of force constants.

There exists Wigner theorem [23] about group-theoretical classification of the linear nor-mal modes. According to this theorem, the modes are classified by irreducible representations (irreps) of the symmetry group \( G _ { 0 } \) of the system in equilibrium. In this way, we can introduce the basis \( \Phi = \{ \phi _ { j } | j = 1 . . N \} \) in the space of all possible atomic displacements, constituted by the complete set of basis vectors of the irreps entering into the mechanical representation of the considered system. Therefore, any vibrational regime \( \mathbf { X } ( t ) = \{ x _ { 1 } ( t ) , x _ { 2 } ( t ) , \ldots , x _ { n } ( t ) \} \) in this system can be decomposed into the above basis with coefficients depending on time t:

\[ \mathbf { X } ( t ) = \sum _ { j = 1 } ^ { N } c _ { j } ( t ) \phi _ { j } \equiv ( C ( t ) , \Phi ) . \eqno ( 3 ) \]

In this equation, each term \( c _ { j } ( t ) \phi _ { j } \) can be considered as nonlinear normal mode according to the definition (1). Indeed, the vector multiplier \( \phi _ { j } \) determines the displacement pattern of all atoms, i.e. the space structure of NNM, while \( c _ { j } ( t ) \) determines time-evolution of the mode. However, for brevity, we often use the term nonlinear normal mode (or vibrational mode) individually for \( \phi _ { j } \), as well as for \( c _ { j } ( t ) \).

The basis vectors \( \phi _ { j } \) correspond to different irreps \( \Gamma _ { n } \) of the group \( G _ { 0 } \) and, therefore, the displacement vector X(t) in (3) can be written as the sum of contributions associated with individual representations of the equilibrium symmetry group \( G _ { 0 } \):

\[ \mathbf { X } ( t ) = \sum ( \mathbf { C } _ { n } ( t ) , \Phi [ \Gamma _ { n } ] ) . \eqno ( 4 ) \]

Here, \( \Phi [ \Gamma _ { n } ] \) is the set of basis vectors of the irrep \( \Gamma _ { n } \).

According to Wigner theorem, the small vibrations of the molecule associated with the different irreducible representations \( \Gamma _ { n } \) are independent from each other. It means that if one excite (by using the appropriate initial conditions when solving linear differential equations) a dynamical regime X(t) corresponding to a given irrep \( \Gamma _ { n } \), this regime can never leads to excitation of the modes belonging to another irreps in the decomposition (4). Therefore, one can ask: “What will happen if we consider large and, therefore, nonlinear vibrations of the molecule?” The theory of the bushes of nonlinear normal modes starts from this question.

The answer was given in Ref. [6] (see also [8], devoted to discussion of the bush theory). It turns out that there exist certain selection rules for excitation transfer from one mode to another. These rules are originate from some group-theoretical restrictions which can be written as a certain system of linear algebraic equations [6]. In particular, one can deduce from this system that excitation from the mode with the given symmetry group G can

transfer only to those nonlinear normal modes whose own symmetry is higher or equal to G. The above selection rules lead to possibility for existence of bushes of NNMs.

Each bush represents a set of NNMs that conserves the energy of initial excitation until it loses stability because of the phenomenon similar to the parametric resonance with some modes outside a given bush. This phenomenon occurs if amplitudes of some bush modes attain sufficiently large values (see details in [8, 10]).

Every bush possesses its own symmetry that is determined by intersection of all symmetry groups of its modes. As was already mentioned, when the given bush loses stability, it transforms into another bush with lower symmetry and with higher dimension.

Let us consider the simplest bushes for nonlinear vibrations of \( S F _ { 6 } \) molecule using some results obtained in [14]. In the equilibrium state, depicted in Fig. 1, the molecule \( S F _ { 6 } \) possesses point symmetry group \( G _ { 0 } = O _ { h } \). All vibrational modes for this molecule, classified by irreps of the group \( O _ { h } \), can be found in Table. 3 in [14] [three translational modes must be excluded since the central atom (S) is supposed to be immovable].

For the present consideration, we need explicit forms of the displacement patterns of NNMs \( \phi _ { 1 } \), \( \phi _ { 2 } \), \( \phi _ { 3 } \), corresponding to one-dimensional irrep \( \Gamma _ { 1 } \), two-dimensional irrep \( \Gamma _ { 5 } \) and three-dimensional irreps \( \Gamma _ { 1 0 } \). These displacement patterns are given in Table I.

TABLE I. Displacement patterns of the NNMs in \( S F _ { 6 } \) molecule

<table><tr><td>Irrep</td><td>NNM</td><td>Pattern</td></tr><tr><td>Γ1</td><td>\( \phi _ { 1 } \)</td><td>1 \( { \underset { 6 } { = } } ( 0 , 0 , - 1 | - 1 , 0 , \) 0|0, −1, 0|1, 0, 0|0, 1, 0|0, 0, 1|)√</td></tr><tr><td>Γ5</td><td>\( \phi _ { 2 } \)</td><td>\( \begin{array} { r } { \frac { 1 } { \sqrt { 1 2 } } ( 0 , 0 , 2 | - 1 , 0 , 0 | 0 , - 1 , 0 | 1 , 0 , 0 | 0 , 1 , 0 | 0 , 0 , - 2 | ) } \end{array} \))</td></tr><tr><td>Γ10</td><td>\( \phi _ { 3 } \)</td><td>\( \begin{array} { r } { \frac { 1 } { \sqrt { 1 2 } } ( 0 , 0 , - 2 | 0 , 0 , 1 | 0 , 0 , 1 | 0 , 0 , 1 | 0 , 0 , 1 | 0 , 0 , - 2 | } \end{array} \))</td></tr></table>

In this table, for each fluorine atom, according to the numbering in Fig. 1, we point out three coordinates \( x , y , z \), which determines displacement of the nucleus of this atom from the equilibrium position. One can see from Table 1 that the molecule shape in the vibrational regime, corresponding to the mode \( \phi _ { 1 } \) represents, at any moment t, the regular octahedron. Its size vibrates in time becoming larger or lesser in comparison with the octahedron corresponding to the equilibrium state. This NNM is called “breathing” mode. It represents one-dimensional bush with symmetry group \( O _ { h } \). The symmetry group of the breathing mode, \( O _ { h } , \), is higher than that of each other vibrational mode and, therefore, according to the above mentioned selection rules, the excitation from this mode cannot