Toward deformation densities for intramolecular interactions without radical reference states using the fragment,atom, localized,delocalized, and interatomic (FALDI) charge density decomposition scheme

A novel approach for calculating deformation densities is presented, which enables to calculate the deformation density resulting from a change between two chemical states, typically conformers, without the need for radical fragments. The Fragment, Atom, Localized, Delocalized, and Interatomic (FALDI) charge density decomposition scheme is introduced, which is applicable to static electron densities (FALDI‐ED), conformational deformation densities (FALDI‐DD) as well as orthodox fragment‐based deformation densities. The formation of an intramolecular NH⋅⋅⋅N interaction in protonated ethylene diamine is used as a case study where the FALDI‐based conformational deformation densities (with atomic or fragment resolution) are compared with an orthodox EDA‐based approach. Atomic and fragment deformation densities revealed in real‐space details that (i) pointed at the origin of density changes associated with the intramolecular H‐bond formation and (ii) fully support the IUPAC H‐bond representation. The FALDI scheme is equally applicable to intra‐ and intermolecular interactions. © 2017 Wiley Periodicals, Inc.     

Correlation effects in EOM-CCSD for the excited states: evaluated by AIM localization index (LI) and delocalization index (DI)

We have extended the evaluation and interpretation of QTAIM (quantum theory of atoms in molecules) localization and delocalization indices lambda (LI) and delta (DI) to electronic excited states by studying ground states (at HF and CCSD levels) and excited states (at CIS and EOM-CCSD) of H2C=CH2, HCCH, H2C=O, H2C=S, CO2, CS2, and SO2. These molecules undergo extensive geometrical changes upon the excitation to the valence adiabatic excited singlet state. The importance of Coulomb correlation effects was demonstrated by comparing the LIs and DIs at none-correlated levels (HF and CIS) and those at correlated levels (CCSD and EOM-CCSD). In interpreting the changes in the magnitudes of the LIs and DIs, we made use of simple molecular orbital and Walsh-diagram analyses. Coulomb correlation is important in determining the magnitude of the LIs and DIs and obtaining geometries that are close to experiment.     

Comparison of localization and delocalization indices obtained with Hartree-Fock and conventional correlated methods: effect of Coulomb correlation

Atomic populations and localization [lambda(A)] and delocalization [delta(A,B)] indices (LIs and DIs) are calculated for a large set of molecules at the Hartree-Fock (HF), MP2, MP4(SDQ), CISD, and QCISD levels with the 6-311++G(2d,2p) basis set. The HF method and the conventional correlation methods [MP2, MP4(SDQ), CISD, and QCISD] yield distinct sets of LIs and DIs. Yet, within the four conventional correlation methods the differences in atomic populations and LIs and DIs are small. Relative to HF, the conventional correlation methods [MP2, MP4(SDQ), CISD, QCISD] yield virtually the same LIs and DIs for molecules with large charge separations while LIs and DIs that differ significantly from the HF values--the LIs are increased and DIs decreased--are obtained for bonds with no or small charge separations. Such is the case in the archetypal homopolar molecules HC(triple bond)CH, H2C=CH2, CH3-CH3, and "protonated cyclopropane" C(3)H(7) (+), in which case the bonding may be atypical. Relative to HF, the typical effect of the conventional correlation methods is to decrease the DI between atoms.     

FALDI‐based decomposition of an atomic interaction line leads to 3D representation of the multicenter nature of interactions

Atomic interaction lines (AILs) and the QTAIM's molecular graphs provide a predominantly two‐center viewpoint of interatomic interactions. While such a bicentric interpretation is sufficient for most covalent bonds, it fails to adequately describe both formal multicenter bonds as well as many non‐covalent interactions with some multicenter character. We present an extension to our Fragment, Atomic, Localized, Delocalized and Interatomic (FALDI) electron density (ED) decomposition scheme, with which we can measure how any atom‐pair's delocalized density concentrates, depletes or reduces the electron density in the vicinity of a bond critical point. We apply our method on five classical bonds/interactions, ranging from formal either two‐ or three‐center bonds, a non‐covalent interaction (an intramolecular hydrogen bond) to organometallic bonds with partial multicenter character. By use of 3D representation of specific atom‐pairs contributions to the delocalized density we (i) fully recover previous notion of multicenter bonding in diborane and predominant bicentric character of a single covalent C C bond, (ii) reveal a multicenter character of an intramolecular H‐bond and (iii) illustrate, relative to a Schrock carbene, a larger degree of multicenter M C interaction in a Fischer carbene (due to a presence of a heteroatom), whilst revealing the holistic nature of AILs from multicenter ED decomposition. © 2018 Wiley Periodicals, Inc.     

On the applicability of QTAIM, Hirshfeld and Mulliken delocalisation indices as a measure of proton spin–spin coupling in aromatic compounds

A correlation between delocalization indices (DIs) and proton spin–spin coupling constants, previously observed for DIs calculated within the QTAIM scheme, is examined using Hirshfeld and Mulliken schemes. The original set of molecules has been extended with a number of five- and six-membered heteroaromatics containing nitrogen, oxygen, sulfur and selenium. QTAIM DIs are found to correlate well with the experimental values, even those containing the electronegative nitrogen and oxygen atoms. In contrast, an acceptable correlation with the Hirshfeld DIs is found only for polybenzenoids, whereas no correlation is found for the Mulliken DIs with any of the systems.     

Performance of the density matrix functional theory in the quantum theory of atoms in molecules

The generalization to arbitrary molecular geometries of the energetic partitioning provided by the atomic virial theorem of the quantum theory of atoms in molecules (QTAIM) leads to an exact and chemically intuitive energy partitioning scheme, the interacting quantum atoms (IQA) approach, that depends on the availability of second-order reduced density matrices (2-RDMs). This work explores the performance of this approach in particular and of the QTAIM in general with approximate 2-RDMs obtained from the density matrix functional theory (DMFT), which rests on the natural expansion (natural orbitals and their corresponding occupation numbers) of the first-order reduced density matrix (1-RDM). A number of these functionals have been implemented in the promolden code and used to perform QTAIM and IQA analyses on several representative molecules and model chemical reactions. Total energies, covalent intra- and interbasin exchange-correlation interactions, as well as localization and delocalization indices have been determined with these functionals from 1-RDMs obtained at different levels of theory. Results are compared to the values computed from the exact 2-RDMs, whenever possible.     

Electron localization and delocalization indices for solids

The electron localization and delocalization indices obtained by the integration of exchange‐correlation part of pair density over chemically meaningful regions of space, e.g., QTAIM atoms are valuable tools for the bonding analysis in molecular systems. However, among periodic systems only few simplest models were analyzed with this approach until now. This contribution reports implementation and evaluation of the localization and delocalization indices on the basis of solid state DFT calculations. A comparison with the results of simple analytical model of Ponec was made. In addition, a small set of compounds with ionic (NaCl), covalent (diamond, graphite), and metallic (Na, Cu) bonding interactions was characterized using this method. Typical features of different types of bonding were discussed using the delocalization indices. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Exploring fundamental differences between red- and blue-shifted intramolecular hydrogen bonds using FAMSEC,FALDI, IQA and QTAIM

We have discovered, using developed by us recently FALDI and FAMSEC computational techniques, fundamentally distinct mechanisms of intramolecular red- and blue-shifted H-bond formation that occurred in different conformers of the same molecule (amino-acid β-alanine) involving the same heteroatoms (O–H???N and N–H???O). Quantitative topological, geometric and energetic data of both H-bonds obtained with well-known QTAIM and IQA methodologies agree with what is known regarding H-bonding in general. However, the FALDI charge and decomposition scheme for calculating in real space 3D conformational deformation densities provided clear evidence that the process of electron density redistribution taking place on the formation of the stronger red-shifted H-bond is fundamentally distinct from the weaker blue-shifted H-bond. Contributions made by atoms of the X–H???Y–Z fragment (IUPAC notation) as well as distinct atoms on the H-bond formation were fully explored. The FAMSEC energy decomposition approach showed that the atoms involved in formation of the red-shifted H-bond interact in a fundamentally different fashion, both locally and with the remainder of the molecule, as compared with those of the blue-shifted H-bond. Excellent correlations of trends obtained with QTAIM, IQA, FAMSEC and FALDI techniques were obtained. Commentary regarding IUPAC recommended definition of an H-bond and validity of observed AILs (or bond paths) of the two H-bond kinds is also discussed.     

Joint QTAIM and Hirshfeld study of the sigma and pi charge distribution and electron delocalization in carbonyl compounds: a comparative study with the resonance model

Atomic charges and delocalization indexes (DIs) for a series of carbonyl compounds comprising dimethyl ketone, acetaldehyde, acetic acid, methyl acetate, acetamide, methyl vinyl ketone, divinyl ketone, and benzoic acid were studied using two different atomic partitionings: the QTAIM and the Hirshfeld (stockholder) scheme. The resonance model, traditionally employed to explain the reactivity of these compounds, is not in line with the total atomic charges and DIs calculated by both methodologies. However, the resonance model is supported to some extent by the pi charges and pi DIs calculated by both schemes, but the calculated values indicate that the pi population delocalizes only to a small degree. Although the absolute values of QTAIM and stockholder atomic charges are significantly different, the pi charges and the values of the DIs show similar trends for all the atoms and molecules of this study; this is especially the case for the pi DIs. A study of the electron density on the level of a single MO performed for CO, H2CO, F2CO, and H2CS reveals that the differences in the atomic sigma charges computed with both partitionings can be traced back to their different treatment of interatomic regions.     

Quantum chemistry aspects of the solvent effects on the ene reaction of 1‐Phenyl‐1,3,4‐triazolin‐2,5‐dione and 2‐methyl‐2‐butene

A density functional theory study was used to investigate the quantum aspects of the solvent effects on the kinetic and mechanism of the ene reaction of 1‐phenyl‐1,3,4‐triazolin‐2,5‐dione and 2‐methyl‐2‐butene. Using the B3LYP/6–311++ G(d,p) level of the theory, reaction rates have been calculated in the various solvents and good agreement with the experimental data has been obtained. Natural bond orbital analysis has been applied to calculate the stabilization energy of N18? H19 bond during the reaction. Topological analysis of quantum theory of atom in molecule (QTAIM) studies for the electron charge density in the bond critical point (BCP) of N18? H19 bond of the transition states (TSs) in different solvents shows a linear correlation with the interaction energy. It is also seen form the QTAIM analysis that increase in the electron density in the BCP of N18? H19, raises the corresponding vibrational frequency. Average calculated ratio of 0.37 for kinetic energy density to local potential energy density at the BCPs as functions of N18? H19 bond length in different media confirmed covalent nature of this bond. Using the concepts of the global electrophilicity index, chemical hardness and electronic chemical potentials, some correlations with the rate constants and interaction energy have been established. Mechanism and kinetic studies on 1‐phenyl‐1,3,4‐triazolin‐2,5‐dione and 2‐methyl‐2‐butene ene reaction suggests that the reaction rate will boost with interaction energy enhancement. Interaction energy of the TS depends on the solvent nature and is directly related to electron density of the bonds involved in the reaction proceeding, global electrophilicity index and electronic chemical potential. However, the chemical hardness relationship is reversed. Finally, an interesting and direct correlation between the imaginary vibrational frequency of the N18? H19 critical bond and its electron density at the TS has been obtained. © 2014 Wiley Periodicals, Inc.     

Electron-pairing analysis from localization and delocalization indices in the framework of the atoms-in-molecules theory

This article presents an overview of recent advances in the study of electron pairing through the use of localization and delocalization indices obtained from double integration over atomic basins of the exchange–correlation density in the framework of the atoms-in-molecules theory. These localization and delocalization indices describe the intra- and interatomic distribution of the electron pairs in a molecule. The main results of the application of these second-order indices to the analysis of molecular structure and chemical reactivity are briefly reviewed. It is shown that localization and delocalization indices represent a powerful tool to describe the electron-pair structure of molecules, which, in turn, provides deeper insight into relevant chemical phenomena such as electron correlation effects and the formation of localized α, β electron pairs. Received: 8 April 2002 / Accepted: 26 June 2002 / Published online: 6 September 2002 Acknowledgements. Financial help was furnished by the Spanish DGES projects no. PB98-0457-C02-01 and BQU2002-04112-C02-02. J.P. thanks the Departament d'Universitats, Recerca i Societat de la Informació de la Generalitat de Catalunya for benefiting from a doctoral fellowship, no. 2000FI-00582. M.S. is indebted to the Departament d'Universitats, Recerca i Societat de la Informació of the Generalitat de Catalunya for financial support through the Distinguished University Research Promotion, 2001. We also thank the Centre de Supercomputació de Catalunya for providing us with computing facilities. Correspondence to: M. Solà e-mail: miquel.sola@udg.es     

An efficient method for computing the QTAIM topology of a scalar field: The electron density case

An efficient method for computing the quantum theory of atoms in molecules (QTAIM) topology of the electron density (or other scalar field) is presented. A modified Newton–Raphson algorithm was implemented for finding the critical points (CP) of the electron density. Bond paths were constructed with the second‐order Runge–Kutta method. Vectorization of the present algorithm makes it to scale linearly with the system size. The parallel efficiency decreases with the number of processors (from 70% to 50%) with an average of 54%. The accuracy and performance of the method are demonstrated by computing the QTAIM topology of the electron density of a series of representative molecules. Our results show that our algorithm might allow to apply QTAIM analysis to large systems (carbon nanotubes, polymers, fullerenes) considered unreachable until now. © 2012 Wiley Periodicals, Inc.     

Relativistic (SR‐ZORA) quantum theory of atoms in molecules properties

The Quantum Theory of Atoms in Molecules (QTAIM) is used to elucidate the effects of relativity on chemical systems. To do this, molecules are studied using density‐functional theory at both the nonrelativistic level and using the scalar relativistic zeroth‐order regular approximation. Relativistic effects on the QTAIM properties and topology of the electron density can be significant for chemical systems with heavy atoms. It is important, therefore, to use the appropriate relativistic treatment of QTAIM (Anderson and Ayers, J. Phys. Chem. 2009, 115, 13001) when treating systems with heavy atoms. © 2016 Wiley Periodicals, Inc.     

Chemical bonding in view of electron charge density and kinetic energy density descriptors

Stalke's dilemma, stating that different chemical interpretations are obtained when one and the same density is interpreted either by means of natural bond orbital (NBO) and subsequent natural resonance theory (NRT) application or by the quantum theory of atoms in molecules (QTAIM), is reinvestigated. It is shown that within the framework of QTAIM, the question as to whether for a given molecule two atoms are bonded or not is only meaningful in the context of a well‐defined reference geometry. The localized‐orbital‐locator (LOL) is applied to map out patterns in covalent bonding interaction, and produces results that are consistent for a variety of reference geometries. Furthermore, LOL interpretations are in accord with NBO/NRT, and assist in an interpretation in terms of covalent bonding. © 2008 Wiley Periodicals, Inc.J Comput Chem, 2009.     

Actinium coordination chemistry: A density functional theory study with monodentate and bidentate ligands

In the current study, the coordination chemistry of nine-coordinate Ac(III) complexes with 35 monodentate and bidentate ligands was investigated using density functional theory (DFT) in terms of their geometries, charges, reaction energies, and bonding interactions. The energy decomposition analysis with naturals orbitals for chemical valence (EDA-NOCV) and the quantum theory of atoms in molecules (QTAIM) were employed as analysis methods. Trivalent Ac exhibits the highest affinities toward hard acids (such as charged oxophilic donors, fluoride), so its classification as a hard acid is justified. Natural population analysis quantified the involvement of 5f orbitals on Ac to be about 30% of total valence electron natural configuration indicating that Ac is a member of the actinide series. Pearson correlation coefficients were used to study the pairwise correlations among the bond lengths, ΔG reaction energies, charges on Ac and donor atoms, and data from EDA-NOCV and QTAIM. Strong correlations and anticorrelations were found between Voronoi charges on donor atoms with ΔG, EDA-NOCV interaction energies and QTAIM bond critical point densities.     

Comparative Analysis of Electron‐Density and Electron‐Localization Function for Dinuclear Manganese Complexes with Bridging Boron‐ and Carbon‐Centered Ligands

Bonding in borylene‐, carbene‐, and vinylidene‐bridged dinuclear manganese complexes [MnCp(CO)₂]₂X (X=B‐tBu, B=NMe₂, CH₂, C?CH₂) has been compared by analyses based on quantum theory of atoms in molecules (QTAIM), on the electron‐localization function (ELF), and by natural‐population analyses. All of the density functional theory based analyses agree on the absence of a significant direct Mn? Mn bond in these complexes and confirm a dominance of delocalized bonding via the bridging ligand. Interestingly, however, the topology of both charge density and ELF related to the Mn‐bridge‐Mn bonding depend qualitatively on the chosen density functional (except for the methylene‐bridged complex, which exhibits only one three‐center‐bonding attractor both in ??²ρ and in ELF). While gradient‐corrected functionals provide a picture with localized two‐center X? Mn bonding, increasing exact‐exchange admixture in hybrid functionals concentrates charge below the bridging atom and suggests a three‐center bonding situation. For example, the bridging boron ligands may be described either as substituted boranes (e.g., at BLYP or BP86 levels) or as true bridging borylenes (e.g., at BHLYP level). This dependence on the theoretical level appears to derive from a bifurcation between two different bonding situations and is discussed in terms of charge transfer between X and Mn, and in the context of self‐interaction errors exhibited by popular functionals.     

Semilocalized approach to investigation of chemical reactivity

Application of the power series for the one‐electron density matrix 36 to the case of two interacting molecules is shown to yield a semilocalized approach to investigate chemical reactivity, which is characterized by the following distinctive features: (1) Electron density (ED) redistributions embracing orbitals of the reaction centers of both molecules and of their neighboring fragments are studied instead of the total intermolecular interaction energy; (2) the ED redistributions are expressed directly in the basis of fragmental orbitals (FOs) without passing to the basis of delocalized molecular orbitals (MOs) of initial molecules; (3) terms describing the ED redistributions due to an intermolecular contact arise as additive corrections to the purely monomolecular terms and thereby may be analyzed independently; (4) local ED redistributions only between orbitals of the reaction centers of both molecules are described by lower‐order terms of the power series, whereas those embracing both the reaction centers and their neighborhoods are represented by higher‐order terms. As opposed to the standard perturbative methods based on invoking the delocalized (canonical) MOs of isolated molecules, the results of the approach suggested are in‐line with the well‐known intuition‐based concepts of the classic chemistry concerning reactivity, namely, with the assumption about different roles of the reaction center and of its neighborhood in a chemical process, with the expectation about extinction of the indirect influence of a certain fragment (substituent) when its distance from the reaction center grows, etc. Such a parallelism yields quantum chemical analogs for the classic concepts and thereby gives an additional insight into their nature. The scope of validity of these concepts also is discussed. Applicability of the approach suggested to specific chemical problems is illustrated by a brief consideration of the S_N2 and Ad_E2 reactions. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 302–316, 2003     

The nature of structure and bonding between transition metal and mixed Si‐Ge tetramers: A 20‐electron superatom system

A novel superatom species with 20‐electron system, Si_xGe_yM⁺ (x + y = 4; M = Nb, Ta), was properly proposed. The trigonal bipyramid structures for the studied systems were identified as the putative global minimum by means of the density functional theory calculations. The high chemical stability can be explained by the strong p‐d hybridization between transition metal and mixed Si‐Ge tetramers, and closed‐shell valence electron configuration [1S²1P⁶2S²1D¹⁰]. Meanwhile, the chemical bondings between metal atom and the tetramers can be recognized by three localized two‐center two‐electron (2c‐2e) and delocalized 3c‐2e σ‐bonds. For all the doped structures studied here, it was found that the π‐ and σ‐electrons satisfy the 2(N + 1)² counting rule, and thus these clusters possess spherically double (π and σ) aromaticity, which is also confirmed by the negative nucleus‐independent chemical shifts values. Consequently, all the calculated results provide a further understanding for structural stabilities and electronic properties of transition metal‐doped semiconductor clusters. © 2016 Wiley Periodicals, Inc.