Properties of the density functional response kernels and its implications on chemistry.

An overview of mathematical properties of the non-local second order derivatives of the canonical, grand canonical, isomorphic, and grand isomorphic ensembles is given. The significance of their positive or negative semidefiniteness and the implications of these properties for atoms and molecules are discussed. Based on this property, many other interesting properties can be derived, such as the expansion in eigenfunctions, bounds on the diagonal and off-diagonal elements, and the eigenvalues of these kernels. We also prove Kato's theorem for the softness kernel and linear response and the dissociation limit of the linear responses as the sum of the linear responses of the individual fragments when dissociating a system into two non-interacting molecular fragments. Finally, strategies for the practical calculation of these kernels, their eigenfunctions, and their eigenvalues are discussed.

Molecular interactions from the density functional theory for chemical reactivity: Interaction chemical potential, hardness, and reactivity principles.

In the first paper of this series, the authors derived an expression for the interaction energy between two reagents in terms of the chemical reactivity indicators that can be derived from density functional perturbation theory. While negative interaction energies can explain reactivity, reactivity is often more simply explained using the "|dµ| big is good" rule or the maximum hardness principle. Expressions for the change in chemical potential (µ) and hardness when two reagents interact are derived. A partial justification for the maximum hardness principle is that the terms that appear in the interaction energy expression often reappear in the expression for the interaction hardness, but with opposite sign.

Molecular Interactions From the Density Functional Theory for Chemical Reactivity: The Interaction Energy Between Two-Reagents.

Reactivity descriptors indicate where a reagent is most reactive and how it is most likely to react. However, a reaction will only occur when the reagent encounters a suitable reaction partner. Determining whether a pair of reagents is well-matched requires developing reactivity rules that depend on both reagents. This can be achieved using the expression for the minimum-interaction-energy obtained from the density functional reactivity theory. Different terms in this expression will be dominant in different circumstances; depending on which terms control the reactivity, different reactivity indicators will be preferred.

Alchemical Normal Modes Unify Chemical Space.

In silico design of new molecules and materials with desirable quantum properties by high-throughput screening is a major challenge due to the high dimensionality of chemical space. To facilitate its navigation, we present a unification of coordinate and composition space in terms of alchemical normal modes (ANMs) which result from second order perturbation theory. ANMs assume a predominantly smooth nature of chemical space and form a basis in which new compounds can be expanded and identified. We showcase the use of ANMs for the energetics of the isoelectronic series of diatomics with 14 electrons, BN doped benzene derivatives (C6-2 x(BN) xH6 with x = 0,1,2,3), predictions for over 1.8 million BN doped coronene derivatives, and genetic energy optimizations in the entire BN-doped coronene space. Using Ge lattice scans as reference, the applicability of ANMs across the periodic table is demonstrated for III-V and IV-IV semiconductors Si, Sn, SiGe, SnGe, SiSn, as well as AlP, AlAs, AlSb, GaP, GaAs, GaSb, InP, InAs, and InSb. Analysis of our results indicates simple qualitative structure property rules for estimating energetic rankings among isomers. Useful quantitative estimates can also be obtained when few atoms are changed to neighboring or lower lying elements in the periodic table. The quality of the predictions often increases with the symmetry of system chosen as reference due to cancellation of odd order terms. Rooted in perturbation theory, the ANM approach promises to generally enable unbiased compound exploration campaigns at reduced computational cost.

A reference-free stockholder partitioning method based on the force on electrons.

We argue that when one divides a molecular property into atom-in-a-molecule contributions, one should perform the division based on the property density of the quantity being partitioned. This is opposition to the normal approach, where the electron density is given a privileged role in defining the properties of atoms-in-a-molecule. Because partitioning each molecular property based on its own property density is inconvenient, we design a reference-free approach that does not (directly) refer atomic property densities. Specifically, we propose a stockholder partitioning method based on relative influence of a molecule's atomic nuclei on the electrons at a given point in space. The resulting method does not depend on an "arbitrary" choice of reference atoms and it has some favorable properties, including the fact that all of the electron density at an atomic nucleus is assigned to that nucleus and the fact all the atoms in a molecule decay at a uniform asymptotic rate. Unfortunately, the resulting model is not easily applied to spatially degenerate ground states. Furthermore, the practical realizations of this strategy that we tried here gave disappointing numerical results. © 2017 Wiley Periodicals, Inc.

Extension of the source-sink potential approach to Hartree-Fock and density functional theory: A new tool to visualize the ballistic current through molecules.

The recent source and sink potential approach by Pickup et al. [J. Chem. Phys. 143, 194105 (2015)] is extended to Hartree-Fock and density functional theory, allowing the calculation of the transmission and the visualization of ballistic currents through molecules at these levels of theory. This visualization allows the study of the transmission process in real-space, providing an important tool to better understand the conduction process.

Chemical transferability of functional groups follows from the nearsightedness of electronic matter.

We establish the physical origins of chemical transferability from the perspective of the nearsightedness of electronic matter. To do this, we explicitly evaluate the response of electron density to a change in the system, at constant chemical potential, by computing the softness kernel, [Formula: see text] The softness kernel is nearsighted, indicating that under constant-chemical-potential conditions like dilute solutions changing the composition of the molecule at [Formula: see text] has only local effects and does not have any significant impact on the reactivity at positions [Formula: see text] far away from point [Formula: see text] This locality principle elucidates the transferability of functional groups in chemistry.

Exploring Electrical Currents through Nanographenes: Visualization and Tuning of the through-Bond Transmission Paths.

In this work, electrical currents through nanographenes, a class of alternant hydrocarbons also known as polycyclic aromatic hydrocarbons, in molecular junctions under small bias are explored. We illustrate that when the π-current dominates, that is, when no quantum interference takes place, the current prefers the direction of the shortest bond (the bond with the highest double bond character) upon entering the molecule from the contacts. As such, the idea of electrons propagating through double bonds from contact to contact, originating from the curly arrow drawings used in a previously established selection rule for transmission, seems to be more deeply rooted in the actual physical process of electron transport than previously anticipated. Furthermore, this work confirms that the σ-current behaves completely differently than the π-current. When this type of current becomes important, that is, when quantum interference takes place, the current generally prefers the shortest path from contact to contact, irrespective of the length of the bonds constituting this path, in accordance with the strong distance dependency of the σ-current. Finally, it is demonstrated that keto groups (and cross-conjugating groups in general) can be used to seal off parts of the molecule for the current. No current flows through the sealed off part of the molecule under small bias and it does not influence the transmission spectrum of the considered system.

Fast and accurate predictions of covalent bonds in chemical space.

We assess the predictive accuracy of perturbation theory based estimates of changes in covalent bonding due to linear alchemical interpolations among molecules. We have investigated σ bonding to hydrogen, as well as σ and π bonding between main-group elements, occurring in small sets of iso-valence-electronic molecules with elements drawn from second to fourth rows in the p-block of the periodic table. Numerical evidence suggests that first order Taylor expansions of covalent bonding potentials can achieve high accuracy if (i) the alchemical interpolation is vertical (fixed geometry), (ii) it involves elements from the third and fourth rows of the periodic table, and (iii) an optimal reference geometry is used. This leads to near linear changes in the bonding potential, resulting in analytical predictions with chemical accuracy (∼1 kcal/mol). Second order estimates deteriorate the prediction. If initial and final molecules differ not only in composition but also in geometry, all estimates become substantially worse, with second order being slightly more accurate than first order. The independent particle approximation based second order perturbation theory performs poorly when compared to the coupled perturbed or finite difference approach. Taylor series expansions up to fourth order of the potential energy curve of highly symmetric systems indicate a finite radius of convergence, as illustrated for the alchemical stretching of H2 (+). Results are presented for (i) covalent bonds to hydrogen in 12 molecules with 8 valence electrons (CH4, NH3, H2O, HF, SiH4, PH3, H2S, HCl, GeH4, AsH3, H2Se, HBr); (ii) main-group single bonds in 9 molecules with 14 valence electrons (CH3F, CH3Cl, CH3Br, SiH3F, SiH3Cl, SiH3Br, GeH3F, GeH3Cl, GeH3Br); (iii) main-group double bonds in 9 molecules with 12 valence electrons (CH2O, CH2S, CH2Se, SiH2O, SiH2S, SiH2Se, GeH2O, GeH2S, GeH2Se); (iv) main-group triple bonds in 9 molecules with 10 valence electrons (HCN, HCP, HCAs, HSiN, HSiP, HSiAs, HGeN, HGeP, HGeAs); and (v) H2 (+) single bond with 1 electron.

The spin polarized linear response from density functional theory: theory and application to atoms.

Within the context of spin polarized conceptual density functional theory, the spin polarized linear response functions are introduced both in the [N, N(s)] and [N(α), N(ß)] representations. The mathematical relations between the spin polarized linear response functions in both representations are examined and an analytical expression for the spin polarized linear response functions in the [N(α), N(ß)] representation is derived. The spin polarized linear response functions were calculated for all atoms up to and including argon. To simplify the plotting of our results, we integrated χ(r, r') to a quantity χ(r, r'), circumventing the θ and ϕ dependence. This allows us to plot and to investigate the periodicity throughout the first three rows in the periodic table within the two different representations. For the first time, χ(αß)(r, r'), χ(ßα)(r, r'), and χ(SS)(r, r') plots have been calculated and discussed. By integration of the spin polarized linear response functions, different components to the polarisability, α(αα), α(αß), α(ßα), and α(ßß) have been calculated.

Evaluating and interpreting the chemical relevance of the linear response kernel for atoms II: open shell.

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.

Conceptual DFT: chemistry from the linear response function.

Within the context of reactivity descriptors known in conceptual DFT, the linear response function (χ(r,r')) remained nearly unexploited. Although well known, in its time dependent form, in the solid state physics and time-dependent DFT communities the study of the "chemistry" present in the kernel was, until recently, relatively unexplored. The evaluation of the linear response function as such and its study in the time independent form are highlighted in the present review. On the fundamental side, the focus is on the approaches of increasing complexity to compute and represent χ(r,r'), its visualisation going from plots of the unintegrated χ(r,r') to an atom condensed matrix. The study on atoms reveals its physical significance, retrieving atomic shell structure, while the results on molecules illustrate that a variety of chemical concepts are retrieved: inductive and mesomeric effects, electron delocalisation, aromaticity and anti-aromaticity, σ and π aromaticity,…. The applications show that the chemistry of aliphatic (saturated and unsaturated) chains, saturated and aromatic/anti-aromatic rings, organic, inorganic or metallic in nature, can be retrieved via the linear response function, including the variation of the electronic structure of the reagents along a reaction path. The connection of the linear response function with the concept of nearsightedness and the alchemical derivatives is also highlighted.

Tuning aromaticity patterns and electronic properties of armchair graphene nanoribbons with chemical edge functionalisation.

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.

Analysis of aromaticity in planar metal systems using the linear response kernel.

The linear response kernel is used to gain insight into the aromatic behavior of the less classical metal aromatic E4(2-) and CE4(2-) (E = Al, Ga) clusters. The effect of the systematic replacement of the aluminum atoms in Al4(2-) and CAl4(2-) by germanium atoms is studied using, Al3Ge-, Al2Ge2, AlGe3+, Ge4(2+), CAl3Ge-, CAl2Ge2, CAlGe3+, and CGe4(2+). The results are compared with the values of the delocalization index (δ(1,3)) and nucleus independent chemical shifts (NICS(zz)). Unintegrated plots of the linear response, computed for the first time on molecules, are used to analyze the delocalization in these clusters. All aromaticity indices studied, the linear response, δ(1,3), and NICS(zz), predict that the systems with a central carbon are less aromatic than the systems without a central carbon atom. Also, the linear response is more pronounced in the σ-electron density than in the π-density, pointing out that the systems are mainly σ-aromatic.

σ, π aromaticity and anti-aromaticity as retrieved by the linear response kernel.

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.

Electronic structure and aromaticity of graphene nanoribbons.

We analyse the electronic structure and aromaticity of graphene nanoribbons and carbon nanotubes through a series of delocalisation and geometry analysis methods. In particular, the six-centre index (SCI) is found to be in good agreement with the mean bond length (MBL) and ring bond dispersion (RBD) geometry descriptors. Based on DFT periodic calculations, three distinct classes of aromaticity patterns have been found for armchair graphene nanoribbons, appearing periodically as the width of the ribbon is increased. The periodicity in the band gap is found to be related to these aromaticity patterns. Also, the appearance of such distinct aromaticity distribution is explained within the framework of the Clar's sextet theory. Both delocalisation and geometry analysis methods are shown to be very fast and reliable tools for easily analysing the aromaticity in carbon nanosystems.

Aromaticity in heterocyclic analogues of benzene: comprehensive analysis of structural aspects, electron delocalization and magnetic characteristics.

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".

How does aromaticity rule the thermodynamic stability of hydroporphyrins?

Several measures of aromaticity including energetic, magnetic, and electron density criteria are employed to show how aromatic stabilization can explain the stability sequence of hydroporphyrins, ranging from porphin to octahydroporphin, and their preferred hydrogenation paths. The methods employed involve topological resonance energies and their circuit energy effects, bond resonance energies, multicenter delocalization indices, ring current maps, magnetic susceptibilities, and nuclear-independent chemical shifts. To compare the information obtained by the different methods, the results have been put in the same scale by using recently proposed approaches. It is found that all of them provide essentially the same information and lead to similar conclusions. Also, hydrogenation energies along different hydrogenation paths connecting porphin with octahydroporphin have been calculated with density functional theory. It is shown by using the methods mentioned above that the relative stability of different hydroporphyrin isomers and the observed inaccessibility of octahydroporphin both synthetically and in nature can be perfectly rationalized in terms of aromaticity.

Multidimensionality of delocalization indices and nucleus-independent chemical shifts in polycyclic aromatic hydrocarbons II: proof of further nonlocality.

In a recent contribution, we examined the effect of 10- and 14-center circuits on the nucleus-independent chemical shifts NICSs using multicenter bond indices (MCBIs) (Fias et al., J Comput Chem 2008, 29, 358). In this study, the nonlocal contributions to the NICS are further investigated for a larger set of polycyclic aromatic hydrocarbons (PAHs). To achieve this, the NICSs are predicted using the MCBI and compared with ab initio results. The NICSs of the central ring of perylene- and benzo-[ghi]perylene-like fragments and of coronene appear to have other nonlocal contributions than the ones previously studied. It is shown that a model based on the MCBI-ring current maps and the inclusion of new circuits proves the existence and shows the nature of these new nonlocal effects on the NICS. This new model leads to a better understanding of the differences between the NICSs and delocalization indices. The results show that the NICS value is not only significantly influenced by the higher order circuits encircling the ring at which it is evaluated but also by the local aromaticity of the surrounding rings, and occasionally, like in the case of coronene, the NICSs are even influenced by currents farther away in the molecule.

Electrostatic Potentials from Self-Consistent Hirshfeld Atomic Charges.

It is shown that molecular electrostatic potentials obtained from iterative or self-consistent Hirshfeld atomic point charges agree remarkably well with the ab initio computed electrostatic potentials. The iterative Hirshfeld scheme performs nearly as well as electrostatic potential derived atomic charges, having the advantage of allowing the definition of the atom in the molecule, rather than just yielding charges. The quality of the iterative Hirshfeld charges for computing electrostatic potentials is examined for a large set of molecules and compared to other commonly used techniques for population analysis.