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.

Bonding and antibonding characters of the first order reduced density matrix. A new look at two charge-shift bonds

This article reports a new way of analyzing chemical bonds based on the use of gradient of a first order reduced density matrix (1-RDM) map to partition bonds into atomic core, valence, and dorsal regions that can be further characterized as bonding or antibonding. These classifications are directly related to the familiar language of molecular orbital analysis. Analyses based on these quantities and the critical points of the 1-RDM performed on two charge-shift bonds show that although they share common features, there are fundamental differences in these interactions that are detected neither by density-based analysis nor by valence bond theory. The results suggest the gradient of 1-RDM as a valuable tool in bond analysis.

Conceptual Insights into DFT Spin-State Energetics of Octahedral Transition-Metal Complexes through a Density Difference Analysis

In this study, an intuitive concept is derived, which explains the characteristic dependence of spin-state energetics on the exact exchange admixture of DFT functionals in the case of octahedral transition metal complexes. The change in electron density distributions upon varying the admixture, c₃ , in the B3LYP functional is analyzed for archetype ionic and covalent systems as well as for the Fe²⁺ ion in an ideal octahedral field. An understanding of how the DFT description of the electronic structure of octahedral complexes changes as a function of c₃ is sought. A systematic spin-state energy analysis of 50 octahedral complexes of various metals and ligands with consistent experimental data is presented, allowing the derivation, in theory, of an optimal c₃ value for each system. The notion that the admixture dependence of DFT spin-state energetics stems from the treatment of nondynamic electrons arising from the mixing of (M-Lz2 )⁰ (dz2 )² and (M-Lx2-y2 )⁰ (dx2-y2 )² configurations into the dominant (M-Lx2-y2 )² (dx2-y2 )⁰ and (M-Lx2-y2 )² (dx2-y2 )⁰ ones in the low(er) spin states is put forward. That is, in the effort to mimic such electron-electron interactions, E_x^LDA overestimates, whereas exact exchange downplays the contribution of this type of electron correlation to the stability of low(er) spin states, leading to the widespread practical observation that the higher the exact exchange admixture, the more stable the high-spin-state configuration.

Analysis of molecular and (di)atomic dual-descriptor functions and matrices

In this work, the dual-descriptor is studied in matrix form [Formula: see text] and both coordinates condensed to atoms, resulting in atomic and diatomic (or where applicable, bond) condensed single values. This double partitioning method of the dual-descriptor matrix is proposed within the Hirshfeld-I atoms-in-molecule framework although it is easily extended to other atoms-in-molecules methods. Diagonalizing the resulting atomic and bond dual-descriptor matrices gives eigenvalues and eigenvectors describing the reactivity of atoms and bonds. The dual-descriptor function is the diagonal element of the underlying matrix. The extra information contained in the atom and bond resolution is highlighted and the effect of choosing either the fragment of molecular response or response of molecular fragment approach is quantified. Graphical Abstract Atom and bond condensed dual descriptor matrices and functions are derived from molecular ones using Hirshfeld-I atoms in molecules weight functions.

Atom and Bond Fukui Functions and Matrices: A Hirshfeld-I Atoms-in-Molecule Approach

The Fukui function is often used in its atom-condensed form by isolating it from the molecular Fukui function using a chosen weight function for the atom in the molecule. Recently, Fukui functions and matrices for both atoms and bonds separately were introduced for semiempirical and ab initio levels of theory using Hückel and Mulliken atoms-in-molecule models. In this work, a double partitioning method of the Fukui matrix is proposed within the Hirshfeld-I atoms-in-molecule framework. Diagonalizing the resulting atomic and bond matrices gives eigenvalues and eigenvectors (Fukui orbitals) describing the reactivity of atoms and bonds. The Fukui function is the diagonal element of the Fukui matrix and may be resolved in atom and bond contributions. The extra information contained in the atom and bond resolution of the Fukui matrices and functions is highlighted. The effect of the choice of weight function arising from the Hirshfeld-I approach to obtain atom- and bond-condensed Fukui functions is studied. A comparison of the results with those generated by using the Mulliken atoms-in-molecule approach shows low correlation between the two partitioning schemes.

Understanding the Highly Varying pKa of Arylamines. A Perspective from the Average Local Ionization Condensed-to-Atom Framework

The highly varying experimental pKa values for 36 arylamines spanning 7 orders of magnitude is carefully examined. Within this framework, a valence condensed-to-atom model for the average ionization energy is introduced and tested. The theoretical approach is connected to orbital Fukui functions directly mapped into semilocal or regional site-specific responses. It is revealed that the average local ionization energies associated with the amino nitrogen atom is linearly correlated to the basicity of the substituted arylamines, properly reproducing the experimental ordering of basicity. The condensed-to-atom descriptor exhibits a high predictive power, providing a new direct reactivity evaluation of significant value.