Information entropy of oxygen allotropes. A still open discussion about the closed form of ozone

Highlights

• Oxygen allotropes with up to 8 atoms are analyzed in terms of Shannon approach.

• Experimentally known species has zero information entropy.

• Cyclic form of ozone also has zero information entropy.

• Ways for its stabilization are briefly discussed.

Abstract

In the present work, information entropies have been calculated for oxygen allotropes, known from experimental and theoretical studies. According to the calculations, only three allotropes (O₂, O₃ (D_3h), and O₈ (D_4h)) have zero information entropies. In terms of the Shannon approach, it indicates high possibility of their formation. Indeed, O₂ is a long-time known compound whereas O₈ (D_4h) cluster is a structural block of the ε-phase of solid oxygen, which has been experimentally studied with X-ray diffraction. The closed form of ozone, the third zero-entropy structure, has not been synthesized at the moment and accompanied with debates around its existence. It has zero information entropy as two other obtained allotropes, so the information approach provides an additional argument in favor of its achievability. A way to increase its stability via encapsulation into molecular cages is discussed.

Introduction

Dioxygen (O₂) and ozone (acyclic or open O₃ molecule) are the long-time known oxygen allotropes. In a classic monograph [1] about molecular forms of oxygen published in 1979, only 7 pages from an overall amount of 304 were devoted to O_n allotropes with n ⩾ 4. The question about possibility of “higher” molecular forms of oxygen is fundamental rather than applied. Being divalent, oxygen does not astonish with the allotropic diversity like, for example, carbon (additionally, higher analogs of H₂O₂ and their organic derivatives are rare and mostly unstable compounds [2]). Exotic oxygen compounds may be scarcely applicable to materials science (as explosives) and synthetic chemistry (as highly effective oxidizing agents or sources of free radicals). However, knowledge about them may provide insights into mechanisms of atmospheric and space processes [3]. In a pure chemical aspect, it is important to define the limits of molecular diversity originating from one element.

Formation of other oxygen allotropes (with larger number of atoms) is discussed (see highlight paper [4], theoretical [5], [6], [7], [8], and experimental works [9], [10], [11], [12]). Thus, tetraoxygen O₄ has been experimentally observed [9], [10], [11] and, more recently, an X-ray study has stated the O₈ cluster as a structural block of the ε-phase of solid oxygen [12]. The last one consists of four dioxygen molecules in a close contact due to the high pressure (11.4 GPa). In addition, exhaustive high-level theoretical study [5] with the CCSD(T) and CAS calculations has demonstrated possible stability of different forms of O₄ and O₆.

Chemists’ interests are shifted from usual conditions (room temperatures) to extreme one (extremely low/high temperatures, giant pressures, or irradiations). As consequence, this leads to discovery of novel allotropes, which commonly are short-lived particles under unusual conditions but nevertheless existing. However, there are examples of the remarkable stability. Fullerenes are one such example. These symmetric polyhedral molecules are produced by laser treatment of graphitic materials. Under the hard conditions of the fullerene synthesis (thousands of Kelvins), simpler and energetically more favorable C₂ molecules are more likely to exist but fullerenes C₆₀, C₇₀, and others are observed [13], [14].

Prevailing “energy-minimum paradigm” is one of the features of current theoretical chemistry. It states that chemical structures with minimal energies (from the set of related compounds) are very likely existing and achievable. Many theoretical works analyze potential energy surfaces (PES) to detect the minima and by comparing their total energies predict the most probable structures. An overwhelming majority of such forecasts predicts true, well describing known experimental data or being proved with further experiments. However, in the case of chemical objects under extreme and non-equilibrium conditions (like the above case of fullerenes), the energy-minimum paradigm sometimes fails as the predicted unfavorable compounds in fact exist. As consequence, additional considerations are invoked to reconcile theory and the observed regularities (such as kinetic stability, special orbital interactions, and symmetry).

The contrary situation also takes place. Many compounds that should be stable according to the quantum-chemical analysis of PES have not been synthesized (see, e.g., work [15] and references therein). It means that some additional factors are more important for their achievability. This is the case of the cyclic form of ozone, one of the title compounds. Its history is well described in previous papers [5], [16], [17]. We just briefly note that after the experimental arguments in favor of the acyclic ozone, Wright has quantum-chemically shown that both acyclic and cyclic forms of O₃ are minima of the corresponding PES [16]. Moreover, the cyclic structure was preferred by 48 kcal/mol that contradicted the experimental observations. The further calculations with higher accuracy, required to take into account a multi-reference nature of trioxides’ wave functions (the CCSD(T) and MRCI calculations [18], [19]), yielded an energy difference of ∼30 kcal/mol in favor of the open form that meets the experiment. However, it was not the end of the story. The PES analysis using an all-valence CASSCF methodology has shown that the barriers surrounding the cyclic ozone are quite high to create substantial handicaps for its dissociation (46.6 kcal/mol) or ring opening (22.7 kcal/mol) [20].

Here arises analogy with fullerenes, a molecular form of carbon. As is known, Fowler and Manolopoulos have proposed an efficient algorithm for generating fullerene structures and collected possible fullerenes from C₂₀ to C₁₀₀ in their Atlas [21]. Most of those structures are hypothetical and have not been obtained but all of them should be minima of PESs (approaches to fullerene stability have been reviewed [22], [23]). Meanwhile, the buckminsterfullerene C₆₀ dominates over the other fullerenes upon the arc-discharge evaporation of graphite (see [13], [14] and references therein). This is additionally amazing if we consider a comparative calorimetric study of C₆₀ and C₇₀ [24]. According to work [24], C₆₀ is thermodynamically less favorable than C₇₀ but it is a fact that the first one is synthesized in substantially greater amounts. Thus, thermodynamic stability does not meet with abundance of the fullerenes. To analyze fullerene structures without considering energy, recent study [25] has used the Shannon approach in its simple form. Note that such approach was previously applied to chemical structures and reactions (see references in [25] and some key works [26], [27], [28], [29], [30], [31], [32], [33]). It operates with information entropy h that is calculated as a logarithmic measure of probability to find in the molecule different sorts of atoms. A smaller information entropy associated with a higher likelihood of formation has been previously postulated for physical [34] and chemical systems [35] but rarely invoked to analyze certain chemical structures. As recently found out [25], only 14 fullerenes (<1% of 2079 studied structures) have h values fitted into the range between the values of the two most abundant fullerenes C₆₀ and C₇₀. Most of these exclusive structures have been currently synthesized. As for C₆₀, it appeared the only member of the fullerene family with zero information entropy.

Oxygen allotropes have never been a subject of the Shannon approach. In the present work, we have calculated the information entropies of the O_n molecules (n = 2–8) to perform a numerical discrimination of the structures and compare with their achievability.