Analytical approximation to the local softness and hypersoftness and to their applications as reactivity indicators.

Local density functional theory derivatives of the electron density have been calculated analytically for the set of canonical hydrogenic orbitals; original solutions have been obtained using the novel density gradient theorem. Results for the first and second derivatives of electron density over N (number of electrons) and over µ (chemical potential) have been demonstrated. Calculations of the state functions ΔN, ΔE, and Δµ disturbed by an external potential Δv(r) have been obtained via the concept of alchemical derivatives. The local softness s(r) and local hypersoftness [ds(r)/dN]v have been proved to provide crucial chemical information on the sensitivity of orbital density to the disturbance of the external potential Δv(r), leading to electron exchange ΔN and the corresponding changes of the state functions ΔE, Δµ. The results are fully compatible with the well-understood character of atomic orbitals in chemistry and open a perspective to applications to atoms, free or bonded.

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.

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.

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.