International Journal of Computational and Theoretical Chemistry
Volume 3, Issue 5, September 2015, Pages: 34-44

Molecular and Ionic Clusters of Rubidium Fluoride: Theoretical Study of Structure and Vibrational Spectra

Ismail Abubakari1, 2, *, Tatiana Pogrebnaya1, 2, Alexander Pogrebnoi1, 2

1The Nelson Mandela African Institution of Science and Technology (NM – AIST), Arusha, Tanzania

2Dept. of Materials, Energy Science and Engineering, The NM - AIST, Arusha, Tanzania

Email address:

(I. Abubakari)
(T. Pogrebnaya)
(A. Pogrebnoi)
(A. Pogrebnoi)

To cite this article:

Ismail Abubakari, Tatiana Pogrebnaya, Alexander Pogrebnoi. Molecular and Ionic Clusters of Rubidium Fluoride: Theoretical Study of Structure and Vibrational Spectra.International Journal of Computational and Theoretical Chemistry.Vol.3, No. 5, 2015, pp. 34-44. doi: 10.11648/j.ijctc.20150305.11


Abstract: In this study, the geometrical structure and vibrational spectra of the trimer molecule Rb3F3 and ionic clusters Rb2F+, RbF2, Rb3F2+, and Rb2F3 were studied by density functional theory (DFT) with hybrid functional B3P86 and Møller–Plesset perturbation theory of second order (MP2). The effective core potential with Def2–TZVP (6s4p3d) basis set for rubidium atom and aug–cc–pVTZ (5s4p3d2f) basis set for fluorine atom were used. The triatomic ions have a linear equilibrium geometric structure of Dh symmetry, whereas for pentaatomic ions Rb3F2+, Rb2F3 and trimer molecule Rb3F3 different isomers have been revealed. For the ions Rb3F2+, Rb2F3 three isomers were confirmed to be equilibrium; the linear (Dh), the planar cyclic (C2v) and the bipyramidal (D3h) while for trimer Rb3F3, two isomers were found; the hexagonal (D3h) and the "butterfly-shaped" (C2v) configuration.

Keywords: Geometrical Structure, Vibrational Spetra, Ionic Clusters, Hybrid Functional, Density Functional Theory,
Møller–Plesset Perturbation Theory, Effective Core Potential, Isomers, Basis Set


1. Introduction

Rubidium halides RbX (X is a halogen) may form molecular and ionic clusters in saturated vapours. These species are characterized by different geometric structures, vibrational spectra, and thermodynamic properties depending on the number of atoms comprising the species [1]. Other properties of clusters like electronic, optical, magnetic, and structural are strongly depend on their size and composition [2,3] thus the possibility that materials with desired properties can be made is accustomed by changing the magnitude and structure of the cluster aggregates [4]. Extensive studies of the properties of alkali halide cluster ions had been done previously using different methods [5–15]. High temperature mass spectrometry is well known as a successful method for investigation of ionic clusters in gaseous phase [16–22]. Some ions M2X+, MX2, M3X2+ and M2X3 were already recorded in vapours over alkali metal halides [16,17,20,22,23]. The cluster ions and neutral molecules over rubidium halides have been detected and studied by mass spectrometric technique [20,23,24]. Theoretical quantum chemical methods have been proved to be useful tools in attaining the characteristics of ions and molecules [1]. For the rubidium halides, quantum chemical methods have been used to study the structure and properties of some ionic clusters over rubidium iodide, Rb2I+, RbI2, Rb3I2+ and Rb2I3 [15] and rubidium chloride, Rb2Cl+, RbCl2, Rb3Cl2+, and Rb2Cl3 [25]. This study aims the theoretical investigation of neutral and ionic clusters of rubidium fluoride.

2. Methodology

The calculations were performed using the density functional theory (DFT) with hybrid functional the Becke–Perdew correlation B3P86 [26–29], and second order Møller–Plesset perturbation theory (MP2) implemented into PC GAMESS (General Atomic and Molecular Electronic Structure System) program [30] and Firefly version 8.1.0 [31]. The effective core potentials with Def2–TZVP (6s4p3d) basis set for Rb atom [32,33] and aug–cc–pVTZ basis set (5s4p3d2f) for F atom [32,34] were used. The bases set were taken from the EMSL (The Environmental Molecular Sciences Laboratory, GAMESS US), Basis Set Exchange version 1.2.2 library [35,36]. For visualisation of the geometrical structure, specification of parameters, and assignment of vibrational modes in infrared spectra the Chemcraft software [37] and MacMolPlt program [38] were applied.

The thermodynamic functions were calculated within the rigid rotator-harmonic oscillator approximation using the Openthermo software [39]. The values of energies rE and enthalpies r(0) of the reactions were computed as follows:

rE = Ei prod – ∑Ei reactant            (1)

Δr(0) = ∆rE + ∆rε                    (2)

rε = 1/2hc(∑ωi prod – ∑ωi reactant)      (3)

where ∑Ei prod, ∑Ei reactant are the sums of the total energies of the products and reactants respectively,rε is the zero point vibration energy correction, ∑ωi prod and ∑ωi reactant are the sums of the vibration frequencies of the products and reactants respectively.

3. Results and Discussion

3.1. Diatomic RbF and Dimer Rb2F2 Molecules

To come up with the more appropriate methods to be used for calculations, three different DFT hybrid functionals, B3LYP5, B3P86 and B3PW91 and Møller–Plesset perturbation theory (MP2) have been used in calculations of properties of the diatomic RbF and dimer Rb2F2 molecules. Then the calculated properties have been compared with the experimental data to choose the best performing method to be used in complex ionic and molecular cluster of rubidium fluoride.

The calculated equilibrium geometrical parameters, normal vibrational frequencies, ionization energy and dipole moments of the RbF molecule are shown in Table 1.

As is seen in Table 1, the internuclear distance (Re) of the diatomic molecule RbF computed using all methods are highly overrated, longer than the reference data by ~0.08–0.09 ; the values of frequency (ωe) are underrated by 14–27 cm–1 (~5–7%) if compared with the experimental data; the values of the dipole moment (μe) are overrated by ~0.6–1.5 D.

The values of the ionization energies, adiabatic IEad and vertical IEvert, were obtained as the energy difference of the RbF+ ion and neutral molecule; for the adiabatic IEad internuclear separation Re(Rb-F) was optimized both for neutral and ionic species, while the vertical IEvert was calculated using the optimized value of Re(Rb-F) in neutral molecule only and accepted the same for the ion.

Table 1. Properties of the RbF molecule.

Property B3LYP5 B3P86 B3PW91 MP2 Ref. data
Re(Rb–F) 2.364 2.347 2.352 2.356 2.2703 [40]
–E 123.97635 124.03424 124.02145 123.31829
ωe,(Σu+) 349 357 353 362 376 [41, 42]
μe, 9.2 9.1 9.1 9.5 8.5131 [43]
IEvert 9.44 9.59 9.45 9.66
IEad 8.96 9.08 8.92    

Notes: Here and hereafter, Re is the equilibrium internuclear distance (), 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).

(a) (b) (c)
Figure 1. Equilibrium geometric structures: (a) Rb2F2; (b) Rb2F+; (c) RbF2–.

The IEad by MP2 method was not found in this study because the optimization procedure by MP2 was not implemented for the species with multiplicity more than 1 in the software [30,31]. The reference data on the IE of RbF to the best of our knowledge are not available. The theoretical values on IEvert as well as IEad found by all four methods are in a good agreement with each other, respectively, IEvert being by ~0.5 eV higher than IEad.

For the dimeric molecule Rb2F2 the structure was confirmed to be planar of D2h symmetry (Fig. 1a); the results are tabulated in Table 2. The geometrical parameters and vibrational frequencies calculated by different methods are in accordance with each other and literature data [44,45] as well.

The IR spectrum of Rb2F2 calculated by the MP2 method is shown in Fig. 2. Three peaks observed are assigned to asymmetrical stretching Rb-F modes at 248 cm–1 and 289 cm–1, and bending out of plane vibration at 91 cm–1. Among these three IR active modes, two have highest intensity and correspond to those which had been measured experimentally in IR spectrum in Ar matrix [45].

The dissociation reaction Rb2F2 2RbF was considered and energy and enthalpy of the reaction was calculated using Eqs. (1)-(3). The values of rH°(0) found are in fair agreement with the reference data obtained from IVTANTHERMO database [46], the best result by the MP2 being overrated by 2 kJ×mol-1.

In conclusion of this section we can state that the data obtained by MP2 and DFT/B3P86 for the diatomic RbF and dimer Rb2F2 molecules, agree better with the available reference experimental data as compared with other DFT hybrid functionals considered. Therefore DFT/B3P86 and MP2 methods were chosen for further computations of other ionic and molecular clusters.

Table 2. Properties of the dimer Rb2F2 molecule of D2h symmetry.

Property B3LYP5 B3P86 B3PW91 MP2 Ref. data
Re(Rb–F) 2.553 2.534 2.542 2.527 2.527 [44]a
αe(F–Rb–F) 83.7 83.7 83.9 83.0 82.2 [44]a
-E 248.02449 248.14023 248.11280 247.47912  
∆rE 188.5 188.4 183.5 192.8  
∆rH°(0) 185.1 185.6 180.8 189.7 187.6 [46]
ω1 (Ag) 250 255 250 265  
ω2 (Ag) 101 101 101 107  
ω3 (B1g) 221 227 221 245  
ω4 (B1u) 86 87 86 91  
ω5 (B2u) 232 238 233 248 230 [45]
ω6 (B3u) 268 275 269 289 266 [45]

Notes: a The MP2 calculation. Here and hereafter, αe is the valence angle (deg). rE and rH°(0) are the energy and enthalpy of the dissociation reaction (kJ×mol-1). The reducible vibration representation reduces to Γ = 2Ag + B1g + B1u + B2u + B3u.

Figure 2. Theoretical IR spectrum of Rb2F2, molecule (D2h); MP2 result.

Table 3. Properties of triatomic Rb2F+ and RbF2 ions of linear symmetry D∞h.

Property Rb2F+ RbF2-
B3P86 MP2 B3P86 MP2
Re (Rb–F) 2.481 2.479 2.540 2.534
-E 148.07834 147.50514 223.96693 223.54366
ω1 (Σg+) 126 128 244 253
ω2 (Σu+) 368 379 257 275
ω3 (Pu) 78 80 35 43
I2 3.10 3.05 3.59 3.62
I3 1.85 1.90 3.62 3.70

Note: Here and hereafter, Ii are the infrared intensities (D2×amu–1×2). The reducible vibration representation reduces to Γ = Σg+ + Σu+ + Pu

3.2. Triatomic Rb2F+ and RbF2–Ions

Two structures were considered, linear of D∞h symmetry and V-shaped of C2V symmetry; the latter converged to linear during optimization and only linear structure was proved to be equilibrium (Figs. 1b, 1c). The calculated characteristics of the triatomic ions, Rb2F+ and RbF2 are summarized in Table 3.

The values obtained by the two methods are generally in agreement with each other. By comparing positive and negative triatomic ions, the internuclear distance in negative ion RbF2 is longer by approximately 0.06due to an extra negative charge; in accordance to this the asymmetrical valence vibrational frequency ω2 is higher for positive ion than for negative. The deformational frequency ω3 of the negative ion is almost twice less compared the positive ion, that indicates the floppy deformational potential of the RbF2 ion. When compared with the similar ions for CsF [14], it may be noted the linear structure for Cs2F+ alike Rb2F+ and floppy V-shaped structure of CsF2- with the angle of about 150°. In the spectra of these ions for CsF a correspondence between frequencies of the positive and negative triatomic ions is similar to that of the ions for RbF.

3.3. Pentaatomic Rb3F2+ and Rb2F3–Ions

A number of possible geometrical configurations were considered for the pentaatomic ions Rb3F2+ and Rb2F3, linear of D∞h symmetry, V-shaped of C2v symmetry, planar cyclic of C2v symmetry, and bipyramidal of D3h symmetry. Among them three structures were confirmed to be equilibrium, linear (D∞h), planar cyclic (C2v), and bipyramidal (D3h), they are shown in Fig. 3. The geometrical parameters and vibrational spectra are recorded in Tables 4-6.

Table 4. Properties of pentaatomic Rb3F2+ and Rb2F3 ions of linear symmetry D∞h.

Property Rb3F2+ Property Rb2F3-
B3P86 MP2 B3P86 MP2
Re1(Rb2–F3) 2.442 2.443 Re1(Rb1–F4) 2.492 2.492
Re2(Rb1–F3) 2.587 2.571 Re2(Rb1–F3) 2.605 2.586
-E 272.16771 271.26450 -E 348.05476 347.30131
ω1 (Σg+) 80 84 ω1 (Σg+) 90 96
ω2 (Σg+) 362 374 ω2 (Σg+) 284 292
ω3 (Σu+) 139 147 ω3 (Σu+) 278 285
ω4 (Σu+) 350 367 ω4 (Σu+) 297 321
ω5 (Pg) 62 64 ω5 (Pg) 21 27
ω6 (Pu) (2) 3 ω6 (Pu) 17 19
ω7 (Pu) 71 77 ω7 (Pu) 52 61
I3 0.03 0.00 I3 4.22 3.32
I4 6.21 6.11 I4 2.42 3.31
I6 (0.02) 0.05 I6 1.54 1.63
I7 3.58 3.63 I7 3.95 3.98

Note: The reducible vibration representation for both positive and negative ions reduces to Γ = 2Σg+ + 2Σu+ + Pg + 2Pu

The linear isomer is specified with two non-equivalent internuclear distances, terminal Re1 and bridged Re2. For both positive and negative ions, the terminal distance is shorter than bridged one by ~0.10-0.15 Å. The linear structure was proved to be equilibrium by the absence of imaginary frequencies for negative ion and for positive according to MP2 calculations. At the same time in the DFT calculation of linear Rb3F2+ ion, the value of ω6 was imaginary. Nevertheless during the optimization procedure the bent structure of Rb3F2+ converged into slightly nonlinear without energy gain compared to linear, hence the linear configuration was accepted for this ion with the estimated value of ω6 » 2 cm-1.

The existence of very low frequencies in the vibrational spectra of the linear pentaatomic isomers implies the floppy nonrigid structure of these species with shallow bending potential. For such species the entropy is assumed to be rather high and, based on the thermodynamic approach, the linear isomers are expected to prevail in saturated vapour compared to others. The IR spectra of the Rb3F2+ and Rb2F3 ions (D¥h) are shown in Fig. 4. For both ions four modes are active in IR spectra, and for the negative ion all of them are observed. For positive ion only two peaks at ω4 = 367 cm–1 and ω7 = 77 cm–1 are seen, while remaining two modes ω3 and ω6 are not displayed due to very low intensities.

The properties for planar cyclic structure are given in Table 5. There are three non–equivalent internuclear distance Re1, Re2 and Re3 and two valence angles αe and βe (Figs. 3c, 3d) to specify the geometric configuration. The equilibrium internuclear distances of the negative ion are slightly greater than those of positive ion. Regarding the vibrational spectra of Rb3F2+ and Rb2F3, a similarity is seen among the respective frequencies and IR intensities as well.

Isomerization reactions from linear into cyclic configurations were considered:

Rb3F2+(D∞h) Rb3F2+(C2v)            (4)

Rb2F3(D∞h) Rb2F3(C2v)            (5)

The energy of each reaction is the relative energy ∆Eiso of the cyclic isomer with respect to linear one. The values of ∆Eiso (Table 5) obtained by DFT and MP2 methods are slightly different; the DFT overrates a bit the energy of the cyclic isomer compared to MP2 method. Relying on the latter results, the energy of the cyclic isomer, is higher by 5.0 kJ×mol-1 for the Rb3F2+ ion and lower by 2.5 kJ×mol-1 for Rb2F3 compared to linear isomer. Hence the cyclic isomer of both ions is close by energy to the linear one.

All in all most of the properties of the cyclic isomers of positive and negative pentaatomic ions look alike.

Results for bipyramidal isomers are shown in Table 6. Only one equilibrium internuclear distance Re(Rb–F) and one valence angle are needed to specify the geometric configuration (Figs. 3e, 3f). The internuclear distances of positive and negative ion are almost the same within the same method of calculation. Valence angle at the vertex of the bipyramidal is obtuse for Rb3F2+ and acute for Rb2F3. This tells us that the positively charged bipyramidal ion is slightly flattened compared to negatively charged one.

For the bipyramidal structure, in the vibrational spectra the lowest frequencies are about 70 cm–1 (Rb3F2+) and ~120 cm–1 (Rb2F3). The absence of low frequencies in spectra and the bipyramidal shape itself indicate the rigidity and compactness of these isomers.

Table 5. Properties of pentaatomic Rb3F2+ and Rb2F3 ions, planar cyclic isomers of C2v symmetry.

Property Rb3F2+ Property Rb2F3-
B3P86 MP2 B3P86 MP2
Re1(Rb2–F5) 2.467 2.464 Re1(Rb2–F4) 2.484 2.481
Re2(Rb2–F4) 2.787 2.748 Re2(Rb1–F4) 2.808 2.768
Re3(Rb1–F4) 2.503 2.505 Re3(Rb1–F3) 2.541 2.546
αe(Rb–F–Rb) 106.2 105.7 αe(F–Rb–F) 91.2 92.4
βe(F–Rb–F) 90.1 91.2 βe(Rb–F–Rb) 95.6 93.6
-E 272.16384 271.26261 –E 348.05431 347.30224
ΔEiso 10.2 5.0 ΔEiso 1.2 -2.5
ω1 (A1) 295 (1.51) 308 (1.64) ω1 (A1) 271 (2.20) 280 (2.11)
ω2 (A1) 73 (0.02) 77 (0.04) ω2 (A1) 78 (0.07) 82 (0.41)
ω3 (A1) 91 (0.01) 99 (0.02) ω3 (A1) 79 (0.36) 82 (0.13)
ω4 (A1) 263 (2.14) 271 (2.05) ω4 (A1) 249 (1.85) 258 (1.95)
ω5 (B1) 105 (1.32) 110 (1.34) ω5 (B1) 123 (1.07) 145 (1.14)
ω6 (B1) 39 (0.19) 38 (0.22) ω6 (B1) 25 (0.12) 26 (0.33)
ω7 (B2) 311 (2.36) 322 (2.34) ω7 (B2) 291 (1.57) 299 (1.71)
ω8 (B2) 134 (0.86) 156 (0.90) ω8 (B2) 158 (1.45) 175 (1.27)
ω9 (B2) 40 (0.17) 42 (0.14) ω9 (B2) 53 (2.11) 52 (2.17)
μe 5.4 5.9 μe 4.8 5.4

Note: Eiso = E(C2v) – E(D∞h) is the relative energy of planar cyclic isomer regarding the linear one (kJ×mol-1). The values given in parentheses near the frequencies are infrared intensities (D2×amu-1×Å-2). The reducible vibration representation reduces into Γ = 4A1 + 2B1 + 3B2

For the isomerization reactions

Rb3F2+(D∞h) Rb3F2+(D3h)         (6)

Rb2F3(D∞h) Rb2F3(D3h)         (7)

the energiesEiso = E(D3h)–E(D∞h) are given in Table 6. The values of Eiso for positive and negative ions are negative, except Rb3F2+ by DFT. The negative values of Eiso indicate that the bipyramidal isomers are energetically more stable than linear isomers.

Thus three isomers for pentaatomic ions were revealed to exist. In order to come up with the conclusion of the most abundant isomer in saturated vapour, the relative concentrations were calculated using the following relation:

rH°(0) = –RTln(piso/p) + TrΦ°(T)       (8)

where piso/p is the ratio of the pressure of the cyclic or bipyramidal isomer to that of linear one, R is the gas constant, T is the absolute temperature, rH°(0) andrΦ°(T) are the enthalpy and change in the reduced Gibbs free energy of the isomerization reactions.

The reduced Gibbs free energy was found by the equation

Φ°(T) =            (9)

The thermodynamic functions, enthalpy increments H°(T)-H°(0), reduced Gibbs energies Φ° (J×mol-1×K-1), and entropies S°(T) were calculated for the temperature range 298-2000 K and gathered in Appendix. The required geometrical parameters and vibrational frequencies were used as obtained by MP2 method. The values of rH°(0) for the isomerisation reactions (4)-(7) were found by Eqs. (1)-(3).

Table 6. Properties of pentaatomic Rb3F2+ and Rb2F3 ions, bipyramidal isomers of D3h symmetry.

Property Rb3F2+ Property Rb2F3-
B3P86 MP2 B3P86 MP2
Re(Rb–F) 2.637 2.619 Re(Rb–F) 2.639 2.622
αe(Rb–F–Rb) 93.7 94.0 αe(F–Rb–F) 86.5 86.4
-E 272.16557 271.26800 –E 348.05871 347.31000
ΔEiso 5.6 –9.2 ΔEiso -10.4 -22.8

ω1 (

264 273

ω1 (

242 252

ω2 (

114 121

ω2 (

125 126

ω3 (

207 216

ω3 (

210 221
ω4 (Eʹ) 219 235 ω4 (Eʹ) 216 229
ω5 (Eʹ) 67 70 ω5 (Eʹ) 117 119
ω6 (Eʹʹ) 157 176 ω6 (Eʹʹ) 142 158
I3 2.43 2.50 I3 3.50 3.62
I4 5.64 5.73 I4 5.57 5.65
I5 0.13 0.16 I5 1.64 1.71

Note: Eiso = E(D3h)–E(D∞h) is the relative energy of bipyramidal isomer regarding the linear one (kJ×mol-1). The reducible vibration representation reduces into Γ = .

Figure 3. Geometrical structure of pentaatomic ions: (a) linear Rb3F2+; (b) linear Rb2F3; (c) planar cyclic Rb3F2+; (d) planar cyclic Rb2F3; (e) bipyramidal Rb3F2+; (f) bipyramidal Rb2F3.

(a) (b)
Figure 4. Theoretical IR spectra of pentaatomic ions (D∞h) calculated by MP2 method: (a) Rb3F2+; (b) Rb2F3–.

The values of piso/p were calculated for the temperature range between 700-1800 K; the plots for positive and negative ions are shown in Figs. 5a and 5b, respectively. As is seen the ratios piso/p are much less than 1 for all four cases. For the positive ion the amount of both cyclic and bipyramidal species is negligibly small and for the negative one it does not exceed 16% at 700 K and decreasing with temperature increase. Worth to note that the bipyramidal isomers are much less abundant in equilibrium vapour compared to linear despite they possess lower energy. It may be explained by prevailing of the competing entropy factor as it was predicted above. The existence of similar three isomeric forms was discussed previously regarding the pentaatomic ions over CsF [14] and RbI [15] where the linear (or close to linear) isomers were proved to be predominant in equilibrium vapours. Therefore the entropy factor is significant to conclude which isomer is predominant in equilibrium vapour.

3.4. Trimer Rb3F3 Molecule

Three possible geometrical configurations were considered for trimer molecule, Rb3F3: linear of D∞h symmetry, hexagonal of D3h symmetry and butterfly-shaped of C2v symmetry. The linear structure appeared to be non–stable as imaginary frequencies were revealed. Two other configurations were confirmed to be equilibrium (Fig. 6). The obtained geometric parameters and vibrational frequencies for the hexagonal and butterfly-shaped isomers are shown in Table 7. For the former, only one internuclear distance Re(Rb–F) and one valence angle are required to describe the structure while for the latter, four internuclear distances and two valence angles are needed. The energy ∆Eiso of the isomerization reaction

Rb3F3(D3h) Rb3F3(C2v)           (10)

(a) (b)
Figure 5. Temperature dependence of relative abundance of cyclic and bipyramidal isomers regarding linear: (a) Rb3F2+; (b) Rb2F3–.

Table 7. Properties of trimer Rb3F3 molecule, hexagonal (D3h) and butterfly-shaped (C2v).

Property Rb3F3 (D3h) Property Rb3F3 (C2v)
B3P86 MP2 B3P86 MP2
Re(Rb–F) 2.536 2.529 Re1(Rb1–F4) 2.484 2.479
Re2(Rb3–F5) 2.616 2.603
Re3(Rb2–F6) 2.576 2.566
Re4(Rb3–F6) 2.897 2.821
αe (Rb–F–Rb) 128.3 129.1 αe(Rb2 –F5–Rb3) 99.7 99.5
βe(F–Rb–F) 111.7 110.9 βe(F4 – Rb1–F6) 89.4 88.1
-E 372.23005 371.23920 -E 372.22917 371.24065
      ΔEiso 2.3 -3.8

ω1 (

178 182 ω1 (A1) 286 (3.56) 295 (3.62)

ω2 (

284 303 ω2 (A1) 118 (0.65) 140 (0.77)

ω3 (

91 92 ω3 (A1) 111 (0.13) 117 (0.08)

ω4 ()

82 84 ω4 (A1) 65 (0.09) 72 (0.04)
ω5 (Eʹ) 313 328 ω5 (A2) 215 226
ω6 (Eʹ) 187 193 ω6 (A2) 49 50
ω7 (Eʹ) 40 38 ω7 (B1) 97 (2.00) 103 (2.02)
ω8 (Eʹʹ) 41 42 ω8 (B1) 15 (0.21) 24 (0.22)
I4 2.38 2.46 ω9 (B2) 312 (2.33) 329 (2.50)
I5 6.71 6.84 ω10 (B2) 269 (0.42) 280 (0.17)
I6 2.42 2.38 ω11 (B2) 220 (2.42) 237 (2.54)
I7 0.64 0.70 ω12 (B2) 79 (0.21) 85 (0.22)
      μe 7.9 8.4

Notes: ∆Eiso = 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 reduce as follows: Γ =  and Γ = 4A1 + 2A2 + 2B1 + 4B2, respectively. For the C2v butterfly-shaped isomer, the values given in parentheses near the frequencies are infrared intensities (D2amu-1Å-2).

Figure 6. Geometrical structures of trimer Rb3F3 molecule: (a) hexagonal of D3h symmetry; (b) butterfly-shaped of C2v symmetry.

Figure 7. Temperature dependence of relative amount of the C2v isomer regarding to D3h isomer of trimer Rb3F3 molecule.

(a) (b)
Figure 8. Theoretical IR spectra of the trimer Rb3F3 molecule calculated by MP2 method: (a) hexagonal isomer (D3h); (b) butterfly-shaped isomer (C2v).

was determined. As compared to hexagonal, the butterfly-shaped configuration has a bit higher energy, by 2.3 kJ×mol-1, according to DFT/B3P86 method, while it has a bit lower energy, by 3.8 kJ×mol-1, according to MP2 method; that is these two isomers are close by energy to each other.

The relative concentrations of isomers have been determined similarly as it is described above in section 3.3. The ratio of p(C2v)/p(D3h) versus temperature is shown in Fig. 7. As is seen the ratio is close to one in a broad temperature range which means that two isomers compete with each other being in comparable amount. At low temperatures C2v  isomer show high abundancce but its abundance decreases with temperature increase hence D3h isomer seems to dominate at elevated temperatures. Worth to mention that similar isomeric structures were revealed for the Cs3F3 molecule [14]: one hexagonal D3h and the other of a "butterfly-shaped" (Cs); the lower symmetry of the second of Cs3F3 compared to Rb3F3 apparently due to the bigger size and higher polarizability of the caesium atom.

The IR spectra of the trimer Rb3F3 isomers are presented in Fig. 8. For the hexagonal isomer, only four modes are active in IR spectrum, and all of them are seen. In the spectrum of C2v isomer all modes, except two, ω5 and ω6 of A2 symmetry, have nonzero intensities but not all of them may be observed due to low intensities. For both isomers, the frequencies at ~200 cm-1 and above correspond to the stretching asymmetric vibrations Rb–F, the frequencies below 100 cm-1 relate to bending vibration modes. The most intensive bands are assigned to the stretching Rb–F modes, 328 cm-1 and 295 cm-1 for the D3h and C2v isomers, respectively.

4. Conclusion

A number of molecular and ionic clusters, including the trimer Rb3F3 molecule, triatomic Rb2F+ and RbF2- and pentaatomic Rb3F2+ and Rb2F3- ions, have been studied by DFT/B3P86 and MP2 methods. These methods were accepted for calculations of geometrical parameters and vibrational frequencies of the clusters because they provided a better agreement with the available reference data for diatomic RbF and dimer Rb2F2 molecules. Alternative configurations have been considered for the molecular and ionic clusters. For the triatomic ions the linear structure (Dh) was confirmed to be equilibrium. The existence of isomers was proved for the pentaatomic ions and trimer molecule. The three isomers of comparable energy were revealed for Rb3F2+ and Rb2F3- ions: linear (Dh), planar cyclic (C2v), and bipyramidal (D3h), the linear one being the most abundant in the equilibrium vapour compared to others. Two isomeric forms of the trimer molecule Rb3F3 were figured out: planar hexagonal (D3h) and butterfly-shaped (C2v); they were shown to have almost equal energy and comparable relative abundance in saturated vapour.

Authors’ Contributions

All authors participate well in all steps including computation, data analysis and manuscript preparation towards production of this work.

Acknowledgment

The authors are thankful to the Tanzania Commission for Science and Technology (COSTECH) and The Nelson Mandela African Institution of Science and Technology (NM–AIST) for support and sponsorship of this work.

Appendix

The thermodynamic functions of ionic and molecular clusters, triatomic ions Rb2F+ and RbF2, pentaatomic ions Rb3F2+ and Rb2F3, and trimer Rb3F3 in gas phase are given in Tables A1–A6. The values of molar heat capacity cp°, entropy S°, Gibbs reduced free energy Φ° are given in J×mol-1×K-1, and enthalpy increment H°(T)–H°(0) is in kJ×mol-1, absolute temperature T in K. The thermodynamic functions of the most abundant isomers only are given for the species existing in different isomeric forms.

Table A1. Thermodynamic functions of Rb2F+ (D∞h).

T cp° H°(T)–H°(0) Φ°
298.15 59.92 304.811 15.398 253.167
700 61.86 356.979 40.013 299.818
800 61.98 365.248 46.205 307.492
900 62.06 372.553 52.407 314.323
1000 62.12 379.094 58.616 320.478
1100 62.15 385.017 64.829 326.081
1200 62.19 390.426 71.047 331.221
1300 62.20 395.405 77.267 335.969
1400 62.23 400.016 83.489 340.381
1500 62.25 404.310 89.713 344.501

Table A2. Thermodynamic functions of RbF2 (D∞h).

T cp° H°(T)–H°(0) Φ°
298.15 60.22 294.437 15.591 242.144
700 61.94 346.756 40.270 289.227
800 62.04 355.034 46.470 296.947
900 62.11 362.345 52.678 303.814
1000 62.16 368.891 58.891 310.000
1100 62.19 374.817 65.108 315.628
1200 62.22 380.230 71.329 320.790
1300 62.23 385.210 77.551 325.555
1400 62.25 389.823 83.776 329.983
1500 62.26 394.119 90.002 334.118

Table A3. Thermodynamic functions of Rb3F2+ (D∞h).

T cp° H°(T)–H°(0) Φ°
298.15 107.67 465.801 27.418 373.840
700 111.32 559.626 71.692 457.209
800 111.53 574.506 82.836 470.961
900 111.68 587.652 93.997 483.211
1000 111.79 599.425 105.171 494.254
1100 111.86 610.083 116.353 504.308
1200 -1492.90 619.819 127.543 513.533
1300 111.96 628.780 138.738 522.058
1400 112.01 637.079 149.937 529.981
1500 112.03 644.808 161.140 537.381

Table A4. Thermodynamic functions of Rb2F3 (D∞h).

T cp° H°(T)–H°(0) Φ°
298.15 108.02 443.330 27.584 350.813
700 111.41 537.331 71.934 434.569
800 111.60 552.222 83.085 448.365
900 111.74 565.375 94.253 460.649
1000 111.84 577.154 105.432 471.722
1100 111.91 587.816 116.619 481.799
1200 111.96 597.556 127.813 491.045
1300 112.00 606.519 139.011 499.587
1400 112.03 614.821 150.213 507.526
1500 112.06 622.551 161.418 514.939

Table A5. Thermodynamic functions of Rb3F3 (D3h).

T cp° H°(T)–H°(0) Φ°
298.15 126.60 467.382 30.557 364.893
700 131.78 578.192 82.868 459.809
800 132.06 595.808 96.061 475.732
900 132.24 611.375 109.278 489.955
1000 132.42 625.319 122.513 502.806
1100 132.52 637.944 135.759 514.527
1200 132.59 649.478 149.015 525.299
1300 132.68 660.095 162.279 535.265
1400 132.74 669.928 175.547 544.537
1500 132.78 679.086 188.821 553.205

Table A6. Thermodynamic functions of Rb3F3 (C2v).

T cp° H°(T)–H°(0) Φ°
298.15 126.48 462.998 30.013 362.283
700 131.76 573.769 82.308 456.186
800 132.06 591.383 95.500 472.008
900 132.25 606.950 108.716 486.154
1000 132.40 620.893 121.950 498.943
1100 132.52 633.518 135.196 510.613
1200 132.60 645.051 148.452 521.341
1300 132.66 655.667 161.715 531.271
1400 132.72 665.500 174.983 540.512
1500 132.75 674.658 188.256 549.154

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