Theoretical Study of Cluster Ions Existing in Vapours over Cesium Bromide and Iodide

This work was carried out in collaboration between all authors. Author SFM performed computations, wrote the first draft of the manuscript and managed literature searches. Author TPP performed corrections and some selected computations regarding the structure and vibrational spectra. Author AMP performed some selected thermodynamic calculations. All authors analyzed and discussed the results and approved the final manuscript. ABSTRACT The properties of ions Cs 2 X + , Cs 3 X 2+ , CsX 2− , and Cs 2 X 3− (X = Br or I) have been studied using the density functional theory and Möller–Plesset perturbation theory of the 2 nd and 4 th order. For all species the equilibrium geometrical configurations and vibration frequencies were determined. Different isomers of pentaatomic ions were found to exist: the linear ( D ∞h ), V-shaped ( C 2v ), kite-shaped ( C 2v ) and bipyramidal ( D 3h ). The relative abundances of isomers were calculated for temperatures between 700 K and 1600 K. It was found that at about 800 K, the amount of different isomers was comparable for Cs 3 Br 2+ , Cs 3 I 2+ and Cs 2 I 3− ions, while for Cs 2 Br 3− the linear isomer was proved to be predominant. The enthalpies of dissociation reactions with elimination of CsX molecules and the enthalpies of formation of ions were determined.


INTRODUCTION
Alkali halide cluster ions form a potential group for researches due to the possibilities of designing and fabricating new materials. Some of these cluster ions have unique properties such as electronic, optical and magnetic which are function of size and composition [1][2][3]. These species can serve as fundamental building blocks for a new class of materials with desired properties [3,4].
Different species composed of cesium and iodine are proved to exist among the fission products that can be released in nuclear power plants [5][6][7][8]. They have major impact on ground contamination and radiation doses in the environment in case of accidents such as containment building leakages. They are highly radioactive in short term for iodine and in middle term for cesium [5,6,9]. Thus, evaluations of their thermodynamic properties are essential for safety features of the nuclear pressurized water reactor.
Considerable studies of alkali halide cluster ions have been done in the past decades [4,[10][11][12][13][14][15]. Different analytical procedures have been employed for the investigation of ionic clusters [16,17]. Mass spectrometry stands as a major experimental technique which is capable of analyzing a broad characterization of their properties [18]. Various positive and negative ions had been identified in equilibrium vapours using high temperature mass spectrometry [19][20][21][22][23][24][25]. For the treatment of experimental data thermodynamic functions of molecules and ions are required and for the calculation of the thermodynamic functions the geometrical parameters and vibrational frequencies are needed. However they are difficult to be measured by available experimental techniques [26].
The ionic species Cs + , Cs 2 I + , Cs 3 I 2 + , I  , CsI 2  , and Cs 2 I 3  have been detected in the saturated vapour over cesium iodide by high temperature mass spectrometry [24,31]. Photoelectron spectroscopy was applied to CsI 2  [32]. The ions Cs 2 I + and CsI 2  were resulted from collisions of Cs 2 I 2 with Xe [33]. In line to this, we expect similar ions to exist in saturated vapour over cesium bromide. Thus the aims of the present work were to determine the characteristics of the cluster ions of cesium bromide and iodide using quantum chemical methods, as well as to calculate the thermodynamic properties of the ions. To verify the reliability of the results obtained, the properties of neutral species CsX and Cs 2 X 2 have been calculated and analyzed through a comparison with the available reference data.

COMPUTATIONAL DETAILS
The calculations were performed by GAMESS (General Atomic and Molecular Electronic Structure System) program [34], Firefly version 8.0.0 [35], employing electron density functional theory (DFT) with the Becke-Lee-Yang-Parr functional (B3LYP5) [36,37] and Becke-Perdew functional (B3P86) [37][38][39], as well as the second order and fourth order Möller-Plesset perturbation theory (MP2 and MP4). The effective core potential (Cs ECP GEN 46 3, 9 electrons in the core) with Def2-QZVP basis set (6s5p4d1f) [40] for Cs and the relativistic effective core potentials (Br ECP GEN 28 4, 7 electrons in the core; I ECP GEN 46 4, 7 electrons in the core) with SDB-aug-cc-pVTZ basis set (4s4p3d2f) [41] taken from EMSL (The Environmental Molecular Sciences Laboratory, U.S.) [42][43][44] were used. The B3LYP5 and MP2 methods were applied to compute the geometrical parameters and vibrational spectra of cluster ions. The geometrical structures determined are confirmed as corresponding to minima energy by the absence of imaginary frequencies.
The dissociation energies ∆ r E were calculated by B3LYP5, B3P86 and MP2 methods. Also more advanced MP4 level was employed, the equilibrium geometrical structure found by MP2 method was used. Furthermore, the correction for basis set superposition error (BSSE) [45] has been taken into account for MP2 and MP4 methods using the procedure proposed in [46]. The methods of calculations with BSSE corrections are denoted hereafter as MP2C and MP4C.

Molecular Properties of CsX and
Cs 2 X 2 (X =Br or I) For  For the dimer molecules Cs 2 Br 2 and Cs 2 I 2 the structure was proved to be planar cycle (rhomb) with symmetry D 2h (Fig. 1a)   Therefore we expect the similar trend to appear in the properties of the ions considered further when the same level of computational method is applied.
The energies of dissociation reactions ∆ r E of Cs 2 X 2 into CsX molecules have been calculated using different theoretical levels: B3LYP5, B3P86, MP2, MP2C, MP4, and MP4C. The enthalpies of reactions ∆ r H(0) were obtained using the energies ∆ r E, and the zero-point vibration energy (ZPVE) correction ∆ε as given in following equations where h is the Plank's constant, c is the speed of light in the free space, Ʃ i prod , and Ʃ i react are the sums of the vibration frequencies of the products and reactants, respectively. are taken as a benchmark and the differences ∆ between the theoretical and reference values are depicted by the bar diagrams. As one can see, B3LYP5 and B3P86 methods give underrated results for both dimeric species while MP2C and MP4 demonstrate rather good agreement. According to our highest level of computation, MP4C, ∆ = −3.5 kJmol −1 and −8.4 kJmol −1 for Cs 2 Br 2 and Cs 2 I 2 respectively. Based on these results, and assuming a factor of 1.5, we estimated uncertainties to be 5 kJmol −1 and 13 kJmol −1 to the corresponding theoretical values of ∆ r H°(0) calculated by MP4C. It should be noted also that our result found by MP4C method for Cs 2 Br 2 is in a good agreement with both reference data [55] and [6]. For Cs 2 I 2 the agreement between the value 14313 and experimental one, 150. 9 [55], is worse, but within the uncertainties, there is no contradiction. If we take into account the data from [6] as a benchmark the agreement appeared to be better as seen in Fig. 2 c. Moreover the bar diagram in Fig. 2 c for cesium iodide now looks alike to that of bromide ( Fig. 2 a).
It is also worth to note here that the difference between the enthalpies of dissociation reactions ∆ r H°(0) for Cs 2 Br 2 and Cs 2 I 2 is 7 kJmol −1 according to our results which agrees well with ~8 kJmol −1 according to [6]. The last value seems to be more feasible than 2.6 kJmol reported recently by Roki et al. [7] is evidently higher than the enthalpy of dissociation of cesium bromide and therefore looks like overrated.

Geometrical Structure and Vibrational Spectra of the Cluster Ions
For the calculation of the properties of the cluster ions two methods; DFT (B3LYP5) and MP2 have been used. As a whole array of the data obtained for the neutral species, triatomic and pentaatomic, positive and negative ions demonstrates alike trends from DFT to MP2 levels, therefore we present hereafter the results found by more reliable MP2 method.

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Linear structures with D ∞h symmetry are proved to exist for the triatomic ions ( Fig. 1 b, c). The properties such as equilibrium internuclear separations R e (CsX), the frequencies of normal vibrations i , and the intensities of vibrations in IR spectra I i have been determined and given in Table 3.
It can be observed that internuclear separation R e (CsX) increases by ~0.08 Å from positive to negative ions for both species. Also an increase of internuclear distances by ~0.25 Å has been featured across both positive and negative cesium bromide to cesium iodide. Clearly the increase of internuclear distance from positive to negative ions is due to an excess negative charge in the CsX 2 − ion, and from cesium bromide to cesium iodide is due to an extra shell. This increase in the internuclear distance corresponds to the decrease in the antisymmetric stretching frequency ω 2 : 138 cm 1 (Cs 2 Br + ) to 109 cm 1 (Cs 2 I + ); and 113 cm 1 (CsBr 2  ) to 98   As concerns the linear configuration for the Cs 3 Br 2 + ion the imaginary frequencies have been revealed. The further optimization showed the bent (V-shaped) structure of C 2v symmetry to exist with the valence angle α e (BrCsBr) = 144 and practically without decrease in energy compared to the linear shape within the accuracy of optimization procedure. Other pentaatomic ions, Cs 2 Br 3  , Cs 3 I 2 + , and Cs 2 I 3 − , were confirmed to be linear. Note these two equilibrium structures were obtained disregarding initial configuration started from: linear, bent or fivemembered ring structures were converted during the optimization into V-shaped for Cs 3 Br 2 + or linear for other three ions. Here and hereafter we call the isomer I as the ion of the bent (V-shaped) structure (C 2v ) for the Cs 3 Br 2 + ion and linear (D ∞h ) for Cs 2 Br 3  , Cs 3 I 2 + , and Cs 2 I 3 − ; their properties are given in Table 4. There are two different internuclear separations, terminal and bridge, which are denoted as R et (Cs−X) and R eb (Cs−X), respectively. From positive to negative ions of cesium bromide, there is a slight increase in the internuclear distance by 0.07 Å and 0.05 Å for R et (CsBr) and R eb (CsBr), respectively. The similar increase is observed for cesium iodide. Regarding terminal and bridge distances, for both ions the first one is shorter than the second, by ~0.16 Å for Cs 3 X 2 + and by ~0.13 Å for Cs 2 X 3  .
In vibrational spectra of the ions with linear or Vshaped structure, several low deformational frequencies are observed, that gives evidence that these structures are floppy regarding to bending of the central moiety.
The isomer II, planar cyclic (C 2v ), and isomer III, bipyramidal (D 3h ), were also found to correspond to minima at the potential energy surface (PES). The optimized geometrical parameters were located and all calculated vibrational frequencies were confirmed to be real. Tables 5 and 6 report the properties of the ions Cs 3 X 2 + and Cs 2 X 3 − for cyclic and bipyramidal structures, their geometrical configurations are shown in Figs. 3 c, d, e, and f. The cyclic structure is described by three different internuclear separations CsX, which are denoted as R e1 , R e2 , and R e3 , and two independent valence angles α e and β e . The internuclear distances of the negative ions are slightly greater than those of positive ions correspondingly. The relative energy of the cyclic isomers with respect to the isomer I is  r E iso = E II E I and for all four ions the values of  r E iso are negative and lie in the range between 8 and 12 kJmol −1 . This result indicates that the cyclic isomer has lower energy on the PES than the isomer I and is more stable energetically. The vibrational spectra look alike for positive and negative ions both in frequencies and IR intensities.
The isomer III, that is of the bipyramidal shape, is specified by two parameters that are internuclear distance R e (CsX) and valence angle α e . The internuclear distances for positive ions are slightly shorter than those for negative ions for cesium bromide or equal for iodide. We refer to the magnitudes of ionic radii of Cs + , Br  , I  ions. The bond angles at the vertices are almost the same and close to right angle. The corresponding vibrational frequencies of Cs 3 X 2 + and Cs 2 X 3  ions are close to each other while the frequencies of positive ions are slightly larger than those of negative. The relative energy of bipyramidal isomer with respect to the isomer I is  r E iso = E III E I , the values of  r E iso are negative and in the range between 30 and 38 kJmol −1 . Therefore the isomer III is more energetically stable than the linear one for Cs 2 Br 3  , Cs 3 I 2 + , Cs 2 I 3  , and V-shaped for Cs 3 Br 2 + . In addition the bipyramidal isomer has lower energy than cyclic one and thus appeared to be the most energetically stable among three alternative isomers.

Relative Concentration of Isomers
To examine the relative concentrations of isomers I, II, and III in saturated vapours over cesium bromide or iodide, thermodynamic calculations were performed. We considered the isomerization reactions I → II and I → III. The relative concentrations x i of the two isomers in equilibrium vapour was calculated using the following formula: where ∆ r H(0) is the enthalpy of the reaction; T is absolute temperature; ∆ r Φ(T) is the change in the reduced Gibbs energy of the reaction, Φ(T) =  [H(T)  H(0)  TS(T)]/T; x i = p i /p I ; p i is the partial pressure of the isomer II or III, and p I is the partial pressure of the isomer I. Hence here we have two ratios to be considered for each of the pentaatomic ions: x II = p II /p I and x III = p III /p I . The enthalpies of the isomerization reactions ∆ r H(0) were evaluated using isomerization energies ∆ r E iso and the ZPVE corrections ∆ε by use of Eqs. (1) and (2); the energies ∆ r E iso were calculated by MP4 method. The values of ∆ r Φ(T) and other thermodynamic functions were calculated in the rigid rotator-harmonic oscillator approximation using the optimized coordinates and vibrational frequencies obtained in the MP2 calculations as the input parameters. The values of reduced Gibbs free energy and other thermodynamic functions are reported in the APPENDIX. The thermodynamic functions and the relative concentration of the isomers were computed for the temperature range between 700 K and 1600 K related to the experimental condition. The results of calculations of energies and enthalpies of the isomerization reactions ∆ r H(0), ZPVE corrections ∆ε, change in the reduced Gibbs free energies ∆ r Φ(T), and relative concentration x i = p i /p I the isomers are reported in Table 7 for T = 800 K.
For each isomerization reaction considered the value of ∆ r H(0) is negative which means that the isomer in the right hand side of the reactions is more favorable by energy regarding the isomer I. Compared to the results found by the MP2 method the magnitudes of ∆ r H(0) by MP4 level are less and are in the range ~2030 kJmol 1 for the bipyramidal isomer and ~58 kJmol 1 for cyclic isomer.  , and Cs 2 I 3  the isomers I and II are found in comparable amount, but the bipyramidal one is not abundant. For the ion Cs 2 Br 3  the linear isomer is the most abundant compared with two others and actually only this one among the three exists.
The temperature dependence of the relative concentration x i has been examined for the temperature range between 700 and 1600 K (Fig. 4). As is seen all relative concentrations of the isomers decrease with temperature increase. For the ion Cs 3 Br 2 + at 800 K, the concentration of cyclic isomer is 27% and decreases slowly to about 18% at 1500 K (Fig. 4 a). The concentrations of bipyramidal Cs 3 Br 2 + is 3% at 800 K and decreases to 0.7% at 1500 K. For the negative ion Cs 2 Br 3  the concentrations of both cyclic and bipyramidal are very low and decrease further with temperature rise (Fig. 4 b). For the ions Cs 3 I 2 + and Cs 2 I 3 − , a very close appearance of the plots is observed in Figs 4 c, d. There is rather big amount of both cyclic and bipyramidal isomers at temperatures around 700800 K, and then the relative concentrations are dropping down rapidly when the temperature rises still remaining essential for the cyclic species.
As the three isomers may occur in a comparable amount, the fraction w i of each isomer out of three was found as well using the following equation: where p i (i = I, II, or III) represents the concentration of the isomer of interest. The fraction w i was expressed through the ratio x i = p i /p I mentioned above: = .
The values of w i at 800 K were found to be as follows: for Cs 3 Br 2 + are 0.77, 0.21 and 0.02; Cs 3 I 2 + are 0.66, 0.27 and 0.07 and Cs 2 I 3 − are 0.70, 0.24 and 0.06 that is of linear, cyclic and bipyramid isomers, respectively. From these values, we can notice that the isomer I dominates for all ions, while the cyclic is less abundant but still significant in its amount, and the fraction of the bipyramidal does not exceed 10%. Fig. 5 elucidates the influence of temperature on the fraction of isomers w i . As it is seen raise in temperature increases the amount of linear isomer and decreases slowly that of cyclic isomers. For the bipyramid the fraction decreases rapidly as temperature elevates. Thus for all ions with temperature increase, the cyclic and bipyramidal isomers are decreasing in their content whereas isomer I is increasing and being predominant.
Concluding this section it is worth to emphasize the importance of the entropy factor on the relative content of the isomers. The effect of entropy and consequently in reduced Gibbs energy is appeared to be essential and prevailing over the energetic factor. In spite of the higher energetic stability of isomers II and III their relative amount is lower than of the isomer I. It is distinctly seen particularly for the bipyramidal isomer as its energy is lower by ~30 kJmol 1 than that of the isomer I but its relative concentration is very small in saturated vapour at elevated temperatures. Therefore the considerable decrease in entropy of the isomerization reactions prevails over the energy factor and result in the predominance of the isomers I.

The Enthalpies of Dissociation Reactions and Enthalpies of Formation of Ions
In this section, we have examined the dissociation reactions of the cluster ions with elimination of CsX molecules. The energies of reactions  r E have been calculated at different theoretical levels, B3LYP5, MP2, MP2C, MP4, MP4C, where in MP2C and MP4C the BSSE correction have been taken into account. The results are presented in Fig. 6. One can see the similar trend in values of ∆ r E with an enhancement of the theoretical level from B3LYP5 to MP4C i.e. the lowest values come from the B3LYP5 method, the highest ones from MP2 following by the further decreasing to the MP4C level. This trend is common either of triand pentaatomic ions. Also the distinct similarity may be observed for the same isomers of pentaatomic ions that is a slight oscillation of ∆ r E from MP2 to MP4C for isomer I and almost monotonic decrease for both cyclic and bipyramidal isomers. From these results the true value of ∆ r E comes when approaching the limit that we suppose is close to the MP4C result. The latter appeared to be an intermediate between B3LYP5 and MP2 results. Regarding a certain pentaatomic ion, the change in ∆ r E is not the same for different isomers that is the change in ∆ r E from MP2 to MP4C is about 10 kJmol 1 for the isomers I or II, while that is ~20 kJmol 1 for the isomers III.
This results in a small decrease of the isomerization energies from MP2 to MP4 and further to MP4C. The BSSE correction itself looks slightly different for three isomers of the same ion; that contributes to the error of computational scheme [46] and hence to the uncertainties of the theoretical values of the enthalpies of the dissociation reactions accepted here.
The values of ∆ r E obtained using the MP4C level were taken for calculation of the enthalpies of the reactions ∆ r H°(0) by Eq. (1). The results on ∆ r E, ∆ r H°(0), ZPVE, and ∆ f H°(0) are listed in Table 8.   It is worth to remind here about the increase of stability of the isomers in the rank IIIIII; the enthalpy of dissociation of the isomer I is less by 611 kJmol −1 than of isomer II and 1727 kJmol −1 compared to isomer III. The reason is that the isomers with compact shape are more stable against dissociation.
The experimental data are available in [24] for ions existing in vapour over cesium iodide where the ion molecular reactions were investigated by mass spectrometric method and the equilibrium constants ° for the heterophase reactions involving the tri-and pentaatomic cluster ions were measured. Using these constants we have computed the values of ∆ r H°(0) by the formula: As an example, an ion molecular reaction for the positive triatomic ion may be expressed as follows: where [CsI] corresponds to condensed phase. The equilibrium constant for this reaction is , and Cs 2 I 3  needed for the calculation of the thermodynamic functions were taken from our MP2 results. The enthalpies of dissociation calculated through experimental data [24] are denoted as "based on experiment" hereafter and given in Table 8. As regards the pentaatomic ions, the existence of isomers had not been considered by Sidorova et al. [24]. In our work, as we have found the isomers may exist at a comparable amount, the fractions of the isomers, w i (i = I, II, or III), are taken into account, that is the measured current is multiplied be the fraction, e.g. the ion current for isomer I is I(Cs 3 I 2 + , I) = I(Cs 2 I 3 + )w I . The values of the ion currents obtained by this way have been used to calculate the equilibrium constant for each isomer and then the enthalpies of reactions ∆ r H°(0) "based on experiment". It is worth to note that for the pentaatomic ions only a few values of currents had been measured and no statistical treatment was done in [24].
As is seen in Table 8, for the triatomic ions, Cs 2 I + and CsI 2  , the theoretical magnitudes of ∆ r H°(0) are in a good agreement with "based on experiment" values but underrated by ~5 kJmol −1 . Similarly the theoretical values for the dimer molecules Cs 2 I 2 are underrated compared to experimental data which is mentioned above. For the pentaatomic isomers the agreement in most cases is worse than for triatomic ions and the discrepancy between theoretical values and "based on experiment" approaches 13 kJmol    (3) and (4)  for III  however the data "based on experiment" show a small systematic increase in stability by 36 kJmol −1 from the positive to negative ions correspondingly. Therefore when we consider the different trends, i.e. from bromides to iodides, from triatomic to pentaatomic, from positive to negative, it is indicative that the experimental values for the cesium iodide species seem to be slightly overrated. Nevertheless, within the uncertainty limits, the theoretical and "based on experiment" values of ∆ r H°(0) are in satisfactory agreement and there is no crucial contradiction between them. The similar agreement is seen for the enthalpies of formation of the ions ∆ f H°(0) presented in Table 8.  ; absolute temperature T in kelvins.