Universal mathematical identities in density functional theory: Results from three different spin-resolved representations

Spin-Polarized Conceptual Density Functional Theory from the Convex Hull

We present a new, nonarbitrary, internally consistent, and unambiguous framework for spin-polarized conceptual density-functional theory (SP-DFT). We explicitly characterize the convex hull of energy, as a function of the number of electrons and their spin, as the only accessible ground states in spin-polarized density functional theory. Then, we construct continuous linear and quadratic models for the energy. The nondifferentiable linear model exactly captures the simplicial geometry of the complex hull about the point of interest and gives exact representations for the conceptual DFT reactivity indicators. The continuous quadratic energy model is the paraboloid of maximum curvature, which most tightly encloses the point of interest and neighboring vertices. The quadratic model is invariant to the choice of coordinate system (i.e., {N, S} vs {Nα, Nβ}) and reduces to a sensible formulation of spin-free conceptual DFT in the appropriate limit. Using the quadratic model, we generalize the Parr functions {P+(r), P-(r)} (and their derivatives with respect to number of electrons) to this new spin-polarized framework, integrating the Parr function concept into the context of (spin-polarized) conceptual DFT, and extending it to include higher-order effects.

Chiral Jahn-Teller Distortion in Quasi-Planar Boron Clusters

In this work, we have observed that some chiral boron clusters (B16-, B20-, B24-, and B28-) can simultaneously have helical molecular orbitals and helical spin densities; these seem to be the first compounds discovered to have this intriguing property. We show that chiral Jahn-Teller distortion of quasi-planar boron clusters drives the formation of the helical molecular spin densities in these clusters and show that elongation/enhancement in helical molecular orbitals can be achieved by simply adding more building blocks via a linker. Aromaticity of these boron clusters is discussed. Chiral boron clusters may find potential applications in spintronics, such as molecular magnets.

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.

Links among the Fukui potential, the alchemical hardness and the local hardness of an atom in a molecule

This paper presents a brief summary of the difficulty that resides in the definition of the elusive concept of local chemical hardness. We argue that a definition of local hardness should be useful to a reactivity principle and not just as a mere definition. We then continue with a formal discussion about the benefits and difficulties of using the Fukui potential, which is interpreted as an alchemical derivative (alchemical hardness), as descriptor of local hardness of molecules. Computational evidence shows that the alchemical hardness is at least as good a descriptor as the combination of other two well-stabilized descriptors of local hardness, such as the Fukui function and grand canonical local hardness. Although our results are auspicious for the alchemical hardness as descriptor of local hardness, we finish by calling the attention of the community on the importance of discussing the raison d'être of a local hardness function and its main characteristics. We suggest that an axiomatic construction of local hardness could be they way of constructing a local hardness which is both useful and free of arbitrariness.

Extending conceptual DFT to include additional variables: oriented external electric field

The extension of the E = E[N, v] functional for exploring chemical reactivity in a conceptual DFT context to include external electric fields is discussed. Concentrating on the case of a homogeneous field the corresponding response functions are identified and integrated, together with the conventional response functions such as permanent dipole moment and polarizability, in an extended response function tree associated with the E = E[N, v, ε] functional. In a case study on the dihalogens F2, Cl2, Br2, I2 the sensitivity of condensed atomic charges (∂q/∂ε) is linked to the polarizability of the halogen atoms. The non-integrated (∂ρ(r)/∂ε) response function, directly related to the field induced density change, is at the basis of these features. It reveals symmetry breaking for a perpendicular field, not detectable in its atom condensed counterpart, and accounts for the induced dipole moment directly related to the molecular polarizability. The much higher sensitivity of the electronic chemical potential/electronegativity as compared to the chemical hardness is highlighted. The response of the condensed Fukui functions to a parallel electric field increases when going down in the periodic table and is interpreted in terms of the extension of the outer contours in the non-condensed Fukui function. In the case of a perpendicular field the (∂f(r)/∂ε) response function hints at stereoselectivity with a preferential side of attack which is not retrieved in its condensed form. In an application the nucleophilic attack on the carbonyl group in H2CO is discussed. Similar to the dihalogens, stereoselectivity is displayed in the Fukui function for nucleophilic attack (f+) in the case of a perpendicular electric field, and opposite to the one that would arise based on the induced density. Disentangling the expression for the evolution of the Fukui function in the presence of an electric field reveals that this difference can be traced back to local differences in the polarization or induced density between the anionic and the neutral system. This difference may be exploited, e.g. for an appropriately substituted H2CO, to generate enantioselectivity.

How predictive could alchemical derivatives be?

The chemical space contains all possible compounds that can be imagined. Its size easily equals the number of fundamental particles in the observable universe. Rational design of compounds aims to find those sectors of the chemical space where compounds optimize a set of desired properties. Then, rational design demands tools to efficiently navigate the chemical space. Ab initio alchemical derivatives offer the possibility to navigate, without empiricism, the energy landscape through alchemical transformations. An alchemical transformation is any process, physical or fictitious, that connects to points in the chemical space. In this work, those transformations are constructed as a perturbative expansion of the energy with respect to perturbations in the stoichiometry. The response functions of that expansion are what is called alchemical derivatives. In this work we assess how effective alchemical derivatives are in predicting energy changes associated to changes in the composition. We do this by including in the expansion, for the first time, electrostatic, polarization and electron-transfer effects. The system we chose is one that challenges alchemical derivatives because none of these effects dominates its behavior. The transmutations studied here correspond to substitutional doping of Al₁₃ with up to four atoms of Si, Al_13-nSi_n. Two types of transformations are considered, those in which the number of electrons remains constant and those in which the number of electrons also changes. It is found that contrary to what has been reported before, polarization cannot be neglected. If polarization is not included, alchemical derivatives fail to predict the change of energy and the relative energy between isomers. For isoelectronic substitution of four or more atoms, the perturbative approach collapses because the strength of the perturbation becomes too strong to guarantee convergence of the series. It is shown, however, that if only one atom is mutated at a time, alchemical derivatives rank pretty well the isomers of Al_13-nSi_n according to their energy. In the case of non-isoelectronic transformations, it is observed that the series rapidly diverges with increasing number of electrons. In this situation, it becomes more important to keep the degree of transmutation of the parent system small.

A 3D visualization of the substituent effect : A brief analysis of two components of the operational formula of dual descriptor for open-shell systems

Six organometallic compounds coming from a basic Mo-based complex were analyzed from the perspective of the dual descriptor in order to detect subtle influences that a substituent group could exert on the reactive core at a long range. Since the aforementioned complexes are open-shell systems, the used operational formula for the dual descriptor is that one defined for those aforementioned systems, which was then compared with spin density. In addition, dual descriptor was decomposed into two terms, each of which was also applied on every molecular system. The obtained results indicated that components of dual descriptor could become more useful than the operational formula of dual descriptor because differences exerted by the substituents at the para position were better detected by components of dual descriptor rather than the dual descriptor by itself.

Towards the rationalization of catalytic activity values by means of local hyper-softness on the catalytic site: a criticism about the use of net electric charges

By means of the Spin-Polarized Conceptual Density Functional Theory (SP-CDFT), three 2,6-bis(imino)pyridine catalysts based on iron(II), used for polymerization of ethylene, were studied. The catalysts differed by the substituent group, bearing either -H, -NO2 or -OCH3. To date, catalytic activity, a purely experimental parameter measuring the mass of polyethylene produced per millimole of iron per time and pressure unit at a fixed temperature, has not been explained in terms of local hyper-softness. The latter is a purely theoretical parameter designed for quantifying electronic effects; it is measured using the metal atom responsible for the coordination process with the monomer (ethylene). Because steric effects are not relevant in these kinds of catalysts and only electronic effects drive the catalytic process, an interesting link is found between catalytic activity and the local hyper-softness condensed on the iron atom by means of four functionals (B3LYP, BP86, B97D, and VSXC). This work demonstrates that the use of local hyper-softness, predicted by the SP-CDFT, is a suitable parameter for explaining order relationships among catalytic activity values, thus quantifying the electronic influence of the substituent group inducing this difference; the use of only net electric charges does not lead to clear conclusions. This finding can aid in estimating catalytic activities leading to a more rational design of new catalysts via computational chemistry.

Koopmans’ multiconfigurational self-consistent field (MCSCF) Fukui functions and MCSCF perturbation theory

Prediction of chemical reactivity has become one of the highest priority tasks of computational chemistry since the development of the methods of modeling electronic structure. Despite the general simplicity of the physical concept of reactivity and the rapid development of modern density functional theory (DFT) methods, this task remains state-of-the-art for systems with wavefunctions that have a multiconfigurational character. In such cases, for the accurate description of reactivity one needs to use multiconfigurational approaches that are much heavier computationally then ordinary single-determinant DFT methods. Moreover, the complexity of the calculation of reactivity is increased by the necessity to calculate ionic and transition states. These computational challenges can be addressed by employing the concepts of Koopmans’ theorem and its extension to a multiconfigurational case. We present a simplified methodology for the calculation of Fukui functions, based on Koopmans’ approximation for multiconfigurational Green’s functions developed in our previous works. Also, an extension of this methodology based on perturbation theory has been developed to improve accuracy.

Global and local chemical reactivities of mutagen X and simple derivatives

Registered by the World Health Organization (WHO), 3-chloro-4-(dichloromethyl)-5-hydroxy-2(5H)-furanone (MX) is one of the strongest bacterial mutagens ever tested, as highlighted by the Ames Salmonella typhimurium TA100 assay. We provide new insights concerning this mutagenic activity on the basis of global and local theoretically defined electrophilicity indices. Our results further support the idea that mutagenicity of MX and its analogues is related more closely to one-electron transfer processes from the electron-rich biological environment than to adduct formation processes. We also stress that, although the Z-open tautomers are intrinsically more electrophilic than furanone ring analogues, the observed mutagenic activity is significantly correlated only to the electrophilicity response of the ring forms. In that context, we also emphasize that it is electrophilicity at the C α in the α-β unsaturated carbonyl moiety that exhibits a strong correlation with the observed mutagenic activity.

The nucleophilicity N index in organic chemistry

The nucleophilicity N index (J. Org. Chem. 2008, 73, 4615), the inverse of the electrophilicity, 1/ω, and the recently proposed inverse of the electrodonating power, 1/ω⁻, (J. Org. Chem. 2010, 75, 4957) have been checked toward (i) a series of single 5-substituted indoles for which rate constants are available, (ii) a series of para-substituted phenols, and for (iii) a series of 2,5-disubstituted bicyclic[2.2.1]hepta-2,5-dienes which display concurrently electrophilic and nucleophilic behaviors. While all considered indices account well for the nucleophilic behavior of organic molecules having a single substitution, the nucleophilicity N index works better for more complex molecules. Unlike, the inverse of the electrophilicity, 1/ω, (R(2) = 0.71), and the inverse of the electrodonating power, 1/ω⁻ (R(2) = 0.83), a very good correlation of the nucleophilicity N index of twelve 2-substituted-6-methoxy-bicyclic[2.2.1]hepta-2,5-dienes versus the activation energy associated with the nucleophilic attack on 1,1-dicyanoethylene is found (R(2) = 0.99). This comparative study allows to assert that the nucleophilicity N index is a measure of the nucleophilicity of complex organic molecules displaying concurrently electrophilic and nucleophilic behaviors.