Atomic Charges Are Measurable Quantum Expectation Values: A Rebuttal of Criticisms of QTAIM Charges

Probing the nature of the Co(III) ion in corrins: comparison of reactions of aquacyanocobyrinic acid heptamethyl ester and aquacyano-stable yellow cobyrinic acid hexamethyl ester with neutral N-donor ligands

Equilibrium constants (log K) for substitution of coordinated H(2)O in aquacyanocobyrinic acid heptamethyl ester (aquacyanocobester, ACCbs) and aquacyano-stable yellow cobyrinic acid hexamethyl ester (aquacyano-stable yellow cobester, ACSYCbs), in which oxidation of the C5 carbon of the corrin interrupts the normal delocalized system of corrins, by neutral N-donor ligands (ammonia, ethanolamine, 2-methoxyethylamine, N-methylimidazole, and 4-methylpyridine) have been determined spectrophotometrically as a function of temperature. Log K values increase with the basicity of the ligand, but a strong compensation effect between ΔH and ΔS values causes a leveling effect. The aliphatic amines with a harder donor atom produce ΔH values that are more negative in their reactions with ACSYCbs than with ACCbs, while the softer, aromatic N donors produce more negative ΔH values with ACCbs than with ACSYCbs. Molecular modeling (DFT, M06L/SVP, and a quantum theory of atoms in molecules analysis of the electron density) shows that complexes of the aliphatic amines with SYCbs produce shorter and stronger Co-N bonds with less ionic character than the Co-N bonds of these ligands with the cobester. Conversely, the Co-N bond to the aromatic N donors is shorter, stronger, and somewhat less ionic in the complexes of the cobester than in those of the SYCbs. Therefore, the distinction between the harder Co(III) in ACSYCbs and softer Co(III) in ACCbs, reported previously for anionic ligands, is maintained for neutral N-donor ligands.

Extending Hirshfeld-I to bulk and periodic materials

In this work, a method is described to extend the iterative Hirshfeld-I method, generally used for molecules, to periodic systems. The implementation makes use of precalculated pseudopotential-based electron density distributions, and it is shown that high-quality results are obtained for both molecules and solids, such as ceria, diamond, and graphite. The use of grids containing (precalculated) electron densities makes the implementation independent of the solid state or quantum chemical code used for studying the system. The extension described here allows for easy calculation of atomic charges and charge transfer in periodic and bulk systems. The conceptual issue of obtaining reference densities for anions is discussed, and the delocalization problem for anionic reference densities originating from the use of a plane wave basis set is identified and handled.

6-bromoindigo dye

6-Bromoindigo (MBI) [systematic name: 6-bromo-2-(3-oxo-2,3-dihydro-1H-indol-2-ylidene)-2,3-dihydro-1H-indol-3-one], C(16)H(9)BrN(2)O(2), crystallizes with one disordered molecule in the asymmetric unit about a pseudo-inversion center, as shown by the Br-atom disorder of 0.682 (3):0.318 (3). The 18 indigo ring atoms occupy two sites which are displaced by 0.34 Å from each other as a result of this packing disorder. This difference in occupancy factors results in each atom in the reported model used to represent the two disordered sites being 0.08 Å from the higher-occupancy site and 0.26 Å from the lower-occupancy site. Thus, as a result of the disorder, the C-Br bond lengths in the disordered components are 0.08 and 0.26 Å shorter than those found in 6,6'-dibromoindigo (DBI) [Süsse & Krampe (1979). Naturwissenschaften, 66, 110], although the distances within the indigo ring are similar to those found in DBI. The crystals are also twinned by merohedry. Stacking interactions and hydrogen bonds are similar to those found in the structures of indigo and DBI. In MBI, an interaction of the type C-Br...C replaces the C-Br...Br interactions found in DBI. The interactions in MBI were calculated quantum mechanically using density functional theory and the quantum theory of atoms in molecules.

Topological partition of the elastic constants of crystals

We present a partitioning of the elastic constants of a crystal into atomic contributions by using the atomic basin concept inherent to Bader's Quantum Theory of Atoms in Molecules. The partition is made by following the evolution of the cell volume and the atomic basin volumes under appropriately defined cell deformations. The method is carefully examined, including internal consistency checks. The transferability of atomic contributions between different crystals is determined by obtaining and comparing the oxygen contribution to the elastic constants of a selection of cubic oxides that includes the rock-salt, perovskite, antifluorite, and cuprite crystal families.

Definition of molecular structure: by choice or by appeal to observation?

There are two schools of thought in chemistry: one derived from the valence bond and molecular orbital models of bonding, the other appealing directly to the measurable electron density and the quantum mechanical theorems that determine its behavior, an approach embodied in the quantum theory of atoms in molecules, QTAIM. No one questions the validity of the former approach, and indeed molecular orbital models and QTAIM play complementary roles, the models finding expression in the principles of physics. However, some orbital proponents step beyond the models to impose their personal stamp on their use in interpretive chemistry, by denying the possible existence of a physical basis for the concepts of chemistry. This places them at odds with QTAIM, whose very existence stems from the discovery in the observable topology of the electron density, the definitions of atoms, of the bonding between atoms and hence of molecular structure. Relating these concepts to the electron density provides the necessary link for their ultimate quantum definition. This paper explores in depth the possible causes of the difficulties some have in accepting the quantum basis of structure beginning with the arguments associated with the acceptance of a "bond path" as a criterion for bonding. This identification is based on the finding that all classical structures may be mapped onto molecular graphs consisting of bond paths linking neighboring atoms, a mapping that has no known exceptions and one that is further bolstered by the finding that there are no examples of "missing bond paths". Difficulties arise when the quantum concept is applied to systems that are not amenable to the classical models of bonding. Thus one is faced with the recurring dilemma of science, of having to escape the constraints of a model that requires a change in the existing paradigm, a process that has been in operation since the discovery of new and novel structures necessitated the extension of the Lewis model and the octet rule. The paper reviews all facets of bonding beginning with the work of Pauling and Slater in their accounting for crystal structures, taking note of Pauling's advocating possible bonding between large anions. Many examples of nonbonded or van der Waals interactions are considered from both points of view. The final section deals with the consequences of the realization that bonded quantum atoms that share an interatomic surface do not "overlap". The time has come for entering students of chemistry to be taught that the electron density can be seen, touched, and measured and that the chemical structures they learn are in fact the tracings of "bonds" onto lines of maximum density that link bonded nuclei. Matter, as we perceive it, is bound by the electrostatic force of attraction between the nuclei and the electron density.

Performance of 3D-space-based atoms-in-molecules methods for electronic delocalization aromaticity indices

Several definitions of an atom in a molecule (AIM) in three-dimensional (3D) space, including both fuzzy and disjoint domains, are used to calculate electron sharing indices (ESI) and related electronic aromaticity measures, namely, I(ring) and multicenter indices (MCI), for a wide set of cyclic planar aromatic and nonaromatic molecules of different ring size. The results obtained using the recent iterative Hirshfeld scheme are compared with those derived from the classical Hirshfeld method and from Bader's quantum theory of atoms in molecules. For bonded atoms, all methods yield ESI values in very good agreement, especially for C-C interactions. In the case of nonbonded interactions, there are relevant deviations, particularly between fuzzy and QTAIM schemes. These discrepancies directly translate into significant differences in the values and the trends of the aromaticity indices. In particular, the chemically expected trends are more consistently found when using disjoint domains. Careful examination of the underlying effects reveals the different reasons why the aromaticity indices investigated give the expected results for binary divisions of 3D space.

Energy additivity in branched and cyclic hydrocarbons

This paper considers the degree to which branched hydrocarbons obey a group additivity scheme for energy and population, extending the study of the known experimental and theoretical transferability of the methyl and methylene groups of the linear hydrocarbons. The chemical groups are defined and their properties are determined using the quantum theory of atoms in molecules (QTAIM). The calculations are carried out with a large basis set at the restricted Hartree–Fock and MP2(full) levels of theory. The deviations from additivity, noted for small ring hydrocarbons leading to the definition of strain energy, are also investigated, showing that the QTAIM energies recover the experimental values. The particular delocalization of the electron density over the surface of the cyclopropane ring, responsible for its “homoaromatic” properties, is discussed in some detail. The calculations reported here satisfy the virial theorem as required for the atomic definition of energy. The problems associated with the use of DFT theory arising from its failure to satisfy the virial theorem are discussed with reference to the study of group transferability.

Comparison of the Hirshfeld-I and iterated stockholder atoms in molecules schemes

Two recently introduced self-consistent Hirshfeld procedures for obtaining atoms in molecules are compared in detail. The Hirshfeld-I scheme introduces self consistency by requiring that the atomic population of the promolecular atom is equal to that of the atom-in-the-molecule. In the iterated stockholder atoms (ISA) approach, self consistency is obtained by requiring that for every value of the radius of a sphere around every nucleus, the average electron density on the surface of this sphere is the same in the promolecular atom and in the atom in the molecule. The relationships between the two schemes are examined, and common backgrounds and differences are discussed. Whereas it can be argued that the Hirshfeld-I approach has a stronger physical background, the ISA scheme avoids having to define what states of the atoms are to be considered when constructing the promolecule.

QTAIM charge-charge flux-dipole flux models for the infrared fundamental intensities of the fluorochloromethanes

The molecular dipole moments, their derivatives, and the fundamental IR intensities of the fluoro-, chloro-, and fluorochloromethanes are determined from QTAIM atomic charges and dipoles and their fluxes at the MP2/6-311++G(3d,3p) level. Root-mean-square (rms) errors of 0.01 D and 5.6 km mol(-1) are found for the dipole moments and fundamental IR intensities calculated using QTAIM parameters when compared with those obtained directly from the MP2/6-311++(3d,3p) calculations and 0.04 D and 23.1 km mol(-1) when compared with the experimental values. Charge, charge flux, and dipole flux contributions are calculated for all the normal vibrations of these molecules. A large negative correlation coefficient of -0.92 is calculated between the charge flux and dipole flux contributions and indicates that electron transfer from one side of the molecule to the other during vibrations is accompanied by relaxation with electron density polarization in the opposite direction. The CF, CCl, and CH stretching normal modes of these molecules are shown to have characteristic sets of charge, charge flux, and dipole flux contributions. Although the FCF and ClCCl deformation normal modes can also be discriminated from one another based on the sizes and signs of these contributions, some HCH deformations have contributions that are similar to those for some of the ClCCl deformations.

On the applicability of resonance forms in pyrimidinic bases. II. QTAIM interpretation of the sequence of protonation affinities

The atomic properties of neutral and protonated forms of uracil and some model compounds, computed from B3LYP/6-31++G//B3LYP/6-31G charge densities with the QTAIM theory, indicate that sigma electron reorganization plays a significant role in the protonation processes. This reorganization is substantially different for O=C-C=C and O=C-C-X (X = N, O) units, involving transfers of electron population between all atoms in the first case but not across the C-X bond in the second unit. O-Protonation is basically favored over the N-protonation because of the lower electron population transferred to the proton. The stability sequence of N-protonated forms can be rationalized in terms of the closer position of the proton, when attached to N3, to regions of larger electron population (carbonyl groups).

Electron sharing indexes at the correlated level. Application to aromaticity calculations

Electron sharing indexes (ESI) have been applied to numerous bonding situations to provide an insight into the nature of the molecular electronic structures. Some of the most popular ESI given in the literature, namely, the delocalization index (DI), defined in the context of the quantum theory of atoms in molecules (QTAIM), and the Fuzzy-Atom bond order (FBO), are here calculated at a correlated level for a wide set of molecules. Both approaches are based on the same quantity, the exchange-correlation density, to recover the electron sharing extent, and their differences lie in the definition of an atom in a molecule. In addition, while FBO atomic regions enable accurate and fast integrations, QTAIM definition of an atom leads to atomic domains that occasionally make the integration over these ones rather cumbersome. Besides, when working with a many-body wavefunction one can decide whether to calculate the ESI from first-order density matrices, or from second-order ones. The former way is usually preferred, since it avoids the calculation of the second-order density matrix, which is difficult to handle. Results from both definitions are discussed. Although these indexes are quite similar in their definition and give similar descriptions, when analyzed in greater detail, they reproduce different features of the bonding. In this manuscript DI is shown to explain certain bonding situations that FBO fails to cope with. Finally, these indexes are applied to the description of the aromaticity, through the aromatic fluctuation (FLU) and the para-DI (PDI) indexes. FLU and PDI indexes have been successfully applied using the DI measures, but other ESI based on other partitions such as Fuzzy-Atom can be used. The results provided in this manuscript for carbon skeleton molecules encourage the use of FBO for FLU and PDI indexes even at the correlated level.

Intermolecular electrostatic energies using density fitting

A method is presented to calculate the electron-electron and nuclear-electron intermolecular Coulomb interaction energy between two molecules by separately fitting the unperturbed molecular electron density of each monomer. This method is based on the variational Coulomb fitting method which relies on the expansion of the ab initio molecular electron density in site-centered auxiliary basis sets. By expanding the electron density of each monomer in this way the integral expressions for the intermolecular electrostatic calculations are simplified, lowering the operation count as well as the memory usage. Furthermore, this method allows the calculation of intermolecular Coulomb interactions with any level of theory from which a one-electron density matrix can be obtained. Our implementation is initially tested by calculating molecular properties with the density fitting method using three different auxiliary basis sets and comparing them to results obtained from ab initio calculations. These properties include dipoles for a series of molecules, as well as the molecular electrostatic potential and electric field for water. Subsequently, the intermolecular electrostatic energy is tested by calculating ten stationary points on the water dimer potential-energy surface. Results are presented for electron densities obtained at four different levels of theory using two different basis sets, fitted with three auxiliary basis sets. Additionally, a one-dimensional electrostatic energy surface scan is performed for four different systems (H2O dimer, Mg2+-H2O, Cu+-H2O, and n-methyl-formamide dimer). Our results show a very good agreement with ab initio calculations for all properties as well as interaction energies.

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.