Energy decompositions according to physical space partitioning schemes

This work describes simple decompositions of the energy of molecular systems according to schemes that partition the three-dimensional space. The components of those decompositions depend on one and two atomic domains thus providing a meaningful chemical information about the nature of different bondings among the atoms which compose the system. Our algorithms can be applied at any level of theory (correlated or uncorrelated wave functions). The results reported here, obtained at the Hartree-Fock level in selected molecules, show a good agreement with the chemical picture of molecules and require a low computational cost in comparison with other previously reported decompositions.     

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.     

A strategy to derive new internal coordinates by partitioning the internal configuration space according to invariance properties

A method to derive optimally orthogonal curvilinear coordinates for N-body systems is proposed. The invariance of certain subspaces under groups of linear transformations is employed to partition the configuration subspace into internal and external components. The construction is initially carried out locally by orthogonalizing typical group invariant vector fields. Integration is performed subsequently by means of integrating factors. Simple examples of orthogonal invariants illustrate the discussion.     

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.     

Energy partitioning scheme based on self-consistent method for subsystems: populational space approach

The refinement of the previously proposed energy partitioning scheme, self-consistent charge and configuration method for subsystems (Korchowiec and Uchimaru, J Chem Phys 2000, 112, 1623), is proposed. Our new realization takes rigorously into account all the interactions between subsystems and guarantees proper symmetry of the intermediate wavefunctions. In addition, the scheme is supplemented with natural orbitals for chemical valence to trace the charge reorganization during polarization and charge transfer steps. The water dimer and ammonia borane are used to illustrate the proposed formalism.     

Determination of energies and electronic densities of functional groups according to partitionings in the physical space

This work formulates our previously reported partitionings of the first-order reduced density matrix and the molecular electronic energy using for both quantities an identical mathematical framework. The procedure provides a consistent and rigorous scheme for extending our algorithms to unions of atomic domains, in order to describe molecular fragments which can be identified as functional groups. Numerical determinations, performed in several series of organic compounds and clusters, support the reliability of our methodology to describe properties of atomic groups.     

Fast density matrix-based partitioning of the energy over the atoms in a molecule consistent with the Hirshfeld-I partitioning of the electron density

For the Hirshfeld-I atom in the molecule (AIM) model, associated single-atom energies and interaction energies at the Hartree-Fock level are efficiently determined in one-electron Hilbert space. In contrast to most other approaches, the energy terms are fully consistent with the partitioning of the underlying one-electron density matrix (1DM). Starting from the Hirshfeld-I AIM model for the electron density, the molecular 1DM is partitioned with a previously introduced double-atom scheme (Vanfleteren et al., J Chem Phys 2010, 132, 164111). Single-atom density matrices are constructed from the atomic and bond contributions of the double-atom scheme. As the Hartree-Fock energy can be expressed solely in terms of the 1DM, the partitioning of the latter over the AIM naturally leads to a corresponding partitioning of the Hartree-Fock energy. When the size of the molecule or the molecular basis set does not grow too large, the method shows considerable computational advantages compared with other approaches that require cumbersome numerical integration of the molecular energy integrals weighted by atomic weight functions.     

Localization maps by orbital partitioning of the electron density

Summary We define a localization measure for one-determinantal wave-functions based on the partitioning of the total electron density to orbital contributions. The set of occupied orbitals is the more localized the fewer terms are necessary to describe the total density. This measure varies from point to point in space which leads to characteristic localization maps for molecules.Supported in part by the grant Hungarian Research Fund OTKA 517/1990     

A study of the relationships between unpaired electron density, spin-density and cumulant matrices

This work describes the derivation of simple relationships between the density matrix of effectively unpaired electrons and the spin-density matrix in N-electron systems. The link between both devices turns out to be the one-electron matrix arising from the diagonal contraction of the cumulant matrix corresponding to the second-order reduced density matrix. We study some features of this contracted matrix, showing its usefulness to describe the electronic correlation. Numerical determinations performed in selected systems with different spin symmetries confirm the theoretical predictions.     

Communication: Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A double-atom partitioning of the molecular one-electron density matrix is used to describe atoms and bonds. All calculations are performed in Hilbert space. The concept of atomic weight functions (familiar from Hirshfeld analysis of the electron density) is extended to atomic weight matrices. These are constructed to be orthogonal projection operators on atomic subspaces, which has significant advantages in the interpretation of the bond contributions. In close analogy to the iterative Hirshfeld procedure, self-consistency is built in at the level of atomic charges and occupancies. The method is applied to a test set of about 67 molecules, representing various types of chemical binding. A close correlation is observed between the atomic charges and the Hirshfeld-I atomic charges.     

Efficient multipole expansion: Choice of order and density partitioning techniques

Two approaches to improve the convergence of the multipole series were considered: 1) an increase in the order of the expansion; 2) decomposition of the molecular charge density into smaller distributions. New decompositions of the molecular electronic density and a computational procedure to generate high-order moments are presented. The accuracy and timing of test calculations on the H₂O ... H₂O system are given and suggestions are made for optimizing the choice of an expansion for more general systems.We express our appreciation to the National Institutes of Health which supported this work under Grant 1 R01 GM 20436-02.     

Atoms-in-molecules in momentum space: a Hirshfeld partitioning of electron momentum densities

A direct application of the Hirshfeld atomic partitioning (HAP) scheme is implemented for molecular electron momentum densities (EMDs). The momentum density contributions of individual atoms in diverse molecular systems are analyzed along with their topographical features and the kinetic energies of the atomic partitions. The proposed p-space HAP-based charge scheme does seem to possess the desirable attributes expected of any atoms in molecules partitioning. In addition to this, the main strength of the p-space HAP is the exact knowledge of the kinetic energy functional and the inherent ease in computing the kinetic energy. The charges derived from HAP in momentum space are found to match chemical intuition and the generally known chemical characteristics such as electronegativity, etc.     

Ultraviolet photodissociation of bromopropane: Energy partitioning and curve potential crossing analysis

The photodissociation mechanism of bromopropane (C₃H₇Br) molecule in the UV region is treated within time-dependent numerical methods. The results and our ab initio calculations indicated that the absorption of C₃H₇Br in the investigated region primarily originated from the excitations to the lowest three repulsive states, as assigned as 2A ₂, 4A ₁, and 3A ₁ in C _s symmetry. Dissociations occurred on the PES surfaces of the three states, giving C₃H₇ + Br (² P _3/2) or C₃H₇ + Br* (² P _1/2) as two final products, and being impacted by an avoided crossing between the PES surfaces of the 3A ₁ and 4A ₁ states. The transition dipole to the 2A ₂ state was perpendicular to the symmetry plane, so perpendicular to the C-Br bond. The transitions to the 3A ₁ state was polarized parallel to the symmetry plane, and also parallel to the C-Br bond. We have also determined the avoided crossing probabilities, which affected the relative fractions of the individual pathways and angular distributions of photofragments, for the photolysis of C₃H₇Br in the UV region.     

Analysis of the Heyd-Scuseria-Ernzerhof density functional parameter space

The Heyd-Scuseria-Ernzerhof (HSE) density functionals are popular for their ability to improve upon the accuracy of standard semilocal functionals such as Perdew-Burke-Ernzerhof (PBE), particularly for semiconductor band gaps. They also have a reduced computational cost compared to hybrid functionals, which results from the restriction of Fock exchange calculations to small inter-electron separations. These functionals are defined by an overall fraction of Fock exchange and a length scale for exchange screening. We systematically examine this two-parameter space to assess the performance of hybrid screened exchange (sX) functionals and to determine a balance between improving accuracy and reducing the screening length, which can further reduce computational costs. Three parameter choices emerge as useful: "sX-PBE" is an approximation to the sX-LDA screened exchange density functionals based on the local density approximation (LDA); "HSE12" minimizes the overall error over all tests performed; and "HSE12s" is a range-minimized functional that matches the overall accuracy of the existing HSE06 parameterization but reduces the Fock exchange length scale by half. Analysis of the error trends over parameter space produces useful guidance for future improvement of density functionals.     

Energy characteristics of adsorbed water in active carbons according to the NMR relaxation data

Nuclear magnetic relaxation measurements were used to determine activation energy E _act of the motion of water molecules adsorbed in active carbons. The E _act value was found to depend on the filling of active carbon pores due to changes in the state of water molecules under adsorption. It was established that the E _act = f(p/p _s) plots, where p/p _s is the relative pressure of water vapor observed for microporous active carbons (FAS-1, 2, N-15, SKT-6A), are similar in form to the corresponding plots of changes in water adsorption heats. In particular, we concluded that the plateau in the E _act = f(p/p _s) dependences, as in the case of adsorption heats, reflects the volumetric filling of active carbon micropores with water. We show that a linear function describes the increase in E _act values for water upon the complete filling of micropores with an increase in the volume of adsorbed water clusters per one primary adsorption center (W ₀/a _m ). We establish that, for water in the FAS-3 sample, the deviation of E _act values from this linear function was due to the contribution from the vapor phase in the mesopores (x ₀ = 0.7−1.2 nm) that make up a considerable part of the active carbon’s porous system.     

The self-interaction correction to the local spin density model: Effect on atomic momentum space properties

Perdew and Zunger showed that the exact energy density functional for the ground state is strictly self-interaction-free. However, the local spin density (LSD) approximation lacks this self-interaction correction (SIC). Perdew and Zunger studied the effect on coordinate-space properties of incorporating the SIC. In the present study we examine the effect of SIC on momentum space properties, viz. the electron momentum distribution, ⊂p″⊃ values, electron momentum densities and the Compton profiles for atoms He to Ar. A remarkable improvement is seen in all the momentum space properties of the SIC LSD model over the LSD model when compared to their near Hartree-Fock counterparts.     

Stockholder projector analysis: a Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A previously introduced partitioning of the molecular one-electron density matrix over atoms and bonds [D. Vanfleteren et al., J. Chem. Phys. 133, 231103 (2010)] is investigated in detail. Orthogonal projection operators are used to define atomic subspaces, as in Natural Population Analysis. The orthogonal projection operators are constructed with a recursive scheme. These operators are chemically relevant and obey a stockholder principle, familiar from the Hirshfeld-I partitioning of the electron density. The stockholder principle is extended to density matrices, where the orthogonal projectors are considered to be atomic fractions of the summed contributions. All calculations are performed as matrix manipulations in one-electron Hilbert space. Mathematical proofs and numerical evidence concerning this recursive scheme are provided in the present paper. The advantages associated with the use of these stockholder projection operators are examined with respect to covalent bond orders, bond polarization, and transferability.     

Hirshfeld partitioning of the electron density: atomic dipoles and their relation with functional group properties

Atomic dipole moments, derived within the Hirshfeld partitioning of the molecular electron density, have been studied for compounds of the type H-X and Cl-X, for a series of functional groups X frequently encountered in organic molecules. In the case of the H-X compounds, the component of the atomic dipole moment on H along the axis connecting H with the central atom in X is found to be linearly correlated with the electronegativity of X, the hardness of X playing no significant role. In the case of the Cl-X compounds, the situation is less clear. However, evidence seems to point to the conclusion that for these compounds, also the group hardness plays an important role.