First prediction of inter-Coulombic decay of C60 inner vacancies through the continuum of confined atoms
Ruma De, Maia Magrakvelidze, Mohamed E Madjet, Steven T Manson, Himadri S Chakraborty
aa r X i v : . [ phy s i c s . a t m - c l u s ] D ec First prediction of inter-Coulombic decay of C inner vacancies through the continuum of confinedatoms Ruma De , Maia Magrakvelidze , ∗ , Mohamed E Madjet ,Steven T Manson , and Himadri S Chakraborty D. L. Hubbard Center for Innovation and Entrepreneurship, Department of NaturalSciences, 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 Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia,USA ∗ Current address: Department of Physics, Kansas State University, Manhattan,Kansas 66502, USAE-mail: [email protected]
Abstract.
Considering the photoionization of Ar@C and Kr@C endofullerenes,the decay of C innershell excitations through the outershell continuum of the confinedatom via the inter-Coulombic decay (ICD) pathway is detailed. Excitations to atom-C hybrid states demonstrate coherence between ICD and electron-transfer mediateddecay (ETMD). This should be the dominant above-threshold decay process for avariety of confined systems, and the strength of these resonances is such that theyshould be amenable for study by photoelectron spectroscopy.PACS numbers: 61.48.-c, 33.80.Eh, 36.40.Cg irst prediction of C -to-confined-atom backward RICD was made in 2006 [21]. Later, such broadeningof atomic Auger lines due to non-local Coulomb-mediated decay in endofullerenes werealso suggested by others [22, 23]. The first detailed calculations of RICD resonancescorresponding to Ar inner 3 s excitations in the photoionization of the levels of theencapsulating C , the atom-to-fullerene forward RICD, was recently performed byus [24]. In addition, a dominant and novel effect was found in the coherence betweenthe Auger and ICD transition amplitudes to produce resonance structures in thephotoionization of atom-fullerene hybridized states [24, 25]. These resonant hybridAuger-ICD (RHA-ICD) features, with their various shapes and widths, bear signaturesof this coherence. However, to the best of our knowledge, no RICD of a C innervacancy producing a purely atomic outer vacancy in the encaged atom, the fullerene-to-atom backward RICD, has been predicted until now. Only, as a reverse analogy, but irst prediction of C -to-confined-atom backward RICD ns subshell ofthe atoms X , Ar and Kr, in X @C , we predict autoionizing resonances from the RICDof C inner excitations. Owing to the hybridization of some of the excited states ofthe compound, the RICD amplitude is also found to admix coherently with the ETMDprocess. The results, along with our previous findings [24, 25], complete the full ICDlandscape in a photon-driven endofullene molecule, highlighting these materials, in gasor condensed phase, as possible candidates for experiments.Kohn-Sham density functional theory is used to describe the ground state electronicstructure of the compounds [27]. The C molecule is modeled by smudging sixtyC ions over a classical spherical jellium shell, fixed in space, with an experimentallyknown C mean radius 3.5 ˚Aand thickness ∆. The nucleus of the confined atom isplaced at the center of the sphere. The Kohn-Sham equations for the system of a totalof 240 + N electrons ( N = 18 for Ar, N = 36 for Kr and 240 delocalized electronsfrom C ) are then solved to obtain the electronic ground state properties in the localdensity approximation (LDA). The gradient-corrected Leeuwen and Baerends exchange-correlation (XC) functional [LB94] [28] is used for the accurate asymptotic behavior ofthe ground state radial potential V LDA ( r ) = V jel ( r ) − z atom r + Z d r ′ ρ ( r ′ ) | r − r ′ | + V XC [ ρ ( r )] , (1)which is solved self-consistently in a mean-field framework. The requirement ofcharge neutrality produced ∆ = 1.3 ˚A, in agreement with the value inferred fromexperiment [29].The time-dependent local density approximation (TDLDA) is employed to simulatethe dynamical response of C to incident photons [30]. The dipole operator, z ,corresponding to light that is linearly polarized in z -direction, induces a frequency-dependent complex change in the electron density arising from dynamical electroncorrelations. This can be written, using the independent particle (IP) susceptibility χ , as δρ ( r ; ω ) = Z χ ( r , r ′ ; ω )[ z ′ + δV ( r ′ ; ω )] d r ′ , (2)in which δV ( r ; ω ) = Z δρ ( r ′ ; ω ) | r − r ′ | d r ′ + " ∂V xc ∂ρ ρ = ρ δρ ( r ; ω ) , (3)where the first and second terms on the right hand side are, respectively, the inducedchanges of the Coulomb and the exchange-correlation potentials. Obviously, δV includesthe dynamical field produced by important electron correlations within the linearresponse regime. In this method, the photoionization cross section corresponding toa bound-to-continuum dipole transition nℓ → kℓ ′ is given by σ nℓ → kℓ ′ ∼ |h kℓ ′ | z + δV | nℓ i| , (4) irst prediction of C -to-confined-atom backward RICD Ar C Figure 1. (Color online) (a) Ground state radial wavefunctions of the Ar@C complex: these are identified as pure Ar 3 s and pure C inner 1 s , 3 d . The scaledradial potential of the system is also shown. (b) The radial wavefunction of an empty p state of the complex that can be excited by dipole transitions from C states inpanel (a). This state is hybridized between Ar 4 p and C p ; the wavefunction of thelater is also shown. The coherence of ICD and ETMD amplitudes in the emission ofAr 3 s @ photoelectrons is schematically illustrated. where the TDLDA matrix element M = D + h δV i , with D being the independent-particle LDA matrix element.It is well-known that in X @C the atomic valence np electrons strongly hybridizewith the energetically shallower p electrons of the host C [24, 25, 31]. The subvalent ns levels of X , however, maintain their purity, as seen in Figure 1(a) which showspredominantly atom-like Ar 3 s @ radial wavefunction from the ground LDA spectrumof Ar@C . Single-electron excitations from a number of C inner levels n ′ ℓ (whoseionization thresholds are indicated in Figure 2) occur at energies higher than the Ar3 s @ and Kr 4 s @ thresholds, 30.1 eV and 26.5 eV respectively. Of these C levels,@3 d and @1 s wavefunctions are presented in Fig. 1(a). (Although these levels are thequantum states of the whole compound, we use nℓ @ and @ nℓ respectively to ascertaintheir atom- or C -dominant character.) Using the well-known approach by Fano [32]to describe the dynamical correlation through the interchannel coupling, the RICDamplitudes of these C photo-vacancies via X ns @ ionization can be expressed by M d-c that denotes the coupling of C discrete (d) excitation channels @ n ′ l → ηλ with the irst prediction of C -to-confined-atom backward RICD -2 -1 Ar 3sAr 3s@C
30 35Photon energy (eV)10 -2 -1 Kr 4sKr 4s@ C r o ss s ec ti on ( M b ) (a)(b) Figure 2. (Color online) Photoionization cross sections of 4 s subshell for free andconfined Kr (a) and 3 s subshell of free and confined Ar (b). The total cross section ofempty C is also presented. Various C ionization thresholds of respective complexesare shown. ns @ → kp continuum (c) channel of X . Following [24], M d-c can be written as: M d-c ns @ → kp ( E ) = X @ n ′ ℓ X ηλ h ψ @ n ′ ℓ → ηλ | | r − r | | ψ ns @ → kp ( E ) i E − E @ n ′ ℓ → ηλ D @ n ′ ℓ → ηλ , (5)where E @ n ′ ℓ → ηλ and D @ n ′ ℓ → ηλ are, respectively, excitation energies and LDA matrixelements of channels @ n ′ ℓ → ηλ and E is the photon energy corresponding to the ns @ → kp transition. In Eq. 5 the ψ are independent-particle (LDA) wavefunctionsthat represent the final states (channels) for transitions to excited/continuum states.Obviously, the Coulomb matrix element in the numerator of Eq. 5 acts as the “corridor”for energy transfer from the C de-excitation across to the atomic ionization process,producing ICD resonances in the @ ns cross sections.These C -to- X ICD resonances are displayed in Figure 2 both for 4 s [Fig. 2(a)] and3 s [Fig. 2(b)] photoionization of, respectively, confined Kr and Ar. As seen, the spectraare rather dramatically structured. Note that the corresponding results for free atomsare flat, since the current energy range does not include any regular autoionizing (Auger)decay of atomic innershell vacancies. The resonances in Figs. 2 are strong, of variedshapes, and should be easily accessible via photoelectron spectroscopy. Furthermore,the narrow width of these resonances, which is very different than characteristic atomicAuger resonances that are generally broad, is directly related to the C excitations.Indeed, like generic cluster wavefunctions, C wavefunctions are typically delocalized. irst prediction of C -to-confined-atom backward RICD /r [Eq. 5], spread-out(delocalized) wavefunctions translate to a decrease in the value of the matrix element,as compared to atomic compact (localized) wavefunctions.Fig. 2 exhibits three particularly notable features: (i) The above-threshold vacancydecay is completely dominated by ICD. (ii) For both Kr 4 s @ and Ar 3 s @, thecharacteristic Cooper minima are moved lower in energy to 35 eV and 36 eV fromtheir well known positions of 41 eV and 42 eV [33], respectively, for free atoms. Thisshift is a consequence of the atom-C dynamical coupling, particularly the coupling of Xns @ → kp ionization with a host of C continuum channels. Note that this couplingwas not included in Eq. 5, which only captures the resonances, but is certainly presentin the full dipole matrix element M (see Eq. 4). (iii) A comparison with the emptyC cross section in Fig. 2(b) reveals a few extra resonances in both the Kr 4 s @ and theAr @3 s results. These are present owing to the additional excited states in the excitedspectrum of the whole compound, since it now also includes the excited states of thecaged atom. In fact, many of these excited states of the compound must be hybridsbetween the atomic and C pure states – a fact addressed below suggests that some ofthese resonances are the result of the coherent mixing of ICD and ETMD amplitudes.As an example, consider a hybridized dipole-allowed excited state 4 p + from C @3 d [Fig. 1(a)] in Ar@C . The radial wavefunction of 4 p +, shown in Fig. 1(b), resultsfrom a symmetric hybridization of free Ar and empty C p excited states as | p + i = √ α | Ar p i + √ − α | C p i . (6)Based on Eq. 6, the hybridizations in ψ and D for this transition are then | ψ @3 d → p + i = √ α | ψ @3 d → Ar p i + √ − α | ψ @3 d → C p i , (7) D @3 d → p + = √ α D @3 d → Ar p + √ − α D @3 d → C p . (8)Using Eqs. 7 and 8 in Eq. 5 for the transition @3 d → p +, and assuming that the overlapbetween Ar 4 p and C p states is negligibly small, we can break up M d-c as M d − c s @ → kp ( E ) = α h ψ @3 d → Ar p | | r − r | | ψ s @ → kp ( E ) i E − E @3 d → p + D @3 d → Ar p + (1 − α ) h ψ @3 d → C p | | r − r | | ψ s @ → kp ( E ) i E − E @3 d → p + D @3 d → C p . (9)The processes that Eq. 9 embodies are schematically shown in Fig. 1. The first termdenotes the release of energy from a transfer de-excitation (blue arrow) of the atomicpart of the 4 p + hybridized electron state, and the subsequent migration (purple curvedarrow) of that energy to Ar knocking out a 3 s electron (red arrow). This process isessentially an ETMD. The second term is the direct de-excitation (black arrow) of theC part of 4 p + followed by the regular ICD. Obviously, ETMD and ICD coherently mixto produce the ensuing resonance structure. Thus, some resonances in Fig. 2 occur fromdecay rates underpinning this coherence; a detailed characterization of the structuresbased on Fano-fitting is forthcoming. irst prediction of C -to-confined-atom backward RICD participant RICDs, where the precursorhole is filled by the excited electron itself, it is of great interest to access the influenceof spectator processes; these could significantly affect the situation and certainly needstudy. Furthermore, with contemporary focus [34] on photoemission phase and timedelay studies by interferometric metrology [35], we hope that the current results willstimulate similar temporal spectroscopy with ICD resonances.Finally, it is important to note that, based upon our explanation of the details ofmulticenter decay, this should be a strong process for any atom or molecule encagedin any fullerene, in any position, central or not. The ICD-ETMD coherence involveshybridization in the final continuum state which should be quite general – all it requiresis that both the fullerene and the trapped atom or molecule have dipole-allowed finalstates, continuum and quasi-discrete, of the same symmetry.
Acknowledgment
The work is supported by NSF and DOE, Basic Energy Sciences, Office of ChemicalSciences.
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