Hyperhardness and hypersoftness of atoms and their ions

CONTEXT

The theory of reactivity based on cDFT has been supplemented with the new method of calculating the atomic and local indices. With the use of previously derived relationship of the electron density gradient to the softness kernel and to the linear response function, we deliver theoretical analysis to obtain significant reactivity indices-the electron density derivatives: local softness and local hypersoftness together with the global hyperhardness index and the derivative of the global softness with respect to the number of electrons. The local derivatives have been applied in the calculation of responses of atoms to perturbation by an external potential by the alchemical approach. The vital role of the local softness has been confirmed; the potential role of the hypersoftness has been indicated.

METHOD

Our original theoretical scheme has been numerically illustrated with the results obtained with electron density calculations with B3LYP method implemented in Gaussian 16 package. The aug-cc-pvqz basis set has been routinely applied, except for the Ca atom (cc-pvqz). Using the pVTZ basis set recommended by Sadlej was necessary for the potassium atom.

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.

The Connectivity Matrix: A Toolbox for Monitoring Bonded Atoms and Bonds

The concept of a connectivity matrix, essential for the reaction fragility (RF) spectra technique for monitoring electron density evolution in a chemical reaction, has been supported with a novel formulation for the diagonal matrix elements; their direct link to the electron density function ρ(r) has been demonstrated. By combining the concept with the atomization energy of a system, the separation of the potential energy into atomic and/or bond contributions has been achieved. The energy derivative diagrams for atoms and bonds that are variable along a reaction path provide new insight into the reaction mechanism. Diagonalization of the connectivity matrix resulted in the eigenvectors that provide information on a role of individual atoms in the development of structural changes along a reaction path.

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.

Evolution of the atomic valence observed by the reaction fragility spectra on the reaction path

The computational fragility spectra of atoms on the reaction path are presented for a selection of canonical processes represented by an amino group rotation around the (X)HC-NH(Y) bond (X = O, S; Y=H, CH₃). Calculated spectra are found to very accurately describe the variation of atomic valence. Significant linear correlation is also demonstrated between the Wiberg bond indices and the corresponding elements of the connectivity matrix, instrumental for calculation of the spectra. Demonstrated atomic fragility spectra contain rich and subtle information on the variation of the bonding status of all atoms, including the weak interacting individual hydrogens. Correlation with the atomic valences confirm the earlier finding that the spectra contain a picture of the electron density flow upon a reaction.

A new way of studying chemical reactions: a hand-in-hand URVA and QTAIM approach

Bond formation and bond cleavage processes are central to a chemical reaction. They can be investigated by monitoring changes in the potential energy surface (PES) or changes in the electron density (ED) distribution ρ(r) taking place during the reaction. However, it is not yet clear how the corresponding changes in the PES and ED are related, although the connection between energy and density has been postulated in the famous Hohenberg-Kohn theorem. Our unified reaction valley approach (URVA) identifies the locations of bond formation/cleavage events along the reaction path via the reaction path curvature peaks and their decomposition into the internal coordinate components associated with the bond to be formed or cleaved. One can also investigate bond formation/cleavage events using the quantum theory of atoms-in-molecule (QTAIM) analysis by monitoring changes in the topological properties of ρ(r) and the associated Laplacian ∇2ρ(r). By a systematic comparison of these two approaches for a series of ten representative chemical reactions ranging from hydrogen migration to cycloaddition reactions and gold(i) catalysis, we could for the first time unravel the PES-ED relationship. In the case of a bond formation, all changes in the ED occur shortly before or at the corresponding curvature peak, and in a bond cleavage, the ED changes occur at or shortly after the curvature peak. In any case, the ED changes always occurred in the vicinity of the curvature peak in accordance with the Hohenberg-Kohn theorem. Our findings provide a comprehensive view on bond formation/cleavage processes seen through the eyes of both the PES and ED and offer valuable guidelines on where to search for significant ED changes associated with bond formation or cleavage events.

Reaction force surface for the hydrogen transfer reaction in malonaldehyde: A classical wavefront-based formulation

This paper investigates the simulation of a modified Hamilton–Jacobi equation in order to provide a classical wavefront-based formulation of reaction dynamics. Here, the wavefront is interpreted as minimizing a particular action functional. The method is rooted in the geometric optics and an eikonal equation is typically used for describing the light wave propagation. We have introduced a general formulation for reaction force surface (RFS) and a host of several force-based properties, which offer a reinvigorated understanding of the occurrence of a reaction event at molecular level. The calculation is performed on a model potential energy surface representing the hydrogen transfer reaction in malonaldehyde.

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.