Analytical application of the recursion and moments methods to the electronic structure of C 60 : Exact solution for the and states

Transcription

1 PHYSICAL REVIEW B VOLUME 53, NUMBER 3 5 JANUARY 996-I Analytical application of the recursion moments methods to the electronic structure of C 60 : Exact solution for the states Yeong-Lieh Lin * Franco Nori Department of Physics, The University of Michigan, Ann Arbor, Michigan Received 8 August 995 The recursion moments methods are applied analytically to study the electronic structure of a neutral C 60 molecule. We employ a tight-binding Hamiltonian which considers both the s p valence electrons of carbon. From the recursion method, we obtain exact results for the eigenvalues eigenfunctions, including the highest occupied molecular orbital the lowest unoccupied molecular orbital states. We also compute the Green s function in analytic closed form obtain the local density of states around several ring clusters, which can be probed experimentally by using, for instance, a scanning tunneling microscope. From the method of moments, identical results for the energy spectrum are also derived. In addition, the local density of states on one carbon atom is obtained; from this we can derive the degree of degeneracy of the energy levels. I. INTRODUCTION AND SUMMARY OF RESULTS Since the discovery of a simple technique for the production in bulk quantities of fullerenes, undoped doped C 60 molecules have generated enormous interest among chemists, physicists, materials scientists. In C 60, carbon atoms sit at the 60 vertices of the pentagons hexagons of a truncated icosahedron. The carbon-carbon bonds are of two different lengths:.46 Å for the single bonds bonds on the pentagons.40 Å for the double bonds bonds on the hexagons not shared by a pentagon. The single double bonds are denoted by solid dotted lines throughout the figures in this paper. The literature on C 60 is vast, here we do not attempt a review. The interested reader is referred to Refs. 7 papers listed therein. In this work, we focus on an analytical study of the electronic structure of a single neutral C 60 molecule. We model this system through a tight-binding Hamiltonian Eq. which considers both the s p valence electrons of carbon. For decades, the recursion method method of moments have been successfully applied to the electronic structure calculations in a variety of physical systems. However, most of these applications have been purely numerical. Furthermore, the relationship between these two methods has rarely been investigated thoroughly. The goal of this paper is to analytically apply these two methods to study the electronic structure of a C 60 molecule. Relations between these two approaches will be discussed. This paper is organized as follows. In Sec. II, we discuss the physical nature types of the interaction between these valence electrons, with a total number of 40. The couplings can be separated into two main contributions: coming from the -bonded -bonded electrons. We first focus on the states. In Sec. III we use the recursion method 8 to analytically solve the Hamiltonian for the electrons. We derive closed-form expressions for their energy eigenvalues eigenfunctions including the highest occupied molecular orbital HOMO the lowest unoccupied molecular orbital LUMO for two cases: equal unequal hopping integrals for the single double bonds. The main results concerning the -state eigenvalues eigenfunctions are summarized in Tables III, IV, V. The physical nature of the wave functions of the HOMO LUMO is discussed in Sec. III D. The beauty of the recursion method for C 60 lies in the fact that the recurrence is simple terminates very quickly e.g., after four iterations only, providing exact very concise expressions for the parameters of the recursion. Thus, the dimensionality of the problem is reduced by a factor of 5, namely from to 4 4. The same results for the energy spectrum of the states are derived in Sec. IV by using the method of moments. The main ingredient in this approach is the analytical computation of sums over contributions from all paths starting ending at the same site, each one with its corresponding kinetic energy. The spirit of this approach follows Feynman s program: to compute physical quantities from sums over paths. In addition, the local density of states LDOS on one carbon atom is obtained; from this we can derive the degree of degeneracy of the energy levels. In the two sections mentioned above, we provide detailed descriptions of the steps e.g., the choice of appropriate starting states incorporating the physical properties of the system, the iteration schemes employed in the analytic application of the recursion moments methods to C 60. Furthermore, every step of either approach is exact. All the calculations are done analytically, either by h or with the assistance of computer symbolic manipulation software. Diagonalizations are achieved by iteratively applying the Hamiltonian on simple initial states. The relations between the recursion moments methods are discussed in Sec. V. There it will be seen that while these two approaches are related to each other, each one has its own advantages. These methods are significantly different from the ones used so far. Furthermore, they are neither numerical nor require the use of group theory. For a lucid clear account of a very different approach, based on group theory focused only on the states, the reader is referred to Ref. 3. In Sec. VI, we present the precise algebraic expressions, derived from the Green s functions, as well as plots of the LDOS around several ring clusters i.e., a carbon atom, a /96/53 3 /64 5 /$ The American Physical Society

2 64 YEONG-LIEH LIN AND FRANCO NORI 53 pentagon ring, a hexagon ring; also for two opposite carbon atoms, pentagons, hexagons. We find that around a pentagon ring the LDOS is large at low energies. This is related to the fact that a pentagon has zero double bonds five single bonds. On the other h, around a hexagon ring the LDOS is large at high energies. This is because a hexagon has three double bonds three single bonds. The LDOS is relevant to the several important experimental techniques which probe the local spectroscopy of molecules; for instance, by using a scanning tunneling microscope, as described in the review in Ref. 9. In Sec. VII we discuss the solutions of the -states Hamiltonian Eq. 3. To describe the states, the coupling between orbitals on the same atom that between orbitals along the same bond are both taken into account. By following a one b two b transformation, 0 the eigenvalues eigenfunctions for the states are then analytically obtained. II. ELECTRONIC STATES To investigate the electronic properties of a single C 60 molecule, we consider the four carbon valence electrons s, p x,p y, p z in the tight-binding Hamiltonian H T c i c i i, ij,, t c i c j. Here, i denotes the carbon site denotes the valence orbitals s, p x,p y, p z. Also, is the orbital energy t is the hopping matrix element between orbitals on the nearest-neighboring sites i j. The 60 p z orbitals, each pointing along the outward radial direction, are hybridized to form states. The other three orbitals s, p x, p y, distributed on the plane tangential to the surface of the molecule, are hybridized along the lattice bonds to form bonding antibonding states. Let us assume that the mixture of these three orbitals at each site produces three sp hybrid orbitals: sp a along the double bond sp b sp c along the two single bonds, respectively. From a physical point of view, the 60 outer orbitals are relevant to the conducting properties of the molecules the 80 orbitals are mainly responsible for the elastic properties. The former are also responsible for the bond dimerization. Also, only states occur around the Fermi energy. As a result of its planar structure, the electrons in graphite contain only pure p z orbitals. Nevertheless, in the curved structure of C 60, an extra small component from s, p x, p y is induced along the radial direction due to the surface curvature. The interaction between the orbitals in C 60 is then slightly increased in comparison to that in graphite. However, since this component is very small, the overlap integral between orbitals is negligible. The nonzero Hamiltonian parameters used here are the same ones as in Ref.. The original Hamiltonian Eq. can thus be written in terms of H H as H T H H, where H ij t ij c i c j H V i, c i c i V c i c j. 3 ij The electronic -states Hamiltonian Eq. describes the kinetic energy of the p z electrons hopping on the 60 vertices of a C 60 fullerene t ij is the hopping integral between nearest-neighboring atoms i j. The hopping integrals are considered unequal for single double bonds. Also c i is the electron operator creating a p z orbital on the atom located at vertex i. A constant term corresponding to the onsite p z orbital energy is omitted in Eq.. Inthe -states Hamiltonian Eq. 3, sp a, sp b, sp c denotes the hybridized orbitals. Also, V sts for the hopping integral between orbitals on the same carbon atom V sts for that between orbitals on different atoms that are associated with the same bond the length difference between single double bonds is neglected here. A more detailed discussion on the origin of V V is presented in Appendix A. Admittedly, this is a simplified model like the Ising model for the electronic structure of C 60, the hydrogen atom of the main fullerene family of C 60n molecules. However, its understing is a convenient stepping stone to the study of more complex models. The spectroscopy of, for instance, the hydrogen atom can be analytically solved by using several different approaches. These lead to the same analytical expressions for the eigenvalues eigenvectors. In spite of the fact that modern computers can easily obtain numerical expressions for them, it is useful to have analytic results. In fact, analytical solutions have always been pursued in the area of spectroscopy of atoms, molecules, clusters. In this paper, we pursue several approaches to the analytical study of the spectroscopy of an important molecule. We would like to emphasize that the approaches described here are not intended to substitute more traditional methods, but to present alternative viewpoints complementary results. III. RECURSION METHOD APPROACH A. Formulation We first focus on the -states Hamiltonian Eq.. Let us begin with a brief outline of the recursion method 8 for obtaining the eigenvalues eigenfunctions. First, one must choose an appropriate normalized starting state f 0. Further states are iteratively generated by the three-term recurrence relation H f n a n f n b n f n b n f n, 4 with the condition b 0 f 0. Here, the real parameters a n b n are determined by a n f n H f n b n f n H f n f n H f n. By convention, the b s are chosen to be positive. The process terminates at f N with the last recurrence H f N a N f N b N f N. The constructed orthonormal 5 6

3 ANALYTICAL APPLICATION OF THE RECURSION AND... states f 0, f,..., f N along with the parameters a 0,a,...,a N b,b,...,b N thus constitute the chain model 8 of a given Hamiltonian. The representation of H in the new basis f n is in a tridiagonal form. The energy levels can be achieved by constructing the following polynomials with P (E) 0 P 0 (E) : P n E E a n P n E b n P n E b n. 7 The eigenvalues E are determined by the N zeros of the last polynomial P N (E) 0, with an arbitrary nonzero b N. It follows then that the eigenfunctions are N N n 0 P n E f n, 8 where N N- n 0 P n (E ) / are the normalization constants for the eigenfunctions, P 0 (E ) always equals. So far, we have briefly outlined the recursion method. We now come to the application of this approach to the electronic structure of a fullerene molecule. For convenience, we work in units of the single-bond hopping integral. The hopping amplitude is therefore for every single bond t for each double bond. It has been pointed out that the value of t is about.. We will use two alternative sets denoted by A B) of starting states. Each set consists of two initial states from which the whole energy spectrum can be derived, the final results do not depend on the choice of initial bases. Both sets lead to the same solutions for the eigenvalues eigenfunctions. From the physical point of view, this consistency ensures the equivalent interpretation of the results from these two sets of initial states. We denote by j the p z orbital centered at the jth atom. For convenience of visualization, in all our figures we flatten the truncated icosahedral structure of C 60 into a plane. Note that labelings vary according to the different choices of initial states. B. Case A The first set of initial states starts from a five-atom pentagon ring see Fig. a a six-atom hexagon ring see Fig. b. Starting from a pentagon ring, we choose the initial state u 0 as a linear combination of the five orbitals on it, u 0 5 j. 5 j Here the labeling is as shown in Fig. a. From now on, H will denote H unless specified otherwise e.g., H ). As an illustration, we present here the calculation of the first recurrence, H u t 6 ) 3 t 7 ) 4 t 8 ) 3 5 t 9 ) 4 t 0 )] 9 FIG.. Flattened C 60 molecule obtained by stretching a a pentagon b a hexagon, with the site labels used in cases A B in the recursion method ) t ) u 0 t u. We then obtain a 0, b t, u / 5 0 j 6 j. Following the same procedure, we can construct further states obtain the parameters a s b s at the same time. The beauty of the recursion method for C 60 lies in the fact that the recurrence terminates very quickly, exactly at u 7. As a result, we have exact very concise formulas for a 0 through a 7, b through b 7.

4 644 YEONG-LIEH LIN AND FRANCO NORI 53 TABLE I. States parameters for case A in the recursion method approach. It is interesting to notice that a n a 7 n b n b 8 n for n 4,5,6,7. Starting from a pentagon ring Starting from a hexagon ring n u n a n b n v n a n b n j j 5 j 6 j 0 t 0 0 j j t j j j j t j j t 6 5 j 6 j j j t 6 6 j ( ) j t 6 j 7( ) j j 4( ) j j 3 j 6 ] t 6 4 j 3( ) j j 3 j 6 ] j 3( ) j (j 3) (j 6) ] 0 t 6 5 j 4( ) j (j 3) (j 6) ] 6 j 7( ) j 0 t 6 6 j ( ) j t Similarly, starting from a linear combination of the six orbitals on a hexagon ring, we have the initial state v 0 as v 0 6 j j, 6 j 0 where the labeling is referred to Fig. b. The choice of alternating signs, ( ) j, meets the threefold symmetry requirement of a hexagon ring in C 60, with alternating single double bonds. Again, the recurrence terminates at v 7. We then obtain another group of parameters a s b s. All these u n v n states, as well as their respective parameters a n b n, are summarized in Table I. Through these two groups of a s b s, we can, respectively, construct two polynomials P 8 p (E) P 8 h (E) which are of eighth degree in E. Here the superscript p for pentagon (h for hexagon refers to the polynomial constructed from the group of a s b s generated by u 0 ( v 0 ). The roots of these two polynomials can be analytically obtained correspond to the electronic energy levels. It follows then that we have a total of 6 distinct eigenvalues. When t, a common root exists for both polynomials P 8 p P 8 h. We thus have 5 distinct energy levels. It is also straightforward to obtain the eigenvectors through Eq. 8. C. Case B In this approach, we exploit the symmetry property that the inversion operator leaves the C 60 molecule invariant. We therefore take the first second starting state 0 ( 0 )as a linear combination of the orbitals on two opposite i.e., antipodes pentagon hexagon rings 0 6 j j P j ), j where the site labels are presented in Fig. a b for the first second starting state. Here P sts for parity, with the value or. Note that atom j atom j are antipodes. Following the same procedure, we find the very convenient remarkable result that the recurrence terminates even faster at 3 ( 3 ) for the first second initial state. Due to the two possible values of P, we have two different results for a 3 for each starting state. In Appendix B we present the calculation of the last recurrence, namely those for 3 3. It can be clearly seen that H 3 generates two possible values for a 3 no further state; so does H 3. As a consequence, we obtain four groups of a 0,a,a,a 3 b,b,b 3. All the states n n, as well as their respective parameters a n b n, are listed in Table II. It follows that we can construct four polynomials P p 4 (E), P p 4 (E), P h 4 (E), P h 4 (E). Each one is of fourth degree in E. The superscript p sts for the polynomial constructed from 0 with P equal to, similarly for the others. With the choice of b 4 /b b b 3, these polynomials can be explicitly written as P 4 p E E 4 t 3 E 3 5 t E t 3 t 8 E t t 3 t t, P 4 p E E 4 3 E 3 t t E 3 t 4t 8 E t 4 t 3 t 4 t 4, 0 5 j P j ) 0j P 4 h E E 4 t E 3 3 t E t 3 t E t t t,

5 ANALYTICAL APPLICATION OF THE RECURSION AND... TABLE II. States parameters for case B in the recursion method approach. The parity P can be or. Starting from two opposite pentagon rings n n a n b n j j P j ] 0 j 6 j P j ] 0 t 0 0 j j P j ] t j j P j ] ( Pt) Starting from two opposite hexagon rings n n a n b n j ( ) j P j ] t j 7( ) j P j ] 0 5 j 4( ) j j 3 j 6 P( j (j 3) (j 6) )] t 4 j 3( ) j j 3 j 6 P( j (j 3) (j 6) )] Pt P 4 h E E 4 E 3 t t E t t E t t t. By analytically solving these four polynomials, we obtain the same 6 eigenvalues obtained above case A). Similarly we can obtain the eigenvectors, which are also equal to those obtained from the alternative set A of initial states. D. Results for the -state eigenvalues eigenfunctions of C 60 In Table III, we summarize the eigenvalues the corresponding eigenvectors for the case t. In Table IV, we present the closed-form eigenvalues including the HOMO LUMO energies explicitly expressed in terms of the single-bond hopping integral t the double-bond hopping integral t. Thus, eigenvalues for the limiting cases t 0or t 0 can also be readily inferred. It is interesting to note that the fivefold degenerate HOMO state is obtained from the initial state 0 with negative parity, made of orbitals on two opposite hexagons with a threefold symmetry axis passing through their centers. On the other h, the threefold degenerate LUMO state is obtained from the initial state 0 with negative parity, consisting of orbitals on two opposite pentagons with a fivefold symmetry axis passing through their centers. As to the eigenfunctions, we present in Table V those for the HOMO LUMO in terms of t t /t, the ratio of the hopping matrix elements for single double bonds. Since the respective degree of degeneracy for each eigenvalue can be acquired from the local density of states on a carbon atom discussed in a later section, the other degenerate eigenvectors can be generated by stard group theory analysis. This is outside the scope of the present paper. We therefore only present these eigenfunctions derived from the pure application of the recursion method. It is instructive to notice that the following points further support the results for the degeneracy. They are the sum of the product of energy its corresponding degeneracy equals the trace of the Hamiltonian, which is zero; the number of states with even parity equals that with odd parity; 3 the behavior of the eigenvalues can be easily studied in the limits when either t 0ort 0. E. Relations between alternatives A B It is worthwhile to point out the following relations between the cases A B presented before. First, P 8 p (E) P 4 p (E)P 4 p (E) P 8 h (E) P 4 h (E)P 4 h (E) upto an overall constant factor. In other words, roots solved from P 8 p (E) P 8 h (E) are identical to those solved from P 4 p (E) P 4 p (E) P 4 h (E) P 4 h (E). Second, from Tables I II we can see that, for n 0,, 3, n u n P u 7 n )

6 646 YEONG-LIEH LIN AND FRANCO NORI 53 n v n P v 7 n ). Third, for those eigenvalues E which are common roots of P 8 p P 4 p, or common roots of P 8 h P 4 h, P 7 n E P n E. Also for those eigenvalues E which are common roots of P 8 p P 4 p, or common roots of P 8 h P 4 h, P 7 n E P n E. Here, all the P n (E ) s refer to the polynomials constructed in the case A n 0,,, 3. Fourth, P (E ), P (E ), P 3 (E ) calculated from the two alternatives A B are the same. Fifth, for the eigenvector with respect to the same eigenvalue, the normalization constants N A N B, calculated in A B, respectively, satisfy N A N B. From the above five properties, the equivalence of results from both alternatives becomes clear. As an example, for an eigenvalue E which is a common root of P 8 p P 4 p,we can obtain the eigenvector from A as 7 N An 0 P n E u n N A P 0 ( u 0 u 7 ) P P ( u u 5 ) P 3 3 N Bn 0 P n E n, u u 6 ) u 3 u 4 ) where the last equality is just the result obtained directly from B. In fact, from the results of alternative A, we can underst the parity property associated with C 60. IV. METHOD OF MOMENTS APPROACH A. Methodology application In this method, the first central task is the computation of moments, defined by M l j H l j, 3 where the order l is a positive integer. Note that M 0 when l 0. The physical meaning of the above quantummechanical expectation value is as follows. The Hamiltonian 3 TABLE III. The eigenvalues E the corresponding eigenvectors (/N ) n 0 P n (E ) f n for C 60 with t. Recall that P 0 (E )is always equal to. Here n ( n ) denotes n ( n ) with P. Also (9 5) (9 5). Notice that the HOMO energy is ( 5)/ the LUMO energy is ( 3 5 )/4. E P (E ) P (E ) P 3 (E ) N f n 3 6 n 6 n ( 3)/.303 (3 3)/ (3 3)/4 / (39 9 3)/4 n ( 3)/.303 (3 3)/ (3 3)/4 / (39 9 3)/4 n ( 3 5 )/4.438 ( 3 5 )/4.80 ( 3 5 )/ ( 3 5 )/ ( 5 ) ( 5 ) ( 5 ) ( 5 ) ( 5 ) ( 5 ) ( 5 ) ( 5 ) (5 5) 6 (5 5) 6 (5 5) 6 (5 5) 6 n n n n 0 3 n 3 5 n (3 5)/.68 ( 5)/ ( 5)/ (3 5)/ (5 5 5)/ n (3 5)/ 0.38 ( 5)/ ( 5)/ (3 5)/ (5 5 5)/ n ( 5)/.68 ( 3 5)/ ( 5)/ ( 5)/ (5 3 5)/ n ( 5)/ 0.68 ( 3 5)/ ( 5)/ ( 5)/ (5 3 5)/ n ( 7)/.56 ( 3 7)/ (5 7)/ n ( 7)/.56 ( 3 7)/ (5 7)/ n

7 ANALYTICAL APPLICATION OF THE RECURSION AND... TABLE IV. The eigenvalues the corresponding degree of degeneracy for C 60 with a single-bond hopping-integral t a double-bond hopping-integral t. Note that t.t. The characteristic polynomials from which those eigenvalues are solved are indicated in the left-h column. Here 6 t 8( 5)t t 0(3 5)t ] 6 t 8( 5)t t 0(3 5)t ]. Also satisfy (6 t 3 4 t t t t 5 t 3 ) 54( 3 3 ) 3 (64t 4 60 t 3 t 88 t t 00 t t 3 5 t 4 )] 8 (3 3 ). Notice that the fivefold degenerate HOMO threefold degenerate LUMO energy, as indicated, is solved from the polynomial constructed from the starting state 0 ( 0 ) with negative parity which consists of orbitals on two opposite hexagons pentagons with a threefold fivefold symmetry axis passing through their centers. P 4 p P 4 p P 4 h P 4 h Energy Degeneracy (t t ) (t t )/3 5 (t t )/3 3 5 (t t )/ k 5 t /4 3 [( 3 5 t / t ]/4 LUMO t /4 3 (t 5t 4t )/ 4 (t 5t 4t )/ 4 t 5 t / 3 t ( 5 t / 3 (t 5t 4t t 4t )/ 5 (t 5t 4t t 4t )/ HOMO 5 (t 5t 8t t 4t )/ 4 (t 5t 8t t 4t )/ 4 TABLE V. The HOMO wave function N 3 n 0 P n (E ) n LUMO wave function N 3 n 0 P n (E ) n for C 60 in terms of t.. Recall that P 0 (E ). Here n sts for n with P n for n with P. Also 6 t (4t 5)( 5 ). HOMO E ( 4 t 4 t 5)/ P (E ) t ( 4 t 4 t 5)/ P (E ) t (3 4 t 4 t 5)/ P 3 (E ) t ( 4 t 4 t 5)/ N 6 t 6 t 5 (3 t ) 4 3 t 4 t 5 E P (E ) P (E ) P 3 (E ) N LUMO ( 3 5 )/4 (5 5 )/4t 4( 5 )t ( 5 ) /6t 4( 5 )t 4 5 ( 5 ) /6t 80 t 40 ( 5 )t 50 (3 5) (4 5t 5 5 5) 6 t H ( t) ( ) 3 ( ) 3 labels as shown in Fig.. For simplicity without any loss of generality, we choose orbital to be our starting state. Now, for l, starting from vertex following the connectivity of C 60 Fig., we write down the factors t,, on the vertices, 3, 3, respectively. A similar procedure holds for l. Starting from the three resulting vertices with respective factors, we then write down ( t)( ) t on the vertices 4, 4, 4, 4 ; ( )( ) on the vertices 5 5 ; ( )( ) ( )( ) ( t)( t) t on the vertex. Our strategy here is as follows: each time the power of the kinetic-energy Hamiltonian increases by one, we move to H is applied l times to an initial p z electron state j, localized at carbon site j. Each time H is applied, the electron gains a certain amount of kinetic energy depending upon the bond single or double it travels. This enables the electron to hop through l bonds, reaching the final state H l j. The moment M l just equals the total kinetic energy gained by the electron returning to the starting site j after hopping l steps. It is obvious that M l will be zero when the l-hops path does not return to the starting site. In other words, M l 0 when there is no path of l hops for which the electron may return to the initial site. For the case t, the absolute value of the moment M l is the total number of closed paths of l steps starting ending at the same site. The spirit of this approach follows Feynman s program: to compute physical quantities from sums over paths. The moments can be calculated analytically by h, as well as by computer using a symbolic manipulation program. Below we describe these two implementations starting from the former. Let us first examine the action of the Hamiltonian on an arbitrary state j. This results in three nearestneighbor atom states with an additional factor accounting for the respective bond hopping energy. For example, FIG.. Site labels used in the moments method. Independent vertices are those with j running from to 4.

8 648 YEONG-LIEH LIN AND FRANCO NORI 53 the adjacent nearest-neighboring vertices. Also the factor on each vertex is the sum of the factors on the three nearestneighbor vertices times the bond hopping integral between vertices. It is then straightforward, with the aid of the flattened C 60 graph, to obtain all the factors on the available vertices for any power of the Hamiltonian. From its definition, it is evident that the moment of order l is just the factor on the vertex for H l. For example, M 0 i.e., the number of closed paths obtained by moving one step is zero M t. By following this strategy, we can generate, one by one, all the moments to any order. It is worthwhile to notice the mirror-symmetry between the left right halves of C 60 for the atom vertices bonds. So we need only concern ourselves with the factors on the vertices in the right part. Furthermore, because of the geometrical equivalence, with respect to vertex, of eight pairs of vertices ( j j in Fig., the total number of independent vertices can be reduced to 4. All calculations up to this point can be done analytically by h. An alternative analogous procedure can be implemented by using symbolic manipulation software on a computer. First, we define an auxiliary quantity, W l ( j), which is the sum over all possible paths of l steps on which an electron may hop from the vertex to the vertex j. From the connectivity of C 60, we can then construct 4 independent recurrence relations. For instance, W l () tw l () W l (3), W l () tw l () W l (4), W l (3) tw l (4) W l () W l (5). The recurrence relations state that the vertex j can be reached by taking the l th step from the three nearest-neighbor vertices. The factors t account for the connecting bond-hopping integral. With the initial conditions W 0 () W 0 ( j) 0 for the rest of j s, we can obtain the moments to any order as M l W l (). We list M 3 through M 9 here: 0, t 4 8 t 6,, t 6 8 t 4 4 t 3 48 t 0, 4 t 4, t 8 3 t 6 6 t 5 84 t 4 48 t 3 56 t 70, 54 t 4 8 t 3 6 t 7, for l 3,4,...,9. The moments M 0 through M 3 the highest order moment needed to obtain the entire energy spectrum are not presented here. The correctness of the calculated moments M l is assured by the consistency of the results from these two approaches. It is evident that through these two implementations we can also obtain the quantities H l j for j which can be appropriately interpreted as the sum-over-paths between sites j. For instance, H l j just equals W l ( j). To obtain the energy spectrum, we again utilize Eq. 7. So the next step is to express the parameters a n b n in terms of the moments. We employ the following formulas: define the auxiliary matrix M with the first row elements defined as M 0,l M l. The other rows are evaluated by using only one immediate predecessor row: M n,l M l n,l M n, M n,l M n, M n, M n,k M n,l k, k 0 n ;l 0,,... 4 The a n s b n s are obtained from the elements of the second third columns as TABLE VI. Parameters a n b n for n 0,,...,9 calculated from the moments M l with t in the moment method for C 60. Numbers in the denominator are relatively prime to those in numerator all numbers in square root are primes. n a n b n a n M n, b n M n, M n,, 6 here n 0,,,... Note that elements in the first column M n,0 are always equal to. We find analytically that b 5 exactly equals 0 for t b 6 exactly equals 0 for an arbitrary t. Below we discuss the t case similar results can be obtained for t. The exact vanishing of b 5, in the case t, indicates the truncation at b 5. Thus, we expect 5 eigenvalues. It also turns out that the highest order of moment we need is l 30. The moments M 4 through M 0 for t are 5,, 9, 8, 607, 306, 474, 3080, 3 7, 9 78, , 79 00, , 57 54, , , The moments M through M 30 were computed but are not shown here. Through the calculated parameters a 0,a,...,a 4 b,b,...,b 4 we only present a 0,...,a 9 b,...,b 9 in Table VI, we can construct the polynomial P 5 (E) by using Eq. 7. By solving P 5 (E) 0, we thus obtain 5 energy levels. The results are exactly identical to those obtained from the recursion method approach. B. Alternative application of the moments method In the above description of the moments approach, the main ingredient is the computation of the moments for a natural choice of state centered at the atom labeled by. However, it is worthwhile to incorporate the inversion symmetry property. Therefore, instead of focusing on a single localized state, we turn to the computation of the moments with respect to states I 4 ), 7 where atoms labeled by 4 in Fig. are antipodes. It is a simple exercise to construct the following identity for the moments defined by I H l I, I H l I H l H l

9 ANALYTICAL APPLICATION OF THE RECURSION AND... TABLE VII. Parameters a n b n computed from the moments I H l I with t. From I H l I From I H l I n a n b n a n b n These moments can then be readily obtained since the quantities H l H l 4 are already available. Note that the lowest order for the appearance of a nonzero H l 4 is l 9. This is because the shortest path for vertex to reach vertex 4 contains 9 steps. We thus obviously have I H l I M l for l 8. For the case t, the highest order we need for I H l I ( I H l I )isl 4 (6), because b 7 (b 8 ) calculated from the moment I H l I ( I H l I ) gives exactly 0. Through the computed parameters a 0,...,a 6 b,...,b 6 a 0,...,a 7 b,...,b 7,weanalytically obtain 7 (8) eigenvalues which are identical to those belonging to the P (P ) category from the recursion method. The moments I H l I, for l 9,...,4, are 3, 439, 378, 3 339, , Also the moments I H l I are 300, 43, 88, 30 5, 6 000, 6 779, 5 080, 57 83, respectively, for l 9,...,6. Their respective set of parameters a n s b n s are presented in Table VII. The result that a n b n for n 0,,, 3 in Table VII are identical to those in Table VI comes from the fact I H l I M l for l 8. For an arbitrary t, we need the moments I H l I up to order 6. The moments I H 9 I through I H 3 I are listed in Table VIII, while I H 4 I, I H 5 I, I H 6 I were computed but are not presented here. We analytically find that b 8 calculated from these two sets of moments exactly equals 0 in both cases. Consequently, consistent results for the eigenvalues are recovered. In this section, we have presented an unconventional choice of initial states concentrated on the moments with respect to these states. It is shown that this approach is even more efficient in analytically obtaining the energy eigenvalues. V. RELATIONSHIP BETWEEN THE RECURSION AND MOMENTS METHODS Generally speaking, the moments method is closely related to the recursion method, especially in the aspect that both methods lead to the same results for the parameters a n b n. In this section, we illustrate this point by showing that the same expressions for the parameters in the recursion method case B can be obtained through the moments method. As the initial states are 0 0 in this case TABLE VIII. Moments sums over paths I H l I, with an arbitrary t, for l 9,...,3. B, the moments we now need to compute are 0 H l 0 0 H l 0. It is straightforward to find that 0 H l 0 H l H l 3 H l 5 Order (l) Moment ( I H l I ) 9 54 t 4 4 t 3 6 t 7 0 t 0 50 t 8 40 t t t t t 3 80 t t 6 76 t t t 3 3 t 330 t 7 t 0 80 t 9 0 t 8 04 t t t t t 3 69 t t 8 78 t 7 4 t t t t t 430 Order (l) Moment ( I H l I ) 9 54 t 4 t 3 6 t 7 0 t 0 50 t 8 40 t t 6 96 t t t 3 80 t t 6 88 t 5 90 t 4 76 t 3 3 t 330 t 7 t 0 80 t 9 0 t t t 6 30 t t 4 00 t 3 69 t t t t 6 76 t t 4 6 t t 430 P H l 4 H l H l 9 0 H l 0 H l H l H l 3 H l 7 H l 4 P H l 4 H l 3 H l H l H l Notice that the site labels on the right-h sides of the above two equations refer to Fig.. Through the techniques for the calculation of these quantities previously discussed, the moments 0 H l 0 0 H l 0 can be readily obtained. Anticipating the termination at b 4, we only need these moments up to order 8. In Table IX we give the moments with respect to 0 0. By utilizing Eqs. 4 6, we thus obtain a n s b n s which are consistent with those in Table II derived directly from the recursion method. To illustrate this consistency, we explicitly present here the results for the b n s. From 0 H l 0, we have

10 650 YEONG-LIEH LIN AND FRANCO NORI 53 TABLE IX. Moments sums over paths 0 H l 0 0 H l 0 for l,...,8. Order (l) Moment 0 H l 0 t t 8 4 t 4 4 t t 4 t 3 40 t 3 6 t 6 30 t 4 8 t 3 0 t t 6 6 t 5 t 4 (36 P)t 3 8 t 8 8 t 8 5 t 6 3 t 5 (396 4P )t 4 (6 P )t t 56 Order (l) Moment 0 H l 0 t t t 3 t 3 3 t 5 t 3 4 t 4 4 t 3 0 t 0 t 5 5 t 5 5 t 4 6 t 3 5 t 0 t 8 6 t 6 6 t 5 4 t 4 48 t 3 60 t 38 t 3 7 t 7 7 t 6 33 t 5 84 t 4 (33 P )t 3 33 t 7 t 8 t 8 8 t 7 44 t 6 3 t 5 (66 P )t 4 (336 4P )t 3 84 t 30 t 34 b t, b, b 3, b 4 ( P )t 0. From 0 H l 0, we have b, b t, b 3, b 4 ( P )t 0. We thus demonstrate the fact that the same results for the parameters a n b n can be obtained by using either the recursion or moments methods. The advantage of the recursion method lies in the fact that we can simultaneously generate the states the parameters. However, it is sometimes difficult to derive the states parameters when the recursion method is applied to some starting state, for example, a single carbon atom state j, while the moment method provides stard procedures to calculate the parameters after the moments are obtained. VI. LOCAL DENSITY OF STATES In the recursion moments methods, the diagonal element of the Green function (E H) can be expressed as a continued fraction 8, G 0 E f 0 E H 0 f E a 0 E a b b E a b N E a N The local density of states (E) for f 0 is related to the imaginary part of G 0 (E) by E lim 0 ImG 0 E i. From the computational point of view, G 0 (E) can be obtained by iteratively applying the following transformation: G n E E a n b n G n E, starting from G N (E) /(E a N ). By substituting the parameters a n b n from either one of the two methods into Eq. using Eq., we thus obtain the local density of states on several initial states. In our two alternative applications of the recursion method, we have used four different starting states: u 0, v 0, 0, 0. Their G 0 (E) s can be written explicitly as G 0 E t E E t E G p 3 G 0 E t E t E E G h, 4 where. G p P t t E {E t [E t E ] } for the state 0 for the state u 0 G h P t t E {E t [E E t ] } for the state 0 for the state v 0.

11 53 65 ANALYTICAL APPLICATION OF THE RECURSION AND... The LDOS around these ring-clusters are plotted in Fig. 3 Fig. 4. Notice that around a pentagon hexagon ring the LDOS is large at low high energies. This is related to the fact that a pentagon hexagon has zero three double bonds five three single bonds. Also from the moments approach, the local density of states on two antipode carbon atoms are plotted in Fig. 5. In principle, they are experimentally accessible by using a scanning tunneling microscope. 9 In one of the moments method approaches, the initial state is ap z orbital on a carbon atom. The local density of states in the case t is plotted in Fig. 6. From Fig. 6, we can obtain the degree of degeneracy for each energy level, which is the respective LDOS value times. The common factor of comes from the consideration that the total number of electrons is 60. For t, conclusions for the degeneracy can be similarly drawn. as the sum of the coefficients for orbitals on the bonds associated with the site i. Similar equations hold for the other A s B s. In the first place, Eqs. 6 9 can be rewritten as V a i V a j V A i V a i V a j V A j VII. SOLUTION FOR THE ELECTRONIC STATES It is apparent that the Hamiltonian H is more complex than H due to the two different couplings. However, an analytic transformation of this Hamiltonian into a simpler one with a single renormalized hopping parameter between sites can be established. 0 The energy states can then be readily solved. Below we first present the transformation of the -states Hamiltonian into a -type one. As a consequence, the resulting solutions of the eigenvalues for the states are obtained. We first write the states eigenfunctions as i C i i,, 5 where C a,b, c represents, respectively, the corresponding coefficient for the orbital sp a,sp b, sp c at each site i. Let us now focus attention on a given atom labeled i. Also let atoms j,k, l be the three adjacent nearest neighbors of the i atom. The energy eigenvalue equation H then reads a i V a j V b i c i, b i V c k V a i c i, c i V b l V a i b i, a j V a i V b j c j, c k V b i V a k b k, b l V c i V a l c l, so on. We then define A i a i b i c i 3 as the sum of the coefficients for orbitals at the site i, B i a j c k b l 33 FIG. 3. The local density of states around ring clusters: a a pentagon, b a hexagon. They are obtained by using the recursion method. Notice that around a pentagon hexagon ring the LDOS is large at low high energies. This is related to the fact that a pentagon hexagon has zero three double bonds five three single bonds.

12 65 YEONG-LIEH LIN AND FRANCO NORI 53 FIG. 4. The local density of states around two opposite i.e., antipodes pentagons with a even parity, b odd parity, two opposite hexagons with c even parity, d odd parity. They are obtained by using the recursion method. Notice that around two antipode pentagon hexagon rings the LDOS is large at low high energies. When the situation A i A j 0 occurs, the secular equation immediately yields V V. 36 In all, there are 30 pairs of similar equation sets. We thus have two energy levels: V V belonging to the bonding states, V V belonging to the antibonding states, each having a degree of degeneracy equal to 30. Furthermore, by respectively summing up Eqs. 6 through 8 Eqs. 9 through 3, we have A i V B i V A i 37 V B i V A i V A j A k A l. By substituting Eq. 37 into Eq. 38, we obtain 38 V V V A i V V A j A k A l. 39 It can be directly recognized that Eq. 39 is entirely equivalent to the problem of a tight-binding Hamiltonian with one state per atom a single nearest-neighbor hopping integral, whose eigenvalue eigenvector solutions are already fully explored in previous sections. Let E (,,...,5) st for the eigenvalues listed in Table III. It follows then that V V V V V E. 40

13 ANALYTICAL APPLICATION OF THE RECURSION AND... FIG. 6. The local density of states for a carbon atom obtained from the moments method with the respective degree of degeneracy shown above each peak. Summarizing, in this section we obtain 90 bonding states 90 antibonding states. Among the bonding antibonding states, 30 states are lumped together at the energy level V V (V V ). The other 60 bonding 60 antibonding states are closely related to the energy spectrum for the states. Finally, in the limit V V, we have V V V E. 4 It is also straightforward to obtain the corresponding eigenvector for an eigenvalue. Since all the A i s are already known, the initial coefficients a i, b i, c i can be calculated. FIG. 5. The local density of states on two opposite carbon atoms antipodes with a even parity, b odd parity. They are obtained by using the moments method. Notice that there are seven evenparity solutions in a eight different odd-parity solutions in b ; for a total of 5 different energy levels for t. Thus, we obtain V V 4 V V V V E V V V E V 9V. 4V 4 The minus sign designates the bonding states, the plus sign designates the antibonding states. VIII. CONCLUSION In conclusion, we use several approaches based on the recursion moments methods in order to explore the electronic structure of a C 60 molecule, obtaining exact closedform expressions for the eigenvalues eigenfunctions, including the HOMO LUMO states, as well as the Green s functions LDOS through alternative methods. These quantities are relevant to the several important experimental techniques which probe the local spectroscopy of molecules; for instance, by using a scanning tunneling microscope, as described in the review in Ref. 9. For comparison purposes, we have also done a direct numerical diagonalization of the full Hamiltonian the results obtained are consistent with those from the previous analytical methods. However, the much more elegant powerful recursion moments methods provide valuable insights closed-form expressions for various quantities characterizing the elec-

14 654 YEONG-LIEH LIN AND FRANCO NORI 53 tronic structure of C 60. Finally, a generalized version of the recursion method can be used to greatly simplify the calculation of the electronic properties of large C 60n fullerenes i.e., C 40,C 540,C 960,C 500,C 60, C 940 ). 3 V i,sp a H j,sp a a s i *a s j t ss a x i *a x j t px p x a y i *a y j t py p y ACKNOWLEDGMENTS a s i *a x j a x i *a s j t spx a s i *a y j a y i *a s j t spy The authors gratefully acknowledge stimulating useful discussions with C. N. Yang F. Guinea. F.N. acknowledges partial support from GE a Rackham grant. APPENDIX A: ORIGIN OF V AND V In this appendix we examine more closely the physical interpretation for the hopping integrals V V. Let i, be a hybrid orbital located at site i. We have i,sp a a s i s a x i p x a y i p y, i,sp b b s i s b x i p x b y i p y, j,sp a a s j s a x j p x a y j p y, A A A3 where i j are nearest neighbors. We also assume that orbitals i,sp a j,sp a lie along the bond connecting i j. It is straightforward to obtain V i,sp a H i,sp b a s i *b s i s a x i *b x i a y i *b y i p A4 a x i *a y j a y i *a x j t px p y. A5 Here s p are the s-level p-level energies the t s are hopping matrix elements between orbitals on nearestneighboring sites. It is evident that, with a suitable choice of a local coordinate system, we can always have i,sp a H i,sp b i,sp a H i,sp c i,sp b H i,sp c A6 i,sp a H j,sp a i,sp b H k,sp c i,sp c H l,sp b. A7 In the above equation, we have assumed that orbitals i,sp b, k,sp c lie along the bond connecting i k orbitals i,sp c, l,sp b along the bond connecting i l. APPENDIX B: CALCULATION OF H 3 AND H 3 In this appendix we present the calculation of the recurrence relations for states 3 3 in the recursion method case B. It is shown that the recurrence terminates at these states two solutions for a 3 are constructed from each state. First, H 3 0 j P j ) j 0 j ( j t j ) P j t j j ) 30 Pt j t P j 0j 30 Pt j Pt P j 0j Pt 3. We thus obtain a 3 ( Pt) b 3. Second, H 3 t ) P t )] Pt ) P )] Pt 3. Therefore, we have a 3 P t b 3.

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