Energy decompositions according to physical space partitioning schemes

Localized operator partitioning method for electronic excitation energies in the time-dependent density functional formalism

We extend the localized operator partitioning method (LOPM) [J. Nagesh, A. F. Izmaylov, and P. Brumer, J. Chem. Phys. 142, 084114 (2015)] to the time-dependent density functional theory framework to partition molecular electronic energies of excited states in a rigorous manner. A molecular fragment is defined as a collection of atoms using Becke's atomic partitioning. A numerically efficient scheme for evaluating the fragment excitation energy is derived employing a resolution of the identity to preserve standard one- and two-electron integrals in the final expressions. The utility of this partitioning approach is demonstrated by examining several excited states of two bichromophoric compounds: 9-((1- naphthyl)- methyl)- anthracene and 4-((2- naphthyl)- methyl)- benzaldehyde. The LOPM is found to provide nontrivial insights into the nature of electronic energy localization that is not accessible using a simple density difference analysis.

Pairing and unpairing electron densities in organic systems: two-electron three center through space and through bonds interactions

Two-electron three-center bonding interactions in organic ions like methonium (CH5(+)), ethonium (C2H7(+)), and protonated alkanes n - C4H11(+) isomers (butonium cations) are described and characterized within the theoretical framework of the topological analysis of the electron density decomposition into its effectively paired and unpaired contributions. These interactions manifest in some of this type of systems as a concentration of unpaired electron cloud around the bond paths, in contrast to the well known paradigmatic boron hydrids in which it is not only concentrated close to the atomic nucleus and the bond paths but out of them and over the region defined by the involved atoms as a whole. This result permits to propose an attempt of classification for these interactions based in such manifestations. In the first type, it is called as interactions through bonds and in the second type as interactions through space type.

Bond energy analysis revisited and designed toward a rigorous methodology

The present study theoretically revisits and numerically assesses two-body energy decomposition schemes including a newly proposed one. The new decomposition scheme is designed to make the equilibrium bond distance equivalent with the minimum point of bond energies. Although the other decomposition schemes generally predict the wrong order of the C-C bond strengths of C(2)H(2), C(2)H(4), and C(2)H(6), the new decomposition scheme is capable of reproducing the C-C bond strengths. Numerical assessment on a training set of molecules demonstrates that the present scheme exhibits a stronger correlation with bond dissociation energies than the other decomposition schemes do, which suggests that the new decomposition scheme is a reliable and powerful analysis methodology.

Energy decompositions according to physical space partitioning schemes: Treatments of the density cumulant

This article is a continuation of our previous paper on schemes of energy decompositions of molecular systems in the real space [D. R. Alcoba et al., J. Chem. Phys. 122, 074102 (2005)] now using correlated state functions. We study, according to physical arguments, the appropriate management of the density cumulant arising from the second-order reduced density matrix at correlated level, whose contributions can be assigned to one-center or to two-center terms in the energy partitioning. Our treatments are applied within two physical space partitioning schemes: the Bader partitioning into atomic basins and the fuzzy atom procedure. The results obtained in selected molecules are analyzed and discussed in detail.

One- and two-center physical space partitioning of the energy in the density functional theory

A conceptually new approach is introduced for the decomposition of the molecular energy calculated at the density functional theory level of theory into sum of one- and two-atomic energy components, and is realized in the "fuzzy atoms" framework. (Fuzzy atoms mean that the three-dimensional physical space is divided into atomic regions having no sharp boundaries but exhibiting a continuous transition from one to another.) The new scheme uses the new concept of "bond order density" to calculate the diatomic exchange energy components and gives them unexpectedly close to the values calculated by the exact (Hartree-Fock) exchange for the same Kohn-Sham orbitals.

Laplacian Field of the Effectively Unpaired Electron Density: Determination of Many-Body Effects on Electron Distributions

This work carries out the study of the Laplacian field function of the electron density L(r) = -nabla2rho(r) splitted in two contributions rho(r) = rho(p)(r) + rho(u)(r), which correspond to the effectively paired and effectively unpaired electron densities, respectively. The visualization of the concentration and depletion of these fields and their spatial localization show no contribution of the effectively unpaired electrons to the conventional bonding among two centers, but the field -nabla2rho(u)(r) provides an interesting structure. We also study the reliability of the information contained in the partitioning of this electron density field function for describing nonclassical bondings as the three-center two-electron ones.

Chemical fragments in real space: definitions, properties, and energetic decompositions

The physical and chemical meaning of real space molecular fragments resulting from arbitrary partitions of the density is reviewed under a common unifying formalism. Both fuzzy (interpenetrating) and non-fuzzy (exhaustive) decompositions are treated on an equal basis. Density decompositions are consistently generalized to compatible density matrix partitions by using Li and Parr's ideas (Li and Parr J Chem Phys 1986, 84, 1704), and these are carried onto an energy partition. It is argued that the merits of a given decomposition should be judged against both the charge and the energetic image it provides. Atomic partitions are used to show how the interacting quantum atoms approach (IQA) allows us to cope with the most important energy cancellations of quantum chemistry. Binding results from a trade-off between atomic (or fragment) energy deformations with respect to a reference and interatomic (interfragment) interactions. Deformation energies are divided into charge transfer and redistribution terms and their relative roles are analyzed. A number of systems are compared against the fuzziness of different density decompositions. The results consistently show that fuzzy partitions tend to give low atomic net charges and enhanced covalency, while exhaustive partitions generate larger net charges and smaller covalencies, across a wide range of bonding regimes.

Grid-based energy density analysis: Implementation and assessment

Grid-based energy density analysis (grid-EDA) that decomposes the total energy into atomic energies by a space-partitioning function is proposed. The kinetic energy, nuclear attraction, and exchange-correlation functional are evaluated on grid points and are split into atomic contributions. To reduce numerical errors in the conventional scheme of numerical integration, the electronic Coulomb and HF exchange interactions are evaluated by the pseudospectral method, which was first applied to an ab initio method by Friesner [Chem. Phys. Lett. 116, 39 (1985)], and are decomposed into atomic contributions. Grid-EDA using the pseudospectral method succeeds in ensuring less than 1 kcalmol error in total energies for small molecules and providing reliable atomic energy contributions for the problematic lithium cluster, which exhibits a strong basis-set dependence for Mulliken-type EDA. Also, site-dependent atomization energies are estimated by grid-EDA for cluster models such as Li(48), C(41)H(60), and Mg(32)O(32). Grid-EDA reveals that these models imitate crystal environments reasonably because atomization energies estimated from the inner atoms of the models are close to the experimental cohesive energies.

Hartree–Fock Energy Partitioning in Terms of Hirshfeld Atoms

A Hirshfeld decomposition scheme of the Hartree-Fock total molecular energy into atomic energies is presented. The calculations are performed by direct numerical integration and the results are compared for a set of 28 molecules containing different kinds of atoms. The calculated atomic energies show a strong dependency on changes of atomic electron population and hybridization. Linear correlations are found between the energy and the population for H, these being related to the electronegativity of this atom and to the external potential created by the remaining atoms. The proposed energy partitioning scheme appears to be useful for studies such as proton acidity, the anomeric effect and group transferability, and allows atomic virial ratios to be obtained. Finally, the atomic potential energies are found to mimic trends based on exact expressions as well as trends displayed by molecular quantities, thus lending credibility to the partitioning scheme used.

An orbital localization criterion based on the theory of “fuzzy” atoms

This work proposes a new procedure for localizing molecular and natural orbitals. The localization criterion presented here is based on the partitioning of the overlap matrix into atomic contributions within the theory of "fuzzy" atoms. Our approach has several advantages over other schemes: it is computationally inexpensive, preserves the sigma/pi-separability in planar systems and provides a straightforward interpretation of the resulting orbitals in terms of their localization indices and atomic occupancies. The corresponding algorithm has been implemented and its efficiency tested on selected molecular systems.

A study of the partitioning of the first-order reduced density matrix according to the theory of atoms in molecules

This work describes a simple spatial decomposition of the first-order reduced density matrix corresponding to an N-electron system into first-order density matrices, each of them associated to an atomic domain defined in the theory of atoms in molecules. A study of the representability of the density matrices arisen from this decomposition is reported and analyzed. An appropriate treatment of the eigenvectors of the matrices defined over atomic domains or over unions of these domains allows one to describe satisfactorily molecular properties and chemical bondings within a determined molecule and among its fragments. Numerical determinations, performed in selected molecules, confirm the reliability of our proposal.

Determination of Three-Center Bond Indices from Population Analyses: A Fuzzy Atom Treatment

This work proposes the use of the treatment referred to as fuzzy atoms to describe three-center bond indices within studies of electron population analysis. A simple manipulation of our algorithms reported previously for describing multicenter bondings enables us to introduce this methodology in our mathematical framework, providing suitable numerical determinations of three-center bond indices, two-center bond ones, and electron atomic populations. The results, obtained in selected systems, are discussed and compared to those arising from other procedures of population analysis.

Density functional energy decomposition into one- and two-atom contributions

The present work provides a generalization of Mayer's energy decomposition for the density-functional theory (DFT) case. It is shown that one- and two-atom Hartree-Fock energy components in Mayer's approach can be represented as an action of a one-atom potential V(A) on a one-atom density rho(A) or rho(B). To treat the exchange-correlation term in the DFT energy expression in a similar way, the exchange-correlation energy density per electron is expanded into a linear combination of basis functions. Calculations carried out for a number of density functionals demonstrate that the DFT and Hartree-Fock two-atom energies agree to a reasonable extent with each other. The two-atom energies for strong covalent bonds are within the range of typical bond dissociation energies and are therefore a convenient computational tool for assessment of individual bond strength in polyatomic molecules. For nonspecific nonbonding interactions, the two-atom energies are low. They can be either repulsive or slightly attractive, but the DFT results more frequently yield small attractive values compared to the Hartree-Fock case. The hydrogen bond in the water dimer is calculated to be between the strong covalent and nonbonding interactions on the energy scale.