Quantification of Aromaticity Based on Interaction Coordinates: A New Proposal Sarvesh Kumar Pandey, † Dhivya Manogaran, † Sadasivam Manogaran,* ,†,‡ and Henry F. Schaefer, III* ,‡ † Department of Chemistry, Indian Institute of Technology, Kanpur 208 016, India ‡ Center for Computational Quantum Chemistry, University of Georgia, Athens, Georgia 30602, United States * S Supporting Information ABSTRACT: Attempts to establish degrees of aromaticity in molecules are legion. In the present study, we begin with a fictitious fragment arising from only those atoms contributing to the aromatic ring and having a force field projected from the original system. For example, in benzene, we adopt a fictitious C 6 fragment with a force field projected from the full benzene force field. When one bond or angle is stretched and kept fixed, followed by a partial optimization for all other internal coordinates, structures change from their respective equilibria. These changes are the responses of all other internal coordinates for constraining the bond or angle by unit displacements and relaxing the forces on all other internal coordinates. The “interaction coordinate” derived from the redundant internal coordinate compliance constants measures how a bond (its electron density) responds for constrained optimization when another bond or angle is stretched by a specified unit (its electron density is perturbed by a finite amount). The sum of interaction coordinates (responses) of all bonded neighbors for all internal coordinates of the fictitious fragment is a measure of the strength of the σ and π electron interactions leading to aromatic stability. This sum, based on interaction coordinates, appears to be successful as an aromaticity index for a range of chemical systems. Since the concept involves analyzing a fragment rather than the whole molecule, this idea is more general and is likely to lead to new insights. 1. INTRODUCTION Concepts like valency, bond order, and aromaticity are exten- sively used in the chemistry literature although quantification of these concepts is difficult if not impossible. In this article, we focus on the quantification of aromaticity. Although several attempts have been made in this direction from the inception of aromaticity in 1856, 1,2 the degree of success has not been entirely satisfactory. 3 Using experimental and theoretical methods, Katritzky 4,5 and Jug 5-7 tried various formulations for the quantification of aromaticity. For this purpose, the energy, structure, and magnetic criteria are primarily used. Using the energy difference between an aromatic cyclic system and its corresponding suitable reference system (usually acyclic olefins or conjugated unsaturated analogues), the “resonance energy (RE)” 8 is defined and extended to an “aromatic stabilization energy (ASE)” based on homodesmotic reactions. 9 The energy criteria lead to difficulties, including selecting the suitable nonaromatic reference compound and a correct value of the heat of formation from the different values given by different authors for the same compound. 4 The cyclic CC bond lengths in aromatic systems fall between the single and double bonds, and Krygowski used this structural information in his “harmonic oscillator model of aromaticity (HOMA)” index. 10,11 The HOMA index based on the geometry criterion has achieved partial success, but the definition of the reference single and double bond lengths limits its applicability. From the early days of NMR, the induced ring current in the presence of an applied magnetic field in benzene and other aromatic systems, the NMR chemical shifts, 12 and magnetic susceptibility (Δ) 13 values were often used to infer aromatic systems. Schleyer and co-workers extended these magnetic criteria to the “nucleus independent chemical shift (NICS)”, 14 which is often thought to be a better criterion and is widely used as the aromaticity index (AI). The reference compounds required for the other methods are not required for NICS. However, NICS has its own limitations. 15,16 There are several other methods reported in the literature based on variations of one of these energetic, structural, or magnetic criteria, but these may be less satisfactory and not widely used. 15 The inverse of the force constant matrix (F) (Hessian) is the compliance constant matrix (C), 17 and the reciprocals of the diagonal compliance matrix elements are the relaxed force constants (RFCs). 18,19 In the literature, 20 it has been shown that the RFC of a bond is a measure of bond strength interaction. Hence, if we add the RFCs of all the ring bonds of the force field, it should be a measure of aromaticity. But the sum of RFCs of all the bonds, or the sum of RFCs of all the internal coordinates (bonds + angles) of the aromatic ring, does not correspond to the expected aromaticity order of furan < pyrrole < thiophene. 21 Because of the electronegativity and size Received: January 9, 2016 Revised: April 13, 2016 Published: April 13, 2016 Article pubs.acs.org/JPCA © 2016 American Chemical Society 2894 DOI: 10.1021/acs.jpca.6b00240 J. Phys. Chem. A 2016, 120, 2894-2901