Bond fukui indices: Comparison of frozen molecular orbital and finite differences through mulliken populations

Bond Fukui functions and matrices are introduced for ab initio levels of theory using a Mulliken atoms in molecules model. It is shown how these indices may be obtained from first‐order density matrix derivatives without need for going to second‐order density matrices as in a previous work. The importance of taking into account the nonorthogonality of the basis in ab initio calculations is shown, contrasting the present results with previous work based on Hückel theory. It is shown how the extension of Fukui functions to Fukui matrices allows getting more insight into the nature of bond Fukui functions. All presently introduced indices respect the necessary normalization conditions and include the classical single atom condensed Fukui functions. © 2013 Wiley Periodicals, Inc.     

Atomic charges,dipole moments,and Fukui functions using the Hirshfeld partitioning of the electron density

In the Hirshfeld partitioning of the electron density, the molecular electron density is decomposed in atomic contributions, proportional to the weight of the isolated atom density in the promolecule density, constructed by superimposing the isolated atom electron densities placed on the positions the atoms have in the molecule. A maximal conservation of the information of the isolated atoms in the atoms-in-molecules is thereby secured. Atomic charges, atomic dipole moments, and Fukui functions resulting from the Hirshfeld partitioning of the electron density are computed for a large series of molecules. In a representative set of organic and hypervalent molecules, they are compared with other commonly used population analysis methods. The expected bond polarities are recovered, but the charges are much smaller compared to other methods. Condensed Fukui functions for a large number of molecules, undergoing an electrophilic or a nucleophilic attack, are computed and compared with the HOMO and LUMO densities, integrated over the Hirshfeld atoms in molecules.     

Critical thoughts on computing atom condensed Fukui functions

Different procedures to obtain atom condensed Fukui functions are described. It is shown how the resulting values may differ depending on the exact approach to atom condensed Fukui functions. The condensed Fukui function can be computed using either the fragment of molecular response approach or the response of molecular fragment approach. The two approaches are nonequivalent; only the latter approach corresponds in general with a population difference expression. The Mulliken approach does not depend on the approach taken but has some computational drawbacks. The different resulting expressions are tested for a wide set of molecules. In practice one must make seemingly arbitrary choices about how to compute condensed Fukui functions, which suggests questioning the role of these indicators in conceptual density-functional theory.     

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.     

Chemical reactivity and selectivity using Fukui functions: basis set and population scheme dependence in the framework of B3LYP theory

The use of Fukui functions for the site selectivity of the formaldehyde molecule for nucleophilic, electrophilic and radical attacks has been made with special emphasis to the dependence of Fukui values on the basis sets as well as population schemes in the framework of B3LYP theory. Out of the five population schemes selected viz., Mulliken population analysis, natural population analysis, CHELP, CHELPG and atoms in molecules (AIM), it is found that the CHELPG and AIM schemes predict precise reactive site with less dependency on the basis sets. Charges derived from Hirshfeld partitioning, calculated using the BLYP/dnd method (implemented in the DMOL³ package), provide non-negative Fukui values for all the molecular systems considered in this study. Supporting results have been obtained for acetaldehyde and acetone molecules at the 6-31+G** basis set level. These results support the fact that high Fukui values correspond to soft–soft interaction sites. On the other hand, the correlation of the low Fukui value to the hard–hard interaction site merits further investigation. Received: 10 November 2001 / Accepted: 6 March 2002 / Published online: 13 June 2002     

Hirshfeld‐based intrinsic polarizability density representations as a tool to analyze molecular polarizability

In this work, a general scheme to visualize polarizability density distributions is proposed and implemented in a Hirshfeld‐based partitioning scheme. This allows us to obtain easy‐to‐interpret pictorial representations of both total and intrinsic polarizabilities where each point of the density is formed by the contribution of any atom or group of atoms in the molecule. In addition, the procedure used here permits the possibility of removing the size dependence of the electric‐dipole polarizability. Such a development opens new horizons in exploring new applications for the analysis of the molecular polarizability tensor. For instance, this visualization shows which atoms or regions are more polarizable distinguishing, moreover, the fine structure of atoms affected by the vicinity, and might extend the dipole polarizability as a tool for aromaticity studies in polycyclic aromatic hydrocarbons. Additionally, this approach can serve us to assess the methods performance in describing the interaction of electric fields with a molecule and local electron correlation effects in intrinsic polarizabilities. © 2015 Wiley Periodicals, Inc.     

Theoretical studies on geometry,solvent effect,and photochromic mechanism of two bis‐heterocyclic compounds containing pyrazolone ring

Two bis‐heterocyclic compounds containing pyrazolone ring, 1‐phenyl‐3‐methyl‐4‐(6‐hydro‐4‐amino‐5‐sulfo‐2,3‐pyrazine)‐pyrazole‐5‐one and 1‐phenyl‐3‐methyl‐4‐(6‐hydro‐4‐methylamino‐5‐sulfo‐2,3‐pyrazine)‐pyrazole‐5‐one, are investigated to gain a deeper insight into their geometries and photochromic mechanism by applying density functional theory. The solvent effects are simulated using the polarizable continuum model of the self‐consistent reaction field theory. Bader's atom‐in‐molecule theory is used to investigate the nature of hydrogen bonds. The data of energy, dipole moments, and the condensed Fukui functions have been calculated to assess the stability and reactivity of the title compounds. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Voronoi deformation density (VDD) charges: Assessment of the Mulliken, Bader, Hirshfeld, Weinhold, and VDD methods for charge analysis

We present the Voronoi Deformation Density (VDD) method for computing atomic charges. The VDD method does not explicitly use the basis functions but calculates the amount of electronic density that flows to or from a certain atom due to bond formation by spatial integration of the deformation density over the atomic Voronoi cell. We compare our method to the well-known Mulliken, Hirshfeld, Bader, and Weinhold [Natural Population Analysis (NPA)] charges for a variety of biological, organic, and inorganic molecules. The Mulliken charges are (again) shown to be useless due to heavy basis set dependency, and the Bader charges (and often also the NPA charges) are not realistic, yielding too extreme values that suggest much ionic character even in the case of covalent bonds. The Hirshfeld and VDD charges, which prove to be numerically very similar, are to be recommended because they yield chemically meaningful charges. We stress the need to use spatial integration over an atomic domain to get rid of basis set dependency, and the need to integrate the deformation density in order to obtain a realistic picture of the charge rearrangement upon bonding. An asset of the VDD charges is the transparency of the approach owing to the simple geometric partitioning of space. The deformation density based charges prove to conform to chemical experience.     

Comparison among four different ways to condense the Fukui function

Four different ways to condense the Fukui function are compared. Three of them perform a numerical integration over different basins to define the condensed Fukui function, and the other one is the most traditional Fukui function using Mulliken population analysis. The basins are chosen to be the basins of the electron density (AIM), the basins of the electron localization function (ELF), and the basins of the Fukui function itself. The use of the last two basins is new and presented for the first time here. It is found that the last three methods yield results which are stable against a change in the basis set. The condensed Fukui function using the basins of the ELF is not able to give information on the reactivity of an acceptor molecule. In general, the condensed Fukui function using the basins of the density or the basins of the Fukui function describe the reactivity trends well. The latter is preferred, because it only contains information about the Fukui function itself and it gives the right information for donor as well as acceptor centers.     

Information distance analysis of molecular electron densities

The information‐theoretic basis of the Hirshfeld partitioning of the molecular electronic density into the densities of the “stockholder” atoms‐in‐molecules (AIM) is summarized. It is argued that these AIM densities minimize both the directed divergence (Kullback–Leibler) and divergence (Kullback) measures of the entropy deficiency between the AIM and their free atom analogs of the promolecule. The local equalization of the information distance densities of the Hirshfeld components, at the local value of the corresponding global entropy deficiency density, is outlined and several approximate relations are established between the alternative local measures of the missing information and the familiar function of a difference between the molecular and promolecule densities. Various global (of the system as a whole) and atomic measures of the entropy deficiency or the displacements relative to the isoelectronic promolecule, defined for densities or probabilities (shape functions) in both the local resolution and the Hirshfeld AIM discretization, are introduced and tested. This analysis is performed also for the valence electron (frozen‐core) approximation. Illustrative results for representative linear molecules, including diatomics, triatomics, and tetraatomics, are reported. They are interpreted as complementary characteristics of changes in the net AIM charge distribution and of the displacements in the information content of the electron distributions of bonded atoms. These numerical results confirm the overall similarity of the stockholder AIM to their free atom analogs and reflect the information displacements due to the AIM polarization and charge transfer in molecules. They also demonstrate the semiquantitative nature of the approximate relations established between the entropy deficiency densities and the related functions involving the density difference function. This development extends the range of interpretations based on the density difference diagrams into probing the associated information displacements in a molecule accompanying the formation of the chemical bonds. © 2002 John Wiley & Sons, Inc. Int J Quantum Chem, 2002     

Negative and Infinite Fukui Functions: The Role of Diagonal Dominance in the Hardness Matrix

The possible genesis of negative atom condensed Fukui functions is discussed based on hardness kernel matrix relationships. The recent hypothesis that diagonal dominance of the hardness matrix is a requirement for positive Fukui functions is proven, and general considerations also predict the possibility of regions with numerically unstable Fukui functions, including discontinuities.     

2‐(Benzoylsulfanyl)acetic acid and 2,5‐dioxopyrrolidin‐1‐yl 2‐(benzoylsulfanyl)acetate by powder X‐ray diffraction studies

The structures of the title compounds, C₉H₈O₃S, (I), and C₁₃H₁₁NO₅S, (II), were determined by X‐ray powder diffraction. Both were solved using the direct‐space parallel tempering algorithm and refined using the Rietveld method. In (I), the C—S—C bond angle is slightly smaller than normal, indicating more p character in the bonding orbitals of the S atom. The carboxylic acid group joins across an inversion centre to form a dimer. The crystal packing includes a weak C—H...O hydrogen bond between an aromatic C—H group and a carboxylic acid O atom to form a two‐dimensional network parallel to (10). The C—S—C bond angle in (II) is larger than its counterpart in (I), indicating that the S atom of (II) has less p character in its bonding orbitals than that of (I), according to Bent's rule. The crystal structure of (II) includes weak C—H...O hydrogen bonds between the H atoms of the methylene groups and carbonyl O atoms, forming a three‐dimensional network.     

Revisiting the calculation of condensed Fukui functions using the quantum theory of atoms in molecules

The analysis of previously reported shortcomings of the condensed Fukui functions obtained making use of the quantum theory of atoms in molecules indicates these drawbacks are due to the inadequacy of the definition employed to compute them and not to the partitioning. A new procedure, which respects the mathematical definition and solves these problems, is presented for the calculation of condensed Fukui functions for atomic basins defined according to the quantum theory of atoms in molecules. It is tested in a set of 18 molecules, which includes the most controversial reported cases.     

5‐Amino‐2,4,6‐tribromoisophthalic acid: the MAD triangle for experimental phasing

The title compound, C₈H₄Br₃NO₄, shows an extensive hydrogen‐bond network. In the crystal structure, molecules are linked into chains by COO—H...O bonds, and pairs of chains are connected by additional COO—H...O bonds. This chain bundle shows stacking interactions and weak N—H...O hydrogen bonds with adjacent chain bundles. The three Br atoms present in the molecule form an equilateral triangle. This can be easily identified in the heavy‐atom substructure when this compound is used as a heavy‐atom derivative for experimental phasing of macromolecules. The title compound crystallizes as a nonmerohedral twin.     

Glycyl‐L‐alanine: a multi‐temperature neutron study

Neutron diffraction data have been collected at 12, 50, 150 and 295 K for the dipeptide glycyl‐L‐alanine, C₅H₁₀N₂O₃, in order to obtain accurate positional and anisotropic displacement parameters for the H atoms. The values of these parameters serve as a benchmark for assessing the equivalent parameters obtained from a so‐called Hirshfeld‐atom refinement of X‐ray diffraction data described elsewhere [Capelli et al. (2014). IUCrJ, 1 , 361–379]. The flexibility of the glycyl‐L‐alanine molecule in the solid and the hydrogen‐bonding interactions as a function of temperature are also considered.     

Halogen‐ and Hydrogen‐Bonded Salts and Co‐crystals Formed from 4‐Halo‐2,3,5,6‐tetrafluorophenol and Cyclic Secondary and Tertiary Amines: Orthogonal and Non‐orthogonal Halogen and Hydrogen Bonding,and Synthetic Analogues of Halogen‐Bonded Biological Systems

Co‐crystallisation of, in particular, 4‐iodotetrafluorophenol with a series of secondary and tertiary cyclic amines results in deprotonation of the phenol and formation of the corresponding ammonium phenate. Careful examination of the X‐ray single‐crystal structures shows that the phenate anion develops a C?O double bond and that the C?C bond lengths in the ring suggest a Meissenheimer‐like delocalisation. This delocalisation is supported by the geometry of the phenate anion optimised at the MP2(Full) level of theory within the aug‐cc‐pVDZ basis (aug‐cc‐pVDZ‐PP on I) and by natural bond orbital (NBO) analyses. With sp² hybridisation at the phenate oxygen atom, there is strong preference for the formation of two non‐covalent interactions with the oxygen sp² lone pairs and, in the case of secondary amines, this occurs through hydrogen bonding to the ammonium hydrogen atoms. However, where tertiary amines are concerned, there are insufficient hydrogen atoms available and so an electrophilic iodine atom from a neighbouring 4‐iodotetrafluorophenate group forms an I???O halogen bond to give the second interaction. However, in some co‐crystals with secondary amines, it is also found that in addition to the two hydrogen bonds forming with the phenate oxygen sp² lone pairs, there is an additional intermolecular I???O halogen bond in which the electrophilic iodine atom interacts with the C?O π‐system. All attempts to reproduce this behaviour with 4‐bromotetrafluorophenol were unsuccessful. These structural motifs are significant as they reproduce extremely well, in low‐molar‐mass synthetic systems, motifs found by Ho and co‐workers when examining halogen‐bonding interactions in biological systems. The analogy is cemented through the structures of co‐crystals of 1,4‐diiodotetrafluorobenzene with acetamide and with N‐methylbenzamide, which, as designed models, demonstrate the orthogonality of hydrogen and halogen bonding proposed in Ho’s biological study.     

Removing electrons can increase the electron density: a computational study of negative Fukui functions

Ab initio and density-functional theory calculations for a family of substituted acetylenes show that removing electrons from these molecules causes the electron density along the C-C bond to increase. This result contradicts the predictions of simple frontier molecular orbital theory, but it is easily explained using the nucleophilic Fukui function-provided that one is willing to allow for the Fukui function to be negative. Negative Fukui functions emerge as key indicators of redox-induced electron rearrangements, where oxidation of an entire molecule (acetylene) leads to reduction of a specific region of the molecule (along the bond axis, between the carbon atoms). Remarkably, further oxidization of these substituted acetylenes (one can remove as many as four electrons!) causes the electron density along the C-C bond to increase even more. This work provides substantial evidence that the molecular Fukui function is sometimes negative and reveals that this is due to orbital relaxation.     

密度泛函理论研究聚烯烃催化剂取代基电子效应与催化活性的关系

 采用密度泛函 BP/DNP 优化了五组 (每组化合物具有相同的框架结构和不同电子效应的取代基) 共 18 个最受关注的烯烃配位聚合催化剂, 分别计算了每个化合物中心金属上的 Hirshfeld、Mulliken 和 QEq 电荷, 中心金属的 Fukui 指数以及化合物的 HOMO 和 LUMO 能量值, 然后将这些结构参数和配合物催化乙烯聚合的活性相关联. 结果发现, 中心金属的 QEq 电荷能正确反映取代基的电子效应, 且与化合物的 HOMO 以及 LUMO 能量值与催化活性之间有良好的相关性, 可用来预测催化剂活性; 而 Hirshfeld 和 Mulliken 电荷不能正确反映取代基的电子效应, 不适合计算这些化合物的中心金属电荷. 中心金属的 Fukui 指数受取代基电子效应影响较小, 和催化剂活性之间的相关性不明显.     

Intramolecular Hydrogen Bonds in Low‐Molecular‐Weight Polyethylene Glycol

We used static DFT calculations to analyze, in detail, the intramolecular hydrogen bonds formed in low‐molecular‐weight polyethylene glycol (PEG) with two to five repeat subunits. Both red‐shifted O?H???O and blue‐shifting C?H???O hydrogen bonds, which control the structural flexibility of PEG, were detected. To estimate the strength of these hydrogen bonds, the quantum theory of atoms in molecules was used. Car–Parrinello molecular dynamics simulations were used to mimic the structural rearrangements and hydrogen‐bond breaking/formation in the PEG molecule at 300 K. The time evolution of the H???O bond length and valence angles of the formed hydrogen bonds were fully analyzed. The characteristic hydrogen‐bonding patterns of low‐molecular‐weight PEG were described with an estimation of their lifetime. The theoretical results obtained, in particular the presence of weak C?H???O hydrogen bonds, could serve as an explanation of the PEG structural stability in the experimental investigation.     

Halogen bonds on demand: I...S contacts in cocrystals of trans‐bis(thiocyanato‐κN)tetrakis(4‐vinylpyridine‐κN)nickel(II) and 2,3,5,6‐tetrafluoro‐1,4‐diiodobenzene

Hydrogen bonds are considered a powerful organizing force in designing supramolecular architectures because they are directional, selective and reversible at room temperature. trans‐Dithiocyanatotetrakis(4‐vinylpyridine)nickel(II) is a popular host for the inclusion of small molecules and 2,3,5,6‐tetrafluoro‐1,4‐diiodobenzene (TFDIB) represents a strong halogen‐bond donor. These constituents cocrystallize in a 1:1 stoichiometry, [Ni(NCS)₂(C₇H₇N)₄]·C₆F₄I₂, in the tetragonal space group I4₁/a. Both residues occupy special positions, i.e. the pseudo‐octahedral Ni^II complex is located on a twofold axis and the TFDIB molecule sits about a crystallographic centre of inversion. The components interact via a short S...I contact of 3.2891 (12) Å between the thiocyanate S atom of the host and the iodine substituent at the perhalogenated aromatic ring of the smaller guest molecule. This interaction meets the commonly accepted criteria for a halogen bond. Such halogen bonds to sulfur are significantly less common than to smaller electronegative atoms.