Theoretical Study of Structure and Vibrational Spectra of Molecular and Ionic Clusters Existing in Vapour over Rubidium Chloride

The geometrical structure and the vibrational spectra of dimer Rb2Cl2, trimer Rb3Cl3, tetramer Rb4Cl4 molecules and heptaatomic Rb4Cl3 , Rb3Cl4 – ions were studied. The cluster molecules and ions had been detected in equilibrium vapour over rubidium chloride previously. The quantum chemical calculations by DFT with hybrid functional B3P86 and MP2 methods were performed. The effective core potential with Def2–TZVP (6s4p3d) basis set for rubidium atom and full electron aug–cc–pVTZ (6s5p3d2f) basis set for chlorine atom were used. The equilibrium configuration was confirmed to be rhomb of symmetry D2h for dimer Rb2Cl2, distorted cube (Td) for tetramer Rb4Cl4 and polyhedral (C3v) for heptaatomic ions Rb4Cl3 + and Rb3Cl4 . For the trimer molecule Rb3Cl3 two isomers have been revealed: hexagonal (D3h) and butterfly-shaped (C2v), the latter has lower energy and is proved to be predominant in equilibrium vapour in a broad temperature range.


Introduction
Several alkali metals including rubidium may chemically combine with halogen to form different neutral and ionic clusters which exist in vapours [1][2][3][4][5][6][7][8][9][10]. These clusters are characterized by different geometrical structures, vibrational spectra, and thermodynamic properties which are mostly depend on the number of atoms composing the species [1]. The formed clusters possess specific electronic, optical, magnetic and structural properties which make them useful in different technical and scientific applications [2,3]. These properties are also strongly depending on size and composition of the clusters. Changing the magnitude and structure of a cluster aggregate allows coming up with new materials of desired properties [11].
Different molecular and ionic associates were detected in vapour over rubidium chloride; they are dimer, trimer, and tetramer molecules and the ions Rb + (RbCl) n (n = 1-4) [22].
Previously the quantum chemical methods has been used to study tri-and pentaatomic ions Rb 2 Cl + , RbCl 2 -, Rb 3 Cl 2 + , Rb 2 Cl 3 - [20]. The ionic clusters were studied by density functional theory (DFT) with the Becke-Lee-Yang-Parr functional (B3LYP5), second order and fourth order Møller-Plesset perturbation theory (MP2 and MP4) using the basis sets aug-cc-pVTZ (5s4p2d1f) for Cl and cc-pVTZ (7s5p4d1f) for Rb [20]. This study aims the investigation of the structure and properties of neutral and heavier ionic clusters of rubidium chloride.

Methodology
The density functional theory (DFT) with hybrid functional the Becke-Perdew correlation B3P86 [23][24][25][26] and second order Møller-Plesset perturbation theory (MP2) implemented into PC GAMESS (General Atomic and Molecular Electronic Structure System) program [27] and Firefly version 8.1.0 [28] were used for calculation of geometrical parameters and frequencies of normal vibrations. The visualization of the geometrical structure, specification of parameters, and assignment of vibrational modes in infrared spectra was done using the Chemcraft software [29] and MacMolPlt program [30]. The effective core potentials with Def2-TZVP (6s4p3d) basis set for Rb atom [31,32] and full electron aug-cc-pVTZ basis set for Cl (6s5p3d2f) atom [32,33] were applied in this study. The basis sets were taken from the EMSL (The Environmental Molecular Sciences Laboratory, GAMESS US) Basis Set Exchange version 1.2.2 library [34,35].
The Openthermo software [36] was used in calculation of the thermodynamic properties including energies ∆ r E and enthalpies ∆ r H°(0) of the reactions which were calculated through the following equations: where ∑E i prod , ∑E i reactant are the sums of the total energies of the products and reactants respectively, ∆ r ε is the zero point vibration energy correction, ∑ω i prod and ∑ω i reactant are the sums of the vibration frequencies of the products and reactants respectively.

Monomer RbCl and dimer Rb 2 Cl 2
The properties of the diatomic RbCl and dimer Rb 2 Cl 2 molecules was computed using three different DFT hybrid functionals, B3LYP5, B3P86 and B3PW91 and Møller-Plesset perturbation theory (MP2) for the purpose of choosing the appropriate method to be implemented in this study and estimation of uncertainties in theoretical values. The results for diatomic molecule RbCl together with the available experimental data are shown in Table 1. Notes: Here and hereafter, Re is the equilibrium internuclear distance (Å), E is the total electron energy (au), ωe is the fundamental frequency (cm -1 ), µe is the dipole moment (D), IEvert and IEad are the ionization energies, vertical and adiabatic, respectively (eV), EA is the electron affinity (eV).
As is seen the calculated internuclear distance (R e ) using three DFT hybrid functionals were overrated by 0.07-0.09 Å when compared with the reference data and that of MP2 was overrated by 0.07 Å. Considering both DFT and MP2 methods, the results obtained show very little variation whereas DFT/B3P86 and MP2 give the best results regarding to the reference. The calculated values of frequency were underrated by 6-9 cm -1 (2.7-3.9%) for DFT methods while MP2 result coincides with the experimental value [38]. The results for dipole moment (µ e ), show high deviation from reference data for the case of MP2, overrated by 0.88 D; and overrated by 0.41-0.47 D for the DFT methods.
The adiabatic ionization energy (IE ad ) was obtained as the energy difference of the RbCl + ion and neutral molecule where the internuclear separation R e (Rb−Cl) was optimized both for neutral and ionic species and vertical ionization energy (IE vert ) was obtained as the energy difference of the RbCl + ion and neutral molecule where the internuclear separation R e (Rb−Cl) was optimized in neutral molecule only and accepted the same for the ion. Electron affinities (EA) were obtained by determining the energy difference between neutral molecule and negatively charged ion. The results obtained for IE and EA using both DFT and MP2 methods agree well with the reference. As the optimization procedure by MP2 was not implemented for the species with multiplicity more than 1 in the software [27,28], the IE ad by MP2 method was not calculated in this study.
For the dimer molecule Rb 2 Cl 2 , a planar cyclic structure of D 2h symmetry ( Fig. 1a) was confirmed to be equilibrium, while linear one (C ∞v ) was found to be a saddle point on the potential energy surface. The calculated properties; internuclear distance, valence angle, vibrational spectra and enthalpy of the dissociation reaction are presented in Table 2. As seen the calculated values of internuclear distance are overrated by 0.04-0.09 Å compared with literature data [45] and the values of enthalpy of dissociation reaction are in agreement with the reference data [22,46].
The absence of low frequencies in the vibrational spectra of Rb 2 Cl 2 indicates the rigidity of the structure; the lowest frequency being about 50 cm -1 .
In the IR spectrum, three vibrational modes ω 4 , ω 5 , and ω 6 have nonzero intensities but due to overlapping of ω 5 and ω 6 only two peaks could be seen which are asymmetrical stretching 171 cm -1 and wagging 52 cm -1 (MP2).
Based on results for the RbCl and Rb 2 Cl 2 molecules, it is worth to mention that the data obtained by MP2 and DFT/B3P86 hybrid functional agree better with the available literature data therefore these two methods were chosen for further computations of heavier molecular and ionic clusters.

Trimers Rb 3 Cl 3 and tetramer Rb 4 Cl 4
For the trimer molecule Rb 3 Cl 3 , three possible geometrical configurations were considered; linear of D ∞h symmetry, hexagonal of D 3h symmetry and butterfly-shaped of C 2v symmetry. The linear configuration was found to be unstable due to presence of imaginary frequencies but the rest two configurations, (Figs. 1b and 1c) were confirmed to be equilibrium. The results are shown in Table 3. Notes: ∆rEiso = E(C2v)-E(D3h) is the relative energy of the butterfly-shaped isomer regarding the hexagonal one (kJ mol −1 ). The reducible vibration representations for Rb3F3 of D3h and C2v symmetry reduces to Γ = 3 + 3 + + and Γ = 6A1 + A2 + B1 + 4B2 respectively. For the C2v butterfly-shaped isomer, the values given in parentheses near the frequencies are infrared intensities (D 2 amu −1 Å −2 ).
The relative concentration of isomers p(C 2v )/p(D 3h ) have been estimated using the same procedure as it was described previously [21]. The enthalpy of the isomerization and relative energy of isomers ∆ r E iso (MP2) for the isomerization reaction Rb 3 Cl 3 (D 3h ) ⇌ Rb 3 Cl 3 (C 2v ) was calculated using Eqs. (2) and (3).
The values of p(C 2v )/p(D 3h ) were obtained for the temperature range between 700−2000 K; the plot is shown in Fig. 2. As is seen the butterfly-shaped isomer is more abundant than hexagonal one and the concentration of the former decreases with temperature increase.  The IR spectra of both isomers of Rb 3 Cl 3 molecule are shown in Fig. 3. The most intensive bands correspond to the stretching vibration modes and bending modes have lower intensities.
When comparing the properties of Rb 3 Cl 3 with Rb 3 F 3 molecule, a similarity may be noted: the same two isomers were revealed for Rb 3 F 3 [21] and the butterfly-shaped isomer was predominant in that case as well.
For the tetramer molecule Rb 4 Cl 4 , the geometrical configuration considered was cube with T d symmetry (Fig. 1d) and it was confirmed to be equilibrium. The geometrical parameters and vibrational frequencies of the tetramer molecule are presented in Table 4.   One internuclear distance R e (Rb-Cl) and one valence angle α e (Rb-Cl-Rb) or β e (Cl-Rb-Cl) are needed to describe the structure. In the IR spectrum, three vibrational modes are active but only one peak could be observed ω 6 (T 2 ) which is assigned to asymmetrical stretching Rb-Cl vibration; other two modes have low intensities.

Heptaatomic Ions Rb 4 Cl 3 + and Rb 3 Cl 4 -
For heptaatomic ions Rb 4 Cl 3 + and Rb 3 Cl 4 two alternative structures were considered, polyhedral of C 3v symmetry and two-cycled with mutually perpendicular planes of D 2d symmetry (Fig. 4); only polyhedral structure of symmetry C 3v was confirmed to be equilibrium. The results for both positive and negative ions are presented in Table 5. The polyhedral structure is described by two internuclear distances R e1 and R e2 and three valence angles, α e (Rb 2 -Cl 5 -Rb 3 ), β e (Rb 1 -Cl 4 -Rb 3 ) and γ e (Cl 4 -Rb 2 -Cl 7 ). Compared to the D 2d structure, the polyhedral configurations possesses lower energy with the energy drop of 87 kJ mol -1 and 83 kJ mol -1 for Rb 4 Cl 3 + and Rb 3 Cl 4 respectively, according to MP2 results.  As is seen in Table 5 the respective geometrical parameters and vibrational frequencies of the positive and negative ions are close to each other. The IR spectra are shown in Fig. 5. In both spectra, the stretching vibrational modes have higher intensities than bending modes.

Conclusion
The properties of the trimer and tetramer molecules and heptaatomic ionic clusters have been investigated theoretically using DFT/B3P86 and MP2 methods. The corresponding parameters of the species calculated by two methods are generally in a good accordance to each other; while the internuclear distances Rb-Cl found with DFT/B3P86 are longer by 0.02-0.04 Å than those found with MP2 method. Among the species considered, for the Rb 3 Cl 3 molecule two isomers, hexagonal C 3v and butterfly-shaped C 2v , were proved to exist; the latter was evaluated to be more abundant in vapour. When compare the Rb 3 Cl 3 with Rb 3 F 3 studied previously, alike features may be noted (including the existence of two isomers). The results obtained for the heptaatomic ions show similarities between the respective properties of positive and negative ions.