Numerical Simulation of Charged Fullerene Spectrum.
Fullerenes are known as one of allotropes of carbon, along with nanotubes, graphite, diamond, graphene and amorphous carbon. The first fullerene [C.sub.60] molecule was experimentally discovered in the 1980-ies using earlier theoretical predictions [18,19]. It represents a nanometer-size hollow spherical structure consisting of 60 carbon atoms, which are located at the vertices of a truncated icosahedron. Later, the fullerene nanotubes and many other fullerenes were discovered, such as [C.sub.20], [C.sub.70] and even larger structures due to started production of fullerenes in large quantities. This gave a birth to an incredible growth of research in the area of nanoscience, the historic development and the current status of which can be found in [9,21,35].
In past decades, the properties of charged fullerenes have been actively investigated, both experimentally and theoretically [12,25,26,32,34,38,41,42,43]. A large amount of works are devoted to the study of their stability (lifetime), mechanisms for their charging and decay [35, Ch.25].
In our article  the structure of the electronic spectrum of the charged fullerenes is investigated numerically. Simple models are used to show the existence of the volume-localized discrete quantum levels for the usual fullerene. The current paper should be considered as supplementary to the recent publication  which provides a general introduction to the subject. With an account of the well-known fact that geometrically the shape of a [C.sub.60] fullerene (which is a truncated icosahedron with carbon atoms located in its 60 equivalent vertices) is very close to a sphere, it is common to study the electronic structure of fullerenes (and, in general, interaction of electrons and photons with fullerenes and fullerene-like systems) by means of the simple model potentials. It is worthwhile to mention the use of the square-well type potentials [10,11,16,40], of the Dirac bubble and Gaussian-type potentials [1,13,24], as well as self-consistent models for the fullerene cage derived from a spherical jellium shell [25,29,36]. Here we focus on the computational issues of the numeric modelling, on the analysis of the corresponding discrete problem, and on the numerical solution of actual spectral problems with an account of peculiar features of the physical system under consideration. Furthermore, the conclusions of  are checked here by making use of the different values for the Lorentz-bubble potential.
A spectral problem is solved for the one-dimensional spherically symmetric Schrodinger equation with the corresponding potential . Numerical algorithm taking into account the features of the problem is based on finite-element approximation . To solve the corresponding matrix spectral problem we use the SLEPc (Scalable Library for Eigenvalue Problem Computations, http://slepc.upv.es/) . We use a Krylov-Schur algorithm, a variation of Arnoldi method, proposed in . The development of the computational algorithm for solving the spectral problem was carried out on the test problem of the hydrogen atom spectrum, which has an exact analytical solution.
2 Problem statement
For modeling of the electronic spectrum of the charged fullerenes spherical symmetry approximations are used. Here we confine our attention to the case of the positively charged fullerenes. In practice, one can multiply ionize [C.sub.60] with the help of the highly charged ions, fast electrons, or photons . Experimentally, charged fullerenes in the range of Z = 0, ..., 9e, , and even up to Z = 10e, [6,7]. Basic notations are as follows: [m.sub.e] and e are electron's mass and the absolute value of electron charge, [[epsilon].sub.0] is the electric constant of vacuum, h is Planck constant, [a.sub.0] = 4[pi][[epsilon].sub.0][h.sup.2][([m.sub.e][e.sup.2]).sup.-1] is the Bohr radius.
Let us formulate the corresponding quantum-mechanical spectral problem.
With an account of the spherical symmetry of a fullerene, for the wave function we use the standard ansatz [psi](r, v, [psi]) = R(r)[Y.sub.lm](v, [psi]). Here R(r) is the radial part of the wave function is determined in the self-consistent potential of the fullerene ion, [Y.sub.lm](v, [psi]) is the spherical harmonic, l is the quantum number of angular momentum, m is the quantum number of its z-projection. The spherically symmetric Schroodinger equation is written  into a second order differential equation
[d.sup.2]X/[dr.sup.2] - l (l+1)/[r.sup.2] X + 2[m.sub.e]/[h.sup.2] (E - U(r))X = 0 (2.1)
for the function x(r) = rR(r), 0 [less than or equal to] r [less than or equal to] [infinity], under the boundary conditions
X(0)=0, x([infinity])=0. (2.2)
The problem consists in finding electron energy levels E for a given potential U(r).
For numerical solution of the spectral problem (2.1)-(2.2) we use the atomic system of units, in which me = [absolute value of e] = h = 1, and [a.sub.0] denotes the length scale. The equation (2.1) in this case is written in the form
[dr.sup.2]X/[dr.sup.2] - l(l+1)/[r.sup.2] X + 2(E - U(r))X = 0. (2.3)
We will discuss the energy levels of an electron by starting from a simple model potential, and then move on to more complicate form of U. In the simplest model, one can describe a charged fullerene by the potential of a sphere with a constant surface charge density:
[mathematical expression not reproducible], (2.4)
where Z = Ne is a positive charge, and R is the fullerene radius. Our attention will be mainly confined to the [C.sub.60] fullerene, when R = 6.665.
The model above provides a rather simplified description in the sense that it does not take into account the actual physical structure of a fullerene. A more realistic single-electron potential U(r) can be constructed in the framework of the jellium model [4,5,15] as a sum of the positive contribution of the carbon atom's nuclei located on the spherical surface of the fullerene radius R and the negative contribution of the electron clouds. The resulting potential is attractive and it has a cusp-shape form with the clear localization in the thin spherical shell. For [C.sub.60], the corresponding Lorentz-bubble potential reads
[U.sub.2](r) = - V/[(r - R).sup.2] + [d.sup.2], (2.5)
where the parameter V determines the depth, d - the width, and R - the position.
Although the electronic structure of fullerenes is widely studied [1,10,11,13, 16, 24, 29, 36] in the framework of the simple model potentials approach, such investigations have obvious limitations. More nontrivial shape of potential can be derived from the photoelectron intensity measurements for [C.sub.60] fullerenes [25,30]. The spectral computations were also done by taking into account of the actual fullerene's nonspherical geometry [33,39] for electrically neutral systems, and for the charged fullerenes the corresponding DFT computations  yield the qualitatively consistent results.
3 Computational algorithm
The radial Schrodinger problem (2.2)-(2.3) is the example of self-adjoint Sturm-Liouville problems . To find eigenvalues and eigenfunctions, we can use various numerical methods . Developed specialized software for these tasks (see, for example, [22,28]) is widely used.
In our study, we focus on the use of standard approximations by finite elements  and general computational algorithms for solving spectral problems for partial differential equations, which are based on the technology of working with sparse matrices .
The spectral problem (2.2)-(2.3) has singularities that must be taken into account when solving it numerically. The first of these is related to the singularity
q(r) = l(l + 1)/[r.sup.2]
at r = 0. We used the simplest approach, when q(r) is replaced by
[q.sub.[epsilon]](r) = l(l + 1)/[r.sup.2] + [epsilon]
with a sufficiently small value [epsilon]. In addition, the problem is considered on a bounded interval [0, [bar.r]] by choosing a sufficiently large value r. Thus, we seek a solution of equation
[d.sup.2]X / [dr.sup.2]= l(l + 1)/[r.sup.2] + [epsilon] X + 2 (E - U(r))X = 0 (3.1)
under the boundary conditions
X(0)=0, x([bar.r]) = 0. (3.2)
An approximate solution of the problem (3.1)-(3.2) is searched for using a uniform grid:
[r.sub.k] = kh, k = 0,1, ..., M, Mh = [bar.r].
When using Lagrange finite elements of the first order (piece-wise linear approximation), we can define the hat function [[psi].sub.k](r) [subset] [V.sup.h], k = 1, 2, ..., M-1, where
[mathematical expression not reproducible]
For v [member of] [V.sub.h], we have
v(r) = [M-1.summation over (k=1)] [V.sub.k][[psi].sub.k](r),
where [v.sub.k] = v([r.sub.k]), k = 1, 2, ..., M-1.
The pair (u, E) (the eigenfunction u [member of] [V.sub.h] and the eigenvalue E) is determined from the following conditions
a(u, v) = Eb(u,v), [for all]v [member of] [V.sub.h].
For bilinear forms we have
[mathematical expression not reproducible]
The problem (3.2) is associated with the matrix eigenvalue problem
Au = E Bu, u = ([u.sub.1], [u.sub.2], ..., [u.sub.M-1]). (3.3)
Here A = ([A.sub.kj]) is matrix with elements [a.sub.kj] = a([[psi].sub.k], [[psi].sub.j]) and B = ([B.sub.kj]), [b.sub.kj] = b([[psi].sub.k], [[psi].sub.j]).
To find the first eigenvalues and eigenfunctions of the problem (3.3) with sparse matrices A and B, one can use various numerical methods . We apply Krylov-Schur algorithm , which is a variant of the Arnoldi method. The software implementation is performed on the FEniCS  computational platform. Spectral problems are solved using the library SLEPc .
The computational algorithm and the corresponding software thus developed was tested on the well-known problem for the hydrogen atom. In this case the potential has the form
U (r) = -1/r. (3.4)
The exact solution of the problem (2.2), (2.3), (3.4) is well known--see, for example, . The energy levels have the form
E = -1/2 1/[(l + i).sup.2], = l = 0, 1, ..., i = 1, 2, ..., (3.5)
where i = m + 1 is the eigenvalue number for a given l.
The numerical solution of the spectral problem (3.1)-(3.2) with the potential (3.3) is problematic because of the singularity of the functional for r [right arrow] 0. Therefore, instead of (3.4) we use the approximate functional
[mathematical expression not reproducible] (3.6)
with a sufficiently small [r.sub.0].
The results of calculations for values of [r.sub.0] = 0.05 and [r.sub.0] = 0.1 in (3.6) are presented in the Table 1 with the parameter [epsilon] = 1 * [10.sup.-5]. This value is fixed from the methodical computations and it does not affect the precision for the approximate solution of spectral problems under consideration. The data for [bar.r] = 200 are given for M = 4000, the exact value is determined according to (3.5). There is good accuracy in calculation of the energy level. The major error is related to the ground state eigenvalue. Using a more detailed grid (see the results for the M = 5000 case) makes it possible to increase the accuracy of calculating of the ground state eigenvalue. The computations are done on such fine grids in order to obtain the three-digit precision for the eigenvalues.
The matrix spectral problem does not completely inherit the properties of the differential problem. In particular, the computed levels 4-6 and so on are not degenerate. The graphical representation of the spectrum at [r.sub.0] = 0.05 is given in Figure 1(a). The corresponding eigenfunctions are shown in Figure 1(b). The following normalization of eigenfunctions is used
[mathematical expression not reproducible]
4 Results of calculations
First, we present the calculated data on the electron spectrum of an neutral fullerene [C.sub.60], when U(r) = [U.sub.2](r). Following , we studied in  the case
V = 0.711, R = 6.665, d = 0.610.
However, the Lorentz-bubble potential with these values of parameters does not yield the correct affinity energy for [C.sub.60]. Here we take the more accurate values which provide the agreement with the detachment energy 2.65 eV found experimentally from the photoelectron spectroscopy. Namely, we use the potential (2.5), where
V = 0.1104, R = 6.627, d = 0.5.
Calculated data are given in Table 2 and in Figure 2. Numeric computations of the spectrum for different fullerene models are done on the grid with M = 4000.
For the case of the charged sphere model, we set U(r) = [U.sub.1](r), where [U.sub.1](r) is defined for different values of Z according to (2.4). Numeric results for the spectrum with Z =1 are presented in Table 3 and in Figure 3. Analogous data for the case Z = 5 is given in Table 4 and in Figures 4.
For a charged fullerene, we set U(r) = [U.sub.1](r) + [U.sub.2](r). The results of calculations for Z = 1 are presented in Table 5 and in Figure 5. The effect of increasing of the charge of fullerene is traced on the calculated data for the case Z = 5. The first 12 energy levels are presented in the Table 6, the spectrum and basic eigenfunctions are shown in Figure 6.
* A simple mathematical model for describing the single-electron groundstate spectrum of a charged fullerene is proposed. It is based on the use of the potential of a charged sphere and the spherically symmetric potential of an neutral fullerene for the Schrodinger equation.
* The computational algorithm for solution of the spectral problem is based on the use of piecewise linear finite elements and tested on the problem of the spectrum of the hydrogen atom. The software implementation is based on the free library for spectral problems solution SLEPc.
* Results of calculations demonstrate the existence of a system of discrete short-lifetime quantum states for electrons in the potential well of the self-consistent Coulomb field of charged fullerenes. Owing to its single-electron nature, the current model can realistically access the low-energy excited states or capture-decays far below the plasmonic energies.
* For neutral fullerenes, attention is usually focused on distinctly edge effects at the position of the carbon atomic cage. A novel feature for charged fullerenes is presence of the volume-localized phenomena. A possible experimental confirmation of the existence of the volume-localized discrete levels would be of considerable interest for the experimental research and practical tasks, including the development of new sources of coherent radiation in a wide range of wavelengths. Preliminary estimates of the lifetimes of the levels and of the inverse population value can be found in .
Please Note: Illustration(s) are not available due to copyright
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Rafael Arutyunyan, Yuri Obukhov and Petr Vabishchevich
Nuclear Safety Institute, Russian Academy of Sciences
B. Tulskaya 52, 115191 Moscow, Russia
Received August 3, 2018; revised February 16, 2019; accepted February 18, 2019
Caption: Figure 1. Results for the hydrogen atom: a) numeric evaluation of the spectrum, b) radial wave functions R(r) for the lowest energy levels.
Caption: Figure 2. Results for neutral fullerene with (2.5): a) the spectrum structure, b) radial wave function R(r).
Caption: Figure 3. Results for the potential (2.4) of charged sphere with Z =1: a) the spectrum, b) radial wave functions [beta](r) for the lowest levels.
Caption: Figure 4. Results for the potential (2.4) of charged sphere with Z = 5: a) the spectrum, b) radial wave functions R(r) for the lowest levels.
Caption: Figure 5. Results The spectrum of charged fullerene for the potential U(r) = [U.sub.1] (r) + [U.sub.2](r) with Z = 1: a) the spectrum of charged fullerene, b) radial wave functions R(r) for the lowest levels.
Caption: Figure 6. Results for for the potential U(r) = [U.sub.1](r) + [U.sub.2](r) of a charged fullerene with Z = 5: a) the spectrum, b) the radial wave function.
Table 1. Energy levels (hydrogen atom) n l i [r.sub.0] [r.sub.0] [r.sub.0] exact = 0.05 '= 0.1 = 0.05 (N = 5000) 1 0 1 -0.50109 -0.49596 -0.49950 -0.50000 2 0 2 -0.12510 -0.12496 -0.12497 -0.12500 3 1 1 -0.12496 -0.12446 -0.12492 -0.12500 4 0 3 -0.05558 -0.05555 -0.05555 -0.05556 5 2 1 -0.05555 -0.05554 -0.05555 -0.05556 6 1 2 -0.05554 -0.05539 -0.05553 -0.05556 7 0 4 -0.03126 -0.03125 -0.03125 -0.03125 8 3 1 -0.03125 -0.03125 -0.03125 -0.03125 9 2 2 -0.03125 -0.03124 -0.03125 -0.03125 10 1 3 -0.03124 -0.03118 -0.03124 -0.03125 11 0 5 -0.02001 -0.02000 -0.02000 -0.02000 12 4 1 -0.02000 -0.02000 -0.02000 -0.02000 13 3 2 -0.02000 -0.02000 -0.02000 -0.02000 14 2 3 -0.02000 -0.02000 -0.02000 -0.02000 15 1 4 -0.02000 -0.01996 -0.01999 -0.02000 16 0 6 -0.01389 -0.01389 -0.01389 -0.01389 17 5 1 -0.01389 -0.01389 -0.01389 -0.01389 18 4 2 -0.01389 -0.01389 -0.01389 -0.01389 19 3 3 -0.01389 -0.01389 -0.01389 -0.01389 20 2 4 -0.01389 -0.01389 -0.01389 -0.01389 Table 2. Lowest energy levels for neutral fullerene in the potential (2.5) with V = 0.1104, R = 6.627, d = 0.5 n 1 i E 1 0 1 -0.09741 2 1 1 -0.07123 3 2 1 -0.02595 Table 3. The energy levels for the potential (2.4) a charged sphere with Z =1 n 1 i E n 1 i E 1 0 1 -0.11245 7 3 1 -0.03109 2 1 1 -0.08060 8 2 2 -0.02909 3 0 2 -0.05497 9 1 3 -0.02465 4 2 1 -0.05134 10 0 4 -0.02036 5 1 2 -0.04076 11 4 1 -0.02000 6 0 3 -0.03165 12 3 2 -0.01988 Table 4. The energy levels for the potential (2.4) of a charged sphere with Z = 5 n 1 i E n 1 i E 1 0 1 -0.69161 7 4 1 -0.40274 2 1 1 -0.63311 8 0 3 -0.39074 3 2 1 -0.56307 9 2 2 -0.39010 4 0 2 -0.53875 10 1 3 -0.33711 5 3 1 -0.48487 11 3 2 -0.33262 6 1 2 -0.45857 12 5 1 -0.32240 Table 5. The energy levels of a charged fullerene in the potential U(r) = [U.sub,1](r) + [U.sub.2](r) for Z = 1 n 1 i E n 1 i E 1 0 1 -0.23745 7 2 2 -0.03989 2 1 1 -0.20944 8 0 3 -0.03604 3 2 1 -0.16068 9 3 2 -0.02868 4 3 1 -0.09567 10 1 3 -0.02834 5 0 2 -0.06295 11 4 1 -0.02545 6 1 2 -0.05092 12 2 3 -0.02341 Table 6. The energy levels in the potential U(r) = [U.sub.1](r) + [U.sub.2](r) of a charged fullerene with Z = 5 n l i E n l i E 1 0 1 -0.81036 7 1 2 -0.49728 2 1 1 -0.77750 8 5 1 -0.44581 3 2 1 -0.72167 9 0 3 -0.41576 4 3 1 -0.64593 10 2 2 -0.41383 5 0 2 -0.60188 11 1 3 -0.36906 6 4 1 -0.55291 12 3 2 -0.35775
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|Author:||Arutyunyan, Rafael; Obukhov, Yuri; Vabishchevich, Petr|
|Publication:||Mathematical Modeling and Analysis|
|Date:||Apr 30, 2019|
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