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Jahn-Teller distortions and excitation energies in C60(n+)

Based on previously computed parameters for the electron-phonon couplings and the Coulomb exchange, we compute and classify the static Jahn-Teller distortions, i.e. the minima of the lowest adiabatic potential energy surface, of C60(n+), for all values of charge 1<= n<= 9 and spin. We compute the intra-band electronic excitation energies in the different optimal geometries in the sudden approximation, and find a spread of the electronic states of roughly 1 eV. We also obtain the leading vibronic quantum corrections to the ground-state energy, equal to zero-point energy lowering due to the softening of the phonons at the adiabatic Jahn-Teller minima: these non-adiabatic corrections are so large that for 4<= n<= 6 states of different spin symmetry turn lower than the high-spin adiabatic ground state.


Introduction
Low-spin states are associated to larger distortions, thus larger energy gains, than high-spin states in degenerate electron-phonon coupled molecules and impurity centers. Electron-electron Coulomb repulsion opposes this tendency, favoring high-spin states instead, in accord to the first of Hund's rules. The Jahn-Teller (JT) systems C n+ 60 are no exception to this rule: if electron-phonon coupling was the only relevant interaction, then the n-holes ground state would be either of spin S = 0 (even n) or S = 1 2 (odd n). As was recently shown [1], in positive fullerene ions the size of Coulomb interaction is sufficiently large to enforce Hund's rule: the ground states of C n+ 60 was calculated to always be high spin (S = n 2 for n ≤ 5, S = 10−n 2 for n > 5) in the adiabatic approximation. This result is confirmed for n = 2 by NMR investigation of solid-state compounds [2]. The JT distortions in C n+ 60 , though strongly counteracted by the larger electron-electron repulsion, yet represent an important, and still largely unexplored, contribution to the energetics of C n+ 60 . Investigation of this contribution, and in particular of the corrections to the adiabatic approximation, is the main subject of this work.
The JT model relevant for C n+ 60 is conventionally indicated as h n ⊗ (A + G + H), where h refers to the fivefold-degenerate highest occupied molecular orbital (HOMO), and A, G, H refer to the 2 nondegenerate A g , 6 fourfolddegenerate G g and 8 fivefold-degenerate H g molecular vibration modes that are linearly coupled to the h u states according to icosahedral symmetry [1,3,4]. We investigate this model by treating the normal coordinates for these vibrational modes as classical variables, and searching the minima of the adiabatic potential energy surface in the 66-fold dimensional space of these distortions. Each of these static JT configurations is characterized by a reduced symmetry from icosahedral to some (usually) lower symmetry. New vibrational frequencies arise at these local minima: we determine these frequencies by evaluation of the Hessian matrix at the minimum [5]. The lowering of the vibrational frequencies gives the leading quantum correction to the adiabatic approximation. The original icosahedral symmetry of the problem is restored once the presence of several equivalent optimal distortions is recognized, and quantum tunneling between these wells is allowed. Proper accounting of tunneling gives the next-order quantum correction, but in the present work, we limit ourselves to the study of the local properties of the wells and the connectivity of the sets of minima in distortion space, for all values of charge n and spin S.
The competing intra-molecular exchange of Coulomb origin and the JT interaction both contribute to the computed spectrum of excitations. Differences in energies of the fully relaxed configurations at different spin compare directly with spin gaps as could be measured in "slow" spectroscopies such as electron or nuclear magnetic resonance. In contrast, the electronic excitation energies computed keeping the molecular geometry fixed in the lowest minimum compare directly with the vertical excitations probed by fast optical spectroscopies. Both these class of quantities are reported in this work. This paper is organized as follows: Sec. 2 introduces the model and the parameters used in this calculation, which is then described in Sec. 3, along with the properties of the JT minima for all values n and S; Sect. 4 contains the vertical excitation spectra. The the zero-point non-adiabatic corrections are described in Sec. 5. The results are discussed in Sec. 6, and connectivity matrices are collected in an Appendix.

The model Hamiltonian
We report here for completeness the model Hamiltonian previously introduced in Ref. [1] to describe the physics of the holes in the h u HOMO of C 60 fullerene: whereĤ are respectively the single-particle Hamiltonian, the vibron contribution (representing the phonon kinetic energy plus the restoring potential expanded to quadratic order around the equilibrium configuration of neutral C 60 ), the electron-vibron coupling (in the linear JT approximation) [4,6], and finally the mutual Coulomb repulsion between the electrons. Theĉ † σ,m denote the creation operators of a hole in the HOMO, described by the single-particle wave function ϕ mσ (r). σ indicates the spin projection; m and n label the component within the fivefold degenerate electronic HOMO multiplet, based on the C 5 quantum number m from the I h ⊃ D 5 ⊃ C 5 group chain [4,7]. i counts the phonon modes of symmetry Λ (2 A g , 6 G g and 8 H g modes). C rΛµ mn are  Table 1: Computed mode eigenfrequencies and e-v linear coupling parameters of the h u HOMO in C 60 [6]. The classical single-mode JT stabilization energies E s are tabulated for both D 5 and D 3 distortions, for one hole in the HOMO.
Clebsch-Gordan coefficients [7] of the icosahedral group I h , for coupling two h u states to phonons of symmetry Λ. r is a multiplicity label, relevant for modes of Λ = H g symmetry only [6,7].Q iΛµ are the molecular normal-mode vibration coordinates (measured from the adiabatic equilibrium configuration of C 60 ), andP iΛµ the corresponding conjugate momenta. Spin-orbit is exceedingly small in C 60 [8] and it is therefore neglected. The electron-vibron (e-v) couplings g r iΛ are conveniently expressed in units of the corresponding harmonic vibron quantum of energyhω iΛ . In this calculation we adopt the numerical values of the e-v coupling parameters, listed in Table 1, from the Density Functional (DF) calculation of Ref. [6], and a second calculation [9] yields couplings in substantial accord with those of Table 1. The numerical factors k Ag = 5 1 2 , k Gg = 5 4 1 2 , k Hg = 1 inĤ e−v have been introduced for compatibility with the normalization of Ref. [6].  [1]. One of the tabulated parameters (e.g. F 1 ) is a linear combination of the five others.
The Coulomb matrix elements are defined by: where u σ,σ ′ (r, r ′ ) is an effective Coulomb repulsion, screened by the other electrons of the molecule. Detailed symmetry analysis shows [1] that, assuming spin-independence of the orbitals, this set of coefficients can be expressed as in terms of a minimal set of independent parameters F r,r ′ ,Λ . A DF calculation of these parameters was carried out in Ref. [1], and for our calculation we adopt those values of the Coulomb parameters, which we report for completeness in Table 2. For the Coulomb parameters we use the shorthands and the combination U defines an average Coulomb repulsion within the n-holes multiplets, so that It should be noted that U differs from the usual definition of the Hubbard U, involving the lowest multiplet in each n-configuration: U min = E min (n + 1) + E min (n − 1) − 2E min (n). This second definition is inconvenient here, since it depends wildly on n.  Table 3: The total adiabatic energy V ad (Q min ) (in meV) of the lowest electronic state for each n and S, including the e-e and e-v contributions from H vib +Ĥ e−v +Ĥ e−e (but excluding the [Un(n − 1)/2] term), for C n+ 60 . The last two columns distinguish the vibrational (Ĥ vib ) and electronic (Ĥ e−v +Ĥ e−e ) contributions.

The adiabatic calculations
We approximate the vibron operatorsQ iΛµ with classical coordinates, in the spirit of the adiabatic approximation. In an orbitally degenerate situation (as for the C n+ 60 ions at hand) the adiabatic approximation usually yields fairly accurate energetics in the limit of large the e-v couplings, so that tunneling between equivalent minima can safely be neglected [10]. The phonon kinetic term in (3) is neglected in the adiabatic approximation. In Sect. 5 we will partly restore this term by taking into account quantum zero-point energies.
In any classical statically JT-distorted configuration, the icosahedral symmetry is broken: therefore states of different icosahedral symmetry representations are inter-mixed. Only the total number of holes n, total spin S and its projection S z are conserved upon distortion. Here, we neglect any change of the Coulomb Hamiltonian upon distortion, and we assume therefore thatĤ e−e is still determined according to Eqs. (5,6,7) by the same parameters F i of Table 2, as in icosahedral symmetry. Also, we assume no change of the phonon frequencies ω iΛ and couplings g r iΛ upon charging. For each n, S and M S , we allow the 64 (6 × 4 G g plus 8 × 5 H g ) phonon coordinates to relax, and determine the optimal distortion, by full minimization of the lowest adiabatic potential sheet V ad (Q) in the space of all the phonons coordinates Q. We leave the A g modes out, since they contribute a trivial spin-and symmetry-independent term to the energetics. Because of particlehole symmetry, charges n > 5 can always be reduced to the computed charges n < 5. In Table 3, we report the resulting optimally-distorted energy in each spin sector, based on the electron-electron (e-e) and e-v couplings of C n+ 60 ions, as previously published in Ref. [1]. The main outcome of the adiabatic calculation is that positive C 60 ions favor high-spin ground states (contrary to the analogous finding for negative ions).
In the present contribution, we extend the previous calculation to obtain the complete set of all the equivalent minima for each (n, S) sector. To this purpose, we generate about a hundred randomly distributed distortions away from the I h high-symmetry point, and let the molecule relax to the closest minimum, by combined standard (simplex and conjugate-gradients) minimization algorithms. We then apply the symmetry operations of the icosahedral group to the each of the minima found, in order to locate any possibly missing minimum. Although the method employed is not deterministic, the symmetry analysis makes the probability that any set of minima is incomplete utterly negligible. Thus, for each n and S we obtain a complete set of equivalent global minima. In the few cases where the minimization leads to non-global minima, we have discarded them based on simple comparison of the adiabatic energies.
In Table 4 we summarize some global properties of the obtained JT minima for all charge and spin states. In these multi-mode JT systems, the local symmetry of an optimal distortion is described in terms of the subgroup G local of symmetry operations which leave that minimum invariant. We remind that the minima in the n = 1 S = 1 2 case, where e-e interaction is unimportant, were found to be 6, of local D 5d symmetry [3,4,6]. For 2 ≤ n ≤ 8, where the role of e-e interaction is crucial, the number of JT minima follows from the local symmetry: it is generally given by the ratio |I h |/|G local | of the orders of the icosahedral group (120) and of the invariant subgroup.
Special care has to be taken for n = 5 holes. Here, in addition to the icosahedral symmetry, the system is particle-hole symmetric, i.e. invariant under exchange of fermion creation and annihilation operators. This transformation leaves the Coulomb HamiltonianĤ e−e invariant, while the the vibron interac-tionĤ e−v is unchanged provided that a sign change of the vibron coordinateŝ Q i,Λ,µ is also performed. Hence, for a given minimum Q min , also its opposite  −Q min is a minimum of the potential energy surface. In the case n = 5 S = 1 2 , this leads to a doubling of the minima: the local C 2v symmetry would lead to 30 minima, but 30 more equivalent minima are added in the opposite positions by particle-hole symmetry. For n = 5 S = 3 2 instead, the number of minima remains 30, since for each minimum there is one of the I h symmetry operations that transforms this minimum into its opposite point. Note that this operation is not the spatial inversion (since all vibrations considered here form even representations), which are invariant under inversion, but a twofold rotation. Finally, for n = 5 S = 5 2 , the electronic state is orbitally nondegenerate, thus no JT distortion takes place. Table 5 collects some quantitative information about the contribution of each mode to the amount of JT distortion at each minimum. As expected, the largest distortion involves always the lowest H g mode, which is the most strongly coupled one (see Table 1). Also, the D 5 distortions receive no contribution of the G g modes, which contribute to all the lower-symmetry minima instead. Table 4 contains some information about the connectivity of the minima in Q space. In some cases, the specification of the number of first, second, etc.  neighbors of a given minimum is sufficient to clarify completely the topology of the minima in the 64-dimensional space. In particular, the D 5d wells of both the n = 2 S = 0 and the n = 4 S = 2 surfaces are located on the six vertices of the five-dimensional regular simplex, the generalization of a tetrahedron, each minimum being equally distant from all the others: this is analogous to the previously determined minima of V ad for n = 1 S = 1 2 [4]. In analogy, the connectivity of the 10 D 3d minima for n = 4 S = 0 is the same of the one depicted in Fig. 1b of Ref. [4].
For the other cases of lower symmetry, the number of neighbors of any given order must be complemented by some extra connectivity information. First, we observe that the minima for n = 2 S = 1 and for n = 3 S = 3 2 are exactly the same. Indeed, these two case are related by a particle-hole symmetry applied only to one spin flavor. For all nonequivalent cases, the complete topological information about the wells is contained in the connectivity matrix C (n, S), whose matrix elements indicate that minima i and j are C (n, S) ij th neighbors. We report those matrices in the Appendix. Careful exam of C (n, S) for n = 2 S = 1 and n = 3 S = 3 2 shows that each of the 15 minima is linked to four nearest neighbor minima, which, in turn, are linked to more minima, forming a completely connected regular polytope. The matrices C 3, 1 2 and C (4, 1), show that, for n = 3 S = 1 2 and n = 4 S = 1, the 30 minima are divided into 6 pentagonal "clusters" of five nearest-neighboring minima. In contrast, for n = 5 S = 1 2 , nearest-neighbor wells come in pairs. Finally, the 30 minima for n = 5 S = 3 2 , show the largest connectivity, and sit at the vertices of a highly symmetric polytope.

Vertical excitation energies
In Table 6 we report the range of "vertical" excitation energies ∆E for all final spin symmetries S ′ , in the frozen minimum configurations Q min (n, S), for all values of n and S. The complete spectrum (available upon request from the authors) is very dense and not much informative. The listed energies give a quantitative prevision of the spectral range where a fast (optical) spectroscopy is likely to locate the intra-band HOMO excitations of the C n+ 60 ions. For the experimentally most accessible case n = 2 S = 1, here follows the complete list of the triplet-triplet excitation energies : 127, 149, 150, 178, 182, 218 Table 6: The lowest and highest vertical excitation energies (in meV) calculated assuming that the C n+ 60 ion remains frozen in one of the adiabatic minima when the electronic state is excited. The first two columns fix the relevant distortion. The third column indicates the spin S ′ of the excited states considered. The excitation energies in the last two columns are referred to the adiabatic energy of each specific minimum, reported in Table 3.
The leading quantum correction to the static JT energetics is given by the zero-point energy gain due to the softening of the vibrational frequency at the JT-distorted minima [5]. To obtain this information, by finite differences we compute the Hessian matrix of the second-order derivatives of the lowest adiabatic potential sheet, at one of the static JT minima Q min The vibrational frequenciesω j at the distorted point are the eigenvalues of the Hessian matrix. There is no square root involved in theω j , as the coordinates are scaled with the harmonic length scale x 0 (ω) defined in Table 5. In the harmonic approximation, these "new" normal mode frequenciesω j contribute a zero-point energy j 1 2hω j to that minimum configuration: the difference between this and the original zero-point energy iΛµ 1 2h ω iΛ gives the leading quantum correction to the classical energy. We rewrite this correction as using the invariance of the trace under change of basis. Table 7 displays the lowest adiabatic energies for given charge and spin in various approximations. The first column reports the adiabatic energy, E class = V ad (Q min ), as in Table 3. The zero-point energy corrections in the following column are comparable in magnitude to the leading adiabatic energies. In particular the very large values of zero-point energy gain for n = 5 S = 1 2 is associated to very shallow minima, connected by low barriers. The lowest vibrational frequency is as small asω 1 ≈ 2.3 meV. In the rather close competition between the Coulomb physics (Hund's rules) and the JT physics (anti-Hund behavior) the zero-point correction is very important, and, as shown by the last column of Table 7, reduces drastically the spin-gap (n = 3), or even changes the ground-state symmetry in favor of an intermediate (n = 4) or low (n = 5) spin state.  Table 7: The total adiabatic energy E class = V ad (Q min ) (in meV) of the lowest electronic state for each n and S, including the e-v and e-e contributions from H vib +Ĥ e−v +Ĥ e−e (but excluding the [Un(n − 1)/2] term), for C n+ 60 . The following column contains the leading non-adiabatic correction E zero , the zeropoint energy defined in Eq. (14). The last column reports the adiabatic energy E class corrected by the zero-point term E zero : for n ≥ 4 it leads to a different ordering of the spin states.
The zero-point correction treated here represents the g 0 i term of a largecoupling expansion, where the adiabatic energy E class is the leading (g 2 i ) term. The next corrections to be considered, of order g −2 i , are associated to tunneling among minima, possibly affected by Berry phases [4,11,12]. Tunneling is likely to be especially important between the pairs of neighboring minima of the n = 5 S = 1 2 adiabatic surface. Tunneling will be dealt with in future work.

Discussion and Conclusions
In the present calculation both e-e and e-v interactions are included for the HOMO shell of C 60 . E-e exchange terms are treated essentially exactly, in the assumptions that (i) inter-band couplings can be neglected, and only act as a renormalization of the Coulomb parameters and that (ii) the latter are independent of the charge n in the HOMO. In principle, due to both orbital and geometrical relaxation, the effective Coulomb interaction (6) will depend on the instantaneous charge state of the fullerene ion. However, this effect, a very important one in single-atom calculations, is expected to be fairly small in such a large molecule as C 60 . In a JT system, the coupled phonons should in principle be treated fully quantum mechanically, as nonadiabatic effects may be important. However for strong electron-phonon coupling, the leading terms (of order g 2 ) are obtained in the adiabatic approximation, by studying the minima of the lowest adiabatic potential surface V ad (Q). Non-adiabatic effects are taken into account to the next order (g 0 ) by the calculation of the new harmonic oscillation frequencies close to the adiabatic JT minima. These zero-point corrections are significantly large, and they can even reverse the theoretical prevision for the symmetry of the ground state of the C n+ 6 ion for 4 ≤ n ≤ 6. Tunneling matrix elements which mix different minima to suitable dynamical combinations restore the original icosahedral symmetry and provide the next-order (g −2 ) quantum correction to the energetics. These terms, which will be the subject of future work, are likely to be especially large for n = 5 S = 1 2 . The present calculation was carried out in the linear e-v approximation. As the coupling and thus the distortions are fairly large, quadratic and higherorder (in Q) couplings and vibrations anharmonicity could be important. Unfortunately, no estimate for those higher-order couplings is available yet.
The parameters used in this calculation, both for e-e and e-v interaction are most likely underestimated by the local density approximation used in their determination, as discussed in Ref. [1,6]. Consequently, both the Coulomb repulsion and the phonon-mediated attraction calculated within the local density approximation are likely to need a rescaling by a similar factor of order two. Indeed, the balance between the two opposing interactions is delicate in C n− 60 ions (as demonstrated by the presence of both high-spin and low-spin local ground states in different chemical environments [13,14,15,16,17,18,19]). In C n+ 60 e-e interaction prevails at the adiabatic level: high-spin states are favored, as experiments confirm for n = 2 [2]. According to our calculation, however, more highly charged states, close to HOMO half filling should favor local low-spin.
An effective local exchange interaction favoring low spin is a crucial ingredient for superconductivity in a strongly-correlated orbitally-degenerate material such as a solid of doped C 60 [20,21]. If screening and retardation effects could be neglected, the present single-molecule calculation suggests that superconductivity should be strongly suppressed in the hole-doped solid at doping 0 ≤ n ≤ 3, but could be recovered close to half filling 4 < n < 6. It is presently unclear if such a high level of hole doping is practically accessible, except possibly by field-induced charging [22].

A Appendix
We report here the connectivity matrices whose matrix element (ij) indicates that minima i and j are C (n, S) ij th neighbors. for all the nontrivial n, S cases. We include up to 9th neighbors, substituting those of higher order with a dash.
The 60 minima for n = 5 S = 1 2 are conveniently split into two sets of 30 minima connected by the I h operations. The minima in the "b" block are one by one ordinately opposite to those in the "a" block. Accordingly, the structure of the C-matrix is as follows: where