Introduction

Practical applications of tin dihalides are manifold. They can be found as intermediates in metal deposition processes and in etching of semiconductor devices [1]. Solid tin diiodide is used as a photorecording medium, due to its high photosensitivity [2]. The crystal is monoclinic, the unit cells contain six SnI2 molecules [2]. Tin diiodide is also a possible ingredient in metal halide lamps [3], and for this reason its vaporization process as well as the knowledge of the possible molecular species present in their vapor phase is essential. It was established by mass spectrometry that the vapors of SnI2 contain both monomeric and dimeric species [4, 5]. The geometry of the monomeric molecule was determined by gas-phase electron diffraction [6] and subsequently it had also been used as one of the examples to check the performance of the joint electron diffraction/vibrational spectroscopic analysis developed by Spiridonov and Gershikov [7] (see also in Refs. 8 and 9). The dimer structure was never determined by experiment. Tin diiodide is not the easiest object for computations and computational techniques have only recently achieved such a level that their results could be compared with experiment. From the computational studies of SnI2, the relatively early study of Benavides-Garcia and Balasubramanian [1] can be mentioned for the monomer and the DFT and MP2 studies of both monomer and dimer by Saloni et al. [5]. Our aim in the present study is to determine the structure of the tin diiodide species present in the vapor and determine their vibrational frequencies and thermodynamic properties by high-level computations at such a level that gives results comparable with the available experimental data.

The structure of tin dihalide dimers is rather different from what they were supposed to be in earlier studies [4, 10]. The general consensus about metal dihalide dimer structures used to be that they have a D 2h-symmetry halogen-bridged structure. However, as recent computational results show, this does not apply to the tin dihalides in that the stereochemical activity of the lone electron pair on tin does have its influence on the dimer structure just as it does on the structure of the monomer [5, 11–14]. Two different structures were found for these dimers, one with C s and the other with C 2v symmetry, usually with the lower-symmetry structure being the ground state.

Computational details

Due to the large size of tin and iodine, quasirelativistic effective core potentials (ECP) of the Stuttgart group were used to describe both atoms, they include scalar relativistic effects but do not treat spin-orbit effects. We checked both the so-called large-core ECPs (covering 46 electrons—[Kr]4d10), and the small-core ECPs (covering 28 electrons—[Ar]3d10), for both Sn and I atoms with different associated basis sets: SDB-cc-pVTZ [15, 16], cc-pVDZ-PP [17, 18], cc-pVTZ-PP [17, 18], cc-pVQZ-PP [17, 18], cc-pV5Z-PP [17, 18]. All computations were carried out with the Gaussian 03 program package [19].

Full geometry optimizations were carried out for the monomer using density functional (B3LYP [20], B3PW91 [21]), second-order Møller-Plesset (MP2) and CCSD(T) level computations. The density functional calculations consistently gave several hundredths of an angstrom larger bond lengths and also somewhat larger bond angles than the MP2 calculations. We had a similar experience with our calculations of the SnBr2 structure [12]. Since the MP2 and CCSD(T) computed geometries of the monomer agreed much better with the experimental results than the density functional ones, our discussion is based on these results. For the dimer, different geometrical arrangements are possible, all of them were calculated at the B3LYP/SDB-ccpVTZ level. The stable structures were further investigated at higher-computational levels; in these calculations the symmetry of the molecule was constrained, while all other parameters were optimized. Frequency calculations for both the monomer and the dimer were carried out at different levels, the best ones will be given in subsequent tables, while all our results with different methods and basis set combinations are given in the Supporting Information.

The rearrangement of the most stable structures of the dimer was also calculated at several computational levels with different basis sets and the transition-state structure between them was also determined together with the relative energies and some thermodynamic parameters. For the dimer basis set superposition error (BSSE) was taken into account with the counterpoise method implemented in Gaussian 03. For the ionization potential we used the adiabatic formalism, where the ions were optimized stable states.

Results and discussion

Molecular geometries

Monomer

The geometrical parameters and vibrational frequencies of the monomeric SnI2 molecule from different levels of computation are given in the supporting information, the highest-level computed data are given in Table 1, together with some relevant computed and experimental data from the literature. Figure 1 shows the variation of the Sn–I bond length and the I–Sn–I bond angle with changes of methods and basis sets. The convergence of the density functional results is good—but their absolute values are about 0.04 Å longer than the estimated experimental equilibrium bond length and that is not acceptable. As for the correlated calculations, even with the quintuple-zeta quality basis set the complete basis set limit (CBSL) is not reached. Although not converged yet, the MP2 results are shorter than the estimated experimental equilibrium bond length and this is in line with the general experience that MP2 bond lengths tend to be too short. The CCSD(T) values are better; if we extrapolate from the trend between the calculated MP2 and CCSD(T) values, we estimate the cc-pV5Z-PP result as 2.696 Å, in good agreement with experiment—although, admittedly, this is not yet a converged CBSL value. On the other hand, we should also mention that the bases we applied only take into account the scalar relativistic effects and not the spin-orbit effect that, in a molecule with such large atoms, might be considerable. For the I2 molecule, e.g., the spin-orbit effect on the bond length was estimated to be as large as 0.020 Å [27], thus, it might also be expected that the Sn–I bond length in SnI2 would increase with the spin-orbit effects taken into consideration. Therefore, an agreement between the CBSL computed bond length and the experimental equilibrium bond length is realistic.

Table 1 Geometrical parameters and vibrational frequencies (IR intensities in parenthesis) of the SnI2 molecule (bond lengths in Å, angles in degrees, frequencies in cm− 1, and intensities in km mol− 1)
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Fig. 1
figure 1

Variation of the computed geometrical parameters of SnI2 with different methods and basis sets. Left: bond lengths, right: bond angles

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The trends for the bond angles follow a similar pattern, although the differences are smaller. We do not attempt to compare them with the experimental bond angle because the accuracy of the latter is doubtful as it does not fit the trend observed for all the other Group 14 dihalides (see Fig. 22 in Ref. 8).

Dimer

Different possible geometrical arrangements were calculated for this structure (see Computational Section) as shown in Fig. 2, where their relative energies and the number of their imaginary frequencies are also indicated. As seen from the figure, only two structures were found stable; with C s and C 2v symmetry, respectively.Footnote 1 These structures were calculated at higher computational levels as well and their geometrical parameters are given in Table 2. These two structures are very similar to the ones found stable for tin dichloride and dibromide [5, 11–13], perhaps one difference can be mentioned: for the C 2v-symmetry isomer the central four-membered ring puckers towards the terminal iodines, while in the tin dichloride and dibromide dimers this puckering is in the opposite direction; obviously the large size of the iodine atoms being the reason. The terminal bond lengths of the dimer are about 0.02–0.04 Å longer than the corresponding monomer bonds, and the bridging bonds are considerably, about 0.16–0.18 Å, longer than the terminal bonds—as expected.

Fig. 2
figure 2

Symmetries, relative energies, and number of imaginary frequencies of different possible isomers of tin diiodide dimers at the B3LYP/SDB-cc-pVTZ level

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Table 2 Geometrical parameters of the two stable isomers of Sn2I4 (bond lengths in Å, angles in degrees)a
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Frequencies

The vibrational frequencies of the monomer molecule were calculated at different levels and are given in Table 1. The symmetric and asymmetric stretching frequencies are very close to each other and therefore their assignment in experiment might be difficult especially in the gas phase where considerable overlap might be expected. There was a matrix isolation study of this molecule [25], in both argon and xenon matrices and there the two lines were separated. The higher frequency with lower intensity was assigned to the symmetric stretching, while the higher intensity with smaller value to the asymmetric stretching. This assignment is in agreement with our results for both the relative values of these frequencies and their intensities, except for some of the lowest, double-zeta, level computations (see Supporting Information). It is also worth mentioning that the early estimation of Brewer et al. [24] and those by the joint electron diffraction/vibrational spectroscopic analyses [7a, c] all had the relative magnitude of the two stretching frequencies wrong. At the same time, there appears to be a remarkable consensus on the low-frequency bending mode to be about 60 cm−1.

The vibrational frequencies of the two stable dimeric molecules are given in Table 3. The frequencies corresponding to the terminal symmetric stretchings of the more stable isomer are about 15 cm−1 smaller than the symmetric stretching frequency of the monomer (MP2 level), in accordance with the longer dimer terminal bond length. The terminal bond length of the C 2v-symmetry isomer is about the same as the bond length of the monomer and the stretching frequencies are also about the same. The frequency corresponding to the ring puckering is especially low, about 7 cm−1, for the C 2v isomer.

Table 3 Vibrational frequencies (cm− 1) and infrared intensities (in parenthesis, km mol− 1) of the Sn2I4 dimers from computationa
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Relative energies from computation

Table 4 shows the relative energies of the two stable forms of the dimer; apparently they do considerably depend on the computational level. The density functional calculations underestimate this difference as the MP2 calculation with triple-zeta bases gives 4 times as large and about 2.5 times as large energy difference as the B3LYP and B3PW91 calculations, respectively. We also calculated the structure of the transition state between these two molecules (see Supporting Information). Figure 3 shows the energy differences; the energy of the transition state seems to be high enough, so we might not expect the second, C 2v-symmetry structure to appear in the gas phase.

Table 4 Relative energies (kJ mol− 1), BSSE corrected dimerization enthalpies (kJ mol− 1), and entropies (J mol− 1 K− 1) of the two stable isomers of Sn2I4 a
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Fig. 3
figure 3

Relative energies of the two stable isomers and their transition state structure, MP2/cc-pVTZ-PP level (B3PW91/cc-pVTZ-PP in parenthesis)

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The computed dimerization enthalpies are underestimated. Again, the computed values are seriously method dependent; with the B3LYP method especially failing badly. The MP2 dimerization energy of the stable C s-symmetry isomer is about 5 kJ mol−1 smaller than the experimental value (see Table 4). It is worth mentioning that with the MP2 method the difference between the applied basis sets (large versus small-core) and the option for the correlation calculation (frozen core versus FC1, in which the outermost core orbitals are retained) can be rather large; in this case about 25 kJ mol−1 in ΔH 0298 —see our results and those of Ref. [5] in Table 4. The computed entropy (MP2 level, Table 4) and the computed ionization potentials, given in Table 5 together with the experimental data, agree well with the experimental results.

Table 5 Calculated adiabatic ionization potentials, measured appearance energies and vertical ionization potentials (in eV)
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Thermodynamic calculations

Detailed thermodynamic calculations have been carried out by Hilpert et al. [4] using experimentally determined molecular parameters for the monomer [6], and assuming a planar structure of D 2h symmetry for the dimer. The molecular geometry and vibrational frequencies of the dimer were estimated on the basis of spectroscopic investigations on metal dihalides. The ratio of the stretching force constant of the bridging Sn–I bond to that of the terminal Sn–I bond was taken as 0.5. We decided to recalculate the thermodynamic functions using the geometry and vibrational frequencies of monomer and dimers (C s and C 2v) obtained by ab initio calculations in the present work in order to see the influence of the dimer structure on the thermodynamic properties of the dimerization process.

Thermodynamic functions for the gaseous monomers and dimers have been calculated in the rigid rotator harmonic oscillator approximation by using the equations given in Ref. [31]. The thermodynamic functions for the monomer and the two dimer structures are listed for different temperatures in Table 6. In the case of monomer there is excellent agreement between the calculated values of the Gibbs energy function ( \( - [G_{\text{m}}^0 (T) - H_{\text{m}}^0 (0)]/T \)) and those given in Table VI of the review paper of Brewer et al. [24] as well as those calculated by Hilpert et al. [4]. For example, at 298.15 K the corresponding values are 292.96, 293.13, and 292.35 J K−1 mol−1, respectively. The enthalpy increments (\( H_{\text{m}}^0 (298.15) - H_{\text{m}}^{\text{0}} (0) \)) are also in good agreement with each other (14.96, 14.77, and 14.71 kJ mol−1). As far as the Gibbs energy function (\( - [G_{\text{m}}^0 (T) - H_{\text{m}}^0 (0)]/T \)) of the Sn2I4 dimer is concerned the values for the C 2v (present work) and D 2h (Hilpert et al. [4]) dimers are about 2% higher and 1% lower, respectively, than that for the C s dimer (present work). It is interesting to note that the thermodynamic functions for the dimer with D 2h symmetry, suggested by Hilpert et al. [4], are surprisingly close to those for the C s dimer. A careful inspection of the calculated thermodynamic functions reveals that the differences in the Gibbs energy function between the dimers with different symmetry result from the entropy values of the low-frequency vibrational modes.

Table 6 Thermodynamic functions of SnI2(g) and Sn2I4(g)
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The Gibbs energy functions and the enthalpy increments of SnI2(g) and Sn2I4(g) can be used in the calculation of the enthalpy of dimerization:

$$ {\text{2 SnI}}_{\text{2}} \;{\text{ = }}\;{\text{Sn}}_{\text{2}} {\text{I}}_{\text{4}} . $$

The enthalpy change of the dimerization reaction at 298 K (\( \Delta _{\text{r}} H_{\text{m}}^{\text{0}} \,(298.15) \)) can be obtained by the third-law method [31] from the experimentally determined equilibrium constant of the dimerization (K p) using the equation

$$ \Delta _{\text{r}} H_{\text{m}}^{\text{0}} (298.15)\; = - T\{ R\ln K_{\text{p}} + \Delta _{\text{r}} [G_{\text{m}}^0 (T) - H_{\text{m}}^{\text{0}} (298.15)]/T\} $$

where K p is the equilibrium constant and Δr on the right hand side of the equation denotes the change of the Gibbs energy function in the dimerization process. The equilibrium constants for the dimerization were taken from Table 6 of the paper of Hilpert et al. [4], who studied the vaporization of solid SnI2 by high-temperature mass spectrometry with a Knudsen cell in the temperature range 474–582 K. To test the reliability of the results obtained by the second-law evaluation method [31] (vide infra) third-law calculations were carried out for the C s dimer by using the values of the thermodynamic functions for the monomer and C s dimer calculated in the same way as the values given in Table 6. The average value for the enthalpy change of the dimerization (\( \Delta _{\text{r}} H_{\text{m}}^0 (298.15) \)) was found to be −101.0 ± 0.6 kJ mol−1, for the C s dimer.

The second-law evaluation method [31] is based on the temperature dependence of the equilibrium constant of the dimerization, which can be calculated from the measured vapor pressures of monomers and dimers over the solid phase (see Table 3 of Ref. 4). The dimer content of the vapor can be estimated from the vapor pressures; it is found to be about 0.2% at the highest experimental temperature of the investigation of Hilpert et al. [4]. On the basis of thermodynamic considerations the appearance of dimers in the vapor phase cannot be excluded [32], since the heat of vaporization of the dimer (42.4 kcal mol−1) is not larger than that of the monomer (34.3 kcal mol−1) by more than 10 kcal mol−1. The logarithm of K p is a linear function of 1/T and from the slope of the straight line an average value of −110.3 ± 4 kJ mol−1 was obtained for \( \Delta _{\text{r}} H_{\text{m}}^0 (500) \). From this value by using the enthalpy increments given in Table 6 one obtains for \( \Delta _{\text{r}} H_{\text{m}}^0 (298.15) \) the values of −101.6, and −97.0 kJ mol−1 for the C s and C 2v dimers, respectively. As can be seen, the agreement between the values obtained by the third-law and second-law evaluation methods in the case of the C s dimers is excellent. The enthalpy of dimerization value for the minimum-energy C s-symmetry dimer is in good agreement with the calculated value of Hilpert et al. (−101.0 ±  4.4 kJ mol−1) for the assumed D 2h dimer [4]. On the other hand, in the case of the C 2v dimer the value obtained for \( \Delta _{\text{r}} H_{\text{m}}^0 (298.15) \) seems to be significantly smaller and therefore this dimer structure can be ruled out with high probability. This conclusion is also supported by the results of the calculation based on the assumption that the dimer terminal bonds are about as strong as the monomer bonds, whereas the bridging bonds can be considered to be about 60% as strong as the terminal bonds [33]. It can be shown that in this case the dimerization enthalpy is equal to \( - 0.2\Delta H_{{\text{atom}}}^0 , \) where \( \Delta H_{{\text{atom}}}^0 \) is the dissociation enthalpy of the gaseous metal dihalide to gaseous atoms [32]. The agreement between the dimerization enthalpy value of −101.3 kJ mol−1 obtained from the dissociation enthalpy (Table VII in Ref. 24) and that for the C s dimer is fortuitously good. It should be noted that the computations carried out in the present work for the minimum-energy C s-symmetry dimer structure yielded, within error limits, the same thermodynamic values for the dimerization reaction as those for the D 2h dimer structure assumed by Hilpert et al. [4]. This suggests that the thermodynamic calculations are rather insensitive to the shape of the halogen-bridged Sn2I4 dimer—probably as long as they have very low-frequency vibrational modes that all the considered structures do.

Conclusion

The molecular geometries of monomeric and dimeric tin diiodide were determined by high-level quantum chemical calculations. We found a good agreement with experiment for the bond length of SnI2—and confirmed earlier suggestions [8] that the bond angle, determined by gas-phase electron diffraction [6], must be at fault. The ground-state geometry of the dimeric Sn2I4 molecule has C s symmetry with a considerably puckered central four-membered ring and strongly bent terminal Sn–I bonds, in accord with the stereochemical activity of the lone electron pair on tin. Our thermodynamic calculations show that, although the geometry of the dimer is very different from the usual D 2h-symmetry arrangement of a metal dihalide dimer, this difference hardly has any effect on the calculated thermodynamic functions of the dimer.

Supporting information: Geometrical parameters and vibrational frequencies of the SnI2 molecule at different computational levels, vibrational frequencies of all Sn2I4 structures from Fig. 2, geometrical parameters and relative energies of the two stable dimer forms at different computational levels, and geometrical parameters of the two Sn2I4 + (C s and C 2v symmetry) ions and of the transition state (TS) structure of Sn2I4. This material is available from the authors upon request.