Photoemission from hybrid states of Cl@ C 60 before and after a stabilizing charge transfer
Dakota Shields, Ruma De, Mohamed El-Amine Madjet, Steven T. Manson, Himadri S. Chakraborty
aa r X i v : . [ phy s i c s . a t m - c l u s ] J u l Photoemission from hybrid states of Cl@C before and after a stabilizing chargetransfer Dakota Shields, Ruma De, ∗ Mohamed El-Amine Madjet, Steven T. Manson, and Himadri S. Chakraborty † Department of Natural Sciences, D.L. Hubbard Center for Innovation,Northwest Missouri State University, Maryville, Missouri 64468, USA Qatar Environment and Energy Research Institute,Hamad Bin Khalifa University, P.O. Box 34110, Doha, Qatar Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA (Dated: July 12, 2019)Photoionization calculations of the endofullerene molecule Cl@C with an open-shell chlorineatom are performed in the time-dependent local density approximation (TDLDA) based on a spher-ical jellium model. Cross sections for atom-fullerene hybrid photoemission studied show the effectsof the hybridization symmetry, the giant plasmon and the molecular cavity. Comparisons with theresults of Ar@C provide insights in the role of a shell-closing electron and its influence on thedynamics. The results for Cl@C are further compared with those of a more stable, lower energyconfiguration that results after a C electron transfers to Cl forming Cl − @C . This comparisonreveals noticeable differences in the ionization properties of the antibonding hybrid state while thebonding hybrid remains nearly unaltered showing a magnification covering the entire giant plasmonenergy range. I. INTRODUCTION
Significant success in the synthesis of endofullerenemolecules – systems of an atom or a smaller moleculeincarcerated within the fullerene cage [1] – has spawneda series of experiments [2–4] using merged beam tech-niques at Berkeley Advanced Light Source. Experimentsaccessed photoionization properties of these materials ingas phase. In addition, endofullerenes, being naturalentrapment of atoms, have led to a series of theoreti-cal studies of the effects of ionizing radiation on thesesystems, i.e., photoionization [5, 6]. Such fundamentalspectroscopic knowledge is particularly useful due to abroad horizon of applied importance of these materials in,namely, (i) solid state quantum computations [7, 8], (ii)improving the superconducting ability of materials [9],(iii) biomedical fields [10], (iv) contrast-enhancement re-search for magnetic resonance imaging (MRI), (v) im-proving organic photovoltaic devices [11], and even in(vi) astrophysics [12].Endofullerenes confining open-shell atoms have poten-tial applied interests of rather exotic nature [13]. For in-stance, N@C has attracted interest due to its uniquelylong spin relaxation times driven by the confinement [14].Electron paramagnetic resonance study [15] of P@C has shown enhancement in hyperfine coupling of thephosphorous’ unpaired electrons with its nucleus whichis attributed to the admixture of excited states acquiringangular momentum from the cage. In contrast, the hy-perfine interaction between the positively-charged muonand the unpaired electrons in the muonium atom in C ,relative to free muonium, is predicted to diminish from ∗ [email protected] † [email protected] the confinement [16]. These discoveries render particu-lar relevance to study open-shell atomic endofullerenes,including assessing the spectroscopy of their more stableconfigurations resulting from electron transfers to atomicvacancies.One unique phenomenon that ubiquitously occursacross endofullerene systems is the emergence of atom-fullerene ground state orbital hybridization. This en-tails the formation of symmetric (bonding) and antisym-metric (antibonding) hybrid states as the eigenstates ofthe whole system from the mixing of an atomic and afullerene orbital of identical angular momentum symme-try. A number of our previous studies has predictedsuch hybrid states in various endofullerenes and theirbroad spectrum of photoionization properties [17–21]. Itis therefore of particular value to scrutinize the sensitiv-ity of the hybridization and the photoemission dynamicsof these hybrid states via gentle changes of configurationfeatures by, for instance, comparing the effects of con-finement upon successive atoms in the periodic table.A prototype case that we examine here is the hybridlevel photoionization of Cl@C versus Ar@C ; Cl hasjust one electron less than Ar in the outer 3 p shell andan atomic number lower by one. Closed-shell Ar, beingchemically inert, almost certainly locates at the center ofthe spherical C . We treat the barely open-shell Cl alsowithin the spherical geometry so as to retain the samespherical calculation as was done for the Ar case. Wethen consider a system of Cl − @C produced by thetransfer of a C electron to fill in the Cl hole. This con-figuration attains lower energy forming closed-shell Cl − .There has been experimental evidence, based on laserdesorption mass spectroscopy, of C with a single Cl − inside [22]. While it is expected that the polarization in-teraction of the ion can induce some offset in its positionfrom the center of C , a density functional theory calcu-lation with Born-Oppenheimer molecular dynamics indi-cates that this offset is quite small within neutral C [23].Therefore, we treat Cl − @C assuming spherical geom-etry as well. We then compare the hybrid photoionizationof this new configuration with Cl@C . Ultimately, thegeneral comparison among these three endofullerene sys-tems unfolds the delicate dependence of the hybridizationand resulting photoionization cross sections on a shell-closing electron as well as on an electron transfer fromthe cage to the atom. II. A BRIEF DESCRIPTION OF THEORY
The details of the theory are described in Ref. [18].Choosing the photon polarization along the z -axis, thephotoionization dipole transition cross section in a frame-work of time-dependent local density approximation(TDLDA) is given by σ nℓ → kℓ ′ ∼ |h ψ k ℓ ′ | z + δV | φ nℓ i| . (1)Here k is the momentum of the continuum electron, z isthe one-body dipole operator, φ nl is the single electronbound wavefunction of the target level, and ψ k l ′ is therespective outgoing dipole-allowed continuum wavefunc-tion, with l ′ = l ± δV represents the complex inducedpotential that accounts for electron correlations withinthe linear response framework.We model the bound and continuum states self-consistently using the independent particle LDA method.The jellium potentials, V jel ( r ), representing 60 C ionsfor C is constructed by smearing the total positivecharge over a spherical shell with known molecular radius R = 3 . A [24] and thickness ∆. A constant pseudopo-tential v is added to the jellium for quantitative accuracy.The Kohn-Sham equations for the system of 240 electrons(four valence (2 s p ) electrons from each carbon atom), plus all electrons of the central atom with atomic number ζ , are then solved in the LDA potential V LDA ( r ) = − ζr + V jel ( r ) + Z d r ′ ρ ( r ′ ) | r − r ′ | + V XC [ ρ ( r )] , (2)to obtain the bound and continuum orbitals in Eq. (1).Eq. (2) uses the Leeuwen-Baerends (LB) exchange-correlation functional V XC [25], which provides an ac-curate asymptotic description of the ground state poten-tial. The parameters v and ∆ are determined by re-quiring both charge neutrality and obtaining the experi-mental value [26] of the first ionization threshold of C .The values of ∆ = 1 . A thus obtained closely agree withthat extracted from measurements [24]. We remark thattreating an open-shell Cl atom in the spherical model isan approximation. For instance, our calculation overesti-mates Cl ionization energy from NIST data database [27]by about 7%. But this should not take away much fromthe main message of this paper, particularly given thata more likely stable configuration Cl − @C is includedin this study which contains a closed-shell Cl − . The TDLDA-derived z + δV ( r ) in Eq. (1) is propor-tional to the induced frequency-dependent changes in theelectron density [28]. This change is δρ ( r ′ ; ω ) = Z χ ( r , r ′ ; ω ) zd r , (3)where the full susceptibility, χ , builds the dynamical cor-relation from the independent-particle LDA susceptibili-ties 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 ) (4)through the matrix equation χ = χ [1 − ( ∂V /∂ρ ) χ ] − involving the variation of the ground-state potential V with respect to the ground-state density ρ . The radialcomponents of the full Green’s functions in Eq. (4) areconstructed with the regular ( f L ) and irregular ( g L ) so-lutions 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(5)as G L ( r, r ′ ; E ) = 2 f L ( r < ; E ) h L ( r > ; E ) W [ f L , h L ] (6)where W represents the Wronskian and h L = g L + i f L .Obviously, TDLDA thus includes the dynamical many-electron correlation by improving upon the mean-fieldLDA description. III. RESULTS AND DISCUSSIONA. Cl@C versus Ar@C
1. Ground State Atom-C Hybridization
In an endofullerene system, an eigenstate of the freeatomic hamiltonian can admix with an eigenstate of theempty C hamiltonian of the same angular momentumsymmetry to produce hybrid states which are eigenstatesof the combined system. Thus, ground state LDA resultsfor Cl@C reveal hybridization between 3 p Cl and 3 p C states which jointly produce symmetrically and antisym-metrically combined states of Cl@C that can be writtenas, | Cl + C i = | φ + i = √ α | φ p Cl i + √ − α | φ p C i (7a) | Cl − C i = | φ − i = √ − α | φ p Cl i − √ α | φ p C i (7b)where the fraction α is the mixing parameter that rendersthe states orthonormal. In Fig. 1, the radial components Cl+C Ar+C ClAr
Radial coordinate (a.u.) -0.8-0.400.4 Cl - C Ar - C ClAr R a d i a l w a v e f un c ti on ( a . u . ) / B i nd i ng e n e r gy ( a . u . ) (a)(b) C ArCl C Ar+C Cl+C ClAr C Ar - C Cl - C FIG. 1. (Color online) LDA radial symmetric (a) and anti-symmetric (b) wavefunctions of Cl@C versus Ar@C . Forboth the molecules the 3 p level of the free atom hybridizeswith the 3 p level of free C ; wavefunctions of these free sys-tems are also displayed. Relevant binding energies are alsoshown to aid the discussion in the text. of these wavefunctions are shown and compared with thecorresponding hybrid wavefunctions of Ar@C . From aperturbation theory viewpoint, the strength of this mix-ing is proportional directly to the overlap of the partic-ipating (free) orbitals and inversely to the separation oftheir binding energies. As Fig. 1 indicates, the energyof 3 p Ar is extremely close to that of 3 p C , while 3 p Clmoves a bit higher, leading to a stronger mixing with avalue of α (Eq. (7)) close to about 0.5 (equal share ofatom-fullerene character) for Ar@C . A somewhat re-duced hybridization in Cl@C with a greater value of α thus implies enhanced Cl and enhanced C characters,respectively, for the symmetric and antisymmetric state.This occurs in spite of a slightly increased wavefunctionoverlap due to a small displacement of 3 p Cl wavefunc-tion toward the shell from that of 3 p Ar (Fig. 1). Alsonote that the resulting symmetric hybrids of the systemsare more separated energetically than the antisymmetrichybrids.
2. Photoionization of Hybrid Levels
Cross sections calculated at the correlated TDLDAlevel for the hybrid states of Cl@C and Ar@C are pre-
10 10010 -2 -1 Cl+C Ar+C
10 100
Photon energy (eV) -3 -2 -1 Cl - C Ar - C C r o ss s ec ti on ( a . u . ) Cl Ar (a)(b)
FIG. 2. (Color online) TDLDA photoionization cross sectionsof symmetric (a) and antisymmetric (b) levels of Cl@C com-pared to Ar@C . Cross sections for 3 p of free Cl and free Arare also plotted for comparisons. sented in Fig. 2 as a function of the photon energy. Com-paring these with 3 p of free Cl and Ar indicates plasmondriven enhancements at low energies [20, 29]. This en-hancement is significantly stronger for symmetric photoe-mission than the antisymmetric one. In the frameworkof interchannel coupling due to Fano, the correlation-modified (TDLDA) matrix element of the photoioniza-tion of X ± C , X being Cl or Ar, can be written as [20], M ± ( E ) = D ± ( E )+ X nℓ Z dE ′ h ψ nℓ ( E ′ ) | | r ± − r nℓ | | ψ ± ( E ) i E − E ′ D nℓ ( E ′ )(8)in which the single electron (LDA) matrix element is D ± ( E ) = h ks ( d ) | z | φ ± i (9)and | ψ nl i in the interchannel coupling integral is the (con-tinuum) wavefunction of the nℓ → kℓ ′ channel. Takingthe hybridization into account, the channel wavefunc-tions in Eq. (8) become | ψ + i = √ α | ψ p X i + √ − α | ψ p C i (10a) | ψ − i = √ − α | ψ p X i − √ α | ψ p C i . (10b)Substituting Eqs. (7), but for a general X, and (10) inEq. (8), and noting that the overlap between a pure Xand a pure C bound state is negligible, we separate theatomic and fullerene contributions to the integral to getthe TDLDA matrix element for X ± C levels as, M + ( E ) = √ α M p X ( E ) + √ − α M p C ( E ) (11a) M − ( E ) = √ − α M p X ( E ) − √ α M p C ( E ) , (11b)where the second terms on the right hand side are re-sponsible for the plasmonic enhancements at the lowerenergies, as seen in Fig. 2. Writing the atomic and C contributions in Eq. (11) respectively as complex quan-tities X r + iX i and C r + iC i , and recalling that the crosssection σ is proportional to the square modulus of thematrix element, we can express the cross sections as, σ + ( E ) ∼ ( √ αX r + √ − αC r ) + ( √ αX i + √ − αC i ) (12a) σ − ( E ) ∼ ( √ − αX r − √ αC r ) + ( √ − αX i − √ αC i ) . (12b)Evidently, the enhancement of the symmetric state crosssection, Eq. (12a), involving the sum of real and imagi-nary components, will be universally larger than that ofthe antisymmetric state which include their differences.Indeed, this is seen for both Cl@C and Ar@C in Fig. 2and a direct consequence of the in-phase versus out-of-phase radial oscillation of the hybrid wavefunctions atC shell (Fig. 1). However, the detailed similarities anddifferences of enhanced cross section between two systemsmust depend on their respective values of α in a rathercomplicated way that involves the interferences betweenthe atomic and the C components of the matrix ele-ments in Eq. (12). We note that while this enhancementis of a comparable size for the symmetric states of bothsystems in Fig. 2(a), it is significantly weaker for anti-symmetric Cl-C compared to Ar-C [Fig 2(b)]. Thesituation further complicates from the effect of loweringbinding energies of Cl@C hybrid levels as is discussedin subsection IIIB below.As the plasmonic effect weakens with increasing en-ergy, the cross sections largely follow their free atomcurves, as seen in Fig. 2. Cooper minimum-like struc-tures develop [30] around 50 eV on both the symmetriccurves with the minimum of Cl@C being deeper due toits higher atomic character. Note that the Cooper min-ima in 3 p cross sections for free atoms are clearly visiblein Fig. 2. In any case, such minima also show up around60 eV for antisymmetric emissions where the structurefor Cl@C is very weak because of its weaker atomiccharacter.Above these energies the cross sections oscillate asa consequence of a well-known multipath interferencemechanism [31] due to the cavity structure of C whichwas modeled earlier [32]. At such high energies the inter-channel coupling in Eq. (8) vanishes to simplify Eqs. (11)to, D + ( E ) = √ α D p X ( E ) + √ − α D p C ( E ) (13a) D − ( E ) = √ − α D p X ( E ) − √ α D p C ( E ) . (13b)The multipath interference model gives [32] D p X ∼ D atom ( k )+ A refl ( k ) h e − ikD o e − iV k − e − ikD i i (14a) D p C ∼ A shell ( k ) e − i V k (cid:2) a i e − ikR i − a o e − ikR o (cid:3) , (14b)where the photoelectron momentum k = p E − ǫ ± ) inatomic units, a i and a o are the values of φ ± at the innerand outer radii R i and R o of C , and V is the averagedepth of the shell potential. In Eq. (14a), while D atom is the contribution from the atomic region, the secondterm denotes the reflection induced oscillations in mo-mentum coordinates with amplitude A refl and frequenciesrelated to D i and D o , the inner and outer diameters ofthe shell. Since, obviously, A refl is proportional to D atom ,the larger the atomic component of a hybrid wavefunc-tion, the stronger is the reflection and the higher is thechances that the oscillations occur about the free atomresult. This is exactly what is seen for the high energycross section of Cl+C in Fig. 2(a). On the other hand,Eq. (14b) presents the portion of the overlap integralfrom the shell region, producing two collateral emissionsfrom shell edges, which oscillate in frequencies related to R i and R o . This part will dominate if a hybrid level has astronger C character, like for Cl-C , which intensifiesoscillations at higher energies but falls significantly lowerthan free 3 p Cl [Fig. 2(b)]. For the Ar@C hybrids, how-ever, due to their almost equal share of atom-C charac-ter (Fig. 1), the strength of high-energy cross sections arecomparable, somewhat below 3 p Ar, but the differences inthe details of their shapes again owe to the interferencebetween reflective and collateral emissions.
B. Cl@C versus Cl − @C A reactive Cl atom is very likely to capture an elec-tron from C which will bring the compound to a morestable configuration Cl − @C . For an empty C , ourLDA ground state structure is insensitive to the locationof the hole among the molecular levels which is not toosurprising for a cloud of 240 delocalized electrons. Like-wise, for the complex Cl − @C the LDA energies andwavefunctions of all pure fullerene states are found inde-pendent of which C orbital the hole is situated at. Infact, these energies and wavefunctions remain practicallyidentical to those of empty C , and therefore of littleinterest. However, the hybridization between 3 p Cl − and3 p C is somewhat modified from that in Cl@C , or, inother words, between after and before the electron makesa transition, as we describe below. Yet it is found thatthe 3 p hybridization in Cl − @C is still insensitive tothe C level the electron transitions from. Therefore,our results presented here are robust being free of a spe-cific choice of the hole level. On the other hand, since Cl - +C Cl+C Cl - Cl Radial coordinate (a.u.) -0.8-0.400.4 Cl - - C Cl - C Cl - Cl R a d i a l w a v e f un c ti on ( a . u . ) / B i nd i ng e n e r gy ( a . u . ) (a)(b) Cl - +C Cl+C Cl - - C Cl - C Cl - C Cl - C C FIG. 3. (Color online) Same as Fig. 1 but for Cl@C in com-parison with a more stable configuration Cl − @C of themolecule. Participating wavefunctions of the free systems Cl,Cl − and C , and relevant binding energies are included. the Cl − ion has a higher binding energy than the neutralCl, the stable configuration Cl − @C exhibits a lowertotal energy than Cl@C , and therefore should be moreabundantly formed. Also, Cl − , being a closed-shell sys-tem, will be more accurately described by our sphericalLDA model than open-shell Cl.The 3 p hybridization in Cl − @C is compared to thatin Cl@C in Fig. 3. Note that for Cl − @C the partic-ipating free levels 3 p Cl − and 3 p @C are significantlyapart from each other (Fig. 3), the former becoming quiteshallower and the latter becoming deeper compared totheir counterparts in Cl@C (Fig. 1). This strongly dis-favors the hybridization. Conversely, as also seen inFig. 3, the radial wavefunction of 3 p Cl − extends moreradially outward than 3 p Cl to increase its overlap with3 p C which favors the hybridization. In the tug-of-war between these two effects, the former wins result-ing in some modification of hybridization for Cl − @C configuration as displayed in Fig. 3. It is seen that theatomic character stays unchanged, but the C characterweakens at the shell for the symmetric hybrid, while thereverse is true for the antisymmetric hybrid. Compar-ing the energy of the hybrid levels, the symmetric hybridmoves energetically higher [Fig. 3(a)] while the antisym-metric hybrids barely separate [Fig. 3(b)] as a result ofthe charge transfer. And this modified hybridization af-fects the resulting photoionization cross sections.
10 10010 -2 -1 Cl - +C Cl+C
10 100
Photon energy (eV) -4 -3 -2 -1 Cl - - C Cl - C C r o ss s ec ti on ( a . u . ) Cl (a)(b) - Cl - Cl FIG. 4. (Color online) Same as Fig. 2 but for Cl@C versus Cl − @C configuration. TDLDA cross sections for 3 p of freeCl and free Cl − are also shown. Fig. 4 compares the TDLDA cross sections for pho-toionizations from hybrid levels of Cl − @C withCl@C . Note that even though there is a slight modifi-cation in hybridization (Fig. 3) post electron transfer, thecross section of the symmetric level [Fig. 4(a)] is hardlymodified, except that Cl − +C starts at a lower pho-ton energy due to the reduction of its binding energy.To be more precise, some reduction in the intensity ofthe structure of Cl − +C wavefunction [Fig. 3(a)] in theshell region results in little effect on the cross section. Inthe energy region of the plasmon, this can be understoodin general from the interchannel coupling contributionof the matrix element in Eq. (8). Specifically, this termin part embodies the overlaps of the “fractional” ion-ization channel emanating from the shell region of thesymmetric hybrid with all C channels which are ac-counted for in M p C in Eq. (12a). The channel overlapincludes both the overlap between the bound and thecontinuum wavefunctions. While the reduction of thestructure in Cl − +C radial wave noted above lowersthe aggregated strength of bound overlaps, the contin-uum overlaps will be slightly favored due to the followingreason. A continuum overlap is more efficient at higherphotoelectron momenta k , since differences between themomenta due to different level energies reduce enablingthe continuum waves to oscillate progressively in phasewith each other. Therefore, the Cl − +C state openingat a lower photon energy benefits continuum overlaps tocause its net increase. This gain must be compensatingthe loss due to weakening hybridization to affect practi-cally no change in the symmetric hybrid level cross sec-tion [Fig. 4(a)] after the electron transfers. Note that theterm M p X in Eq. (12a) has no effect here due to aboutthe same atomic character of hybrid wavefunctions be-fore and after the electron switches locations. For thehigher energy emission of this hybrid [Fig. 4(a)], a ratherminiscule weakening of the multipath oscillations follow-ing the electron transfer traces to the reduction of itsC character that slightly reduces the collateral emis-sion in Eq. (14), while keeping the reflective emissionunchanged. The strong atomic character of this hybrid,which remains unchanged in spite of the electron tran-sition, keeps the average strength of both cross sectionsclose to 3 p of Cl and Cl − which are practically equal atthese energies.Differences between the cross sections of asymmet-ric hybrid states of Cl − @C and Cl@C , shown inFig. 4(b), are rather strong. The differences at plasmonicenergies up to 40 eV are due to the reduction of theatomic character [Fig. 3(b)], particularly via the interfer-ence effects illustrated in Eq. (12b). Note that since thebinding energies of this hybrid state suffer a very littlechange upon the charge transfer, the continuum overlapeffect described above does not apply. However, at higherenergies, 80 eV and above, the difference in cross sectionsgrows progressively stronger. This must be due a cumu-lative effect of the reduction of the atomic component ofdirect ionization, as a result of the reduced atomic char-acter in Cl − -C , the subsequent reduction of reflec-tive amplitude in Eq. (14), and the interference betweenthem. To this end, even after the molecule evolves toa stable configuration Cl − @C , the strong plasmonicmagnification of the emission from the symmetric hybridremains, but the high energy response of the asymmetricstate substantially changes. IV. CONCLUSIONS
Using a fairly successful methodology of the time-dependent local density approximation based on a spher-ical jellium modeling of C ’s ion core, we compute thephotoemission cross sections of the atom-fullerene hy-brid levels of Cl@C . A comparison of the results withthose of Ar@C probes the modification effects of ashell-closing electron on the properties of these hybridphotoemissions. However, Cl@C must be an unstablesystem and will likely induce an electron transfer fromC to Cl to reach a more stable and lower energy con-figuration. We compared the results between these twoconfigurations of Cl endofullerenes to assess the effectsof this electron transition on the hybrid photodynam-ics. Tuning the configuration gently along the sequenceof Ar@C to Cl@C to Cl − @C , a systematic evolu-tion of the ionizing response properties of hybrid statesis uncovered. This is the first study of the photoioniza-tion properties of a halogen endofullerene to the best ofour knowledge. A strong magnification of the low en-ergy emission of the symmetric hybrid over the entiregiant plasmon resonance region and an enhanced multi-path interference effects for the high energy antisymmet-ric hybrid are found to be the most robust features in ourstudy. ACKNOWLEDGMENTS
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