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Most of the work done on the linear response kernel χ(r,r') has focussed on its atom-atom condensed form χAB. Our previous work [Boisdenghien et al., J. Chem. Theory Comput., 2013, 9, 1007] was the first effort to truly focus on the non-condensed form of this function for closed (sub)shell atoms in a systematic fashion. In this work, we extend our method to the open shell case. To simplify the plotting of our results, we average our results to a symmetrical quantity χ(r,r'). This allows us to plot the linear response kernel for all elements up to and including argon and to investigate the periodicity throughout the first three rows in the periodic table and in the different representations of χ(r,r'). Within the context of Spin Polarized Conceptual Density Functional Theory, the first two-dimensional plots of spin polarized linear response functions are presented and commented on for some selected cases on the basis of the atomic ground state electronic configurations. Using the relation between the linear response kernel and the polarizability we compare the values of the polarizability tensor calculated using our method to high-level values.

Tuning the band gap of graphene nanoribbons by chemical edge functionalisation is a promising approach towards future electronic devices based on graphene. The band gap is closely related to the aromaticity distribution and therefore tailoring the aromaticity patterns is a rational way for controlling the band gap. In the present work, it is shown how the three distinct classes of aromaticity patterns already found for armchair graphene nanoribbons can be rationally tuned by chemical edge functionalisation to modify their electronic arrangement and band gap. The electronic structure and the aromaticity distribution are studied using DFT calculations and through a series of delocalisation and geometry analysis methods, like the six-centre index (SCI) and the mean bond length (MBL) geometry descriptor. Novel aromaticity patterns are found for fluorine and nitrogen functionalisation characterised as inverted incomplete-Clar, and broken-Kekulé classes, while oxygen and nitrogen functionalisation is found to cut and extend the aromatic system, respectively. All these different arrangements of aromatic rings along the structure of graphene nanoribbons are explained using Clar's sextet theory, and a mesomeric effect mechanism for fluorine and nitrogen. In all cases, the changes in the aromaticity patterns are related to changes in the band gap. The energy and stability of the different edge functionalised graphene nanoribbons are also studied. An overall picture of edge effects, aromaticity patterns, and band gap tuning is provided.

The chemical importance of the linear response kernel from conceptual Density Functional Theory (DFT) is investigated for some σ and π aromatic and anti-aromatic systems. The effect of the ring size is studied by looking at some well known aromatic and anti-aromatic molecules of different sizes, showing that the linear response is capable of correctly classifying and quantifying the aromaticity for five- to eight-membered aromatic and anti-aromatic molecules. The splitting of the linear response in σ and π contributions is introduced and its significance is illustrated using some σ-aromatic molecules. The linear response also correctly predicts the aromatic transition states of the Diels-Alder reaction and the acetylene trimerisation and shows the expected behavior along the reaction coordinate, proving that the method is accurate not only at the minimum of the potential energy surface, but also in non-equilibrium states. Finally, the reason for the close correlation between the linear response and the Para Delocalisation Index (PDI), found in previous and the present study, is proven mathematically. These results show the linear response to be a reliable DFT-index to probe the σ and π aromaticity or anti-aromaticity of a broad range of molecules.

The degree of aromaticity of six-membered monoheterocycles with IV-VI group heteroatoms (C(6)H(5)X, where X = SiH, GeH, N, P, As, O(+), S(+), Se(+)) was analyzed using the results of ab initio calculations at the MP2/cc-pvtz level. Values of common aromaticity indices including those based on electronic delocalization properties, structural-dynamic features and magnetic properties all indicate high aromaticity of all considered heterocycles. A decrease in aromaticity is observed with increasing atomic number of the heteroatom, except in the case of the pyrylium cation. However, not all types of indices or even different indices within the same type correlate well among each other. Ring currents have been obtained at the HF/cc-pvdz level using the ipsocentric formulation. Ring current maps indicate that in the case of cationic heterocycles the ring current persists in all molecules under consideration. The different conclusions reached depending on the type of index used are a manifestation of the fact that when not dealing with hydrocarbons, aromaticity is ill-defined. One should always express explicitly which property of the molecules is considered to establish a degree of "aromaticity".