1. Photoionization of endohedral atoms in fullerene cages C. Y. Lin and Y. K. Ho Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei, Taiwan

2. Outline Introduction Theoretical methods –Finite-element discrete variable representation –The method of complex-coordinate rotation Results –Energy spectrum –Photoionization cross sections Summary

3. The geodesic dome designed by visionary architect R. Buckminster Fuller in 1967 for the world’s fair in Montreal, Canada. After its inspiration, the first fullerene, also named as buckminsterfullerene or buckyball, was discovered in 1985 at Rice University in Huston by Curl, Kroto and Smalley, who shared a Nobel Prize in Chemistry 1996. In 1990 physicists Krätschmer and Huffman for the first time produced isolable quantities of C 60 by electric arc discharge between graphite electrodes immersed in a noble gas.

4. Buckminsterfullerene molecules are composed of 60 carbon atoms arranged in interlocking hexagons and pentagons. 12 x 20 x Radius: 3.54 Ǻ [1] C-C 5-6 bond lengths: 1.46 Ǻ [2] C-C 6-6 bond lengths: 1.39 Ǻ [2] Thickness of the shell: ~1 Ǻ [1] Ionization potential: 7.54 eV [3] Electron affinity: 2.65 eV [4] [1] Xu et al, Phys. Rev. Lett. 76, 3538 (1996). [2] Darzynkiewicz and Scuseria, J Phys. Chem. A 101, 7141 (1997). [3] Hertel et al, Phys. Rev. Lett. 68, 784 (1992). [4] Wang et al, Chem. Phys. Lett. 182, 5 (1991).

5. Endohedral fullerenes are fullerenes with atoms/ions trapped inside their cages, which are given the appealing designation A@C 60, where A is the trapped atoms/ions and C 60 could be any fullerene. Endohedral fullerene production: –Producing the fullerenes in the presence of the element to be encapsulated. –Window mechanism: breaking a C-C bond to open a window for atoms going in and coming out. –Collisional capture: exposing C 60 to an intense beam of atoms at an energy chosen so that the atoms can penetrate the carbon cage but cannot destroy it. Applications: –The fundamental units in designing a quantum computer. –The agents for improved superconductivity. –The contrast agent for magnetic resonance imaging (MRI). –Drug delivery, such as the delivery of radioisotopes to cancer cells.

6. Photoionization of endohedral atoms – In a sense, the inner atom A in A@C 60 serves as a “lamp” that shines “light” in the form of photoelectron waves that “illuminates” the fullerene C 60 from the inside. Methodology Jellium model Δ-potentialDirac δ-potential Gaussian-type potential The jellium potential representing 60 C 4+ ions is constructed as a uniform charge density over a spherical shell with the inner radius R and the thickness Δ. All valence electrons, four (2s 2 2p 2 ) from each carbon atom in the fullerene, form the delocalized charge cloud. The thickness Δ is determined by requiring the charge neutrality with a given number of valence electrons. The C 60 shell is modeled at various levels of approximations Infinitely thin spherical shell with radius R. The modulation factor S(ω) is a strongly oscillating function of the photon energy. Square well potential with the inner radius R and the thickness Δ. For a given Δ, the determination of U 0 is associated with the electron affinity of C 60. Interpret the observed oscillations in the photoionization cross sections of the C 60 cage. A smooth form for modeling the fullerene cage. A more realistic description of shell boundaries without introducing numerical instability. r c is the radius from the cage center to the shell center. w characterizes the half- width of the spherical Gaussian shell. Amusia et al, Phys. Rev. A 62, 012701 (2000). Puska and Nieminen, Phys. Rev. A 47, 1181 (1993). Xu et al, Phys. Rev. Lett 76, 3538 (1996). Nascimento et al, J. Phys. B 44, 015003 (2011). Atom (Coulomb) + Cage (Gaussian)

7. Finite-element discrete variable representation (FEDVR) is a hybrid computation scheme combing the finite- element approach and the discrete variable representation. This method has been implemented to investigate a variety of interesting physical problems: –Quantum-mechanical scattering problems [Rescigno and McCurdy, Phys. Rev. A 62, 032706 (2000)]. –Bright solitons in Bose-Einstein Condensates and ultracold plasmas [Collins et al., Phys. Scr. T110, 408 (2004)]. –Non-equilibrium Green’s function calculations [Balzer et al, Phys. Rev. A 81, 022510 (2010)]. –Photoionization of impurities in spherical quantum dots [Lin and Ho, Phys. Rev. A 84, 023407 (2011)]. In each element, the interval [χ i, χ i+1 ] is further subdivided by a set of Gauss-Lobatto quadrature points χ m with weights w m to form generalized Gauss-Lobatto points and weights, The wave functions are expanded in terms of local basis functions, Lobatto shape functions,

8. Using this hybrid approach, the kinetic-energy matrix is block diagonal with matrix elements in compact expressions, and the potential-energy matrix elements are diagonal given by the potential values at the grid points. Facilitated by the generalized Gauss-Lobatto points and weights, the integrals can be approximated by the Gauss-Lobatto quadrature, The potential-energy matrix elements: The kinetic-energy matrix elements:

9. The method of complex-coordinate rotation is based on the mathematical developments by Aguilar, Balslve and Combes in 1971, and Simon in 1972 advocating it as a direct method to derive resonances in many-body systems. –J. Aguilar and J. M. Combes, Commun. Math. Phys. 22, 269 (1971). –E. Balslev and J. M. Combes, Commun. Math. Phys. 22, 280 (1971). –B. Simon, Commun. Math. Phys. 27, 1 (1972). Applications of this method to atomic and molecular structure and dynamics: –W. P. Reinhardt, Annu. Rev. Phys. Chem. 33, 223 (1982). –B. R. Junker, Adv. At. Mol. Phys. 18, 208 (1982). –Y. K. Ho, Phys. Rep. 99, 1 (1983). Within the framework of the method of complex-coordinate rotation, the coordinates are transformed under the mapping In the electric dipole approximation, the photoionization cross sections are given by Negative frequency component of the polarizability:

10.  T he avoided crossings are observed for the energy spectrum varying with the potential depth of the fullerene cage. U 0 =0.2U 0 =0.74U 0 =0.76U 0 =0.8  T he 1s state bound in the inner well collapses into the outer well, while the 2s state bound in the outer well simultaneously collapses into the inner well.

11.  The reduced matrix elements for Li 2+ ions endohedrally confined by fullerene cages.  The present result is calculated using the Gaussian-type potential with U 0 =3.0, r c =6.3 and w=0.5 a.u. to mimic the fullerene cage.  Comparisons are made to the case of free Li 2+ ions and the model of square well potential with the well depth U 0 =3.0 and the well width Δ=1.0 a.u.  The reduced dipole matrix elements of free Li 2+ ions (Z=3) are given analytically by Li 2+ @Gaussian Li 2+ @Δ * J. P. Connerade et al, J Phys B: At. Mol. Opt. Phys. 33, 2279 (2000).

12.  The reduced matrix elements for Li 2+ ions endohedrally confined by fullerene cages.  The present result is calculated using the Gaussian-type potential with U 0 =3.0, r c =6.8 and w=1.0 a.u. to mimic the fullerene cage.  Comparisons are made to the case of free Li 2+ ions and the model of square well potential with the well depth U 0 =3.0 and the well width Δ=2.0 a.u. R=5.8 * J. P. Connerade et al, J Phys B: At. Mol. Opt. Phys. 33, 2279 (2000).  The intensities of confinement resonances are greatly reduced for the Gaussian confining shell.  For higher photoelectron energies, the fluctuations are damped for the both Gaussian and square well models. Li 2+ @Gaussian Li 2+ @Δ

13.  The reduced matrix elements for Li 2+ ions endohedrally confined by fullerene cages.  The present result is calculated using the Gaussian-type potential with U 0 =3.0, r c =8.8 and w=3.0 a.u. to mimic the fullerene cage.  Comparisons are made to the case of free Li 2+ ions and the model of square well potential with the well depth U 0 =3.0 and the well width Δ=6.0 a.u. R=5.8 * J. P. Connerade et al, J Phys B: At. Mol. Opt. Phys. 33, 2279 (2000).  The confinement resonances are due to interference between three waves: the original outgoing wave, and the waves reflected at each of the inner and outer cavity boundaries.  The phase difference between the returning reflected waves is modified by the thickness of the confining shell. Li 2+ @Δ Li 2+ @Gaussian

14.  The photoionization cross sections of H@C 60 are calculated using the Gaussian- type potential with the cage radius r c =6.6896 a.u. for a broad range of the well depths and half-widths.  Each set of parameters, U 0 and w, is determined by fitting the electron affinity to the experimental one of C 60 fullerene.  Comparisons are made to the results of the Dirac δ-potential by Baltenkov*. * A. S. Baltenkov, Phys Lett. A 254, 203 (1999).

15.  The photoionization cross sections of Li@C 60 are calculated using the Gaussian-type potential with the cage radius r c =6.6896 a.u. for a broad range of the well depths and half- widths.

16.  The photoionization cross sections of Na@C 60 are calculated using the Gaussian-type potential with the cage radius r c =6.6896 a.u. for a broad range of the well depths and half- widths.

17. Summary –The energy spectrum and photoionization cross sections of endohedral atoms in fullerene cages are investigated using the finite-element discrete variable representation combined with the method of complex-coordinate rotation. –The avoided crossings of the energies varying with the confining well depth are due to the “mirror collapse” from the switch of near degenerate states. –Comparisons of the Gaussian-type potential are made to the square well potential for the reduced dipole matrix elements of endohedral Li 2+ ions. The effects of smooth shell boundaries are demonstrated. –The variation of the confinement resonance with the well depths and half-widths of the Gaussian-type potential are presented for H@C 60. Comparisons are made to the results of the Dirac δ-potential. –The multiple Cooper minima are observed in the photoionization cross sections of Li@C 60 and Na@C 60. In contrast to the diminished cross sections of Li@C 60 near the threshold, the cross sections of Na@C 60 are significantly enhanced under the influence of fullerene cages. Acknowledgments Financial support from the National Science Council of Taiwan is gratefully acknowledged.

18. The convergence of level energies (in atomic units) with respect to NE, the number of finite elements, and NG, the number of Gauss quadrature points, for a hydrogen atom endohedrally confined by the Gaussian potential with U 0 =1.0 Ryd and w=0.26 Ǻ. A(B) denotes A x 10 B. Energies (in atomic units) of 1s-4s and 2p-4p states for a hydrogen atom encapsulated by the Gaussian potential with several specific U 0 (in units of Ryd) and w (in units of Ǻ). Nascimento et al, J. Phys. B 44, 015003 (2011).

19.  The photoionization cross sections of Li@C 60 are calculated using the Gaussian-type potential with the cage radius r c =6.6896 a.u. and the half-width w=1.077 a.u. for several well depths.

20.  The photoionization cross sections of Na@C 60 are calculated using the Gaussian-type potential with the cage radius r c =6.6896 a.u. and the half-width w=1.077 a.u. for several well depths.

21. Model potentials for alkali metal atoms –The basic idea of model potential is to simulate the multi-electron core interaction with the single valence electron by an analytic modification of the Coulomb potential. –The optimized parameters are derived by a least-square fit to experimental energies. –Wave functions and oscillator strengths obtained by R-matrix quantum defect and model potential computations are in excellent agreement. Schweizer et al, Atomic Data and Nuclear Data Tables 72, 33 (1999). Parameters for the model potential AtomZa1a1 a2a2 a3a3 Li33.3953.2123.207 Na117.90223.512.688 K193.49110.5911.730 Rb373.43110.0981.611 Cs553.29411.0051.509