Two theorems about C₂ and some more

ABSTRACT

The story of C₂ continues to fascinate chemists spinning around a possibility of quadruple bonds for p-elements and discussing a wealth of options for the nature of an unconventional fourth bond. This led to lively discussions about the ways of counting or measuring bonds, and the interplay between the bond strength (energy), its length, and rigidity/stiffness (elasticity). Even old concerns about the possibility of theorems in chemistry and thus of the place of chemistry among ‘true’ sciences had been revived. We show that under some mild conditions certain exact statements (lemmas and theorems) about the C₂ molecule can be relatively easily proven which, if not resolves the controversy entirely, at least provides some theoretical reference points to the discussion. Some more general consequences from this experience are discussed as well.

GRAPHICAL ABSTRACT

KEYWORDS:

• C₂
• chemistry theorems
• bonding measures
• analytical models

Acknowledgments

Knowing the addressee of this Festschrift as someone who is not much interested in ‘how much’ we hope that present piece will interest him. We are thankful to the Referee for the benevolent comments. Prof. I. Mayer and Prof. Dr W.H.E. Schwarz are acknowledged for valuable discussions. Dr A.M. Tokmachev contributed to this study at an earlier stage and took part in the discussion which is gratefully acknowledged. We are thankful to Prof. R. Ponec and Dr. B. Braida for sending us reprints of their works.

Note added in proof

Upon acceptance, S. Shaik et al. published ‘The Quadruple Bonding in C₂ Reproduces the Properties of the Molecule’ [Chem. Eur. J., 22, n/a (2016)], as reached us through Dr B. Braida. Although their numerical results conform to our analytical study, the interpretation diverges. (1) Lewis was apparently not right. Either Boutleroff (1865) or Alexejeff (1901) would unanimously say that C₂ possesses a quadruple bond, as published in French, German, and Russian languages [38]. Rumer (1938), by contrast, disliked tetravalence and thus C–C quadruple bonding since the C ground state has only two unpaired electrons and thus must be divalent. (2) Bonding measures are useless for counting bonds. The problem becomes trivial if one uses the ‘tendency to “break”’ the characteristic of multiple bonds, as stressed by Lewis. Four H₂ molecules are to be added to C₂, that is, four bonds to be broken, in order to completely separate two carbon atoms. (3) The presence of the low-lying ${}^3{\Sigma }_g^{+}$ state by itself does not imply whatever reactivity since no concrete reaction is specified, as seen from the triplet O₂ ground state. This problem is discussed in our recent publication [52].

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. The presence of either σ or ι in the notation allows one to uniquely determine which type of banana geminal is meant.

2. The importance of the Pythagorean theorem had been recently recognised on the highest international economical level [22].

3. In a book [23], a boy has to get three banana from an enigmanic planet. We also undertook an expedition to a mostly unstudied planet of DMM/AIMM trying to find them there, but, as we see, they do not exist.

4. Of course, there is an option to receive some gain through the exchange between single electrons in the HO's pointing outwards looking of the Structure 5 (see below).

5. In qualifying a smaller s-weight as an acceptor one, we follow the Bent rule recently proven [17] in the DMM context.

6. It is worth to notice that the nodal structure of the AOs recently regained a considerable interest under the new nickname of ‘nonuclear density maxima’. In the older times, every second year chemistry student in order to become a third year one had to know that the maximum of the electronic density of a 2p-AO lays outside the nucleus. In our days, this ancient wisehood seems to be forgotten as manifested in recent publications [29].

7. Mr Kaushik Basu is a senior vice-president and chief economist of the World Bank and professor of economics at Cornell. According to Wiki, he ‘believes strongly that good moral qualities are essential for growth and development within the economy’.