The h_u hole spectral intensity for C60 -> C60+ molecular photoemission is
calculated at finite temperature by a parameter-free Lanczos diagonalization of
the electron-vibration Hamiltonian, including the full 8 H_g, 6 G_g, and 2 A_g
mode couplings. The computed spectrum at 800 K is in striking agreement with
gas-phase data. The energy separation of the first main shoulder from the main
photoemission peak, 230 meV in C60, is shown to measure directly and rather
generally the strength of the final-state Jahn-Teller coupling.Comment: 5 pages, 3 figure

Erio Tosatti

I. B. Bersuker

M. Lüders

N. Manini

Nicola Manini

P. H. Butler

Paolo Gattari

R. Lindner

English

arXiv

Crossref

Jahn-Teller Spectral Fingerprint in Molecular Photoemission:
                    
                      
                        C

The h(u) hole spectral intensity for C-60-->C-60(+) molecular photoemission is calculated at finite temperature by a parameter-free Lanczos diagonalization of the electron-vibration Hamiltonian, including the full 8 H-g, 6 G(g), and 2 A(g) mode couplings. The computed spectrum at 800 K is in striking agreement with gas-phase data. The energy separation of the first main shoulder from the main photoemission peak, 230 meV in C-60, is shown to measure directly and rather generally the strength of the final-state Jahn-Teller coupling

P. Gattari

E. Tosatti

AIR Universita degli studi di Milano

Jahn-Teller spectral fingerprint in molecular photoemission : C-60

he h(u) hole spectral intensity for C-60--&gt;C-60(+) molecular photoemission is calculated at finite temperature by a parameter-free Lanczos diagonalization of the electron-vibration Hamiltonian, including the full 8 H-g, 6 G(g), and 2 A(g) mode couplings. The computed spectrum at 800 K is in striking agreement with gas-phase data. The energy separation of the first main shoulder from the main photoemission peak, 230 meV in C-60, is shown to measure directly and rather generally the strength of the final-state Jahn-Teller coupling

MANINI N.

GATTARI P.

Tosatti, Erio

Sissa Digital Library

P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19Jahn-Teller Spectral Fingerprint in Molecular Photoemission: C60Nicola Manini* and Paolo GattariDipartimento di Fisica, Università di Milano, Via Celoria 16, 20133 Milano, Italyand INFM, Unità di Milano, Milano, ItalyErio TosattiInternational School for Advanced Studies (SISSA) and INFM Democritos National Simulation Center,Via Beirut 4, 34014 Trieste, Italyand International Centre for Theoretical Physics (ICTP), P.O. Box 586, 34014 Trieste, Italy(Received 28 July 2003; published 3 November 2003)196402-1The hu hole spectral intensity for C60 ! C60 molecular photoemission is calculated at finitetemperature by a parameter-free Lanczos diagonalization of the electron-vibration Hamiltonian,including the full 8 Hg, 6 Gg, and 2 Ag mode couplings. The computed spectrum at 800 K is instriking agreement with gas-phase data. The energy separation of the first main shoulder from the mainphotoemission peak, 230 meV in C60, is shown to measure directly and rather generally the strength ofthe final-state Jahn-Teller coupling.DOI: 10.1103/PhysRevLett.91.196402 PACS numbers: 71.20.Tx, 33.20.Wr, 33.60.–q, 36.40.Cg0 400 800Binding energy [meV]Intensity [arb. units]Bruhwiler et al.Canton et al.theory - 800 K0 400Hg(7) & Hg(8)Hg(1)FIG. 1 (color online). Experimental PE spectra by Cantonet al. [4] and by Brühwiler et al. [3] compared with thecalculation based on the ab initio parameters of Ref. [5] (T 800 K, Ntier  31, Nmax  105, Nsample  245). Intensities arenormalized to unity. Inset: T  800 K spectra including thesingle H1g mode, and H7g and H8g . All spectra are shifted tomove the main peak to zero energy and broadened with a10 meV HWHM Gaussian. The error bar gives the largeststatistical error introduced by random sampling of the initialently general, and since the upward shift tracks the cou- states.Photoemission (PE) from a closed-shell high-symmetry molecule, involving Jahn-Teller (JT) effectin the final state, is accompanied by characteristic vi-bronic structures in the measured hole spectrum [1].Although accurate spectral calculations have recentlyappeared in the chemical literature, e.g., for benzene[2], there still is a strong need for a qualitative under-standing of a more general nature, and informative on thenature and the strength of the JT-coupled problem. Arecent case in point is that of gas-phase C60, where animportant side peak was reported 230 meV above themain PE peak [3,4] (see Fig. 1). As this excitation energyis in fact larger than all vibrational frequencies of C60(32–197 meV), its nature could not be straightforwardlyinterpreted, leading to a debate [6].In this Letter we present first of all a detailed under-standing of the PE structures that are accurately repro-duced for C60 ! C60 (Fig. 1) through calculations thatfully include temperature and that contain no adjustableparameters. By taking further this calculation to pieces,we identify the important ingredients that determine thespectrum. The first is symmetry selection: the initialorbital symmetry dictated by the suddenly injected barehole extends to the whole spectrum. The second is aproperty of weakly coupled JT vibronic multiplets thatrequires the first excitation energy of the same symmetryas the ground state (GS) to rise above the bare vibrationenergy by an amount strictly proportional to the squaredJT coupling. Specifically in C60, symmetry requires the230 meV peak to correspond to an hu vibronic excitationabove the hu GS. JT transforms the highest H8g vibration( h! < 200 meV) of C60 into an hu vibron near 230 meV.Interestingly, the 20% magnitude of this upward shiftfingerprints quite accurately the medium-sized overallJT coupling of C60. As both properties above are appar-0031-9007=03=91(19)=196402(4)$20.00 pling intensity well beyond the weak-coupling limit, thiskind of analysis appears of much more general value. Inparticular, the first vibronic excitation energy as seen inPE of other weak to medium JT-coupled moleculesshould and does gauge their respective couplings as well.We adopt for C60 the simplest model [7–9] describingthe JT coupling of the orbitally degenerate hu hole with2003 The American Physical Society 196402-1P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19the molecular vibrations. The linear JT coupling is non-zero for two nondegenerate Ag, six fourfold-degenerateGg and eight fivefold-degenerate Hg vibration modes, or66 vibrations altogether [7,8,10]. The resulting h  AGH JT Hamiltonian thus consists of the fivefold huhighest occupied molecular orbital (HOMO), of 66 har-monic oscillators, and of the electron-vibration (e-v) cou-pling term [5,8]:Ĥ  Ĥ0  Ĥvib  Ĥe-v; (1)Ĥ 0  HOMOXmĉymĉm; (2)Ĥ vib 12Xjh!jP̂2j  Q̂2j; (3)Ĥ e-v Xr jmm0kgrj h!jCrm 	m0Q̂iĉymĉm0: (4)Here m,  label components within the degenerate mul-tiplets [8,11], j counts modes of symmetry , Crmm0 areClebsch-Gordan coefficients [11] of the icosahedral groupIh that couple the hu fermion operators ĉym to a vibration, Q̂i are the dimensionless normal-mode vi-bration coordinates in units of x0!j h=!j mCqwhere mC is the mass of the C atom, and P̂i thecorresponding conjugate momenta. The additional multi-plicity r  1; 2, needed for Hg vibrations only, labels thetwo separate kinds of Hg coupling allowed under thesame symmetry [5,11].In the present calculation we adopt the fully ab initionumerical values of the e-v coupling parameters grjobtained by a density functional (DF) calculation [5,12].We also use the calculated vibrational frequencies ofRef. [5], that are close to the experimental frequenciesof the Ag and Hg modes, but also cover the experimen-tally unavailable or unreliable Gg modes. e-v couplingsare dimensionless, each of them normalized to its respec-tive vibration quantum, so as to reflect directly the rela-tive coupling strength. Numerical factors kAg  51=22 ,kGg  51=24 , kHg  12 are included to make contact withRef. [5].In the sudden approximation, the angle-integrated PEspectrum IE is given by Fermi’s golden rule [2]IE 110XmiIimEPi; (5)IimE Xfjhfjĉymjiij2E	 Ef  Ei; (6)where jii represents the starting vibrational eigenstate ofneutral C60, of energy Ei, and jfi represents a vibronic JTeigenstate of C60, of energy Ef. ĉymjii is the initial barehole, where the spin- electron has been ejected from oneorbital m of the fivefold-degenerate hu HOMO, but themolecule is still unrelaxed. The factor 1/10 and the sum196402-2over m and  signify averaging over the five orbital andtwo spin states of the hole. Assuming thermal equilib-rium we generate the boson numbers fvjg of the initialneutral molecule by randomly sampling the probabilitydistribution Pi  Z	1 exp	Ei=kBT; where Z Pi exp	Ei=kBT and Ei Ph!jvj. In the presentmedium-coupling regime we compute the final-state vi-bronic energies and matrix elements in (6) by numericalLanczos diagonalization of the Hamiltonian (1) on theproduct basis of the five hu states times the harmonicoscillator ladders. Since these contain an infinite numberof states, some truncation is necessary. As we generallywant to address high temperatures of the molecular beam(T  800 K used for C60, kBT * 2 h! of the stronglycoupled H1g mode) a standard truncation would involvea far too large Hilbert space size. We generate a smarterbasis starting from the initial excitation ĉymjii, and iter-atively adding sets of states (‘‘tiers’’) directly coupled tothose of the previous tier by matrix elements of Ĥe-v, in ascheme inspired by Ref. [14]. This procedure is iteratedNtier times so that states relevant at up to 2Ntierth order inperturbation theory are included [15]. The application ofabout 350 Lanczos steps [16] generates a tridiagonalmatrix, which provides a well converged spectrum.The basis for the calculation of Fig. 1 includes about720 000 states, in particular, all the states generated by upto three applications of Ĥe-v to ĉymjii: the convergence ofthe truncated basis is quite good. We repeat the wholeprocedure of basis generation/spectrum calculation foreach of the Nsample initial states, and average the obtainedspectra. The above procedure is carried out including allHg andGg modes. The Ag modes separate out, as they canbe solved analytically as a simple displaced oscillators,and are included exactly by convolution.Figure 1 compares the theoretical PE spectrum calcu-lated at T  800 K with the measured spectra by twoseparate groups. All experimental features, includingthe characteristically broad and asymmetric main PEpeak, the strongest satellite peak around 230 meV, asecond weaker peak near 400 meV, and a slow decay upto 600 meV, are remarkably reproduced, based purely onab initio parameters. The model and its ingredients thusappear to describe quite accurately the PE spectrum ofC60. We can now take it apart, so as to understand theunderlying physics to satisfaction.We address the two main features of the spectrum,namely, (i) the large broadening and (ii) the 230 meVpeak. By eliminating Gg and Ag modes from the calcu-lation there is no large change in the spectrum. Thus onlythe Hg modes are important. Among Hg modes, thelargest couplings belong to the low-frequency H1g32 meV quadrupolar radial mode, and to the twohigh-frequency C-C stretch modes H7g and H8g at h! 181 and 197 meV. When we include H1g alone we find(inset of Fig. 1) that the thermally broadened asymmetricmain peak is well reproduced. Therefore H1g is mainlyresponsible for the large spectral broadening.196402-2FIG. 2 (color online). Vibronic excitation energies for a singleHg mode coupled to the hu level, as a function of coupling, for aratio g2=g1 characteristic of the H7=8g modes. The averagerefers to the multiplet originating from the one-phonon states.Bold: hu states, the only visible states in T  0 PE.P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19When only H1g is included, all satellite structuresabove 200 meV are absent, and do not reappear even ifwe arbitrarily increase the H1g coupling to larger andlarger values. This rules out the possibility to attribute the230 meV structure to an ‘‘electronic’’ splitting [6]. As itturns out, electronic JT splittings become visible in ex-perimental PE spectra, such as those of FeCO5 [17], andin theoretical spectra, like those of Martinelli et al. [18]only for very strong e-v coupling, whereas in C60 couplingis intermediate at most. This suggests that the 230 meVpeak in C60 should rather be associated to the high-frequency modes H7g and H8g . However, the connectionbetween their frequencies <200 meV and the 230 meVpeak position is far from obvious. To clarify this, wecalculate the spectrum by including H7g and H8g alone(see inset of Fig. 1). The result is a much narrower spec-trum (H1g is now omitted), showing a high-energy sat-ellite at a significantly (10%) higher energy than h!Hg8.This increase in vibronic excitation energy above thepurely vibrational h! for the v  1 phonon excitationof a trivial displaced oscillator is in fact a characteristicsignature of the dynamic JTeffect, and can be understoodon the basis of symmetry. For H7g and H8g , h! is almost3 times the thermal energy: the main initial-state contri-bution to the spectrum comes from jii  the neutralmolecule GS, of symmetry Ag. The bare hole ĉymjiitherefore carries the orbital hu symmetry of the HOMO.This state has nonzero overlap strictly only with finalstates jfi of the same symmetry hu. This single symmetrychannel thus represents the only contributor to the low-temperature PE spectrum according to (6) [19]. The blue-shift of the vibronic peak relative to the bare vibration isnow due to two general properties (in fact common to alldynamical JT systems): (i) the coupling independence ofthe average excitation energy in any vibronic multiplet atweak coupling (the leading change is of order g4) [20],and (ii) the increasing ‘‘repulsion’’ that the GS level exertsfor increasing g on all excited states of the same symme-try, which pushes them upward.Both concepts are illustrated in Fig. 2, depicting thevibronic excitation energies (with special attention to thehu states) for the simplest case of a single Hg mode, as afunction of the e-v coupling strength g2. The approximatecoupling independence of the average excitation energyof the multiplet of states derived from the v  1phonon manifold is apparent. Within this multiplet(made of hu Hg  au  t1u  t2u  2gu  2hu), thehu vibronic states are the only ones that have anotherstate of the same symmetry (the GS itself) lower inenergy ‘‘pushing’’ them upward. As a result, for smallbut increasing coupling, the hu excited vibronic statesmove necessarily upward in energy above h!Hg .Moreover, the amount of shift is at weak couplingproportional to g2 and reflects the strength of the cou-pling. None of these results seem specific to the h H JTmodel of C60. For example, identical conclusions apply tothe e  E problem [20]. Indeed, a 15% blueshift of the196402-3first vibronic satellite is clearly observable in the PEspectra of benzene [21].In C60 we must consider the two high-frequency modesH7g andH8g at h!  181 and 197 meV. The couplings forthese two modes alone account for a 10%, or roughly20 meV, shift of the main H7;8g vibronic satellite aboveh!. This is about half of the total observed and calculatedshift of 40 meV of the 230 meV satellite relative to h!.The remainder is a collective effect of all other modessimultaneously interacting with the HOMO, and pushingtheir respective v  1 satellites (weak and thus invisibleat large temperature) upward. All goes as if effectivelythe coupling of the H7=8g modes were larger, i.e.,effectively moving to the right of the vertical dashedline in Fig. 2.We have thus attained the following understanding ofthe 800 K PE spectrum of C60: (i) the strong 230 meVsatellite (and the weak 400 meV) is due to hu-symmetryvibrons derived from H7=8g modes, blueshifted by repul-sive coupling to the hu GS, and (ii) the large spectralbroadening is due mainly toH1g . Clearly, this broadeningof the 800 K spectrum prevents direct extraction offurther detailed information about the actual e-v cou-plings of the fullerene cation. Our calculations, whenrepeated for lower temperatures, indicate a wealth ofstructures that should emerge. Figure 3 shows that alreadya measurement carried out at 300 K would provideenough detail to get precise indications about the cou-plings of individual modes. For example, the intensityratio between the main peak and the subsequent peaks at35 and 65 meV is a direct measure of gHg1. A coolmolecular beam experiment would be extremely useful.196402-30 200 400Binding energy [meV]Log(Intensity) [arb. units]0-00 K800 KHg(1)1Hg(1)2Hg(3)1 & Hg(4)1 & Hg(1)3Hg(2)1 Hg(7)1 & Hg(8)1300 KHg(1)-1{×10×100Hg(7/8)2{0 500 1000IntensityexperimDF / 2½DFDF×2½FIG. 3 (color online). Temperature dependence of the PE,computed including all modes of C60 for T  0, T  300 K,and T  800 K. Labels jv indicate the leading component inthe vibronic states. Inset: computed spectra (T  800 K) basedon the same DF coupling parameters grj divided by2p,original, and multiplied by2p, compared to experiment [3].P H Y S I C A L R E V I E W L E T T E R S week ending7 NOVEMBER 2003VOLUME 91, NUMBER 19Also in view of future cool-beam data it is meaningful toaddress the question of how accurate the calculated DFe-v couplings used here really are. For t1u electronscoupled to Hg vibrations, the calculated DF couplingswere argued, for example, to be too weak by a substantialamount, by comparison to the larger couplings needed inorder to fit C	60 PE [22]. In our HOMO case, the excellentagreement of calculations with experiment suggests thatthere is no such failure. Figure 3 (inset) shows several PEcalculations with rigidly rescaled e-v parameters.Comparison with experiment indicates that the DF cou-plings are, on the whole, quite close (if slightly on theweak side) to the actual couplings of C60. Much larger JTcoupling values—like those obtained using different ap-proaches [23]—are ruled out, as they would produceexcessive blueshift of the H7=8g satellite. This is verygood news, for it gives further confidence in the futurepossibility to rely solidly on DF for a good description ofthe intrafullerene chemical bond, and of its perturbations.In conclusion, we present here a detailed calculation ofthe PE spectrum of a high-symmetry molecule, C60,which is JT active in its one-hole state. We find excellentagreement with experiment, confirming for the first timethe accuracy of the e-v couplings as extracted from DFcalculations. Perhaps more importantly, we reach a freshunderstanding of the PE spectrum, with its width,its vibronic peak, and the reason for its blueshiftrelative to the bare vibrations. The blueshift of the one-196402-4vibron satellite is argued to represent a more generalcharacteristic fingerprint of JT systems in the weak-to-intermediate coupling regime, and is found, for example,in the e  E JT of benzene as well.We are indebted to G. P. Brivio, P. Brühwiler, A. DelMonte, D. Galli, M. Lueders, L. Molinari, G. Onida,F. Parmigiani, A. Parola, and G. E. Santoro for usefuldiscussion. This work was supported by the EuropeanUnion, Contracts No. ERBFMRXCT970155 (TMRFULPROP) as well as by MIUR COFIN01, and FIRBRBAU017S8R operated by INFM.*Electronic address: nicola.manini@mi.infm.it[1] R. Lindner et al., Science 271, 1698 (1996).[2] H. Köppel et al., J. Chem. Phys. 117, 2657 (2002).[3] P. Brühwiler et al., Chem. Phys. Lett. 279, 85 (1997).[4] S. E. Canton et al., Phys. Rev. Lett. 89, 045502 (2002).[5] N. Manini et al., Philos. Mag. B 81, 793 (2001).[6] N. Manini and E. Tosatti, Phys. Rev. Lett. 90, 249601(2003); S. E. Canton et al., ibid. 90, 249602 (2003).[7] A. Ceulemans and P.W. Fowler, J. Chem. Phys. 93, 1221(1990).[8] N. Manini and P. De Los Rios, Phys. Rev. B 62, 29(2000).[9] C. P. Moate et al., Phys. Rev. Lett. 77, 4362 (1996).[10] M. Lüders et al., Philos. Mag. B 82, 1611 (2002).[11] P. H. Butler, Point Group Symmetry Applications(Plenum, New York, 1981).[12] Another DF calculation [13] based on a different func-tional yielded couplings roughly in the same range.However, we could not recompute the PE spectrumwith these couplings due to their lack of distinctionbetween coupling contributions of type r  1 and r  2.[13] M. Saito, Phys. Rev. B 65, 220508 (2002).[14] A. A. Stuchebrukhov and R. A. Marcus, J. Chem. Phys.98, 6044 (1993).[15] The tier size increases quickly, but we cut it offby including up to some given number Nmax ofstates per tier, according to their perturbative weightwb PafhbjĤe-vjaiEa 	 Eb	1wa where fx 1 x	2	1=2, the sum extends over all the states jaiin the previous tier, and the weight of ĉymjii is unity.[16] J. Jaklic and P. Prelovsek, Adv. Phys. 49, 1 (2000).[17] J. L. Hubbard and D. L. Lichtenberger, J. Chem. Phys. 75,2560 (1981).[18] L. Martinelli et al., Phys. Rev. B 43, 8395 (1991).[19] This, incidentally, rules out the possibility to observetunnel split states [4] in PE, as they necessarily possessdifferent symmetries.[20] I. B. Bersuker and V. Z. Polinger, Vibronic Interactions inMolecules and Crystals (Springer-Verlag, Berlin, 1989),Sec. 4.1.[21] P. Baltzer et al., Chem. Phys. 224, 95 (1997).[22] O. Gunnarsson et al., Phys. Rev. Lett. 74, 1875 (1995);O. Gunnarsson, Phys. Rev. B 51, 3493 (1995).[23] R. D. Bendale et al., Chem. Phys. Lett. 194, 467 (1992).196402-4

Jahn-Teller spectral fingerprint in molecular photoemission: C60

Jahn-Teller Spectral Fingerprint in Molecular Photoemission: C60

Jahn-Teller Spectral Fingerprint in Molecular Photoemission: C60

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