaden the scope of thecomparison. Significant differences from the choice ofthe xc treatment, both in ground and photoionizationdescriptions, are uncovered. Improved agreement of C plasmonic spectrum from the gradient-corrected xc ap-proach with the measured data is found.This paper is structured as follows. Section II in-cludes three subsections: A) the description of jelliumcore ground state structures with brief accounts of two xcparametrization schemes, B) comparison of ground state numerical results between two schemes and between twofullerenes, and C) the essentials of the method that in-corporates electron correlations in responding to the ra-diation; Section III compares the results of the valence(subsection A) and total (subsection B) photoemissions,as well as a comparisons with available measurements forC (subsection C). Conclusions are presented in SectionIV. II. ESSENTIALS OF THE METHODA. LDA exchange-correlation functionals
The details of the method follow the framework asdescribed in Ref. [14]. The jellium potentials, V jel ( r ),representing 60 and 240 C ions, respectively for C and C , are constructed by smearing the total positivecharge over spherical shells with radius R and thickness∆. R is taken to be the known radius of each molecule:3.54 ˚A for C and 7.14 ˚A for C . A constant pseu-dopotential V is added to the jellium for quantitativeaccuracy [34]. The Kohn-Sham equations for systems of240 and 960 electrons, made up of four valence (2 s p )electrons from each carbon atom, are then solved to ob-tain the single electron ground state orbitals in the localdensity approximation (LDA). The parameters V and∆ are determined by requiring both charge neutralityand obtaining the experimental value (for C ) and theknown theoretical value (for C ) of the first ionizationthresholds. The values of ∆ and the binding energiesof the highest occupied molecular orbital (HOMO) andHOMO-1 levels of both systems are given in Table 1.Using the single-particle density ρ ( r ) the LDA poten-tial can be written as, V LDA ( r ) = V jel ( r ) + Z d r ′ ρ ( r ′ ) | r − r ′ | + V XC [ ρ ( r )] , (1)where the 2nd and 3rd terms on the right are the directand xc components. In one scheme, V XC is parametrizeddirectly from ρ ( r ) by the following formula [30]: V XC [ ρ ( r )] = − (cid:18) ρ ( r ) π (cid:19) / − . " . (cid:18) πρ ( r )3 (cid:19) / , (2)in which the first term on the right is exactly derivableby a variational approach from the HF exchange energyof a uniform electron system with a uniform positivelycharged background and the second term is the so calledcorrelation potential, a quantity not borne in HF for-malism. In addition, we include an appropriate correc-tion to eliminate unphysical electron self-interactions forthe i -th subshell that renders the LDA potential orbital-specific [31, 35], V i ( r ) = V jel ( r ) + Z d r ′ ρ ( r ′ ) − ρ i ( r ′ ) r − r ′ + ( V XC [ ρ ( r )] − V XC [ ρ i ( r )]) . (3)This correction approximately captures the electron’slong range properties. We use the acronym SIC to re-fer this model.The other alternative account for xc-functional thatutilizes Eq. (2) but further refines it by adding aparametrized potential [33] in terms of the density andits gradient ∇ ρ as follows, V LB = − β [ ρ ( r )] / X βX sinh − ( X ) , (4)where β is adjustable and X = [ ∇ ρ ] /ρ / . This scheme,termed as LB94, is known to have lead to a consider-able improvement in the asymptotic behavior of the elec-tron when compared to the exact Kohn-Sham potentialscalculated from correlated densities. Consequently, thismodel is expected to also significantly improve the qual-ity of both the excited and continuum spectra. B. Ground states of C and C : SIC versus LB94 We show the ground state radial potentials of C and C obtained via both SIC and LB94 in Fig. 1(b)where the SIC curves, labeled as SIC-av, are occupancy-weighted average over all the subshells. This particularshape of the potentials earlier interpreted multiple fre-quencies in the Fourier transform of the measured pho-toelectron spectra of C [10]. Yet, note the differencesin details from SIC to LB94: (i) For C , to retain theexact same configuration of occupied states optimizedearlier [14] based on a number of experimental find-ings [36, 37], the LB94 potential gets slightly narrower(see Table 1) and deeper but with more widening of thewings on either side of the shell. (ii) While these gen-eral shapes also hold good for C , we note the follow-ing. In the absence of enough experimental information,C ground states were optimized by requiring identicalwidths and similar first ionization energies (Table 1) forboth SIC and LB94. This alters some properties of oc-cupied configuration that includes the LB94 HOMO tobe of σ character (a level with no radial node) 1 w witha very high angular momentum ℓ = 18 as opposed to a π level (with one node) 2 m in SIC of a lower ℓ = 10 [seeFig. 1(a)]. The direct repercussion of this change on theirphotoionization cross sections will be discussed below insection III A. Fig. 1(a) illustrates the general differences,SIC versus LB94, of some valence radial wavefunctions.(iii) Finally, the potential depth decreases from C toC even though the latter accommodates four timesmore electrons than the former. Why does this happen?To answer, we need to bear in mind that the effective TABLE I. Molecular shell-widths, and quantum characters( nℓ ) in harmonic oscillator notations and binding energies(BE) of HOMO and HOMO-1 levels of C and C . Thevalues in parenthesis correspond to LB94 results only whendifferent from SIC.∆ (˚A) HOMO BE H (eV) HOMO-1 BE H-1 (eV)C h − .
51 2 g − . m (1 w ) − .
47 ( − .
43) 2 l (2 m ) − .
98 ( − . radial potential also includes the angular momentum de-pendent centrifugal barrier part ℓ ( ℓ + 1) / r which variesslower as a function of r over the C shell region that isradially farther from C , creating more “energy-room”for larger C . Indeed, a far denser angular momentummanifold of π and σ energy bands are generally found forC as seen for LB94 bands presented in Fig. 1(b).Our SIC and LB94 descriptions of C also producedstatic dipole polarizability (SDP) values of respectively92.8 ˚A and 114 ˚A which are reasonably close to themeasured value of 76.5 ± [38], particularly giventhat the jellium model disregards the molecular core vi-bration. Likewise, our calculated values of SDP for C are 565 ˚A and 638 ˚A , respectively for SIC and LB94.The slight increase in SDP from SIC to LB94 for bothfullerenes is due to a somewhat higher spill-out electrondensity in LB94. This spill-out can be recognized bynoticing the LB94 potential in Fig. 1 being a bit widerat the top causing slight outward spreads of the radialwavefunctions. C. TDLDA dynamical Response
A time-dependent LDA (TDLDA) approach [14] isused to calculate the dynamical response of the com-pounds to the external dipole field z . In this method, thephotoionization cross section corresponding to a bound-to-continuum dipole transition nℓ → kℓ ′ is σ nℓ → kℓ ′ ∼ |h kℓ ′ | z + δV | nℓ i| , (5)where the matrix element M = D + h δV i , with D beingthe independent-particle LDA matrix element; obviously, D solely yields the LDA cross section. Here δV representsthe complex induced potential that accounts for electroncorrelations. In the TDLDA, z + δV is proportional tothe induced frequency-dependent changes in the electrondensity [14, 28]. This change is δρ ( r ′ ; ω ) = Z χ ( r , r ′ ; ω ) zd r , (6) -0.400.40.8 R a d i a l o r b it a l ( a . u . ) H(2h), C LB94H(1w), C
LB94H-1(2m), C
LB94
Radial co-ordinate (Angstrom) -50-40-30-20-100 R a d i a l po t e n ti a l ( e V ) C LB94C SIC-avC
LB94C
SIC-avC σ C π C σ C π H(2h), C SICH(2m), C
SICH-1(2l), C
SIC (a)(b)
FIG. 1. (Color online) (a) Ground state radial wavefunctions for C HOMO (H) levels and both C
HOMO and HOMO-1levels calculated in SIC and LB94. (b) Corresponding radial potentials are shown. Shell widths are identified. Energy bandsof σ and π characters (see text), obtained only in LB94, are illustrated where the full susceptibility χ builds the dynamical cor-relation from the LDA susceptibilities, χ ( r , r ′ ; ω ) = occ X nl φ ∗ nl ( r ) φ nl ( r ′ ) G ( r , r ′ ; ǫ nl + ω )+ occ X nl φ nl ( r ) φ ∗ nl ( r ′ ) G ∗ ( r , r ′ ; ǫ nl − ω ) (7)via the matrix equation χ = χ [1 − ( ∂V /∂ρ ) χ ] − involv-ing the variation of the ground-state potential V withrespect to the ground-state density ρ . The radial com-ponents of the full Green’s functions in Eq. (7) are con-structed with the regular ( f L ) and irregular ( g L ) solu-tions of the homogeneous radial equation (cid:18) r ∂∂r r ∂∂r − L ( L + 1) r − V LDA + E (cid:19) f L ( g L )( r ; E ) = 0(8)as G L ( r, r ′ ; E ) = 2 f L ( r < ; E ) h L ( r > ; E ) W [ f L , h L ] (9)where W represents the Wronskian and h L = g L + i f L .Obviously, TDLDA thus includes the dynamical correla-tion by improving upon the mean-field LDA basis. C LB94C SIC Photon energy (eV) -1 C LB94C
SIC HO M O c r o ss s ec ti on ( M b )
20 30 40 50 60 70 (a)(b)
FIG. 2. (Color online) Photoionization cross sections for theHOMO level calculated in SIC and LB94 for C (a) and C (b). C LB94C SIC Photon energy (eV) -1 C LB94C
SIC HO M O - c r o ss s ec ti on ( M b )
20 30 40 50 60 70 (a)(b)
FIG. 3. (Color online) Photoionization cross sections for theHOMO-1 level calculated in SIC and LB94 for C (a) andC (b). III. RESULTS AND DISCUSSIONA. Photoionization of valence electrons
The photoionization cross sections of the HOMO levelcalculated in TDLDA, both in LB94 and SIC schemes,are presented in Fig. 2. Let us first note that the hostof narrow spikes that appears represents single-electronautoionizing resonances. The positions and shapes ofthese resonances largely vary between two xc schemesThis happens mainly because of their significantly differ-ent descriptions of the unoccupied excited states (thatdepend on the potential’s asymptotic behavior), eventhough their occupied spectra are by and large similar. Infact, it is expected that owing to the better long-rangeaccounts of electronic properties, LB94 resonances aremore accurate in all current results. Neglecting thesesingle-electron features, broad build-ups of the oscillatorstrength above 10 eV are due to the two collective plas-mon resonances. The general shape of the curves is quali-tatively similar between LB94 and SIC for C [Fig. 2(a)],largely because the HOMO levels are of the same π sym-metry in both the schemes [Fig. 1(a)]. In contrast, dueto the different symmetries of HOMO for C [Fig. 1(a)],the broad shapes of the LB94 and SIC curves in Fig. 2(b)noticeably differ from each other. In all the curves, thereappear some imposing oscillation-type structures above30 eV that somewhat mask the second (40-eV) plasmon.Further, comparing Fig.1(a) with (b), we note a generalshift of the plasmonic enhancements toward lower ener-gies for larger C similar to the known trend in the size dependence of plasmons in noble metal cluster stud-ies [39].TDLDA cross sections for HOMO-1 level are shown inFig. 3. Since for each fullerene the HOMO-1 level retainsthe same π symmetry going from SIC to LB94, the broadshapes of the curves obtained from these approximationscompare well, barring the mismatch in details includingin the single-electron resonances. We also note here thesuperposed oscillatory structures at higher energies andthe red-shift of the plasmon resonances in C comparedto C as in the case of HOMO.Cleaner shapes of the plasmon resonances are morereadily captured in the total cross sections that we dis-cuss in the next subsection. We address at this point aphotoelectron diffraction-driven phenomenon that beginsto surface from the waning region of the higher energyplasmon where the collective effect starts to weaken. Aninterference between photoelectron waves, predominantlyproduced at the boundaries of the fullerene shell, under-pins this process. This essentially single-electron effect isthe root cause of the oscillations seen at higher energiesin Figs. 2 and 3 that has been observed before in photo-electron spectroscopy [10, 11] and theoretically discussedat great lengths [40]. Following Ref. [40], one can simplymodel these oscillations in a nℓ -level cross section by σ nℓ → kℓ ′ ∼ A ( k )2 (cid:2) B + ( a o h o ) cos(2 kR o − ℓ ′ π )+ ( a i h i ) cos(2 kR i − ℓ ′ π ) − a o a i h o h i { cos(2 kR − ℓ ′ π ) + cos( k ∆) } ] , (10)where A is a steady energy-dependent part, a o and a i arethe values of the radial bound wavefunction at inner ( R i )and outer ( R o ) radii of the fullerene shell, B = a o + a i , h i and h o are respectively proportional to the deriva-tives of the radial potential [Fig. 1(b)] at R o and R i , and∆ = R o − R i . Obviously, the oscillations in photoelectronmomentum ( k ) depend on the potential shape that alsoincludes the angular momentum dependent centrifugalbarrier. Therefore, it is not surprising that the higher en-ergy sub-structures in Fig. 2(a) and Fig. (3) qualitativelymatch between LB94 and SIC which have identical an-gular momentum symmetry. In Fig. 2(b), however, thismatching worsens. This is the consequence of increasedcentrifugal barrier from much higher angular momentumof LB94 HOMO level 1 w for C that obliterates theinner radius in the effective potential to effectuate h i = 0in Eq. (10), qualitatively altering the net shape of the os-cillations. The details of this angular momentum effectwere discussed earlier [40]. We must also note in Figs. (2)and (3) that these higher energy oscillations are in gen-eral smaller for C as a consequence of the larger radiusof this system leading to higher oscillation “frequencies”in Eq. (10).Equation (10) unravels some further insights. Notethat the first three oscillatory terms in this equation carrya constant phase shift ℓ ′ π , where the dipole selected fi-nal angular momentum ℓ ′ = ℓ ±
1. The implication isthat each of these oscillations for ionization from twoneighboring ℓ states will be 180 o out-of-phase to eachother [41]. However, the oscillation from ∼ a o a i cos( k ∆)in Eq. (10) is independent of ℓ . But note that betweenthe ionization of a π and a σ electron this oscillation isroughly opposite, since the product a o a i is negative fora π radial wave, but positive for a σ – an implication ofwhich will be discussed in the following subsection. C LB94C SIC
10 100
Photon energy (eV) -1 C LB94C
SIC T o t a l c r o ss s ec ti on ( M b ) C LDA (LB94)C
LDA (LB94) (a)(b)
FIG. 4. (Color online) Total cross sections calculated in SICand LB94 for C (a) and C (b). The corresponding single-electron (LDA) results using LB94 are also shown. B. Total and band-differential cross sections
Figure 4 presents the total TDLDA photoionizationcross sections and compares them with respective single-electron LDA results (shown only for LB94) for both thefullerenes. The sum over ℓ largely cancels out oscillationsdue to the reason discussed above (in the last paragraphof subsection III A) and makes the broad higher energyplasmon (HEP) emerge clearly. In fact, both the plasmonresonances in TDLDA stand out in Fig. 4 against the rel-atively smooth LDA curves. Unlike to the lower energyplasmon (LEP) resonances, HEPs exhibit far weaker ef-fects of single-electron resonances but rather long decaytails. Energy red-shifts of the resonances in C com-pared to those in C are noted along with the fact thatC plasmons are utilizing significantly higher oscilla-tor strength density due to its much larger electron poolto collectivize. For each fullerene, significant differencesin the resonance widths between LB94 and SIC are alsonoted. Values of various resonance parameters are givenin Table II.From a non-perturbative many body theory, the emer-gence of plasmon resonances can be thought of as origi-nating from the formation of collective excited states un-der the influence of external electromagnetic field [42].Since the collective excitations are energetically em-bedded in the single-electron ionization channels, theyprovide alternative ionization pathways degenerate withsingle-electron channels. Thus, the autoionization ofthese collective excited states induce resonant enhance-ments in the subshell cross sections as shown in Figs. 2and 3. However, from a perturbative approach the plas-mon mechanism can be best modeled by Fano’s inter-channel coupling formalism [43]. To include the effectsof channel-coupling upon the final state wave functionof each of the perturbed dipole matrix elements M nℓ ( E )one can write [14], M nℓ ( E ) = D nℓ ( E ) + X n ′ ℓ ′ = nℓ Z dE ′ h ψ n ′ ℓ ′ ( E ′ ) | | r nℓ − r n ′ ℓ ′ | | ψ nℓ ( E ) i E − E ′ D n ′ ℓ ′ ( E ′ ) (11)where D nℓ is the unperturbed (LDA) nℓ matrix element, ψ nℓ ( E )’s are the unperturbed final continuum channelwave functions of the single-electron channels, and thesum is over all of the photoionization channels exceptthe nℓ channel. The matrix element within the integralof Eq. (11) is known as the interchannel coupling ma-trix element; the fact that each of nℓ initial state orbitaloverlaps strongly with all other fullerene-orbitals insuresthat these interchannel coupling matrix elements will bestrong. Further, this also justifies the existence of bothlow and high energy plasmons at exactly the same en-ergies for all the subshells for a given fullerene and im-plies the various dipole matrix elements are “in phase” over the two energy regions (bands) of each fullerene[Fig. 1(a)]. Consequently, the various terms in the sum inEq. (11) will add up coherently , leading to the dramaticenhancement.Equation (11) reveals one further important correla-tion feature. Since a π (or a σ ) bound orbital will havenear-perfect overlaps with other π (or σ ) orbitals dueto their almost identical shape and spatial extent, theinterchannel coupling matrix element in Eq. (11) will bestronger for a π - π or a σ - σ self-coupling than a π - σ cross-coupling. Therefore, it is expected that the π electronswill show a preferred participation for building LEP andthe σ for HEP. Figs. 5 and 6 respectively show the only- π C LB94C SIC
10 100
Photon energy (eV) C LB94C
SIC π c r o ss s ec ti on ( M b ) X C total, LB94 X C total, LB94 (a)(b) FIG. 5. (Color online) Total π -band cross sections in SIC andLB94 for C (a) and C (b). Curves are scaled at higherenergies to illustrate a strong minimum (see text). The totalcross sections are also displayed for comparisons. -1 C LB94C SIC
10 100
Photon energy (eV) -1 C LB94C
SIC σ c r o ss s ec ti on ( M b ) C total, LB94C total, LB94 (a)(b) FIG. 6. (Color online) Total σ -band cross sections in SIC andLB94 for C (a) and C (b). The total cross sections arealso displayed for comparisons. and only- σ band cross sections in TDLDA calculated inboth LB94 and SIC. For each fullerene, if we compare the π band result in LB94 with the total cross sectionin LB94 (also shown), a dominant contribution of the π -cloud to LEP and of the σ -cloud to HEP are indeednoted. In general, however, it is also obvious from thesecomparisons that either of LEP and HEP in a fullereneare of both π and σ mixed character, it is just that oneis dominant on the other.A discussion on the red-shift of TDLDA plasmon res-onances from C to C [Fig. 4] may now be in order.Classical plasmon-model [44] of a spherical dielectric shellwith symmetric and antisymmetric vibrations betweenthe inner and outer surfaces suggests that the midpointenergy between the two resonances to be about the samefor C and C , since they have approximately same ini-tial electron densities [45]. According to this model theplasmons are then formed below and above this midpointenergy shifted equally both ways, and this shift growswith the increasing radius, suggesting that the plasmonswill be more separated out for C [45]. Clearly, that isnot seen in Fig. 4, in which both the plasmons red-shiftfor C and in fact move close to each other comparedto their C results (see Table. II for the actual val-ues), suggesting that quantum effects play an importantrole. One possible way to understand this phenomenonquantum mechanically is to recall in Fig. 1(b) that theC ground state potential is shallower while accom-modating a number of electrons four times that of C producing far compact energy levels. This suggests adecrease of the average ground state binding energy forC . Therefore, since in the spirit of Eq. (11) the plas-mons can be interpreted as the coherence in close-packedsingle-electron excitations, it is only expected that theplasmons will begin to excite at lower photon energiescausing their early onsets for C , as seen in Fig. 4. Infact, this trend of red-shifting plasmons with increasingfullerene size should be rather generic, at least in thejellium based quantum calculations. An insight in thephenomenon can be motivated by perceiving a collectivemode as having a natural oscillation frequency p κ/ρ ofa mass density ( ρ ) on a spring of stiffness κ [25]. Thus,a shallower binding potential with higher electron popu-lation for C translates to the loosening of the springdecreasing κ and thereby its resonant frequencies.Let us now compare between the predictions of LB94and SIC for the band-cross sections. For the π -band,LB94 retains a contribution approximately similar tothat of SIC at LEP, but shows depletion at HEP which ismore prominent for C [Fig. 5(a)] than C [Fig. 5(b)].For the σ -band, on the other hand, a notably higher con-tribution to LEP and some increase at HEP by LB94for both the systems are found [Fig. 6]. There are more.Our discussions following Eq. (10) indicate that the ℓ -sum over π or σ cross sections will significantly weakenthe diffraction oscillations coming from the first three os-cillatory terms in Eq. (10), while the fourth oscillation ∼ cos( k ∆) will survive being free of ℓ . As a result, inthe band-cross sections this ∆-dependent oscillation willdominate. Since ∆ slightly shortened in LB94 than SICfor C [Fig. 1(b)], π -band LB94 curve in Fig. 5(a) pro-duces a longer wavelength in k to induce its first min-imum above 100 eV at an energy higher than that inSIC. The equality of ∆ in LB94 versus SIC for C , onthe other hand, justifies the occurrence of these minimaat about the same energy as in π -band results for thisfullerene [Fig. 5(b)]. However, this effect is not so intu-itive for the σ -band case. As seen, while the minimum inLB94 for C [Fig. 6(a)] does appear at higher energiesthan the SIC minimum, they do not seem to coincide forC [Fig. 6(b)] as they did for C . The latter is due tothe fact that the σ states for C , reaching very high in ℓ values compared to their counterparts in C , producesuch strong centrifugal repulsions that the effective po-tentials for high ℓ considerably deform rendering the roleof ∆ less meaningful [40].
10 20 30 40 50
Photon energy (eV) T o t a l c r o ss s ec ti on ( M b ) C C Lorentzian fit, C Lorentzian fit, C C ; Exp. Hertel et alC ; Exp. Reinkoester et al Fit to SIC results
FIG. 7. (Color online) Lorentzian fits to the total cross sec-tions (also shown) obtained from LB94. These are comparedwith the fits of the corresponding SIC results. Two sets ofexperimental data, appropriately red-shifted, are included toaid the comparison between plasmonic responses obtain viatwo xc schemes.
C. Plasmon resonances and comparison withexperiments
Measurements [6, 18, 19] of plasmon resonances in thephotoionization of neutral and ionic C produced rel-atively smooth curves without any evidence of autoion-izing resonances, which exist in our theoretical TDLDAresults. As discussed in details in our earlier study [14],this is likely because the coupling of electronic motions TABLE II. Resonance positions ( E o ), FWHM (Γ) and oscil-lator strength density (OSD) of the lower energy (LEP) andthe higher energy (HEP) plasmon. The values in parenthesisare corresponding LB94 results. E o (eV) Γ (eV) OSDC LEP 15.8 (16.8) 2.5 (3.5) 136 (184)C HEP 37.5 (38.5) 10.0 (13.0) 35 (30)C
LEP 11.9 (12.4) 0.9 (2.0) 642 (601)C
HEP 33.8 (31.5) 10.5 (9.5) 281 (241) with the temperature-induced vibration modes of thecore [46] and the fluctuation of the cluster shape aroundthe shape at absolute zero [47, 48]. In addition, the in-herent over-delocalization of jellium models predicts au-toionizing resonances that are too narrow, as seen inour results. Therefore, in Fig. 7, we fit the non-spikybackground parts of our TDLDA total cross sections ob-tained via LB94 for both the fullerenes using a formulathat includes two Lorentzian line profiles. We furtherpresent in Fig. 7, two similar fitting curves for the SICresults of the fullerenes. For both fullerene systems, Ta-ble 2 presents the positions ( E o ) and full-widths at half-maxima (FWHM), Γ, and oscillator strength densities(OSD) corresponding to each plasmon resonances calcu-lated in LB94 and SIC; LB94 results are displayed inparenthesis.From Table 2 and Fig. 7, going from SIC to LB94, bothLEP and HEP of C move up in energy by 1 eV, whileC LEP by a half eV. We recall the spirit of a clas-sical oscillator model of dielectric shell that the plas-mon frequencies are proportional to the square-root ofthe ratio of rigidity to density ( p ∼ κ/ρ , in analogy tothe oscillation frequency of a mass on a spring of stiff-ness constant κ ) introduced in the previous subsection.Note, LB94 radial waves being slightly more spread outthan their SIC counterparts occupy a larger space effec-tively decreasing the density. This explains the blue-shiftof LB94 plasmons. This trend in LB94 is an improve-ment, since jellium based prediction of C plasmon res-onance energies are known to be below their measuredvalues [14]. However, this trend is reversed for C HEP where LB94 moves this plasmon lower in energyby more than 2 eV from its SIC prediction, ascertainingthe importance of quantum effects to capture the detailsof these resonances. Furthermore, the LB94 width of C LEP is found to be 3.5 eV, which is an increase of 40%over its SIC value of 2.5 eV, while this increase is 30%for C HEP. More than a double increase of width forC
LEP is found going from SIC to LB94, while again,this trend reverses by a small amount for C
HEP. Sig-nificant variations in the OSD utilized by each plasmonsfor either system between two xc approximations are alsonoted in Table 2, accounting for the detailed differencesthat the two calculation schemes generate.Comparisons of the results between the two fullerenesin Fig. 7 as well as in Table 1 indicate a generic red-shiftof plasmon energies for the larger fullerene C , as notedand discussed earlier. We also find in Table 2 a generaltrend of the width Γ to decrease with the increasing sizeof fullerene, except for C
HEP in SIC. Further notethat while for C LEP the OSD value increases from SICto LB94, the trend is found opposite for this resonanceof C . For the HEP, either fullerene exhibits decreasein OSD going from SIC to LB94.Fig. 7 further includes two sets of experimental mea-surements for C , where the data from Hertel et al [13]are red-shifted by 3 eV and those from Reink¨oster etal [19] by 1 eV to match respectively with the energies ofLEP and HEP calculated in LB94. As evident, the mod-ifications in Γ and OSD, as brought about by the LB94scheme, indicate an improved agreement with experimen-tal results compared to what SIC achieves. We must alsonote that in a jellium model, the plasmon resonances onlydecay via the degenerate single-electron channels. In thereal system, however, there would be additional effectsfrom the independent local ion sites positioned based onan appropriate atomistic symmetry, at least for relativelymore tightly bound electrons. As shown in detailed withSIC results in Ref. [14], in order to account for these ad-ditional decay channels, the theoretical cross section ina jellium frame must be convoluted with a small widthin order for a more meaningful comparison with mea-surements. With the already improved agreement of thecurrent “zero-width” results of LB94, it is only expectedthat such a convolution will further better the agreement with the experiment. IV. CONCLUSIONS
In conclusion, the work accounts for various robustsimilarities but detailed differences between the resultsobtained via two standard xc schemes, SIC and LB94, inthe framework of density functional description of delo-calized valence electrons of the fullerene molecule wherethe ionic core is treated as a jellium shell. The focushas been applied to understand both the ground stateand single-photoionization properties of the system. Forthe ionization study, the ultraviolet energy range of plas-mon activities and above-plasmon soft x-ray range wereconsidered. The comparison between the results of twoprototypical spherical fullerenes, C and C , furtherunravels the scopes of validity of these two theoreticalschemes. A natural next step is to consider the influ-ence of xc formalism on the photospectroscopy of non-spherical fullerenes, which, however, is a topic for our fu-ture research. To this end, within the known limitationof the jellium description of the molecular ion-core, thegradient corrected LB94 formalism seems to bring the re-sults closer to the measurements on C over the plasmonresonance energy region. We hope that with possible fu-ture experiments with C the success of LB94 schemecan be verified for larger fullerene systems as well. ACKNOWLEDGMENTS
The work is supported by the National Science Foun-dation, USA. [1] W. Harneit, C. Boehme, S. Schaefer, K. Huebner, K.Fortiropoulos, and K. Lips, Phys. Rev. Lett. , 216601(2007).[2] C. Ju, D. Suter, and J. Du, Phys. Lett. A , 1441(2011).[3] A. Takeda, Y. Yokoyama, S. Ito, T. Miyazaki, H. Shi-motani, K. Yakigaya, T. Kakiuchi, H. Sawa, H. Takagi,K. Kitazawa, and N. Dragoe, Chem. Commun. , 912(2006).[4] J.B. Melanko, M.E. Pearce, and A.K. Salem, in Nan-otechnology in Drug Delivery , edited by M.M. de Villiers,P. Aramwit, G.S. Kwon (Springer, New York, 2009), p.105.[5] R.B. Ross, C.M. Cardona, D.M. Guldi, S.G. Sankara-narayanan, M.O. Reese, N. Kopidakis, J. Peet, B.Walker, G.C. Bazan, E.V. Keuren, B.C. Holloway, andM. Drees; Nature Materials , 208 (2009).[6] S.W.J. Scully, E.D. Emmons, M.F. Gharaibeh, R.A. Pha-neuf, A.L.D. Kilcoyne, A.S. Schlachter, S. Schippers, A.M¨uller, H.S. Chakraborty, M.E. Madjet, and J.M. Rost,Phys. Rev. Lett. , 065503 (2005).[7] C. Xia, C. Yin, and V.V. Kresin, Phys. Rev. Lett. ,156802 (2009). [8] E. Maurat, P.-A. Hervieux and F. L´epine, J. Phys. B ,105101 (2008).[9] T. Barillot, C. Cauchy, P -A. Hervieux, M. Gisselbrecht,S. E. Canton, P. Johnsson, J. Laksman, E. P. Mans-son, J. M. Dahlstr¨om, M. Magrakvelidze6, G. Dixit,M. E. Madjet, H. S. Chakraborty, E. Suraud, P. M.Dinh, P. Wopperer, K. Hansen, V. Loriot, C. Bordas,S. Sorensen and F. L´epine, Phys. Rev. A , 033413(2015).[10] A. R¨udel, R. Hentges, U. Becker, H.S. Chakraborty, M.E.Madjet, and J.M. Rost, Phys. Rev. Lett. , 125503(2002).[11] S. Korica, D. Rolles, A. Reink¨oster, B. Langer, J.Viefhaus, S. Cvejanovic, and U. Becker, Phys. Rev. A , 013203 (2005).[12] M. Magrakvelidze, D. M. Anstine, G. Dixit, M. E. Mad-jet, and H. S. Chakraborty, Phys. Rev. A , 053407(2015).[13] I.V. Hertel, H. Steger, J. de Vries, B. Weisser, C. Menzel,B. Kamke, and W. Kamke, Phys. Rev. Lett. , 784(1992).[14] M.E. Madjet, H.S. Chakraborty, J.M. Rost, and S.T.Manson, J. Phys. B , 165105 (2009). [15] Z. Chen and A.Z. Msezane, Eur. Phys. J. D , 184(2012), Phys. Rev. A , 063405 (2012).[16] M. Sch¨uler, J. Berakdar, and Y. Pavlyukh, Phys. Rev.A , 021403(R) (2015).[17] H.S. Chakraborty and M. Magrakvelidze, in FromAtomic to Mesoscale: the Role of Quantum Coherence inSystems of Various Complexities , ed. S. Malinovoskayaand I. Novikova (World Scientific, Singapore, 2015), pp.221-237.[18] S.W.J. Scully, E.D. Emmons, M.F. Gharaibeh, R.A. Pha-neuf, A.L.D. Kilcoyne, A.S. Schlachter, S. Schippers, A.Mueller, H.S. Chakraborty, M.E. Madjet, and J.M. Rost,Phys. Rev. Lett. , 179602 (2007).[19] A. Reink¨oster, S. Korica, G. Pruemper, J. Viefhaus, K.Godehausen, O. Schwarzkopf, M. Mast, and U. Becker,J. Phys. B , 2135 (2004).[20] M.E. Madjet, H.S. Chakraborty and S.T. Manson, Phys.Rev. Lett. , 243003 (2007).[21] H.S. Chakraborty, M.E. Madjet, J.M. Rost, and S.T.Manson, Phys. Rev. A , 013201 (2008).[22] M.E. Madjet, T. Renger, D.E. Hopper, M.A. McCune,H.S. Chakraborty, Jan-M Rost, and S.T. Manson, Phys.Rev. A , 053201(2012).[24] M.H. Javani, R. De, M.E. Madjet, S.T. Manson, and H.S.Chakraborty, J. Phys. B , 175102 (2014).[25] M.A. McCune, R. De, M.E. Madjet, H.S. Chakraborty,and S.T. Manson, J. Phys. B , 241002 (2011).[26] G. Dixit, H.S. Chakraborty, and M.E. Madjet, Phys.Rev. Lett. bf 111 203003 (2013).[27] M.H. Javani, J.B. Wise, R. De, M.E. Madjet, S.T. Man-son, and H.S. Chakraborty, Phys. Rev. A , 063420(2014).[28] M. Magrakvelidze, R. De, M. H. Javani, M. E. Madjet,S. T. Manson, and H. S. Chakraborty, Eur. Phys. J. D
96 (2016).[29] R. De, M. Magrakvelidze, M. E. Madjet, S. T. Manson,and H. S. Chakraborty, J. Phys. B , 11LT01 (2016). [30] O. Gunnerson and B. Lundqvist, Phys. Rev. B , 4274(1976).[31] M.E. Madjet, H.S. Chakraborty, and J.M. Rost, J. Phys.B , L345 (2001).[32] J.P. Perdew and A. Zunger, Phys. Rev. B , 5048(1981).[33] R. Van Leeuwen and E. J. Baerends, Phys. Rev. A ,2421 (1994).[34] M.J. Puska and R.M. Nieminen, Phys. Rev. A , 1181(1993).[35] M.E. Madjet and P.A. Hervieux, Eur. Phys. J. D ,1309 (1997).[37] J.H. Weaver, J.L. Martins, T. Komeda, Y. Chen, T.R.Ohno, G.H. Kroll, and N. Troullier, Phys. Rev. Lett. ,1741 (1991).[38] I. Compagnon, R. Antoine, M. Broyer, P. Dugourd, J.Lerme, and D. Rayane, Phys. Rev. A , 025201 (2001).[39] E. Cottancin, G. Celep, J. Lerm´e, M. Pellarin, J.R.Huntzinger, J.L. Vialle, and M. Broyer, Theor. Chem.Acc.
514 (2006).[40] M.A. McCune, M.E. Madjet, and H.S. Chakraborty, J.Phys. B , 201003 (2008).[41] O. Frank and J.M. Rost, Chem. Phys. Lett. , 1561(1980).[43] U. Fano, Phys. Rev. A , 1866 (1961).[44] Ph. Lambin, A.A. Lucas, and J.-P. Vigneron, Phys. Rev.B , 1794 (1992).[45] A.V. Korol and A.V. Solov’yov, Phys. Rev. Lett. ,179601 (2007).[46] G.F. Bertsch and D. Tom´anek, Phys. Rev. B , 2749(1989).[47] Z. Penzar, W. Ekardt, and A. Rubio, Phys. Rev. B ,5040 (1990).[48] J.M. Pacheco and R.A. Broglia, Phys. Rev. Lett.62