ORIGINAL PAPER Is hyper-hardness more chemically relevant than expected? Christophe Morell & André Grand & Alejandro Toro-Labbé & Henry Chermette Received: 31 August 2012 / Accepted: 21 January 2013 # Springer-Verlag Berlin Heidelberg 2013 Abstract In this work, the third derivative of the energy with respect to the number of electrons, the so-called hyper-hardness, is investigated to assess whether this quantity has a chemical meaning. To achieve this goal a new working expression for hyper-hardness is devel- oped and analyzed. It transpired from this analysis that hyper-hardness, just like hardness, can measure the reac- tivity or the stability of electron systems. Interestingly, positive values of hyper-hardness point to quite stable species such as noble gases and molecules. On the other hand, radicals almost always display large negative values of hyper-hardness. Keywords Conceptual DFT . Dual descriptor . Hardness derivatives . Principle of maximum hardness Introduction One of the main goals of theoretical chemistry is to under- stand and predict the chemical behavior of chemical com- pounds. Along the years, different concepts have been developed to provide the necessary tools to rationalize the outcome of a chemical reaction. Among these tools, one of the most important is electronegativity [1]. The concept of electronegativity has strongly evolved from its early word- ing by Berzelius [2] in the nineteenth century to its absolute definition in the 1960s by Izckowski and Margrave [3]. Eventually, a quantum justification of the concept of elec- tronegativity has been put forward by Parr and co-workers [4] and constitutes the cornerstone of the conceptual density functional theory [5, 6]. Electronegativity is nowadays de- fined as the opposite of the first derivative of the energy with respect to the number of electrons. Following the definition of absolute electronegativity, the next step was to focus on the second derivative of the energy with respect to the number of electrons. Thus, this index has been identified in a seminal contribution by Parr and Pear- son [7] to the chemical hardness. This identification has open wide a new field of investigation. For instance, the hard and soft acids and bases principle [8, 9] has provided a new paradigm [10–12]. Another example is the proposal for a maximum hardness principle [13]. Actually, the impor- tance of hardness, through its reciprocal, has been recog- nized earlier by the work of Huheey [14, 15] and later by Politzer [16, 17] and was called at that time charge capacity. Charge capacity has been successfully used to understand the chemical behavior of fluorine and classical organic groups such as methyl, ethyl, amino, nitro and others [18]. It is quite interesting to notice that the next derivative, called the hyper-hardness, has been paid very little attention so far. Presumably because it is generally admitted that third derivatives of the energy, at the notable exception of the dual descriptor [19, 20], are supposedly [21], not that relevant for understanding chemical reactivity. Actually, the title of the present article refers to a paper by Geerlings and De Proft that reviewed this topic [22]. Thus, following the pioneering spirit of Huheey and Politzer, it has been decided to investigate the third derivative of the energy with respect to the number of C. Morell : A. Grand CEA Grenoble -INAC/SCIB/LAN (UMR-E n°3 CEA-UJF)), CEA-Grenoble, 17, rue des Martyrs, 38054 Grenoble Cedex 9, France A. Toro-Labbé Laboratorio de Quimica Teorica Computacional (QTC), Facultad de Quimica, Pontifica, Universidad Catôlica de Chile, Casilla 306, Correo 22, Santiago, Chile C. Morell (*) : H. Chermette Université de Lyon; Université Lyon 1(UCBL) et UMR CNRS 5280 Sciences Analytiques, 69622 Villeurbanne Cedex, France e-mail: christophe.morell@univ-lyon1.fr J Mol Model DOI 10.1007/s00894-013-1778-z