Are noble gas molecules able to exhibit a superhalogen nature?

Abstract

Superhalogens are a class of highly electronegative molecules whose electron affinities even exceed those of the halogen elements. Since such species can serve as new oxidizing agents, biocatalysts, and building blocks of unusual salts, they are important to the chemical industry. The ability of noble gas (Ng) atoms to form stable mononuclear (NgF₇⁻) and dinuclear (Ng₂F₁₃⁻) superhalogen anions has been reported. Theoretical considerations supported by ab initio calculations revealed that Ng atoms (i.e. Kr, Xe, Rn) should form strongly bound anionic systems when combined with fluorine atoms (acting as ligands). Particularly, those anionic molecules exhibit larger vertical electron detachment energies (6.97–9.56 eV) than that of the chlorine atom (3.62 eV), confirming their superhalogen nature. We believe that the results provided in this contribution will not only provide evidence of a new type of superhalogen molecule but also stimulate more research interest and efforts in the amazing superatom realm.

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1. Introduction

Electron affinity is one of the major factors that govern molecular reactivity in chemistry. In particular, molecules with high EA usually act as strong oxidizing agents in chemical reactions and tend to form very stable negative ions. It is well known that halogen atoms possess the highest electron affinities (EA) among the elements (fluorine 3.40 eV, chlorine 3.62 eV).¹ However, the EA of a polyatomic system may exceed the 3.62 eV limit due to collective effects. Although such species have been attracting chemists' attention since the early 60s, it was only in 1981 that Gutsev and Boldyrev² proposed to term them superhalogens and introduced a simple formula for one class of these compounds, MX_k+1, where M is a main group or transition metal atom, X is a halogen atom, and k is the maximal formal valence of the central atom (M). Once the extra electron is attached (which results in forming a superhalogen anion), it can delocalize over k + 1 halogen atoms, as opposed to one halogen atom. Consequently, the EA of the superhalogen is higher than that of the constituent halogen. Theoretical as well as experimental studies,^3–10 which have been made to estimate the vertical electron detachment energies (VDEs) of various anions having superhalogens as their neutral parents, have shown that the presence of metals or halogens is not mandatory for a cluster to exhibit superhalogen properties. Examples include B(OF)₄⁻ (VDE = 7.08 eV)⁴ anion which follows the MX_n+1⁻ scheme but has no metal atom and halogen atoms have been replaced with fluoroxyl groups. The discovery of such species conduces to the development of oxidizing agents of new type.^11,12

The main purpose of exploring various novel superhalogen species is to provide reliable data and predictions considering the possible use of such compounds as electron acceptors (oxidizing agents). Superhalogens with strong oxidizing ability can be used to access the high oxidation states of metal atoms otherwise inaccessible in conventional chemistry. Explicitly, superhalogen can even oxidize counterparts with relatively high ionization potential, such as C60 nanoparticles¹² and noble gas atoms.¹³ Moreover, the superhalogens exhibit great application potential in the synthesis of new chemical compounds, and in serving as promising building blocks of novel functional materials, such as nonlinear optical materials and magnetic materials.

Since the presence of metals or halogens is not mandatory for a cluster to exhibit superhalogen properties, the present study examines whether noble gas atoms are capable of forming stable superhalogen anions. Thus far, it has been proven that noble gas fluorides are able to act as fluoride acceptors themselves, and will add one or two fluorine anions. Examples includes the addition of F- to XeF₄ to give the pentagonal planar XeF₅⁻ anion, or XeF₆ to give either XeF₇⁻ or XeF₈²⁻, depending on the amount of fluoride added.^14,15 In 2010, Vasdev et al. reported the spectroscopic evidence for the formation of the XeF₃⁻ anion in gas-phase suggesting its Y-shaped (C_2v-symmetry) structure.¹⁶ The X-ray structural analyses performed by Ellern et al. revealed the existence of dinuclear Xe₂F₁₃⁻ anion which structure resembles a XeF₆ molecule bridged by two long Xe–F bonds to a XeF₇⁻ anion.¹⁷ More recently, the theoretical studies on the issue of the possible existence and stability of XeF₇⁻ anions were performed by Grant et al. (employing the coupled cluster methods at the CCSD(T) level) and resulted in the predicting the D_5h-symmetry (pentagonal bipiramidal) structure of XeF₇⁻ anion.¹⁸ Even though the present knowledge considering noble gas atom compounds and their daughter anions seems substantial, the available description of some of their relatively structurally simple representatives is not satisfactory. In particular, most of the research in the chemistry of noble-gases has been limited to the compounds of the lighter element xenon. The chemistry of the heavier radon has not attached much attention probably due to its radioactivity. The ability to form stable anionic species, however, are expected to be very pronounced in the heavy element radon (with Z = 86). This lack od data has motivated us to take a closer look at the various anionic fluorides containing Kr, Xe or Rn as a central atom and to calculate their vertical detachment energies.

In this contribution, to examine the behavior of Ng in superhalogen design, we theoretically constructed and studied a series of NgF₇⁻ (Ng = Kr, Xe, Rn), NgF₅⁻ (Ng = Xe, Rn), NgF₃⁻ (Ng = Xe, Rn), and dinuclear Ng₂F₁₃⁻ (Ng = Xe, Rn) anions. Also, we provide the Gibbs free energies of the most probable fragmentation reactions that the NgF₇⁻ and Ng₂F₁₃⁻ anions might be susceptible to. The results show that these studied anions exhibit much larger VDE values than EAs of halogen elements, confirming their superhalogen identities. This finding provides a new way to design and synthesize superhalogens by decorating noble gases cores with fluorine ligands. The purpose of these efforts is to provide reliable predictions of their physical–chemical properties, considering the possible use of such compounds as electron acceptors (oxidizing agents) in various chemical processes as well as the role they can play in synthesis (e.g. in oxidation of counterpart systems with high ionization potentials (IP)).^11,12

2. Computational methods

Molecular structures of anionic systems Ng_nF_6n+1⁻ (Ng = Kr, Xe, Rn; n = 1, 2) were combinatorial generated with the SuperhalogenCreator software.¹⁹ Since our main goal was to estimate the electronic stability (i.e. the vertical electron detachment energies, VDEs) for the KrF₇⁻, XeF₇⁻, RnF₇⁻, Xe₂F₁₃⁻, and Rn₂F₁₃⁻​ anions, we limited our geometry optimization calculations to the closed-shell anionic species for which we also obtained harmonic vibrational frequencies at their minimum energy structures. For this purpose, we applied second-order Møller–Plesset (MP2) perturbational method with the 6-311++G(3df,3pd) basis sets.^20,21 We applied the MP2/6-311+G(3df,3pd) approach (instead of more accurate but significantly more expensive methods (i.e. MP4 and CCSD(T))^22–24) since the earlier report on the performance of selected ab initio methods and basis sets in estimating the VDEs of superhalogen anions confirms the ability of this method to reproduce the electronic properties of such anions with high accuracy.²⁵ For the noble gas atoms, the effective core potential (ECP) Stuttgart/Dresden ECP-X-MWB basis set (incorporated in GAUSSIAN09 software)²⁶ was employed (where X is the number of core electrons). For calculations involving krypton, xenon, and radon respectively ECP-28-MWB, ECP-46-MWB, and ECP78MWB basis sets were used. The final values of the vertical electron binding energies were obtained by employing the outer valence Green function OVGF method (B approximation)^27–35 with the 6-311++G(3df,3pd) basis sets.^20,21 The Green function methods are related to and essentially the same, when carried to the same order of analysis, as the propagator^35–37 and equations of motion (EOM)^32,38,39 methods. Since the OVGF approximation remains valid only for outer valence ionizations for which the pole strengths (PSs) are greater than 0.80–0.85,⁴⁰ we verified that the PS values were sufficiently large to justify the use of the OVGF method for all states studied here (the smallest PS found for the states studied in this work was 0.84). The partial atomic charges (required for estimating polarity of Ng–F bonds) were fitted to the electrostatic potential according to the Merz–Singh–Kollman scheme.⁴¹

To approximate the effect of hydration on the binding energies, we employed the polarized continuum (PCM) solvation model^42–44 within a self-consistent reaction field treatment, as implemented in the Gaussian 09 program. The PCM calculations with a dielectric constant of water (ε = 78.39) were carried out with the same basis set as that for the isolated species.

All calculations were performed with the GAUSSIAN09 software package.²⁶ In order to avoid erroneous results from the default direct SCF calculations, the keyword SCF = NoVarAcc was used and the two-electron integrals were evaluated (without prescreening) to a tolerance of 10⁻²⁰ au. The optimizations of the geometries were performed using relatively tight convergence thresholds (i.e. 10⁻⁵ hartree/bohr (or radian) for the root mean square first derivative).

3. Results and discussion

To test the hypotheses whether noble gas atoms are capable of forming stable superhalogen anions, a systematic study of NgF₇⁻ (where Ng = Kr, Xe, and Rn) molecules was performed. The fluorine atoms were chosen as ligands since they are small in size (which allows one to perform high-level ab initio calculations), and their corresponding EA (3.40 eV)¹ is high. It should be mentioned here that NgF₇⁻ systems' electron detachment energy have not been reported thus far.

3.1. Equilibrium structures and thermodynamic stability of NgF₇⁻ (Ng = Kr, Xe, Rn) anions

The exploration of the potential energy surface (PES) of the anionic NgF₇⁻ systems (M = Kr, Xe, and Rn) leads to the minimum energy structures depicted in Fig. 1. The corresponding geometrical parameters along with harmonic vibrational frequencies for these structures are collected in Table 1. According to our findings KrCl₇⁻ and XeCl₇⁻ anions are geometrically unstable. The minimum energy structures of KrF₇⁻ and XeF₇⁻ anions possess D_5h-symmetry. In addition, we checked that lower symmetry structures are not geometrically stable (i.e. we confirmed that the C_3v- and C_5v-symmetry NgF₇⁻ (Ng = Kr, Xe) systems do not correspond to the minimum energy structures and the deformation along their imaginary modes leads to pentagonal bipyramidal species).

Fig. 1 The equilibrium structures (on the left) and corresponding highest occupied molecular orbitals (on the right) of the NgF₇⁻ anions (Ng = Kr, Xe, Rn) obtained at the MP2/6-311++G(3df,3pd)+ECPs level.

Table 1 The MP2/6-311++G(3df,3pd)+ECPs geometrical parameters, corresponding harmonic vibrational frequencies (in cm⁻¹), and the OVGF/6-311++G(3df,3pd)+ECPs vertical electron detachment energies (VDE) for the KrF₇⁻, XeF₇⁻, and RnF₇⁻ anions^a

Species (symmetry)	VDE (PS)	Geometrical parameters and atomic charges	Vibrational frequencies [cm^−1]
a VDE in eV (pole strengths (PS) in parenthesis), bond lengths (r) in Å, atomic charges (q) in a.u., valence angles (α) and dihedral angles (ω) in degrees.
KrF7^− (D5h)	7.508 (0.835)	r(Kr–Fax) = 1.880	υ1,2 = 57 (e′′2)	υ8 = 283 (a′1)	υ15 = 393 (a′1)
		r(Kr–Feq) = 2.007	υ3,4 = 198 (e′′1)	υ9,10 = 305 (e′2)	υ16,17 = 546 (e′1)
		q^Fax = −0.23	υ5,6 = 201 (e′1)	υ11,12 = 317 (e′1)	υ18 = 629 (a′′2)
		q^Feq = −0.35	υ7 = 257 (a′′2)	υ13,14 = 383 (e′2)
		q^Kr = 1.22
XeF7^− (D5h)	6.967 (0.893)	r(Xe–Fax) = 1.987	υ1,2 = 39 (e′′2)	υ8,9 = 287 (e′1)	υ15 = 521 (a′1)
		r(Xe–Feq) = 2.026	υ3,4 = 158 (e′1)	υ10,11 = 381 (e′2)	υ16,17 = 532 (e′1)
		q^Fax = −0.36	υ5 = 193 (a′′2)	υ12,13 = 430 (e′2)	υ18 = 565 (a′′2)
		q^Feq = −0.43	υ6,7 = 199 (e′′1)	υ14 = 448 (a′1)
		q^Xe = 1.89
RnF7^− (Cs)	8.136 (0.908)	r(Rn–F1) = 2.067	α(F1RnF2) = 89.21	υ1 = 4 (a′′)	υ10 = 323 (a′′)
		r(Rn–F2) = 2.078	α(F1RnF3) = 81.52	υ2 = 27 (a′)	υ11 = 323 (a′)
		r(Rn–F3) = 2.087	α(F1RnF4) = 108.67	υ3 = 160 (a′′)	υ12 = 471 (a′)
		r(Rn–F4) = 2.092	α(F1RnF5) = 171.37	υ4 = 167 (a′)	υ13 = 476 (a′′)
		r(Rn–F5) = 2.069	ω(F1RnF2F3) = 81.41	υ5 = 186 (a′)	υ14 = 489 (a′)
		q^F1,F5 = −0.42	ω(F1RnF3F4) = 112.55	υ6 = 225 (a′′)	υ15 = 523 (a′)
		q^F2,F3 = −0.46	ω(F1RnF2F5) = 174.76	υ7 = 231 (a′)	υ16 = 523 (a′′)
		q^F4 = −0.42		υ8 = 241 (a′′)	υ17 = 532 (a′)
		q^Rn = 2.17		υ9 = 243 (a′)	υ18 = 535 (a′)

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According to our findings, in the pentagonal bipyramidal D_5h-symmetry structure of KrF₇⁻ anion, the axial (ax) and equatorial (eq) Kr–F bond lengths read 1.880 and 2.007 Å, respectively. The lengths of the Xe–F bonds in the XeF₇⁻ anion were calculated as equal to 1.987 Å (ax) and 2.026 Å (eq); see Fig. 1, which is in good agreement with those predicted earlier by Grant et al. (ax 1.978 Å; eq, 2.016 Å)¹⁸ and the Xe–F separations observed in crystals (1.932–2.100 Å).¹⁷ The minimum energy structure of RnF₇⁻ anion was found to exhibit C_s-symmetry with the Rn–F bond lengths within the 2.067–2.092 Å range, as depicted in Fig. 1.

As far as the thermodynamic stability of the NgX₇⁻ anions is concerned, we calculated the free enthalpies (ΔH²⁹⁸_r), entropies (ΔS²⁹⁸_r), and Gibbs free energies (ΔG²⁹⁸_r) for the reactions in gas phase (for T = 298.15 K and p = 1 atm) corresponding to the most likely fragmentation channels (see Table 2, where the ΔH²⁹⁸_r, ΔS²⁹⁸_r, and ΔG²⁹⁸_r are collected). The KrF₇⁻ anion is stable with respect to fragmentations leading to the KrF₆⁻, KrF₄⁻, and Kr⁻, whereas it was found to be thermodynamically unstable when Kr or KrF₂⁻ loss is considered (the smallest ΔG²⁹⁸_r value was predicted for the KrF₇⁻ → Kr + 3F₂ + F⁻ reaction and reads −19.96 kcal mol⁻¹). From the other hand, the XeF₇⁻ and RnF₇⁻ species were found to be stable and not susceptible to any fragmentation. In particular, in XeF₇⁻ and RnF₇⁻ case, the smallest ΔG²⁹⁸_r value was predicted for the F₂ detachment (i.e. XeF₇⁻ → XeF₅⁻ + F₂ and RnF₇⁻ → RnF₅⁻ + F₂); however, both these Gibbs free energies were found to be positive (16.08 and 48.64 kcal mol⁻¹, for XeF₇⁻ and RnF₇⁻ respectively), which confirms the thermodynamic stability of the XeF₇⁻ and RnF₇⁻ anions. According to the results gathered in Table 2, the fragmentations leading to the NgF₆, NgF₆⁻, NgF₄, NgF₄⁻, NgF₃, NgF₃⁻, NgF₂, NgF₂⁻, Ng, or Ng⁻ (where Ng = Xe, Rn) are even less likely as the calculated ΔG²⁹⁸_r values span the 66.20–556.99 kcal mol⁻¹ range. Therefore, we conclude that the XeF₇⁻ and RnF₇⁻ anions are not susceptible to any fragmentations.

Table 2 Free enthalpies (ΔH²⁹⁸_r in kcal mol⁻¹), entropies (ΔS²⁹⁸_r in cal (mol⁻¹ K⁻¹)), and Gibbs free energies (ΔG²⁹⁸_r in kcal mol⁻¹) of the fragmentation reactions (for T = 298.15 K, p = 1 atm) considered in this paper. The results are obtained at the MP2/6-311++G(3df,3pd)+ECPs level

Fragmentation path	ΔH^298r	ΔS^298r	ΔG^298r
KrF7^− → KrF6^− + F	34.42	31.90	24.91
KrF7^− → KrF4^− + F2 + F	27.85	71.18	6.63
KrF7^− → KrF2^− + 2F2 + F	30.55	102.87	−0.12
KrF7^− → Kr + 3F2 + F^−	16.31	121.65	−19.96
KrF7^− → Kr^− + 3F2 + F	134.24	124.41	97.15
XeF7^− → XeF6 + F^−	73.43	24.24	66.20
XeF7^− → XeF6^− + F	98.34	28.00	90.00
XeF7^− → XeF5^− + F2	40.30	81.23	16.08
XeF7^− → XeF4 + F2 + F^−	98.78	61.75	80.37
XeF7^− → XeF4^− + F2 + F	114.70	65.77	95.09
XeF7^− → XeF3 +F^− + F2 + F	228.28	137.25	187.36
XeF7^− → XeF3^− + 2F2	80.26	76.72	57.39
XeF7^− → XeF2 + 2F2 + F^−	124.08	94.09	96.03
XeF7^− → XeF2^− + 2F2 + F	149.58	102.05	119.15
XeF7^− → Xe + 3F2 + F	150.60	120.94	114.54
XeF7^− → Xe^− + 3F2 + F^−	308.86	123.70	271.98
RnF7^− → RnF6 + F^−	84.38	17.55	79.15
RnF7^− → RnF6^− + F	74.66	22.70	67.89
RnF7^− → RnF5^− + F2	55.86	24.20	48.64
RnF7^− → RnF4 + F2 + F^−	123.38	54.32	107.19
RnF7^− → RnF4^− + F2 + F	133.18	63.03	114.38
RnF7^− → RnF3 +F^− + F2 + F	279.98	132.71	240.41
RnF7^− → RnF3^− + 2F2	136.12	66.81	116.20
RnF7^− → RnF2 + 2F2 + F^−	161.04	85.80	135.46
RnF7^− → RnF2^− + 2F2 + F	190.99	94.20	162.91
RnF7^− → Rn + 3F2 + F^−	202.14	112.05	168.73
RnF7^− → Rn^− + 3F2 + F	591.22	114.80	556.99

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Analysis of the ΔG²⁹⁸_r values calculated for the most likely fragmentation reactions leads to following conclusions: (i) krypton atom forms thermodynamically unstable KrF₇⁻ anion; (ii) xenon and radon atoms are capable of forming thermodynamically stable respectively XeF₇⁻ and RnF₇⁻ anions. Albeit special attention should be paid to the species, which are not susceptible to spontaneous fragmentations (because their future experimental applications are much more likely), to provide the completeness of the results we decided to discuss the structure and electronic stability of all the anions (including the thermodynamically unstable anions). Furthermore, one should also keep in mind that even the thermodynamically unstable species discussed are both geometrically and electronically stable and any fragmentation process would likely require overcoming certain kinetic barriers (corresponding to bond breaking and structure reorganization). For instance, the kinetic barrier leading from the KrF₇⁻ to the fragmentation products was calculated to be 2.5 kcal mol⁻¹ (Fig. 2) which means that this anion might exist at very low temperatures in the gas phase (assuming lack of any perturbing factors).

Fig. 2 The CCST(T)/6-311++G(3df,3pd)+ECPs//MP2/6-311+G(3df)+ECPs energy diagram for various fragmentation paths of the KrF₇⁻ anion. Transition state (TS) is depicted as structure connecting the KrF₇⁻ anion and the fragmentation products. The corresponding geometrical parameters along with harmonic vibrational frequencies for the TS structure are provided in the ESI (Table ESI 1†).

3.2. The formation of the XeF₇⁻ superhalogen anions

The formation of the NgF₇⁻ superhalogen anion can be explained by choosing one representative system and determining the eventual kinetic (i.e., activation) barriers that have to be overcome to create the resulting compound. We decided to discuss this issue by performing the additional calculations (followed by the results analysis) for arbitrarily chosen anionic XeF₇⁻ system.

While analyzing the energy profile for the XeF₆ + F⁻ → XeF₇⁻ process, the excess electron has to be assigned to the F rather than XeF₆ (for the separated F and XeF₆ species) because of the significantly larger electron affinity of the fluorine atom. Indeed, as indicated by the localization of the highest energy molecular orbital (HOMO), the extra electron is located in the vicinity of the F species (as depicted in see Fig. 3). However, the part of the excess electron density is being transferred to the remaining F atoms as the originally distant F approaches the XeF₆ to form the XeF₇⁻ (see Fig. 3 where also the HOMO for the equilibrium XeF₇⁻ structure is plotted). As a consequence, the excess electron density is equally distributed (due to symmetry reasons) among the seven F ligands in the XeF₇⁻ anion. The energy smoothly decreases as the “additional” F approaches the neutral XeF₆ molecule and there is no kinetic barrier that has to be surmounted. Since the energy of the separated XeF₆ and F⁻ is much larger (by ca. 73 kcal mol⁻¹, see the asymptote in Fig. 3) than the energy of XeF₇⁻ at its equilibrium structure, and taking into account that the XeF₆ + F⁻ → XeF₇⁻ process is predicted to be barrier-free, one may expect the XeF₇⁻ anions to be formed spontaneously in the gas phase (whenever the negatively charged fluorine atoms find themselves in the vicinity of XeF₆ molecules).

3.3. Excess electron binding energies of NgF₇⁻ (Ng = Kr, Xe, Rn) anions

The vertical electron detachment energies of NgF₇⁻ (Ng = Kr, Xe, Rn) anions calculated at the OVGF/6-311++G(3df,3pd)+ECPs level are collected in Table 1 (in each case, the lowest VDE arises from OVGF improvements to the energy of HOMO). All of the calculated VDEs greatly exceed the electron affinity of the chlorine atom (3.62 eV), and thus the studied NgF₇⁻ species should be classified as the superhalogen anions. The largest VDE among systems considered was found for RnF₇⁻ (8.136 eV). The vertical electron detachment energies of the remaining anions are always exceeding 7 eV (see Table 1 and Fig. 1), which suggests that each of these species can potentially act as strong electron acceptors.

In order to support our discussion, we present the three-dimensional pictures of the highest energy molecular orbitals (HOMO) for NgF₇⁻ anions (see Fig. 1). It is shown that for each anion, the HOMO reveals the antibonding character with respect to the ligand-central atom interaction. From the other hand the corresponding highly negative HOMO eigenvalues of −8.5, −7.8, and −9.1 eV for KrF₇⁻, XeF₇⁻ and RnF₇⁻, respectively, reflect the large electronic stabilities of these anions. The large electronic stability of NgF₇⁻ systems has to be related to the high polarity of Ng–F bonds with Ng^δ+F^δ− charge distribution. Indeed, the population analysis performed (based on the partial atomic charges (q^ESP) fitted according to the Merz–Singh–Kollman scheme⁴¹ to reproduce the electrostatic potential) indicates the substantial charge separation (polarity) between the noble gas atom and each of F atoms. Namely, the ESP charges on Ng span the 1.22–2.17 a.u. range for NgF₇⁻ anions with the greatest q^ESP value for the RnF₇⁻ anion (q^Rn = 2.17 a.u.). Due to Ng–F bonds' high polarity, though the HOMOs are antibonding with respect to the central atom–ligand interaction the real contribution of the central atom AOs to the HOMO is small and the HOMOs of the NgF₇⁻ anions are practically nonbonding ligand MOs. Such situation was observed for the superhalogen anions studied in the past⁹ and enables the large stability of those systems.

As it is well established,⁴⁵ the resulting VDE of a superhalogen anion depends mainly on the electronegativity of the ligands this anion is decorated with. In addition, the higher vertical detachment energy of an anion is expected for a large number of such ligands due to so called ‘collective effects’. Thus the unexpectedly large VDE value we obtained for the NgF₇⁻ anions is related to the fact that it contains seven strongly electronegative fluorine atoms serving as ‘ligands’. Explicitly, the excess electron in the NgX₇⁻ species occupies a molecular orbital that extends over all ligands and such an effective delocalization increases the electronic stability of this anion. The dependence of a number of ligands on an electronic stability of resulting compounds is yet to be discussed in the following section.

Fig. 3 The MP2/6–311++G(3df,3pd)+ECPs energy profile for the formation of the XeF₇⁻ anion according to the XeF₆ + F⁻ → XeF₇⁻ reaction. The asymptote corresponds to the sum of the energies of isolated fragments [XeF₆; F⁻] and reads 73.4 kcal mol⁻¹. The highest occupied molecular orbitals (HOMO) holding the excess electron are depicted for the structures corresponding to r = 2.026 Å (equilibrium geometry) and r = 12 Å.

3.4. Electronic stability of the anions matching the NgF₃⁻ and NgF₅⁻ formulas (Ng = Xe, Rn)

Due to the large electronic stabilities of anions designed to match the NgX₇⁻ formula we decided to verify if other negatively charged noble gas fluorides are also characterized by large VDEs (at the same level of theory, i.e. OVGF/6-311++G(3df,3pd)+ECPs). To investigate the possible electronic stability of the Ng's anions, we focus on the closed-shell anionic system utilizing Xe and Rn elements as central atoms and F atoms as ligands (as depicted in Fig. 4). As it found out, the vertical detachment energies calculated for NgF₃⁻ and NgF₅⁻ (Ng = Xe, Rn) anions span the ca. 5.2–7.8 eV range with the smallest (5.237 eV) values for the RnF₃⁻ and the largest (7.769 eV) for the XeF₅⁻ anion. Similar to NgX₇⁻ systems, despite of the fact that HOMOs reveal the antibonding character with respect to the ligand-central atom interaction, the corresponding highly negative HOMO eigenvalues of −6.2, −8.7, −5.9, and 9.0 eV for XeF₃⁻, XeF₅⁻, RnF₃⁻, and RnF₅⁻, respectively, reveal the large electronic stabilities of these anions (Fig. 4). Moreover, partial atomic charge on central atom reads 0.92, 1.46, 1.01, and 1.69 a.u. for respectively XeF₃⁻, XeF₅⁻, RnF₃⁻, and RnF₅⁻, which results in high polarity of Ng-F bonds and enables large electron binding energies of NgF₃⁻ and NgF₅⁻ (Ng = Xe, Rn) anions.

Fig. 4 The equilibrium structures (on the left) and corresponding highest occupied molecular orbitals (on the right) of the NgF₃⁻ and NgF₅⁻ anions (Ng = Kr, Xe) obtained at the MP2/6-311++G(3df,3pd)+ECPs level.

As expected,^9,45 higher VDEs are predicted for larger (i.e., containing a larger number of fluorine ligands) species matching the RnF_n⁻ (n = 3, 5, 7) formula. Namely, in the series of the mononuclear fluorides RnF₃⁻–RnF₅⁻–RnF₇⁻ the VDE grows sharply when going from RnF₃⁻ (5.237 eV) to RnF₅⁻ (7.698 eV) and continuously increases when going to RnF₇⁻ (8.136 eV). In contrary, in the series of the XeF₃⁻–XeF₅⁻–XeF₇⁻ anions the VDE significantly increases when going from XeF₃⁻ (5.467 eV) to XeF₅⁻ (7.769 eV) and decreases when going to XeF₇⁻ (6.967 eV). This behavior indicates that an optimal combination of stabilization and destabilization factors for attaining the largest VDE of mononuclear XeF_n⁻ (n = 3, 5, 7) anions occurs in the pentacoordinate fluorides. The further increase of the number of ligands does not lead to increasing the VDE. In this case (i.e. XeF₇⁻ whose VDE is smaller than the vertical electron detachment energy of XeF₅⁻ anion by 0.802 eV), destabilization due to interligand repulsion prevails over the stabilization due to delocalization of the additional electron over a large number of the ligand atoms.

3.5. Effect of solvent on stability of NgF₇⁻ anions (Ng = Kr, Xe, Rn)

Although the gas phase calculations give a good measure of the raw stability of species, the geometric and energetic properties of compounds may be somewhat different in a dielectric environment where ε ≠ 1. To estimate these effects, the polarizable continuum model (PCM) calculations (at the MP2/6-311++G(3df,3pd)+ECPs level) were carried out in simulated solvent environments with fully optimized structures (Fig. 5). The free energies (ΔH²⁹⁸_PCM), entropies (ΔS²⁹⁸_PCM), and Gibbs free energies (ΔG²⁹⁸_PCM) for the reactions in the liquid phase corresponding to the most likely fragmentation channels are collected in Table 3. For KrF₇⁻, the effects of the polarizable surroundings decrease significantly Gibbs free energies of considered fragmentation channels (the smallest ΔG²⁹⁸_PCM value was predicted for the KrF₇⁻ → Kr + 3F₂ + F⁻ reaction and reads −54.7 kcal mol⁻¹). Thus, thermodynamic instability of KrF₇⁻ is observed both in the gas phase (Table 2) as well as in liquid solution. In contrary, the XeF₇⁻ and RnF₇⁻ species were found to be stable and not susceptible to any fragmentation. Similarly to gas phase, in liquid phase for XeF₇⁻ and RnF₇⁻ anions, the smallest Gibbs free energy value was predicted for the F₂ detachment (i.e. XeF₇⁻ → XeF₅⁻ + F₂ and RnF₇⁻ → RnF₅⁻ + F₂); but both these Gibbs free energies were found to be positive (24.75 and 39.14 kcal mol⁻¹, for XeF₇⁻ and RnF₇⁻ respectively), which confirms the thermodynamic stability of the XeF₇⁻ and RnF₇⁻ anions in water solution. Since the fragmentations leading to the NgF₆, NgF₆⁻, NgF₄, NgF₄⁻, NgF₃, NgF₃⁻, NgF₂, NgF₂⁻, Ng, or Ng⁻ (where Ng = Xe, Rn) are even less likely (calculated ΔG²⁹⁸_PCM values span the 29.5–542.3 kcal mol⁻¹ range), we conclude that the XeF₇⁻ and RnF₇⁻ anions are not susceptible to any fragmentations in water solution. Consequently, analysis of the ΔG²⁹⁸_PCM values calculated for the most likely fragmentation reactions confirms both (i) thermodynamic instability of KrF₇⁻ anion and (ii) xenon and radon atoms capability of forming thermodynamically stable NgF₇⁻ (Ng = Xe, Rn) anions.

Fig. 5 The equilibrium structures of the KrF₇⁻ (on the left), XeF₇⁻ (on the right), and RnF₇⁻ (on the bottom) anions obtained within polarizable continuum model (PCM) at the at the MP2/6-311++G(3df,3pd)+ECPs level.

Table 3 Free enthalpies (ΔH²⁹⁸_PCM in kcal mol⁻¹), entropies (ΔS²⁹⁸_PCM in cal (mol⁻¹ K⁻¹)), and Gibbs free energies (ΔG²⁹⁸_PCM in kcal mol⁻¹) of the fragmentation reactions (for T = 298.15 K, p = 1 atm) calculated within polarizable continuum model (PCM) at the MP2/6-311++G(3df,3pd)+ECPs level of theory

Fragmentation path	ΔH^298PCM	ΔS^298PCM	ΔG^298PCM
KrF7^− → KrF6^− + F	32.25	27.91	23.93
KrF7^− → KrF4^− + F2 + F	5.79	96.13	−22.87
KrF7^− → KrF2^− + 2F2 + F	19.44	98.01	−9.78
KrF7^− → Kr + 3F2 + F^−	−19.80	117.01	−54.69
KrF7^− → Kr^− + 3F2 + F	19.44	98.01	−9.78
XeF7^− → XeF6 + F^−	35.19	19.24	29.45
XeF7^− → XeF6^− + F	169.03	24.39	161.76
XeF7^− → XeF5^− + F2	33.88	30.64	24.75
XeF7^− → XeF4 + F2 + F^−	56.67	56.51	39.83
XeF7^− → XeF4^− + F2 + F	102.77	68.77	82.27
XeF7^− → XeF3 +F^− + F2 + F	189.83	123.42	153.03
XeF7^− → XeF3^− + 2F2	81.71	73.94	59.67
XeF7^− → XeF2 + 2F2 + F^−	82.12	88.54	55.72
XeF7^− → XeF2^− + 2F2 + F	136.86	96.79	108.00
XeF7^− → Xe + 3F2 + F	113.34	115.16	79.00
XeF7^− → Xe^− + 3F2 + F^−	300.56	117.92	256.41
RnF7^− → RnF6 + F^−	45.24	19.70	39.37
RnF7^− → RnF6^− + F	73.54	29.75	64.67
RnF7^− → RnF5^− + F2	47.77	28.95	39.14
RnF7^− → RnF4 + F2 + F^−	79.69	55.76	63.06
RnF7^− → RnF4^− + F2 + F	126.77	56.73	109.86
RnF7^− → RnF3 +F^− + F2 + F	155.11	99.48	125.45
RnF7^− → RnF3^− + 2F2	116.60	71.74	95.21
RnF7^− → RnF2 + 2F2 + F^−	117.49	86.84	91.60
RnF7^− → RnF2^− + 2F2 + F	176.68	96.01	148.05
RnF7^− → Rn + 3F2 + F^−	164.43	112.82	130.79
RnF7^− → Rn^− + 3F2 + F	576.72	115.58	542.26

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An interaction with polarizable surroundings (represented by a water as a solvent) remarkably improves the electron bind ability of NgF₇⁻ (VDE increases up to 11.90, 11.13, and 12.25 eV for respectively KrF₇⁻, XeF₇⁻, and RnF₇⁻), which is accompanied by Ng–F bonds decreased (by about 0.001–0.008 Å, Fig. 5). Those large VDE values imply that in the presence of the polarizable surroundings, NgF₇⁻ systems (Ng = Kr, Xe, Rn) are less sustainable to electron detachment.

In summary, the PCM results confirm that XeF₇⁻ and RnF₇⁻anions are thermodynamically and geometrically stable compounds in water and their electronic stability increases (by ca. 4 eV) when the gas phase is replaced with the liquid phase. The KrF₇⁻ anion was found to be geometrically and electronically stable system, however, due to its thermodynamic instability, the KrF₇⁻ ends up as only a locally stable system.

3.6. Dinuclear anions Ng₂F₁₃⁻ (Ng = Xe, Rn)

Gutsev and Boldyrev have concluded that designing polynuclear superhalogen anions give the possibility of increasing the number of ligands and avoiding the increase of interligand repulsion.⁹ Hence, construction dinuclear structures with more ligands will lead to achieving species with larger VDEs. In this contribution, dinuclear Ng₂F₁₃⁻ (Ng = Xe, Rn) anions, were designed and investigated to verify whether such species have larger VDEs that the above-discussed mononuclear NgF₇⁻ anions. The exploration of the potential energy surface (PES) of the anionic Ng₂F₁₃⁻ systems (M = Xe and Rn) leads to the minimum energy structures depicted in Fig. 6. The corresponding geometrical parameters along with harmonic vibrational frequencies for these structures are provided in the ESI (Table ESI 2†). The Rn₂F₁₃⁻ anion was found to possess C₂-symmetry and its structure resembles two RnF₇ fragments oriented in an anti manner and sharing three fluorine atoms. The Rn–F bond lengths are in the 2.016–2.202 Å range except those in the Rn–F–Rn bridging fragments whose lengths are longer and read 2.305–2.460 Å. Recalling that the Rn–F separations in the C_s-symmetry RnF₇⁻ anion span the 2.067–2.092 Å range, it can be concluded that the Rn–F bonds in the dinuclear Rn₂F₁₃⁻ anion are of similar lengths whereas the elongated bonds (by ca. 0.2–0.9 Å) are observed in the Rn–F–Rn bridging fragments. In addition, we checked that single- and double-bridged structures are not geometrically stable (i.e. we confirmed that the single- and double-bridged Rn₂F₁₃⁻ systems do not correspond to the minimum energy structures and the deformation along their imaginary modes leads to triple-bridged species).

Fig. 6 The equilibrium structures of the Ng₂F₁₃⁻ anions (Ng = Xe, Rn) obtained at the MP2/6-311++G(3df,3pd)+ECPs level. The VDE values (in eV) and the relative energies (E_R, in kcal mol⁻¹) estimated for Xe₂F₁₃⁻ isomers with respect to its corresponding global minimum are also provided. For all cases, the lowest VDE corresponds (from OVGF improvements) to the energy of HOMO.

The structure of Xe₂F₁₃⁻ anion is more complicated as four different isomers can be formed (see Fig. 6 where zero-point corrected energies of isomers are denoted as E_R). Similarly to the Rn₂F₁₃⁻, the lowest energy structure (of C₂-symmetry) resembles two XeF₇ subunits (each of which contains the xenon atom surrounded by seven fluorine atoms) oriented in an anti manner and sharing three fluorine atoms. The values of the F–Xe–F bridging bond lengths were calculated to read 2.025–3.010 Å whereas the remaining Xe–F separations span relatively narrow 1.947–2.025 Å range. The resulting structure, albeit deficient in symmetry elements, corresponds to the global minimum on the ground state Xe₂F₁₃⁻ anionic potential energy surface. The isomer having higher but similar energy (within 0.3 kcal mol⁻¹) possesses C_2v-symmetry and corresponds to the double-bridged (DB) form with the xenon atoms linked to each other through two fluorine (F₁) atoms (Fig. 6). As expected,^46–48 the Xe–F separation in DB connecting fragments are larger (by ca. 0.1–0.5 Å) than the Xe–F distances in the Xe–F terminal bonds (Table SI2†). The two other Xe₂F₁₃⁻ isomers found are much higher in energy than the C₂-symmetry lowest energy structure. Namely, the D_5d- and D_5h-symmetry single-bridged (SB) isomers were estimated to be energetically less stable by 18.7 kcal mol⁻¹ (with respect to the C₂-symmetry Xe₂F₁₃⁻ structure) and as conformational isomers may likely rapidly interconverting at room temperature. In the SB isomers, the quasi-pentagonal bipiramidal XeF₇ subunits can be distinguished (with the ω(F₁KrF₂F₂) = 93.9° dihedral angles) oriented in the anti (D_5d-symmetry isomer) or syn (which results in the D_5h-symmetry configuration of the anion) manner (as depicted in Fig. 6). In conformational isomers the Xe central atoms are linked with fluorine atoms and the corresponding Xe–F bond lengths in bridging fragment were found to be equal to 2.308 Å. The remaining Xe–F bond lengths were estimated to be shorter (1.865–1.981 Å) in the both Xe₂F₁₃⁻ local minima. Finally, the small energy gap (within 1 kcal mol⁻¹) between double-bridged and triple-bridged isomers of Xe₂F₁₃⁻ indicates that this anion may likely exist in both isomeric forms (the remaining single-bridges structures possess much higher energies and differences exceed 18 kcal mol⁻¹), although adopting C₂-symmetry structure is energetically more favorable.

Since the Ng₂F₁₃⁻ (Ng = Xe, Rn) anionic systems thermodynamic stability needs to be confirmed, we decided to verify whether those species are susceptible to various fragmentations. In order to do this, we considered various most likely fragmentation channels in the case of each anionic system (see Table 4 for all the fragmentation reactions investigated). In particular, we tested the susceptibility of each Ng₂F₁₃⁻ (Ng = Xe, Rn) anion to the closed-shell anionic NgF_n⁻ (n = 3, 5, 7) molecule loss. The analysis performed used the standard expressions for an ideal gas in the canonical ensemble and the results are gathered in Table 4. In general, the Gibbs free energies (ΔG²⁹⁸_r) values for the reactions involving the systems considered are positive which indicates the thermodynamic stability thereof. The only negative ΔG²⁹⁸_r value (−12 kcal mol⁻¹) was found for the higher energy isomers (D_5d- and D_5h-symmetry) of the Xe₂F₁₃⁻ anion, which indicates that the Xe₂F₁₃⁻ → XeF₇⁻ + XeF₆ process should be thermodynamically favorable. However, as discussed in the preceding paragraph, the D_5d- and D_5h-symmetry Xe₂F₁₃⁻ isomers are much higher in energy (by 18.7 kcal mol⁻¹) than the corresponding C₂-symmetry isomer, hence its thermodynamic instability seems unimportant (as the formation of those particular isomers is not likely). Thus, we are confident that none of the anions studied (at its lowest-energy structure) is susceptible to fragmentation, as indicated by the positive ΔG²⁹⁸_r values for the fragmentation channels considered.

Table 4 Free enthalpies (ΔH²⁹⁸_r in kcal mol⁻¹), entropies (ΔS²⁹⁸_r in cal (mol⁻¹ K⁻¹)), and Gibbs free energies (ΔG²⁹⁸_r in kcal mol⁻¹) of the fragmentation reactions (for T = 298.15 K, p = 1 atm) calculated at the MP2/6-311++G(3df,3pd)+ECPs level of theory

Species (symmetry)	Fragmentation path	ΔH^298r	ΔS^298r	ΔG^298r
Xe2F13^− (C2)	Xe2F13^− → 2XeF6 + F^−	87.27	60.03	69.37
	Xe2F13^− → XeF7^− + XeF6	13.84	35.78	3.17
	Xe2F13^− → XeF5^− + XeF6 + F2	54.15	68.69	33.67
	Xe2F13^− → XeF3^− + XeF6 + 2F2	94.10	112.50	60.56
Xe2F13^− (C2v)	Xe2F13^− → 2XeF6 + F^−	86.88	57.36	69.78
	Xe2F13^− → XeF7^− + XeF6	13.45	33.11	3.58
	Xe2F13^− → XeF5^− + XeF6 + F2	53.75	66.02	34.07
	Xe2F13^− → XeF3^− + XeF6 + 2F2	93.71	109.83	60.97
Xe2F13^− (D5d)	Xe2F13^− → 2XeF6 + F^−	68.12	46.05	54.39
	Xe2F13^− → XeF7^− + XeF6	−5.31	21.81	−11.81
	Xe2F13^− → XeF5^− + XeF6 + F2	35.00	54.71	18.68
	Xe2F13^− → XeF3^− + XeF6 + 2F2	74.95	98.53	45.58
Xe2F13^− (D5d)	Xe2F13^− → 2XeF6 + F^−	68.13	46.90	54.15
	Xe2F13^− → XeF7^− + XeF6	−5.30	22.66	−12.05
	Xe2F13^− → XeF5^− + XeF6 + F2	35.01	55.56	18.44
	Xe2F13^− → XeF3^− + XeF6 + 2F2	74.96	99.38	45.33
Rn2F13^− (C2)	Rn2F13^− → 2RnF6 + F^−	106.93	65.86	87.30
	Rn2F13^− → RnF7^− + RnF6	22.55	48.31	8.14
	Rn2F13^− → RnF5^− + RnF6 + F2	78.40	73.30	56.55
	Rn2F13^− → RnF3^− + RnF6 + 2F2	158.67	115.12	124.35

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The calculated values of the excess electron binding energies of the Ng_nF_6n+1⁻ (Ng = Xe, Rn) anions reveal a strong dependence on the number of central noble gas atoms (n). Namely, the vertical excess electron binding energy predicted for the Rn₂F₁₃⁻ anion (9.559 eV) is larger by 1.423 eV than that of the RnF₇⁻ system (Table 1). Correspondingly, the most stable isomer (C₂-symmetry) of the Xe₂F₁₃⁻ exhibits the VDE of 8.127 eV, which is larger by 1.160 eV than the electronic stability of the XeF₇⁻ species. In addition, our results indicate that the VDE of the dinuclear superhalogen Xe₂F₁₃⁻ system depends on its geometrical structure. Namely, the VDEs estimated for four Ng₂F₁₃⁻ isomeric structures span the 7.997–8.302 eV range (Fig. 6). This observation is in agreement with previous theoretical studies indicating that VDE strongly depends on superhalogen's geometrical structure (if various isomeric structures are possible and stable).⁴⁹

4. Summary

To conclude, this paper confirms the existence of hypervalent species utilizing noble gas elements as central atoms. The calculations performed at the OVGF/6-311++G(3df,3pd)+ECPs level of theory show that the vertical electron detachment energies of NgF₇⁻ anions (Ng = Kr, Xe, Rn) always exceed 7 eV. The electronic stability of the Ng's anions with decreased number of electronegative F ligands is also large as the VDEs calculated for NgF₃⁻ and NgF₅⁻ (Ng = Xe, Rn) anions span the 5.2–7.8 eV range. The VDEs of dinuclear Ng₂F₁₃⁻ (Ng = Xe, Rn) anions are in the 8.1–9.6 eV range, which is significantly larger than this of corresponding mononuclear anions. Three main factors determining high VDEs of studied anions are: (i) the electronegative character of the ligands; (ii) the relatively large number of the ligands leading to extent of the delocalization of an extra electron; (ii) positive charge on the central atom, which provides an additional stabilization of an extra electron which is delocalized over ligands.

Although this contribution focuses on Kr, Xe, and Rn elements as potential central atoms for superhalogen systems, the concept of using noble gas atoms to design strong electron acceptor is general and could be extended to other candidates. An important implication of these results is that if the right ligands could be identified, chemical compounds of smaller Ng's such as He and Ne could also be formed.

Acknowledgements

This work was supported by the Polish Ministry of Science and Higher Education (MNiSW) Grant No. 0560/IP3/2013/72. Calculations have been carried out using resources provided by Wroclaw Centre for Networking and Supercomputing (http://wcss.pl), grant No. 378.

References

1. H. Hotop and W. C. Lineberger, J. Phys. Chem. Ref. Data, 1985, 14, 731–750.
2. G. L. Gutsev and A. I. Boldyrev, Chem. Phys., 1981, 56, 277–283.
3. C. Sikorska and P. Skurski, Mol. Phys., 2012, 110, 1447–1452.
4. C. Sikorska, Chem. Phys. Lett., 2015, 638, 179–186.
5. A. K. Srivastava and N. Misra, RSC Adv., 2014, 4, 41260–41265.
6. K. Pradhan, G. L. Gutsev, C. A. Weatherford and P. Jena, J. Chem. Phys., 2011, 134, 144305.
7. W. M. Sun, D. Hou, D. Wu, X. H. Li, Y. Li, J. H. Chen, C. Y. Li and Z. R. Li, Dalton Trans., 2015, 44, 19901–19908.
8. L. P. Ding, X. Y. Kuang, P. Shao, M. M. Zhong and Y. R. Zhao, RSC Adv., 2013, 3, 15449–15456.
9. G. L. Gutsev and A. I. Boldyrev, Adv. Chem. Phys., 1985, 61, 169–221.
10. G. N. Reddy and S. Giri, RSC Adv., 2016, 6, 47145–47150.
11. C. Sikorska and P. Skurski, Chem. Phys. Lett., 2010, 500, 211–216.
12. C. Sikorska, Phys. Chem. Chem. Phys., 2016, 18, 18739–18749.
13. N. Bartlett, Proc. Chem. Soc., London, 1962, 218.
14. R. D. Peacock, I. Sheft and H. Selig, Proc. Chem. Soc., London, 1964, 285.
15. R. D. Peacock, H. Selig and I. Sheft, J. Inorg. Nucl. Chem., 1966, 28, 2561–2567.
16. N. Vasdev, M. D. Moran, H. M. Tuononen, R. Chirakal, R. J. Suontamo, A. D. Bain and G. J. Schrobilgen, Inorg. Chem., 2010, 49, 8997–9004.
17. A. Ellern, A. R. Mahjoub and K. Seppelt, Angew. Chem., Int. Ed., 1996, 35, 1123–1125.
18. D. J. Grant, T. H. Wang, D. A. Dixon and K. O. Christe, Inorg. Chem., 2010, 49, 261–270.
19. K. Sikorska and C. Sikorska, SuperhalogenCreator, Gdansk, 2015, http://celinasikorska.pl/superhalogencreator/.
20. A. D. Mclean and G. S. Chandler, J. Chem. Phys., 1980, 72, 5639–5648.
21. R. Krishnan, J. S. Binkley, R. Seeger and J. A. Pople, J. Chem. Phys., 1980, 72, 650–654.
22. K. Raghavachari and J. A. Pople, Int. J. Quantum Chem., 1978, 14, 91–100.
23. G. E. Scuseria, C. L. Janssen and H. F. Schaefer, J. Chem. Phys., 1988, 89, 7382–7387.
24. G. E. Scuseria and H. F. Schaefer, J. Chem. Phys., 1989, 90, 3700–3703.
25. C. Sikorska, D. Ignatowska, S. Freza and P. Skurski, J. Theor. Comput. Chem., 2011, 10, 93–109.
26. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery Jr, J. E. Peralta, F. Ogliaro, M. J. Bearpark, J. Heyd, E. N. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. P. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, N. J. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski and D. J. Fox, Gaussian 09, 2009.
27. D. J. Rowe, Rev. Mod. Phys., 1968, 40, 153–166.
28. J. Simons, J. Chem. Phys., 1972, 57, 3787.
29. L. S. Cederbaum, J. Phys. B: At., Mol. Opt. Phys., 1975, 8, 290–303.
30. J. V. Ortiz, J. Chem. Phys., 1988, 89, 6353–6356.
31. J. Simons, J. Chem. Phys., 1971, 55, 1218-&.
32. J. Simons and W. D. Smith, J. Chem. Phys., 1973, 58, 4899–4907.
33. V. G. Zakrzewski and J. V. Ortiz, Int. J. Quantum Chem., 1994, 23–27.
34. V. G. Zakrzewski and J. V. Ortiz, Int. J. Quantum Chem., 1995, 53, 583–590.
35. V. G. Zakrzewski, J. V. Ortiz, J. A. Nichols, D. Heryadi, D. L. Yeager and J. T. Golab, Int. J. Quantum Chem., 1996, 60, 29–36.
36. J. Linderberg and Y. Ohrn, Int. J. Quantum Chem., 1977, 12, 161–191.
37. J. V. Ortiz, J. Chem. Phys., 1988, 89, 6348–6352.
38. N. W. Winter, A. Laferrie and V. Mckoy, Phys. Rev. A, 1970, 2, 49–60.
39. V. Mckoy, Phys. Scr., 1980, 21, 238–241.
40. V. G. Zakrzewski, O. Dolgounitcheva and J. V. Ortiz, J. Chem. Phys., 1996, 105, 5872–5877.
41. B. H. Besler, K. M. Merz and P. A. Kollman, J. Comput. Chem., 1990, 11, 431–439.
42. M. Cossi, V. Barone, R. Cammi and J. Tomasi, Chem. Phys. Lett., 1996, 255, 327–335.
43. S. Miertus, E. Scrocco and J. Tomasi, Chem. Phys., 1981, 55, 117–129.
44. S. Miertus and J. Tomasi, Chem. Phys., 1982, 65, 239–245.
45. C. Sikorska, Chem. Phys. Lett., 2015, 625, 157–163.
46. B. Yin, J. L. Li, H. C. Bai, Z. Y. Wen, Z. Y. Jiang and Y. H. Huang, Phys. Chem. Chem. Phys., 2012, 14, 1121–1130.
47. B. Yin, T. Li, J. F. Li, Y. Yu, J. L. Li, Z. Y. Wen and Z. Y. Jiang, J. Chem. Phys., 2014, 140, 094301.
48. J. F. Li, Y. Y. Sun, H. C. Bai, M. M. Li, J. L. Li and B. Yin, AIP Adv., 2015, 5, 067143.
49. C. Sikorska and P. Skurski, Chem. Phys. Lett., 2012, 536, 34–38.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra21933k

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