Electronic and nuclear chemical reactivity

Extending the definition of atomic basis sets to atoms with fractional nuclear charge

Alchemical transformations showed that perturbation theory can be applied also to changes in the atomic nuclear charges of a molecule. The alchemical path that connects two different chemical species involves the conceptualization of a non-physical system in which an atom possess a non-integer nuclear charge. A correct quantum mechanical treatment of these systems is limited by the fact that finite size atomic basis sets do not define exponents and contraction coefficients for fractional charge atoms. This paper proposes a solution to this problem and shows that a smooth interpolation of the atomic orbital coefficients and exponents across the periodic table is a convenient way to produce accurate alchemical predictions, even using small size basis sets.

Connection between nuclear and electronic Fukui functions beyond frontier molecular orbitals

Based on the relationship between average local ionization energy Ī(r) and average local electron affinity Ā(r) with the electronic Fukui functions, i.e., f-(r) and f+(r), respectively, in this paper, we establish a connection between nuclear and electronic Fukui functions beyond frontier molecular orbitals. As a consequence of this connection, we obtain expressions of average nuclear Fukui functions interpreted as a variation of average nucleophilicity or electrophilicity (weighted by the electronic orbital Fukui functions) with respect to nuclear displacements, which goes beyond the highest occupied molecular orbital/or lowest unoccupied molecular orbital consideration. Furthermore, from this connection and considering the frontier molecular orbital approximation, we derive expressions of nuclear Fukui functions in terms of the atom-condensed electronic Fukui functions, which imply a locality in the chemical reactivity and could be used to study the variation of local nucleophilicity or electrophilicity with respect to nuclear displacements. Finally, this new way to interpret the nuclear Fukui function could be useful in the future to study the chemical reactivity related to molecular vibrations, internal rotations, bond dissociation, chemical reaction along the model of reaction coordinate, and so on.

Photosensitized Activation of Diazonium Derivatives for C-B Bond Formation

Aryl diazonium salts are ubiquitous building blocks in chemistry, as they are useful radical precursors in organic synthesis as well as for the functionalization of solid materials. They can be reduced electrochemically or through a photo-induced electron transfer reaction. Here we provide a detailed picture of the ground and excited-state reactivity of a series of 9 rare and earth abundant photosensitizers with 13 aryl diazonium salts, which also included 3 macrocyclic calix[4]arene tetradiazonium salts. Nanosecond transient absorption spectroscopy confirmed the occurrence of excited-state electron transfer and was used to quantify cage-escape yields, i.e. the efficiency with which the formed radicals separate and escape the solvent cage. Cage-escape yields were large; increased when the driving force for photo-induced electron transfer increased and also tracked with the C-N2 + bond cleavage propensity, amongst others. A photo-induced borylation reaction was then investigated with all the photosensitizers and proceeded with yields between 9 and 74%.

Electrophilicity index revisited

This review aims to be a comprehensive, authoritative, critical, and accessible review of general interest to the chemistry community; because the electrophilicity index is a very useful global reactivity descriptor defined within a conceptual density functional theory framework. Our group has also introduced electrophilicity based new global and local reactivity descriptors and also new associated electronic structure principles, which are important indicators of structure, stability, bonding, reactivity, interactions, and dynamics in a wide variety of physico-chemical systems and processes. This index along with its local counterpart augmented by the associated electronic structure principles could properly explain molecular vibrations, internal rotations and various types of chemical reactions. The concept of the electrophilicity index has been extended to dynamical processes, excited states, confined environment, spin-dependent and temperature-dependent situations, biological activity, site selectivity, aromaticity, charge removal and acceptance, presence of external perturbation through solvents, external electric and magnetic fields, and so forth. Although electrophilicity and its local variant can adequately interpret the behavior of a wide variety of systems and different physico-chemical processes involving them, their predictive potential remains to be explored. An exhaustive review on all these aspects will set the tone of the future research in that direction.

Activated layered double hydroxides: assessing the surface anion basicity and its connection with the catalytic activity in the cyanoethylation of alcohols

Layered double hydroxides (LDHs) act as catalysts in several reactions like in the cyanoethylation of alcohols with acrylonitrile to produce alkoxypropionitriles. Here we report an experimental and theoretical study in which it is shown that the experimental catalytic activity of LDHs in the cyanoethylation of 2-propanol and methanol correlates with the predicted strength of the basicity of the adsorbed surface species. First, it is shown that using activated LDHs containing Mg²⁺ and Al³⁺ (MgAl-LDH), Mg²⁺ and Ga³⁺ (MgGa-LDH), and Mg²⁺, Al³⁺ and Ga³⁺ (MgAlGa-LDH) great conversions to alkoxypropionitriles in high yields are obtained. Next, the basicity of these LDHs is estimated by means of the local softness, a local reactivity index calculated using density functional theory and appropriate surface models. For that, the adsorption of hydroxide and methoxide anions at the (001) surface of MgAl and MgGa-LDHs is investigated. We include LDHs containing Zn²⁺ and Al³⁺ (ZnAl-LDH) and Zn²⁺ and Ga³⁺ (ZnGa-LDH) in this part of the study to account for the effect of changing the divalent and trivalent metal composition on the basicity. It is found that hydroxide anions adsorbed on the MgGa-LDH surface and methoxide anions adsorbed on the MgAl-LDH surface are the most basic ones. This basicity trend correlates with our experimental findings about the catalytic activity of the activated LDHs. Further analyzing the connection between the LDH composition and the anion basicity, it is argued that the key steps dictating the LDH catalytic activity are the alcohol deprotonation in the cyanoethylation of 2-propanol, as it has been previously suggested, and the methoxide anion attack to the acrylonitrile double bond in the methanol cyanoethylation reaction.

Alchemical geometry relaxation

We propose to relax geometries throughout chemical compound space (CCS) using alchemical perturbation density functional theory (APDFT). APDFT refers to perturbation theory involving changes in nuclear charges within approximate solutions to Schr\"odinger's equation. We give an analytical formula to calculate the mixed second order energy derivatives with respect to both, nuclear charges and nuclear positions (named "alchemical force"), within the restricted Hartree-Fock case.We have implemented and studied the formula for its use in geometry relaxation of various reference and target molecules.We have also analysed the convergence of the alchemical force perturbation series, as well as basis set effects.Interpolating alchemically predicted energies, forces, and Hessian to a Morse potential yields more accurate geometries and equilibrium energies than when performing a standard Newton Raphson step. Our numerical predictions for small molecules including BF, CO, N2, CH$_4$, NH$_3$, H$_2$O, and HF yield mean absolute errors of of equilibrium energies and bond lengths smaller than 10 mHa and 0.01 Bohr for 4$^\text{th}$ order APDFT predictions, respectively.Our alchemical geometry relaxation still preserves the combinatorial efficiency of APDFT: Based on a single coupled perturbed Hartree Fock derivative for benzene we provide numerical predictions of equilibrium energies and relaxed structures of all the 17 iso-electronic charge-netural BN-doped mutants with averaged absolute deviations of $\sim$27 mHa and $\sim$0.12 Bohr, respectively.

From the Electron Density Gradient to the Quantitative Reactivity Indicators: Local Softness and the Fukui Function

Important reactivity measures such as the local softness, the Fukui function, and the global hardness have been calculated directly from first principles with the use of the electron density function, beyond the finite difference approximation. Our recently derived density gradient theorem and the principle of nearsightedness of the electronic matter have been instrumental in obtaining the original, albeit approximate, result on the local softness of an atom. By integration of the local softness s(r), we obtain the global softness S and the Fukui function f(r) = s(r)/S. Local and global softness values have also been calculated analytically for the basic hydrogenic orbitals; the general relation to the atomic number S = σZ ^-2 has been demonstrated, with constants σ characteristic for each orbital type. Global hardness η = 1/S calculated for atoms and ions has been favorably tested against its conventional measure given by the finite difference approximation: (I - A). Calculated test results for atoms and ions in rows 1-4 of the periodic table have been presented.

Reactivity of Single Transition Metal Atoms on a Hydroxylated Amorphous Silica Surface: A Periodic Conceptual DFT Investigation

The drive to develop maximal atom-efficient catalysts coupled to the continuous striving for more sustainable reactions has led to an ever-increasing interest in single-atom catalysis. Based on a periodic conceptual density functional theory (cDFT) approach, fundamental insights into the reactivity and adsorption of single late transition metal atoms supported on a fully hydroxylated amorphous silica surface have been acquired. In particular, this investigation revealed that the influence of van der Waals dispersion forces is especially significant for a silver (98 %) or gold (78 %) atom, whereas the oxophilicity of the Group 8-10 transition metals plays a major role in the interaction strength of these atoms on the irreducible SiO₂ support. The adsorption energies for the less-electronegative row 4 elements (Fe, Co, Ni) ranged from -1.40 to -1.92 eV, whereas for the heavier row 5 and 6 metals, with the exception of Pd, these values are between -2.20 and -2.92 eV. The deviating behavior of Pd can be attributed to a fully filled d-shell and, hence, the absence of the hybridization effects. Through a systematic analysis of cDFT descriptors determined by using three different theoretical schemes, the Fermi weighted density of states approach was identified as the most suitable for describing the reactivity of the studied systems. The main advantage of this scheme is the fact that it is not influenced by fictitious Coulomb interactions between successive, charged reciprocal cells. Moreover, the contribution of the energy levels to the reactivity is simultaneously scaled based on their position relative to the Fermi level. Finally, the obtained Fermi weighted density of states reactivity trends show a good agreement with the chemical characteristics of the investigated metal atoms as well as the experimental data.

Orbital energies and nuclear forces in DFT: Interpretation and validation

The bonding and antibonding character of individual molecular orbitals has been previously shown to be related to their orbital energy derivatives with respect to nuclear coordinates, known as dynamical orbital forces. Albeit usually derived from Koopmans' theorem, in this work we show a more general derivation from conceptual DFT, which justifies application in a broader context. The consistency of the approach is validated numerically for valence orbitals in Kohn-Sham DFT. Then, we illustrate its usefulness by showcasing applications in aromatic and antiaromatic systems and in excited state chemistry. Overall, dynamical orbital forces can be used to interpret the results of routine ab initio calculations, be it wavefunction or density based, in terms of forces and occupations.

Establishing the Relationship between Quantum Capacitance and Softness of N-Doped Graphene/Electrolyte Interfaces within the Density Functional Theory Grand Canonical Kohn-Sham Formalism

The joint density functional theory (JDFT) is applied in the context of the grand canonical Kohn-Sham theory to calculate the global and local softness of pristine and N-substituted graphene structures. A comparison is established between the different theoretical approaches to evaluate total capacitance, revealing that the JDFT approach presents the closest result of this property with experimental data. A model of series capacitors is used to determine the quantum and nonquantum contributions of total capacitance, which enables us to determine the limitations of the rigid band approximation for the studied systems. It is found that global chemical softness is proportional to the total capacitance measured in the experiments, when the geometry relaxation is neglected. In this context, it is possible to obtain quantum and total capacitance (and consequently softness) from an average number of electrons vs applied potential plots and the model of series capacitors. Likewise, the relation of capacitance and softness gives rise to a new definition of local capacitance within the JDFT formalism. The evaluation of global and local softness paves the way to analyze electrochemical surface reactivity as a function of applied potential for a solid-electrolyte interface.

Bond Softening Indices Studied by the Fragility Spectra for Proton Migration in Formamide and Related Structures

Computational scheme to obtain bond softening index λ, defined within the conceptual DFT, has been obtained with the use of the reaction fragility (RF) concept. Numerical results were obtained with the RF spectra for the proton transfer reaction in formamide molecule (H₂NCHO) and the water assisted proton migration in H₂NCHO·H₂O complex. Double proton transfer reaction in the formamide dimer, (H₂NCHO)₂, and its analogues, (H₂NCHS)₂ and (H₂NCHO)·(H₂NCHS), have also been studied. The atomic and bond RF spectra clearly describe the density reorganization in the backbone of each molecule, resulting from proton displacement in the systems. The obtained softening indices have been calculated for hydrogen atoms in the reactant state (RS) and product state (PS) configuration. These indices provide fine characteristics for the local sensitivity of the reacting system to a disturbance of the position of a chosen atom.

Bond Fragility Spectra for the Double Proton-Transfer Reaction in the Formic Acid-Type Dimers

The newly developed method of fragility spectra for observation of bond breaking and formation upon a reaction has been applied to the canonical reaction series of the double proton transfer (DPT). Formic acid and its thio-analogues HCXYH (X, Y = O, S) have been chosen for the analysis. Very accurate linear correlations have been determined between the nondiagonal elements of the connectivity matrix, essential for the method, and the Wiberg bond orders for the corresponding bonds. Relation of the slope of this correlation to the global softness and to the atomic numbers of the bonded atoms has been proved, thus corroborating the c-DFT formula describing the fragility spectra. The electron density changes in bonds, as observed by the fragility spectra, are in harmony with the curvature diagrams reported by other authors.

Using local softness to reveal oxygen participation in redox processes in cathode materials

In this paper, the use of chemical local softness s(r) is proposed as an alternative way of analyzing the initial redox processes that occur in cathode materials used for lithium-ion batteries. It is shown that the chemical local softness is a quantity able to capture the same effects as the standard analysis based on the projected density of states. Because of its own nature, the local softness reveals the atomic sites involved in charge-transfer events and allows a quantitative comparative analysis between different materials. As pointed out by Johannes et al. (Solid State Ion 286:83-89, 2016), this analysis can be used as an indicator of stability of cathode materials for Li-ion batteries.

Exploring Chemical Space with Alchemical Derivatives: BN-Simultaneous Substitution Patterns in C60

With the idea of using alchemical derivatives to explore in an efficient, computer- and cost-effective way Chemical Space was launched several years ago. In the context of Conceptual DFT response functions, these energies vs nuclear charge derivatives permit the estimatation of the energy of transmutants of a given starting or reference molecule showing different nuclear compositions. After an explorative study on small and planar molecules ( Balawender et al. J. Chem. Theory Comput. 2013 , 9 , 5327 ) by the present authors of this paper, the present study fully exploits the computational advantages of the alchemical derivatives in larger three-dimensional systems. Starting from a single reference calculation on C₆₀, the complete BN substitution pattern, from single substituted C₅₈BN via the belt (C₂₀(BN)₂₀ and the ball C₁₂(BN)₂₄ structures to the fully substituted (BN)₃₀, is explored. Successive and simultaneous substitution strategies are followed and compared, indicating that both techniques yield identical results up to 13 substitutions but that for higher substitutions the simultaneous approach needs to be taken. Due to the cost-efficiency of the algorithm this path can indeed be followed as opposed to earlier work in the literature where for each step a full SCF calculation was at stake leading to prohibitively large computational demands for adopting the simultaneous approach. Previously formulated rules governing the substitution pattern by Kar and co-workers are scrutinized in this context and reformulated giving chemical insight in the gradual substitution process and the relative energies of the isomers. In its present form the method offers an interesting venue to study BN substitution patterns in higher fullerenes and graphene and in general paves the way for more efficient exploration of the Chemical Space.

Diagrams for comprehensive molecular orbital-based chemical reaction analyses: reactive orbital energy diagrams

Reactive orbital energy diagrams resting on the reactive orbital energy theory correct conventional frontier orbital diagrams and make it possible to perform comprehensive orbital-based analyses of reactions.

Introducing a new bond reactivity index: Philicities for natural bond orbitals

In the present work, a new methodology defined for obtaining reactivity indices (philicities) is proposed. This is based on reactivity functions such as the Fukui function or the dual descriptor, and makes it possible to project the information from reactivity functions onto molecular orbitals, instead of onto the atoms of the molecule (atomic reactivity indices). The methodology focuses on the molecules' natural bond orbitals (bond reactivity indices) because these orbitals have the advantage of being localized, allowing the reaction site of an electrophile or nucleophile to be determined within a very precise molecular region. This methodology provides a "philicity" index for every NBO, and a representative set of molecules has been used to test the new definition. A new methodology has also been developed to compare the "finite difference" and the "frontier molecular orbital" approximations. To facilitate their use, the proposed methodology as well as the possibility of calculating the new indices have been implemented in a new version of UCA-FUKUI software. In addition, condensation schemes based on atomic populations of the "atoms in molecules" theory, the Hirshfeld population analysis, the approximation of Mulliken (with a minimal basis set) and electrostatic potential-derived charges have also been implemented, including the calculation of "bond reactivity indices" defined in previous studies. Graphical abstract A new methodology defined for obtaining bond reactivity indices (philicities) is proposed and makes it possible to project the information from reactivity functions onto molecular orbitals. The proposed methodology as well as the possibility of calculating the new indices have been implemented in a new version of UCA-FUKUI software. In addition, this version can use new atomic condensation schemes and new "utilities" have also been included in this second version.

Conceptual DFT analysis of the fragility spectra of atoms along the minimum energy reaction coordinate

Theoretical justification has been provided to the method for monitoring the sequence of chemical bonds' rearrangement along a reaction path, by tracing the evolution of the diagonal elements of the Hessian matrix. Relations between the divergences of Hellman-Feynman forces and the energy and electron density derivatives have been demonstrated. By the proof presented on the grounds of the conceptual density functional theory formalism, the spectral amplitude observed on the atomic fragility spectra [L. Komorowski et al., Phys. Chem. Chem. Phys. 18, 32658 (2016)] reflects selectively the electron density modifications in bonds of an atom. In fact the spectral peaks for an atom reveal changes of the electron density occurring with bonds creation, breaking, or varying with the reaction progress.

On the trends of Fukui potential and hardness potential derivatives in isolated atoms vs. atoms in molecules

In the present study, trends of electronic contribution to molecular electrostatic potential [Vel(r¯)(r=0)], Fukui potential [v(+)f|(r=0) and v(-)f|(r=0)] and hardness potential derivatives [Δ(+)h(k) and Δ(-)h(k)] for isolated atoms as well as atoms in molecules are investigated. The generated numerical values of these three reactivity descriptors in these two electronically different situations are critically analyzed through the relevant formalism. Values of Vel(r¯) (when r → 0, i.e., on the nucleus) are higher for atoms in molecules than that of isolated atoms. In contrast, higher values of v(+)|(r=0) and v(-)|(r=0) are observed for isolated atoms compared to the values for atoms in a molecule. However, no such regular trend is observed for the Δ(+)h(k) and Δ(-)h(k) values, which is attributed to the uncertainty in the Fukui function values of atoms in molecules. The sum of Fukui potential and the sum of hardness potential derivatives in molecules are also critically analyzed, which shows the efficacy of orbital relaxation effects in quantifying the values of these parameters. The chemical consequence of the observed trends of these descriptors in interpreting electron delocalization, electronic relaxation and non-negativity of atomic Fukui function indices is also touched upon. Several commonly used molecules containing carbon as well as heteroatoms are chosen to make the investigation more insightful.

Computational Nutraceutics: Chemical Reactivity Properties of the Flavonoid Naringin by Means of Conceptual DFT

The M06 family of density functionals has been assessed for the calculation of the molecular structure and properties of the Naringin molecule. The chemical reactivity descriptors have been calculated through Conceptual DFT. The active sites for nucleophilic and electrophilic attacks have been chosen by relating them to the Fukui function indices and the dual descriptorf(2)(r). A comparison between the descriptors calculated through vertical energy values and those arising from the Koopmans' theorem approximation has been performed in order to check for the validity of the last procedure.

In pursuit of negative Fukui functions: examples where the highest occupied molecular orbital fails to dominate the chemical reactivity

In our quest to explore molecules with chemically significant regions where the Fukui function is negative, we explored reactions where the frontier orbital that indicates the sites for electrophilic attack is not the highest occupied molecular orbital. The highest occupied molecular orbital (HOMO) controls the location of the regions where the Fukui function is negative, supporting the postulate that negative values of the Fukui function are associated with orbital relaxation effects and nodal surfaces of the frontier orbitals. Significant negative values for the condensed Fukui function, however, were not observed.

Stability conditions for density functional reactivity theory: an interpretation of the total local hardness

The second-order Taylor series expansions commonly used in the density functional chemical reactivity theory are used to define local stability conditions for electronic states. Systems which satisfy these conditions are stable to infinitesimal perturbations due to approaching chemical reagents. The basic formalism considered here supersedes previous variational approaches to chemical reactivity theory like the electrophilicity, potentialphilicity, and chargephilicity. The total local hardness emerges naturally in this analysis, and can be clearly interpreted. When the total local hardness is small, the system is relatively insensitive to perturbations. Furthermore, minus the total local hardness is an energetically favorable perturbation of the external potential.

Potentialphilicity and potentialphobicity: Reactivity indicators for external potential changes from density functional reactivity theory

In analogy to the electrophilicity, we define potentialphilicity indicators that represent energetically favorable ways to change the external potential of a molecule at fixed electron number. Similarly, we define a potentialphobicity to represent the least favorable way to change the external potential of a molecule. The resulting indicators should be useful for describing how molecular geometries change and predicting favorable and unfavorable ways for a reagent to approach a molecule. The linear response function enters plays a very important role in this approach, analogous to the role of the hardness for the electrophilicity or the hardness kernel for the Fukui function. The mathematical properties of the response function and its implications for these reactivity indicators are discussed in depth.

Do the local softness and hardness indicate the softest and hardest regions of a molecule?

In this work, we will show that the largest values of the local softness and hardness do not necessarily correspond to the softest and hardest regions of the molecule, respectively. Based on our results, we will argue that it is more useful to interpret the local softness and the local hardness as functions that measure the "local abundance" or "concentration" of the corresponding global properties. This new point of view helps reveal how and when these local reactivity indices are most useful.