On the influence of clustering processes in the liquid structure on Raman scattering

A mathematical relation has been obtained that makes it possible to calculate the polarizability of a polyatomic molecule in the structure of a cluster. It is shown that the scattered frequencies in the Raman effect are proportional to the square root of the number of particles in the most probable (or average) cluster in the liquid structure. The appearance of frequencies in the far part of the Raman spectrum region is caused by the processes of intermolecular interactions in clusters and the processes of disintegration or formation of cluster systems in the structure of disordered condensed media. According to the proposed model and experimental data in the frequency range 20–1300 cm-1, it has been carried out the comparison of the values of the calculated frequencies of the Raman spectrum and their mutual position, which has shown the adequacy of the proposed model. The cluster model of liquid structure and the methods of mathematical statistics and statistical thermodynamics make it possible to expand the capabilities of the classical theory of Raman scattering in liquids and to predict the position of spectral bands in Raman spectra in the far long-wavelength region of the spectrum. It is revealed that the formation and breakdown of the most probable clusters is associated with the correlations of the most stable clusters (in terms of the number of particles) in a condensed medium with the Fibonacci numbers.


Introduction
At present the classical theory of Raman scattering (Raman effect) is well tested and generally agrees with experimental data, the essence of this theory is as follows. The dipole moment of a molecule in a substance is induced by an electric field of an incident electromagnetic wave, which distorts the general electron cloud of the molecule, and the induced dipole moment will be [1][2][3][4][5] 0 0 μ α cos(ω ) E t = . (1) The molecular system has its own vibration modes k Q , which are usually described by the relations (2) Such vibrations can affect the polarizability of the molecule, which is determined by the formula [1][2][3][4][5] 0 0 α α α ... ( For the induced dipole moment of the molecule, within the framework of the classical theory of the Raman effect, we obtain the relation The first term in relation (4) corresponds to elastic scattering of light without changing the frequency (Rayleigh scattering), the second -anti-Stokes scattering of light with frequency ( ) ω ω k   +   , the third represents Stokes scattering of light with frequency ( )  . Within the framework of quantum-mechanical consideration of the phenomenon of Raman scattering, it is possible to obtain a relation for the ratio of the intensities of the Stokes and anti-Stokes light scattering, which satisfies the experimental data. In Placzek theory, for example, the following formula was obtained [6] ( ) ( ) It is confirmed by experiments under conditions of thermal equilibrium of a thermodynamic system. Disordered condensed media as randomly oriented systems, such as liquids, isotropic polymers and quasicrystalline systems are the most preferable for studying physical objects by the Raman scattering method. The classical explanation of the phenomenon of Raman scattering (RS) currently retains its fundamental importance [1][2][3][4][5][6].

Influence of clustering processes on Raman scattering in liquids
The dipole moment of a dimer in the cluster structure is determined by the number of particles in this cluster. For example, in [8,9,27], a formula was proposed according to which the dipole moment of a molecule in a water cluster is proportional to the square root of the number of particles in the cluster.
The induced dipole moment of the dimer in the cluster structure under the action of a harmonic electromagnetic wave is represented by the relation The average polarizability of a dimer formation α depends on the number of particles in the middle cluster, which includes the dimer, and this dependence can be described by the formula [10] where Ζ is the average number of particles in a cluster, max Ζ is the maximum possible number of particles in a cluster for given parameters of the state of matter, 0 α is the polarizability of a molecule in gases.
The average number of particles in a cluster in the liquid structure is determined by the state parameters, in particular, by the density of the liquid, and this dependence can be represented by the relation [11] ρ exp ρ C where ρ, ρ C are the density and critical density of the liquid-vapor transition. The maximum value of the average number of particles in the cluster is observed in the liquid at the maximum value of the ratio ρ ρ 3.06 C = (for benzene), then max 21 Ζ = . Relation (7) was used to calculate the refractive index for simple and organic liquids on the liquid-vapor equilibrium line in a wide range of state parameters variation within the framework of the theory of effective polarizability of Frenkel L.I. and Gubanov A.I. [12] 2 0 max 4π Calculations of the refractive index according to relation (9) and comparison of theoretical calculations with experimental data for liquid noble gases, cyclic hydrocarbons (arenes) and their halogen-substituted ones have shown that relation (7) is quite acceptable for assessing the average polarizability of a dimer formation in the middle cluster structure [10].
The value enclosed in a square bracket is always less than one, therefore this square bracket in relations (7) and (9) can be considered as an expansion in a power series of some function, for example tg( ) x , and then we write where 2n B are the Bernoulli numbers. Considering only the first two terms in the expansion (10), we obtain 2 max tg( ) Having carried out obvious mathematical transformations, taking into account the fact that it is possible to take the approximation MPM 2021 https://doi.org/10.1051/matecconf/202134401010 we get the ratio Consequently, the average polarizability of a dimer in the middle cluster is determined by the formula The induced dipole moment according to the relations (11) and (14) is determined by the relation On the one hand the value of x is determined by the average number of particles in a cluster; on the other hand, it is related to the frequencies of libration vibrations of dimer formations in the cluster structure, and the relation should be satisfied Using the trigonometric identity ( 2sin sin [cos( The obtained relation (17) coincides in its form with formula (4) of the classical theory of the Raman effect, but formula (17) has several important consequences. First, the integration of clustering processes in liquids leads to the dependence of the maxima of the spectral bands for the observed frequencies in the Raman spectra on the composition of cluster formations, and this dependence is determined by the formula min .
ω ω The minimum frequency of libration vibrations of a dimer formation in the cluster structure min . ω k is determined by the formation enthalpy dim.k H ∆ and the inertia moment dim.k J of the dimer configuration [13] dim. min . dim.
Polyatomic molecules of complex geometrical shape can form several possible configurations of dimers, differing in the inertia moment about the main axes and the energy of configuration formation. It has been established now that benzene molecules can form four types of dimeric configurations: Sandwich (S), T-shaped, parallel-displaced (PD), and Chain-configuration (CC) -configurations differing in the value of equilibrium distances between the centers of molecule mass, energy formation of the configuration (binding energy of the dimer) and the inertia moment about the axis passing through the center of configuration mass (Figure 1) [7, 14, 15]. Thus, according to formula (19), for liquids with polyatomic molecules, there is a set of minimum frequencies, each of which forms a spectral series in the Raman spectrum of the liquid.
With a spontaneous breakup of a cluster, the number of particles in its composition changes and becomes equal j Ζ , and then the frequency of libration vibrations of the dimer in the newly formed cluster will be dim. dim.
2 ω The change in frequency is accompanied by the emission or absorption of a quantum of energy with a frequency To calculate the frequency of the emitted energy quantum during the spontaneous decay of the initial cluster, it is necessary to establish the decay law of cluster systems in a condensed medium, on the basis of which the correction root in the parenthesis of formula (21) can be calculated.
Assuming that the root in the parenthesis of this formula remains constant, we can put this value equal to the square root of the constant «golden ratio», i.e. put dim.
As a result, a cluster is formed with the number of particles equal to three times the Fibonacci number, but such a cluster is unstable and splits into three identical Fibonacci clusters. For example, the most probable number of particles in an argon cluster near the melting point is 8 n F = , therefore, according to formula (23), it is necessary to write A cluster with a number of particles equal to 24 is not among the most stable clusters according to the data of mass spectroscopic experiments [16] and splits into three identical Fibonacci clusters.
The merging of four or more Fibonacci clusters at the same time has a small probability; however, the scheme for implementing such a process can be easily obtained by generalizing formula (23) and presenting the decay process in the form A cluster with 32 particles, according to mass spectroscopy data, is a «magic» cluster [16], and it is relatively stable and may not decay.
Let us note one more scheme for the formation of cluster systems, when Fibonacci clusters are formed with the number of particles with consecutive Fibonacci numbers [17,18]

Results of experimental verification of the cluster model using spectroscopic data
To test the obtained relations, which take into account clustering processes in Raman scattering in liquids, we have chosen liquid benzene and toluene, which have great practical application; moreover, these liquids are model substances in theoretical studies and they are well studied.
Raman and IR spectra of liquid benzene in the frequency range 400-3300 cm -1 have been studied in detail; it suffices to mention the work of Bertie and Keefe in the frequency range 11.5-6200 cm -1 [19], Chelli at al. [20], Badoglu and Yurdakul [21]. In the crystalline state, Raman and IR spectra of benzene in the frequency range 55-3131 cm -1 were studied by Kearley, Jonson, Tomkinson [22]. Within the total errors, the spectral measurements of the authors of [19][20][21][22] are in good agreement with each other. Table 2 shows data on Raman spectra in the far spectral region for benzene, obtained by various authors.
The formation energy of various benzene dimeric configurations has been calculated by many authors using quantum chemistry methods using various interaction potentials. Some of the results of such calculations are shown in Table 1.
Having data on the formation energy of various dimeric configurations and their inertia moments with respect to the principal axes, one can calculate the minimum frequencies of libration vibrations of dimers in the cluster structure using the formulas We have noted that the ratio of frequencies is ,that is actually observed in the Raman spectra of the liquid. Table 1. Binding energies and equilibrium distances of configurations of benzene dimers (E in kcal/mol and R in А) [23][24][25][26].

Configurations of the benzene dimer / Basis [26]
Sandwich S  Table 2 shows the values of the minimum frequencies of libration vibrations of dimers in the structure of benzene clusters for various configurations. Arranging the entire set of frequencies obtained by formulas (22) and (26) in ascending order, we obtain the Raman spectrum of benzene, recorded in experimental studies. Table 3 shows the results of theoretical calculations of the Raman spectrum frequencies according to the proposed method and comparison with the experimental data of various authors. In the frequency range 30-350 cm -1 , which corresponds to cluster formations with the number of particles from 2 to 55 in the structure of liquid benzene, there is a good agreement between the theoretical and experimental values of frequencies within the total comparison errors. In the frequency range 350-2000 cm -1 , which corresponds to cluster formations of up to 1500 particles in their composition, the calculation error can reach 10%, which can be explained by the fact that, firstly, vibration degrees of freedom of molecules can be excited at these frequencies, and secondly, the formation of clusters with more than 100 particles is unlikely in organic liquids, for such liquids the formation of small clusters is typical.

Conclusion
In the structure of organic liquids within the framework of the classical theory of Raman scattering the integration of clustering processes leads to the fact that the experimentally observed frequencies in the Raman spectrum are proportional to the square root of the number of particles in the most stable clusters. The appearance of spectral bands in Raman spectra (30-350 cm -1 ) can be explained by libration vibrations of dimer formations of various configurations in the cluster structure.
From a wide variety of clusters in the structure of liquids, a special class of clusters can be distinguished, called Fibonacci clusters, which differ in a number of features of their composition and internal structure. The sequential formation or collapse of a cluster with the number of particles from the Fibonacci series occurs according to the rule when the newly emerged cluster has the same number of particles from the series of Fibonacci numbers, and the ratio of the number of particles in two successively formed clusters, equal to the «golden» ratio 1.618... Φ = , is preserved.