1. Introduction A very simple answer to the above question was found in [4,5]. It turns out that initial excitation can spread from the Vibrations of many fullerenes and fullerites were invesroot mode only to those modes whose symmetry is higher tigated by different experimental and theoretical methods than or equal to the symmetry group GD of the root mode.

(see [1,2] and references in these papers). Although We call the complete collection containing the root mode the majority of such studies are based on the harmonic and all secondary modes corresponded to it a bush of modes.

approximation only, some nonlinear (anharmonic) effects Since no other modes are excited, the full energy is trapped also were discussed in a number of papers. For example, in the given bush. As a consequence of the above idea, we combination modes of second order brought about by can ascribe the symmetry group GD (remember that this is anharmonicity of interactions in C60 fullerene are discussed a group of the root mode) to the whole bush, and in this and the appropriate lines in infrared transmission spectra sense we can consider the bush as a geometrical object.

are reported in [3]. The above effects do not exhaust the It was proved in [4–6] that all modes belonging to a given influence of anharmonicity on the fullerene and fullerite bush B [GD] are coupled by force interactions. It is very vibrational spectra, and we want to consider an application important that the structure of a given bush is independent for these objects of the consistent group-theoretical approach of the type of interactions between particles of our physical for studying nonlinear vibrations in arbitrary physical syssystem.

tems with discrete (point or space) symmetry developed A bush of normal modes can be considered as a dynamical in [4–7]. This approach reveals the existence of new object, as well. Indeed, the set of modes corresponding to nonlinear dynamical objects (or new type of anharmonic a given bush B [GD] does not change in time, while the excitations) in systems with discrete symmetry which we amplitudes of these modes do change. We can write exact call bushes of normal modes. The concept of the bush of dynamical equations for the amplitudes of the modes conmodes can be explained as follows.

tained in the bush B [GD], if interactions between particles In the frame of the harmonic approximation, the set of of our physical system are known. Thus, the bush B [GD] normal modes can be introduced which are classified by represents a dynamical system whose dimension can be irreducible representations (irreps) of the symmetry group G essentially less than that of the original physical system.

of the considered physical system in equilibrium. In this The above properties of bushes of normal modes can harmonic approximation, normal modes are independent of be summarized in the following manner. A normal mode each other, while interactions between them appear when represents a specific dynamical regime in a linear physical some anharmonic terms in the Hamiltonian are taken into system which, upon being exciting at the initial instant t0, account. Let us note that a very specific pattern of atomic continues to exist for any time t > t0. Similarly, a bush of displacements corresponds to each normal mode. As a normal modes represents a specific dynamical regime in a consequence, we can ascribe to a given mode a definite nonlinear system which can exist as a certain object for any symmetry group GD which is a subgroup of the symmetry time t > t0.

group G. The group GD is a symmetry group of the instantaneous configuration of our system in its vibrational 2. Some mathematical aspects of bushes state.

Let us excite at the initial instant t0 only one, arbitrarily of normal modes chosen mode which will be called the root mode. We suppose that all other modes at the initial moment have Let us examine a nonlinear mechanical system of N mass zero amplitude. Let the symmetry group GD and irrep 0 points (atoms) whose Hamiltonian is described by a point correspond to this root mode. Then we can pose the or space group G. Let three-dimensional vectors xi(t) following question: to which other modes can this initial (i = 1, 2,..., N) determine the displacement of the excitation spread from the root mode We will refer to i-th atom from its equilibrium position at time t. The these initially ”sleeping” modes, belonging to the different 3 N-dimensional vector X(t) ={x1(t), x2(t),..., xN(t)}, irreps j ( j = 0), as secondary modes. describing the full set of atomic displacements, can be New approach to nonlinear dynamics of fullerenes and fullerites decomposed into the basis vectors (symmetry-adapted co- F1u, F2g, F2u), 4 (Gg, Gu) and 5 (Hg, Hu) associated with ordinates) of all irreps j of the group G contained in the the group Ih. The infrared (IR) active modes belong to mechanical representation1 the irrep F1u, and the modes, which are active in Raman (R) experiments, belong to irreps Ag or Hg. We found all X(t) = µji(t)( j) = j. (1) bushes of modes for C60 fullerene. There are 22 different i ji j bushes for this fullerene. Let us consider the bush Bcorresponding to the symmetry group GD = C5v Ih.

Here ( j) is the i-th basis vector of the nj-dimensional Only four irreps Ag, Hg, F1u and F2u contribute to it (the i irrep j. The time dependence of X(t) is contained only appropriate invariant vectors are zero for all other irreps of in the coefficients µji(t) while the basis vectors are time the icosahedral group G = Ih)independent. Thus, a given nonlinear dynamical regime of the mechanical system described by the concrete vector X(t) B7: [symmetry C5v]:

can be written as a sum of the contributions j from the Ag(a) - Ih, Hg(a, 0.577a, 0, 0.516a, -0.258a)-D5d individual irreps j of the group G.

Each vibrational regime X(t), can be associated with a F1u(0, 0, a) - C5v, F2u(a, 0.258a, 0.197a) - C5v. (4) definite subgroup GD (GD G) which describes the symmetry of the instantaneous configuration of this system. Now The arbitrary constants entering into the description of the following essential idea is proposed. The subgroup GD different invariant vectors are not connected with each other.

is conserved in time; its elements cannot disappear during As all invariant vectors listed in Eq. (4) are one-parametric time evolution except for the case of spontaneous breaking (their arbitrary constants are denoted by the same symbol a of symmetry which we will not consider in the present only for clarity), it is clear that the bush B7 depends paper. This is the direct consequence of the principle of on four arbitrary constants (one constant for each of the determinism in classical mechanics.

four irreps). The structure of the bush B7 (see Eq.(4)) Introducing the operators acting on the 3N-dimenshows that there exist only four contributions j to the sional vectors X(t), which correspond to the elements g G appropriate dynamical regime X(t). We denote them3 as acting on the three-dimensional vectors xi(t), we can write [Ag], [Hg], [F1u], [F2u]. The invariant vectors listed the above condition of conservation of GD as a condition of in Eq. (4) permit us immediately to write the explicit form invariance of the vector X(t) under the action of the elements of the dynamical regime X(t) corresponding to the bush Bof the group GD by replacing the arbitrary constants with the four functions of time µ(t), (t), (t) and (t) X(t) =X(t), g GD. (2) X(t) =[Ag] +[Hg] +[F1u] +[F2u] Combining Eqs. (1) and (2) one obtains (for the details see [6]) the following invariance conditions for individual = µ(t)[Ag] +(t) 1[Hg] +0.5772[Hg] irreps j (j GD)cj = cj. (3) + 0.5164[Hg] - 0.2585[Hg] + (t)3[F1u] Here j GD is the restriction of the irrep j of the + (t) 1[F2u] +0.2582[F2u] +0.1973[F2u]. (5) group G to the subgroup GD, i.e. the set of matrices of j which correspond to the elements g GD only. The Eq. (5) is a consequence of the relation of the group G nj-dimensional vector cj in Eq. (3) is the invariant vector in and its subgroup GD only, and now we should take into the carrier space of the irrep j corresponding to the given account the concrete structure of our physical system to subgroup GD G. Note that each invariant vector of a find the explicit form of the basis vectors ( j) of the irreps i given irrep j determines a certain subspace of the carrier entering into Eq. (5). They can be obtained by conventional space of this representation, and the total number of arbitrary group-theoretical methods, for example, by the projection constants upon which the vector depends is equal to the operation method. The basis vectors of the irreps determine dimension of this subspace. If in solving Eq. (3) we find that the specific patterns of the displacements of all 60 atoms of cj = 0, then the irrep j does contribute to the dynamical the C60 fullerene structure.

regime X(t) with the symmetry group GD. Moreover, the It is important to note that each of the irreps Ag, Hg, F1u invariant vector cj determines the explicit form of the mode and F2u is contained in the vibrational representation of of the irrep j belonging to the bush of modes associated C60 fullerene several times, namely, 2, 8, 4 and 5 times, with the given nonlinear dynamical regime.

respectively. (These numbers are equal to the numbers of We shall illustrate the general statements of bush theory fundamental frequencies of normal modes associated with with C60 fullerene having the buckyball structure and the icosahedral symmetry group G = Ih. There are 10 irreSince some elements of the matrices of multidimensional irreps of the group G = Ih are irrational numbers, we keep only three digits after the ducible representations of dimensions 1 (Ag, Au), 3 (F1g, decimal point when we write the invariant vectors.

1 Considering vibrational regimes only, we can treat as a 3N - 6 Hereafter we write the symbol j of the irrep j generating the vibrational representation of the group G. contribution j in square brackets next to symbol.

Физика твердого тела, 2002, том 44, вып. 556 G.M. Chechin, O.A. Lavrova, V.P. Sakhnenko, H.T. Stokes, D.M. Hatch the considered irreps). As a consequence, we must treat above discussed infrared active bush B7 we have the time-dependent coefficients in Eq. (5) as vectors of [F1u](root) =O(), [F1u](secondary) =O(3), the appropriate dimensions. Because of this we ascribe a new index (k) to the basis vectors determining the [F2u] =O(3), [Ag] =O(2), [Hg] =O(2).

number of times (mj) the irrep j enters into the vibrational representation. Each contribution j ”splits” into mj Here is an appropriate small parameter characterizing the copies jk, where k = 1, 2,..., mj and, therefore, value of the root mode.

Thus, in the case of weak nonlinearity, the contributions mj of different irreps can be of essentially different value. This X(t) = j = jk. (6) property seems to be important for the interpretation of the j j k=vibrational spectra of bushes of modes.

For the case of the bush B7 we have [Ag] = 1[Ag] +2[Ag], [F1u] =1[F1u]+2[F1u]+3[F1u]+4[F1u], 4. Conclusion etc.

The bush B7 in the C60 fullerene structure forms a In the present paper, we consider a new type of possible 19-dimensional dynamical object: its evolution is denonlinear excitations — bushes of normal modes — in scribed by the dynamical variables listed below as comvibrational spectra of fullerenes and fullerites, using as an ponents of the four vectorial variables µ(t), (t), (t) and example the C60 buckyball structure. We believe that special (t): µ(t) = [µ1(t), µ2(t)], (t) = [1(t),..., 8(t)], experiments for revealing the bushes of vibrational modes in (t) =[1(t),..., 4(t)], (t) =[1(t),..., 5(t)].

their pure form will be important for further elucidation of Thus, although only four of the ten irreps contribute to the role of these fundamental dynamical objects in various the bush B7, its dimension is equal to 19 because several phenomena in fullerenes and fullerites. It seems that such copies of each of these four irreps are contained in the experiments may be similar to those by Martin and others full vibrational representation of C60 fullerene. We cannot reported in [3]. However, unlike these experiments, we must predict the concrete evolution of the amplitudes of the use the monochromatic incident light with frequency close bush modes without specific information of the nonlinear to that of the root mode and with polarization along the interactions in the considered physical systems, but we can symmetry axis of the chosen bush.

assert that there does exist an exact nonlinear regime which The first-principle calculations are desirable for obtaining involves only the modes belonging to a given bush.

the coefficients of the anharmonic terms in C60 fullerene for a more detailed description of the bush dynamics.

3. Optical bushes for C60 fullerene The concept of bushes of normal modes and the appropriate mathematical methods for their analysis are valid for As was already noted, there are 22 bushes of vibrational both molecular and crystal structures. Such a possibility modes for C60 fullerene structure. Five of them are infrared can simplify the assignment of the different optical lines in active and six are Raman active. We call these bushes by fullerites brought about by both intra- and inter-vibrations of the term ”optical”. The root modes of the optical bushes the C60 molecular clusters.

belong to the infrared active irrep F1u or to the Raman active It will be very interesting to study interactions between irreps Ag and Hg. We want to emphasize that some modes bushes of vibrational modes and electron subsystems in associated with the irreps which are not active in optics can fullerenes and fullerites.

be contained in a given optical bush.

All optical bushes with their symmetry groups (in square References brackets), numbers of irreps contributing to them, and their dimensions (in parentheses) are listed below.

[1] C.H. Choi, M. Kertesz, L. Mihaly. J. Phys. Chem. A104, Infrared active bushes:

(2000).

[2] H. Kuzmany, R. Winkler, T. Pichler. J. Phys.: Condens. Matter.

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