Study of hydrogen confined in onion shells
EEPJ manuscript No. (will be inserted by the editor)
Study of hydrogen confined in onion shells
A. L. Frapiccini and D. M. Mitnik IFISUR, Universidad Nacional del Sur, CONICET, Departamento de Física - UNS, Av. L. N. Alem 1253, B8000CPB - BahíaBlanca, Argentina. IAFE, (UBA - CONICET), C1428EGA Buenos Aires, Argentina.
Abstract.
In this work we present a theoretical study of the photoionization for atomic hydrogen confinedin onion fullerene compared with the bare H atom and the single fullerene case. We obtained the expectedconfinement resonances for the integrated energy spectrum, finding different trends for the main peak andthe first ATI peak integrations. We perform this calculations with a recently developed methodology usingGeneralized Sturmian Functions to numerically solve the time-dependent Schrödinger equation.
In recent years, the study of the response to an electro-magnetic field of an atom encapsulated in a fullerene cagehas been the subject of several theoretical works (see forexamples [1, 2]). Examples of situations and phenomenaof direct relevance to confined atoms are helium droplets[3], nano-size bubbles formed in liquid helium [4], atomsA encapsulated in hollow cages of carbon based nano-materials, such as endohedral fullerenes A@C [5–7], thechemical reactivity and chemical valence of atoms [9], thereversible storage of ions in certain solids [10, 11], the ap-pearance of helium bubbles under high pressure in thewalls of nuclear reactors [12], and ESR, NMR and mag-netic moments of compressed atoms [13], etc. Confinedatoms are also very close in principle to the concepts un-derpinning confinement within “quantum dots” [14–16].Thus, the concept of a confined atom provides insight intovarious problems of interdisciplinary significance.A special case of the fullerene is that of the so-calledfullerene onions or buckyonions, which consists of hollowcarbon cages in which smaller buckyballs are encapsu-lated into larger ones (Cn@Cm@. . . ). [17–19]. Interest-ingly, fullerene onions have been detected in the inter-stellar medium [20, 21], and were produced by synthesisby Ugarte in 1992 [22]. Although confinement resonancesin photoionization spectra of A@C ’s have been studiedextensively, little is known about confinement resonancesin photoionization spectra of atoms confined in fullereneonions.The model-potential method was a first step to studyand qualitatively predict the confining effects of the cageon the spectral and dynamic properties of the atom. Theexternal environment imposed by the fullerene cage can,in many instances, be described quite well by a simple, lo-cal, spherically symmetric, attractive cage potential that a e-mail: [email protected] is generally taken to be of constant depth in the regionof the fullerene. Photoionization of atoms which are influ-enced by this model potentials has been treated using anumber of methodologies, such as Hartree-Fock (HF) [1,23], random-phase approximation (RPA) [1, 24, 25], time-dependent close-coupling (TDCC) [26, 27], R-matrix [28],and matrix iterative method [29].In a previous work [31], we proposed to use the Gen-eralized Sturmian Functions (GSF) [32] to numericallysolve the time-dependent Schrödinger equation of a cagedatom interacting with a laser pulse. The adaptability ofthe GSF is key in this methodology, since we can useexponential decaying (real) Sturmians to solve the time-dependent portion of the problem, and the outgoing wave(complex) Sturmians for the time-independent term. Topropagate the time-dependent wave packet during the in-teraction with the pulse, we use and explicit integratingscheme known as Arnoldi [33, 34], which is a Krylov sub-space method. The methodology presented proved to bea useful approach to solve the TDSE avoiding the use ofintegrals to obtain ionization amplitudes.Following this work, we propose to use the GSF method-ology to study the photoionization of atomic H encapsu-lated in fullerene onions. We compare the photoionizationspectra for the H@C and H@C with the onion struc-ture H@C @C , analyzing the confinement resonancesfor the main peak and the first ATI peak separately.In Sec.2 we present an outline of the methodology toextract the ionization amplitude and the spectral densityfor a one-electron atomic system. In Sec. 3 we present re-sults for the study of avoided crossings due to the changein the depth of the spherical well model potential usedto represent the effect of the fullerene. To our knowledge,no research on the avoided crossing due to two sphericalwells has been done before. We also present results for thephotoinization of the caged H, and concentrate the in theanalysis of the confinement resonances in the photoion- a r X i v : . [ phy s i c s . a t m - c l u s ] O c t A. L. Frapiccini, D. M. Mitnik: Study of hydrogen confined in onion shells ization spectra, calculating separately the contribution forthe main (L=1) peak and first ATI (L=0) peak.Atomic units are used throughout unless otherwise in-dicated.
The effect of the confinement of the hydrogen atom by afullerene shell is modeled by a spherical well V ( i ) w ( r ) = (cid:40) − U ( i )0 if r ( i ) c ≤ r ≤ r ( i ) c + ∆ ( i ) otherwise (1)where r ( i ) c is the inner radius of the well and ∆ ( i ) its thick-ness. In the case of the onion shells, the potential can bewritten as V w ( r ) = n (cid:88) i =1 V ( i ) w ( r ) (2)for an n -walled fullerene cage. For our study we will re-strict the potential to n = 2 .The bound states of the caged atom are obtained bysolving the time-independent Schrödinger equation (TISE) [ H + V w ] Ψ ν ( r ) = E ν Ψ ν ( r ) (3)with H = T − Zr (4)where T is the operator of the kinetic energy and Z is theatomic charge.To solve the TISE Eq. (3) we expand the wave func-tion in spherical coordinates, and use generalized Stur-mian Functions (GSF) [32, 35] Ψ ν ( r ) = (cid:88) jlm a νjl S jl ( r ) r Y ml ( (cid:98) r ) . (5)The generalized eigenvalue problem to solve is [ H + V w ] a νl = E ν,l Ba νl (6)for each angular momemtum l and m = 0 . All matrices arereal and symmetric, and the overlap term is also positivedefinite. The size of the matrices depend on the number N max of GSF in the expansion (5), so the dimension ofthe matrices will be N max × N max . We write the time-dependent Schrödinger equation (TDSE)for a two particle system interacting with an external fieldin the form ı ∂∂t Ψ ( r , t ) = H ( r , t ) Ψ ( r , t ) (7) where the Hamiltonian can be written as H ( r , t ) = H + V w + H int (8)with H + V w the unperturbed Hamiltonian, and H int theinteraction with the field. The interaction with the fieldof an electromagnetic pulse of finite duration is H int = (cid:40) f ( r , t ) for t ≤ t ≤ t final for t > t final (9)with r being the electronic coordinates.The interaction with the pulse, within the dipole ap-proximation, is written using the velocity gauge, and weconsider here linear polarization in the (cid:98) z axis, thus H int ( r , t ) = − ı A ( t ) · ∇ r = − ı | A ( t ) | ∂∂z (10)For a photon energy ω and a pulse of duration τ we write | A ( t ) | = A g ( ω, t ) sin( ωt ) for t ∈ [0 , τ ] . (11)To solve the TDSE Eq. (7) for t < t final we expandthe wave packet in spherical coordinates, and use GSF Ψ ( r , t ) = (cid:88) nl a nl ( t ) S nl ( r ) r Y l ( (cid:98) r ) (12)with a nl ( t ) the expansion coefficients that depend on timeand Y ml the spherical harmonics. The set of GSF used tosolve the TDSE are real and with exponential decayingbehavior at large distances. Details of the methodologyinvolved in solving Eq. (7) with the GSF can be found ina previous article [31].The differential probability for an electron having theenergy E is determined in terms of the spectral density D ( E, t ) : dP = D ( E, t ) dE, (13)where D ( E, t ) = √ E (cid:90) | C ( k ) | dΩ k (14)with Ω k denoting the solid angle under which the electronis emitted and the coefficient C ( k ) the ionization ampli-tude.To obtain the coefficients C ( k ) , we find the evolution ofthe wave packet at t > t final , which is equivalent to solvethe time-independent Schrödinger equation given by ( E − H − V w ) Ψ sc ( r ) = Ψ ( r , t final ) (15)where H + V w is the atomic (time-independent) Hamil-tonian, Ψ sc is a scattering term with outgoing boundaryconditions, and Ψ ( r , t final ) is the wave packet at the endof the pulse [36].To see how to extract the coefficients C ( k ) from Eq. (15),we write this equation by means of the Green’s function Ψ sc ( r ) = 1( E − H ) Ψ ( r (cid:48) , t final ) = G + ( r , r (cid:48) ) Ψ ( r (cid:48) , t final ) (16) . L. Frapiccini, D. M. Mitnik: Study of hydrogen confined in onion shells 3 Using the properties of the Coulomb Green’s function, wecan see that the asymptotic form of the scattering functionis Ψ sc ( r ) r →∞ −−−→ −√ πC ( k (cid:98) r ) e ı [ kr +( Z/k ) ln 2 kr ] r (17)for an electron ejected with momentum k = √ E . Thismeans that if the scattering function has the correct (out-going wave) asymptotic behavior, the ionization ampli-tude can be extracted from the function at sufficientlylarge values of the radius r . For this study, we fix the values of r (1 , c and ∆ (1 , andcalculate the energy eigenvalues in Eq.(6) for U (1 , in arange from to a.u.. We use the data provided by Xuet al [37] for a fullerene molecule C , which is r (1) c = 5 . a.u. and ∆ (1) = 1 . a.u.. The second, outer fullerene islocated in place of the C molecule, with values r (2) c =12 . a.u. and ∆ (2) = 1 . a.u. [23]. Fig. 1. (Color online) Energies of the H atom confined intwo spherical well with r (1) c = 5 . a.u., ∆ (1) = 1 . a.u., r (2) c = 12 . a.u. and ∆ (2) = 1 . a.u., as a function of U (1)0 forfour different values of U (2)0 . In Fig. 1 we show the results for the firsts s, p and d bound state energy levels. We fixed the value of the secondwell U (2)0 and plotted the energy as a function of the firstwell U (1)0 . We can see here how the energies behave asthe depth of the second well is increased. The first we cannotice is that, unlike in the case of a single fullerene cage,we have avoided crossings in the p and d levels. We alsoobserve that the s − s crossing remains unchanged untilthe depth of the second well reaches a value of U (2)0 ≈ . a.u., and then this crossing starts to ’move’ to highervalues of U (1)0 .In Fig. 2 we can see the radial probability density (cid:82) r | Ψ ν ( r ) | dΩ r for the s , s and s states in the vicinityof the avoided crossings. In the top panel in Fig. 2, for U (2)0 = 0 . a.u., there is a ’mirror collapse’ between the s − s wave functions first and then between the s − s .As the depth of the second well is increased, for U (2)0 = 0 . a.u. in the center panel in Fig. 2 , the ’mirror collapse’ oc-curs only between the s − s levels, while the s wavefunction remains unchanged. For U (2)0 = 1 . a.u. in thebottom panel in Fig. 2 we go back to the same ordering asbefore, s − s crossing and then s − s crossing. We canalso see how, for the first two plots (up to u (2)0 = 0 . ), wealways start with the s state in the Coulomb well, the s state in the second fullerene well and the s state in thefirst fullerene well. For higher values of U (2)0 , the s and s are exchanged and now then s starts in the second wellwhile the s starts in the Coulomb well.In Fig. 3 we plotted the radial probability density forthe p and p states near the avoided crossings for thesame cases as the s states. We can see here how, for lowervalues of U (1)0 , the p state is always located in the secondfullerene well, while the p is in the first fullerene well. Asthe depth of the first well increases, we observe the mirrorcollapse between these bound states. It is also observedhow the energy crossing for the p − p levels ’follows’first that of the s − s levels, and after U (2)0 = 0 . followsthat of the s − s levels.The avoided crossing phenomena is a mechanism forthe state energy reordering, manifested by the energy levelrepulsion: neighboring energy levels with the same sym-metry do not cross each other, but rather come close andrepel each other in an avoided crossing. An additional in-dicator of the external effects resides in the informationalcharacter. In information theory, entropy is a measure ofthe uncertainty associated with a random variable.In this field, the term usually refers to the Shannonentropy, which measures the expected value of the infor-mation contained in a message, usually in units such asbits, i.e., it is a measure of the average information con-tent that is missing when the value of the random variableis unknown.The Shannon information entropy of one-normalizedelectron density in the coordinate space [38] is defined as S r = (cid:90) ρ ( r ) ln [ ρ ( r )] d r (18)where the electron atomic density is defined as ρ ( r ) = | Ψ ν ( r ) | (19)This quantity is an information measure of the spa-tial delocalization of the electronic cloud. So, it gives theuncertainty of the localization of the electron. The lowerthis quantity, the more concentrated the wave function ofthe state, the smaller the uncertainty, and the higher theaccuracy in predicting the localization of the electron. A. L. Frapiccini, D. M. Mitnik: Study of hydrogen confined in onion shells
Fig. 2. (Color online) Radial s − state probability of the H atomconfined in two spherical well with r (1) c = 5 . a.u., ∆ (1) = 1 . a.u., r (2) c = 12 . a.u. and ∆ (2) = 1 . a.u. near the energycrossing for U (2)0 = 0 . (top), U (2)0 = 0 . (center) and U (2)0 =1 . (bottom). The variation of the Shannon entropy of states with anexternal potential strength may lead to gaining a deeperphysical insight into the dynamics of the system throughthe avoided crossing region [39].In Fig. 4 is shown the Shannon information entropyas a function of the first fullerene cage strength for twodifferent values of the outer fullerene cage depth. The sameenergy levels as in Fig. 1 are plotted. We can see here howthe states exchange their informational properties as theygo through an avoided crossing.
We now turn to the study of the photoionization of theconfined atoms, using the methodology described in Sec.2.2.The interaction with the pulse is as described in Eq. 11with a sine square envelope g ( ω, t ) = sin (cid:0) πτ t (cid:1) with a to- Fig. 3. (Color online) Radial p − state probability of the Hatom confined in two spherical well with r (1) c = 5 . a.u., ∆ (1) = 1 . a.u., r (2) c = 12 . a.u. and ∆ (2) = 1 . a.u. nearthe energy crossing for U (2)0 = 0 . (top), U (2)0 = 0 . (center)and U (2)0 = 1 . (bottom). tal duration of τ = 2 πn c /ω , where n c is an integer givingthe number of optical cycles.We consider photoionization from initial state s . Thevalues for r (1) c , r (2) c , ∆ (1) and ∆ (2) are the same as in theprevious section, and for the the C cage, we use theparameters of the model potential found in [37, 40]: U =8 . eV, and for the C [23]: U = 10 eV.In all the calculations presented now the peak inten-sity × W/cm and n c = 16 optical cycles. Allthe calculations shown in this section were performed us-ing N max = 300 and l max = 10 , and the size of theKrylov space was n Krylov = 30 , with a fixed time stepof δt = 0 . .As a first test we performed calculations for photonenergies of ω = 0 . and . a.u.. The endohedral atomsH@C and H@C @C were compared to the bare Hatom. . L. Frapiccini, D. M. Mitnik: Study of hydrogen confined in onion shells 5 Fig. 4. (Color online) Shannon entropy of the H atom confinedin two spherical well with r (1) c = 5 . a.u., ∆ (1) = 1 . a.u., r (2) c = 12 . a.u. and ∆ (2) = 1 . a.u. for U (2)0 = 0 . (top) and U (2)0 = 0 . (bottom). The results for the ejected electron energy spectrumare shown in Fig. 5. It is observed here how the intensityof the line is affected by the presence of the cage(s). Inthe case of ω = 0 . a.u. for example, we see that theendohedral atom H@C has a lower intensity than thatof the H atom, as well as the H@C @C . On the otherhand, for ω = 0 . , both intensities are higher than that ofthe H.This oscillations in intensity of the spectrum as a func-tion of the photon (photoelectron) energy is due to the‘confinement resonances’. As discussed by Connerade etal [41], they studied the origin and properties of confine-ment resonances considering hydrogenic ions placed at thecentre of a spherical shell as in Eq. (1) for a single well.They assume that the resonances are due to features infinal (continuous) electronic states owing to the confine-ment.In Fig.(6) we show the results for the integration of theenergy spectrum dP/dE around the main peak as a func-tion of the photon energy, for the H@C and H@C @C .We can see here how the presence of the cage(s) affectsthe intensity of the main peak. The range of photon en-ergy in Fig.(6) corresponds to electrons emitted with ener-gies lower than 3 a.u. in order to ensure that the electronwavelength is larger than the distance between the car-bon cage atoms [42]. The addition of the second well forthe H@C @C increases the number of resonances, asexpected.In Fig.(7) we see the time evolution of the L=1 waveof the radial probability density for the H atom and the Fig. 5. (Color online) Energy spectrum of the ejected electronfor photon energies of ω = 0 . (top) and . (bottom) a.u., forthe confined and bare H atoms. Fig. 6. (Color online) Integrated energy spectrum for the mainpeak as a function of the photon energy. encapsulated H@C . The top panel in Fig.(7) correspondsto a photon energy of ω = 0 . a.u. near a maximum inFig. (6) and the bottom panel to ω = 1 . a.u. near aminimum in Fig.(6). It is clear here the effect of the cagein the constructive (top panel) and destructive (bottompanel) interference in the inner region < r < r c .In Fig.(8) we see the time evolution of the L=1 waveof the radial probability density for the H atom and the A. L. Frapiccini, D. M. Mitnik: Study of hydrogen confined in onion shells
Fig. 7. (Color online) Radial probability density for the L=1wave for the bare atom H and endohedral H@C near a max-imum resonance (top) and minimum resonance (bottom), forfour different time stages during the propagation of the wavepacket. Dotted (blue) line shows the location for the C cage. encapsulated H@C @C . The top panel in Fig.(8) cor-responds to a photon energy of ω = 0 . a.u. near a max-imum in Fig. (6) and the bottom panel to ω = 0 . a.u.near a minimum in Fig.(6). Here we see the effect of thecage in the constructive (top panel) and destructive (bot-tom panel) interference in the inner region < r < r c andalso in the region between the two cages.In [23], the study of multiwalled fullerenes suggesteda trend in the confinement oscillations as the atom wasencapsulated in larger fullerenes. In Fig. (9) we see howwe can draw the same conclusions as in [23] for the in-tegrated main line (P peak), comparing the three casesof H@C @C /H with the single walled H@C /H andH@C /H. For the caged H atom, in the top panel of Fig.(9) we see how the single walled atoms oscillate around thedotted line (equal to 1), but the H@C @C actuallyseems to oscillate on the H@C line. To test this, in thebottom panel of Fig.(9) we plotted H@C @C /H@C (red broken line), and it clearly shows a behavior similarto the H@C /H, except for lower photon energies.To see if the trend observed for the integrated mainline extends to the first ATI peak (S peak), we calculatedthe same ratios as in Fig. (9) for this peak. If Fig.(10)we can see the same behavior for the ATI peak as for the Fig. 8. (Color online) Radial probability density for the L=1wave for the bare atom H and endohedral H@C @C near amaximum resonance (top) and minimum resonance (bottom),for four different time stages during the propagation of thewave packet. Dotted (blue) line shows the location for the C and C cages. main line, meaning that the ratio of H@C @C /H@C behaves as the H@C /H for high photon energies. Wesee here that the behavior of the P line is not the same asthe S line, the location and number of peaks differ. Thisis mainly due to the intermediate states involved in thetwo-photon process, which is the primary source of the Speak. We apply an ab-initio methodology to solve the time-dependent Schrödinger equation of an atom interactingwith an electromagnetic pulse of finite duration. The ap-proach is based on the Generalized Sturmian Functions[31], and their adaptability to define different asymptoticbehaviors.We present an application by studying the influenceof the confinement of the H atom in a fullerene cage C and C compared with the fullerene onion C @C .First we perform a study of the avoided crossing for theonion with a simple spherical well potential to representthe fullerenes. Since we have now two parameters to vary . L. Frapiccini, D. M. Mitnik: Study of hydrogen confined in onion shells 7 Fig. 9. (Color online)Integrated energy spectrum for the mainpeak (P peak) as a function of the photon energy. Top panelshows the ratio with respect to the bare H atom, bottom panelshows the single walled H@C and H@ with respect to H,and the double walled H@C @C with respect to H@C . Fig. 10. (Color online) (Color online) Integrated energy spec-trum for the first ATI peak (S peak) as a function of the photonenergy. Top panel shows the ratio with respect to the bare Hatom, bottom panel shows the single walled H@C and H@ with respect to H, and the double walled H@C @C withrespect to H@C . (the depth of each well), we fix the depth of in the locationof the C cage at several different values, and plot theenergy eigenvalues as a function of the depth of the wellof the C . For the cases shown, we observe crossings notonly for the s states, as was the case for a single cage,but also crossings in the p and d states. We also showcalculations for the Shannon information entropy for theelectron density near some of the crossings, which clearlydisplay the exchange of the informational properties asthey go through the avoided crossing.Finally, we calculate the ionization of the bare H andcaged atom interacting with an electromagnetic pulse inthe range of 0.6 a.u. to 3.4 a.u. of photon energy. Wepresent the results for the energy spectrum integratedaround the main peak as a function of the photon en-ergy for the H@C and H@C @C with respect of thebare H atom. We obtained the expected confinement res-onances, observing the increase in the number of oscil- lations in the onion fullerene with respect to the singlewell fullerene. The plots of the radial probability for theL=1 wave in the vicinity of a maximum of minimum inthe energy spectrum show the constructive and destruc-tive interference which give origin to the resonances. Theresults for the main peak show clearly that at high photonenergies, we can separate the effect of the inner C andouter C cages for the onion fullerene, which is not thecase for lower photon energies.We perform the same calculations for the first ATIpeak, observing the confinement resonances, but at dif-ferent locations. However, we confirm the same behaviorfor high photon energies, in which the outer fullerene andinner fullerene effects can be separated.The calculations performed in encapsulated hydrogenare a useful starting point to study the influence of theonion fullerenes in other atomic systems. The conclusionsdrawn here are applicable to other atomic systems whoseinitial state has a similar probability distribution to thatof the 1s. While our focus here is to present results for theconfined H, other multielectronic atoms could be repre-sented by means of the one-active electron approximation. Acknowledgments
We acknowledge the support by Grant No. PIP 201301/607CONICET (Argentina) also thank the support by GrantNo. PGI (24/F059) of the Universidad Nacional del Sur.
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