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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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We report an original method that provides a new insight into the reaction mechanism by direct observation of bond breaking and formation. Variations of the diagonal elements of the Hessian along the IRC are shown to reflect the anharmonic properties of the system that are induced by electron density modifications upon the reaction. This information is presented in the form of the reaction spectrum, demonstrating how particular atoms engage in the reorganization of bonds. The test reactions are: HCOF synthesis and HONS isomerization.

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Definite algorithms for calculation of the atomic contributions to the reaction force Fξ and the reaction force constant kξ (the first and the second derivatives of the energy over the reaction path step) are presented. The electronic part in the atomic and group contributions has been separated, and this opened the way to identification of the reactive molecule fragments on the consecutive stages of the reaction path. Properties have been studied for the two canonical test reactions: CO + HF → HCOF and HONS → ONSH.

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The reaction force and the electronic flux, first proposed by Toro-Labbé et al. (J Phys Chem A 103:4398, 1999) have been expressed by the existing conceptual DFT apparatus. The critical points (extremes) of the chemical potential, global hardness and softness have been identified by means of the existing and computable energy derivatives: the Hellman-Feynman force, nuclear reactivity and nuclear stiffness. Specific role of atoms at the reaction center has been unveiled by indicating an alternative method of calculation of the reaction force and the reaction electronic flux. The electron dipole polarizability on the IRC has been analyzed for the model reaction HF + CO→HCOF. The electron polarizability determined on the IRC α e (ξ) was found to be reasonably parallel to the global softness curve S(ξ). The softest state on the IRC (not TS) coincides with zero electronic flux.

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The concept of the polarization justified Fukui functions has been tested for the set of model molecules: imidazole, oxazole, and thiazole. Calculations of the Fukui functions have been based on the molecular polarizability analysis, which makes them a potentially more sensitive analytical tool as compared to the classical density functional theory proposals, typically built on electron density only. Three selected molecules show distinct differences in their reactivity patterns, despite very close geometry and electronic structure. The maps of the polarization justified Fukui functions on the molecular plane correctly identify important features of the molecules: the site for the preferential electrophilic attack in imidazole (-NH, see the TOC image) and oxazole (5-C), as well as uniquely aromatic character of the thiazole molecule and the acidic forms XH(+) of all three species.

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The Fukui functions based on the computable local polarizability vector have been presented for a group of simple molecules. The necessary approximation for the density functional theory softness kernel has been supported by a theoretical analysis unifying and generalizing early concepts produced by the several authors. The exact relation between local polarizability vector and the derivative of the nonlocal part of the electronic potential over the electric field has been demonstrated. The resulting Fukui functions are unique and represent a reasonable refinement when compared to the classical ones that are calculated as the finite difference of the density in molecular ions. The new Fukui functions are strongly validated by their direct link to electron dipole polarizabilities that are reported experimentally and by other computational methods.

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Existing approximation to the softness kernel, successfully explored in earlier work, has been extended; the normal Gauss distribution function has been used instead of the Dirac delta. The softness kernel becomes continuous functions in space and may be used to calculate the linear response function of the electron density. Three-dimensional visualization of the softness kernel and the linear response function are presented for a nitrogen atom as a working example. By using a single parameter of the spatial Gauss distribution, the novel softness kernel has been adjusted to be consistent with the standard form of the hardness kernel, representing the leading fraction of the electronic interactions in the system.

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New Fukui functions have been derived within the conceptual density functional theory by the analysis of the polarization effect of a system in static electric field. Resulting Fukui functions accurately reproduce the global softness and electronic dipolar polarizability; they meet the condition integral[f(r)/r]dr = -(partial differential mu/partial differential Z)(N) and lead to very reasonable values of the global hardness for atoms for the group of 29 main group elements. Computational clarity makes the new Fukui functions a promising tool in studies of molecular reactivity.

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The amino group in adenine plays a key role in formation of hydrogen bonds in nucleic acids and in other molecular systems. Thus, the structure of this group is of fundamental importance in the molecular recognition phenomena. Ab initio MP2 and density functional B3LYP methods with various basis sets have been used to calculate the optimized structure and the infrared spectrum of adenine (the N9-H tautomer). Calculations at the MP2 level with larger basis sets tend to decrease the degree of pyramidalization of the C-NH2 group, whereas the B3LYP method consistently yields the planar or nearly planar structure of adenine. MP2 complete basis set (CBS) limit method with the aug-cc-pVTZ --> aug-cc-pVQZ (aTZ --> aQZ) extrapolation scheme has predicted very small planarization barrier of adenine, 0.015 kcal/mol, which is in very good agreement with the MP2-predicted planarization barrier of 0.020 kcal/mol, reported by S. Wang and H. F. Schaefer III, J. Chem. Phys. 2006, 124, 044303. Similar results were obtained in calculations by the coupled cluster CCSD(T) CBS method. Thus, it can be concluded that the amino group in adenine, in the gas phase, is very flexible with a small degree of nonplanarity. Extremely low planarization barrier implies that adenine requires very little energy to conform the structure of the amino group to formation of the complementary hydrogen bonds with other molecules. This fact is very important for base pairing in nucleic acids or other polymers containing adenine residues. The anharmonic frequencies of adenine have been calculated at the B3LYP/6-311++G(df,pd) level of theory. The theoretical results show excellent agreement with the available experimental data. The revised assignment of the infrared spectrum of adenine in Ar matrix has been made. The predicted anharmonic frequency of the NH2 inversion, 181 cm(-1), is supported by the experimental data. It is demonstrated that the vibrational frequencies and potential energy distribution (PED) obtained from the B3LYP calculations are more reliable than those obtained at the MP2 level.

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