Local Kinetic Energy Density Research Articles

Study the Chemical Bonding of Heterometallic Trinuclear Cluster Containing Cobalt and Ruthenium: [(Cp*Co) (CpRu)2 (μ3-H) (μ-H)3] using QTAIM Approach

The topological parameters of the metal-metal and metal-ligand bonding interactions in a trinuclear tetrahydrido cluster [(Cp*Co) (CpRu)2 (μ3-H) (μ-H)3]1 (Cp* = η5 -C5Me4Et), (Cp = η5 -C5Me5), was explored by using the Quantum Theory of Atoms-in-Molecules (QTAIM). The properties of bond critical points such as the bond delocalization indices δ (A, B), the electron density ρ(r), the local kinetic energy density G(r), the Laplacian of the electron density ∇2ρ(r), the local energy density H(r), the local potential energy density V(r) and ellipticity ε(r) are compared with data from earlier organometallic system studies. A comparison of the topological processes of different atom-atom interactions has become possible thanks to these results. In the core of the heterometallic tetrahydrido cluster, the Ru2CoH4 part, the calculations show no existence of any bond critical points (BCP) or identical bond paths (BPs) between Ru-Ru and Ru-Co. Electron densities are determined by the position of bridging hydride atoms coordinated to Ru-Ru and Ru-Co, which significantly affects the bonds between these transition metal atoms. On the other hand, the results confirm that the cluster under study contains a 7c–11e bonding interaction delocalized over M3H4, as shown by the non-negligible delocalization index calculations. The small values for electron density ρ(b) above zero, together with the small values, again above zero, for Laplacian ∇2ρ(b) and the small positive values for total energy density H(b), are shown by the Ru-H and Co-H bonds in this cluster is typical for open-shell interactions. Also, the topological data for the bond interactions between Co and Ru metal atoms with the C atoms of the cyclopentadienyl Cp ring ligands are similar. They show properties very identical to open-shell interactions in the QTAIM classification.

A Theoretical Investigation of Chemical Bonding of a Heterometallic Trinuclear Cluster Containing Iridium and Ruthenium: [(Cp*Ir) (CpRu)2 (μ3-H) (μ-H)3] by QTAIM Approach

Numerous integral and local electron density’s topological parameters of significant metal-metal and metal-ligand bonding interactions in a trinuclear tetrahydrido cluster [(Cp* Ir) (Cp Ru)2 (μ3-H) (μ-H)3]1 (Cp = η5 -C5Me5), (Cp* = η5 -C5Me4Et) were calculated and interpreted by using the quantum theory of atoms in molecules (QTAIM). The properties of bond critical points such as the delocalization indices δ (A, B), the electron density ρ(r), the local kinetic energy density G(r), the Laplacian of the electron density ∇2ρ(r), the local energy density H(r), the local potential energy density V(r) and ellipticity ε(r) are compared with data from earlier organometallic system studies. A comparison of the topological processes of different atom-atom interactions has become possible thanks to these results. In the core of the heterometallic tetrahydrido cluster, the Ru2IrH4 part, the calculations showed that there are no bond critical points (BCPs) or identical bond paths (BPs) between Ru-Ru and Ru-Ir. The distribution of electron densities is determined by the position of bridging hydride atoms coordinated to Ru-Ru and Ru-Ir, which significantly affects the bonds between these transition metal atoms. On the other hand, the results confirm that the cluster under study contains a 7c–11e bonding interaction delocalized over M3H4, as shown by the non-negligible delocalization index calculations. The small values for ρ(b) above zero, together with the small values, again above zero, for Laplacian ∇2ρ(b) and the small positive values for total energy density H(b), are shown by the Ru-H and Ir-H bonds in this cluster is typical for open-shell interactions. Also, the topological data for the bond interactions between Ir and Ru metal atoms with the C atoms of the cyclopentadienyl Cp ring ligands are similar. They show properties very identical to open-shell interactions in the QTAIM classification.

Accurate Kohn-Sham auxiliary system from the ground-state density of solids

The Kohn-Sham (KS) system is an auxiliary system whose effective potential is unknown in most cases. It is in principle determined by the ground-state density and it has been found numerically for some low-dimensional systems by inverting the KS equations starting from a given accurate density. For solids, only approximate results are available. In this work, we determine accurate exchange-correlation (xc) potentials for Si and NaCl using the ground-state densities obtained from auxiliary field quantum Monte Carlo calculations. We show that these xc potentials can be rationalized as an ensemble of a few local functions of the density, whose form depends on the specific environment and can be well characterized by the gradient of the density and the local kinetic energy density. The KS band structure can be obtained with high accuracy. The true KS band gap turns out to be larger than the prediction of the local density approximation, but significantly smaller than the measurable photoemission gap, which confirms previous estimates. Finally, our findings show that the conjecture that very different xc potentials can lead to very similar densities and other KS observables is true also in solids, which questions the meaning of details of the potentials and, at the same time, confirms the stability of the KS system.

Laplacian-level meta-generalized gradient approximation for solid and liquid metals

Modern meta-GGAs based on the local kinetic energy density can predict properties of diverse systems with near experimental accuracy, but unexpectedly describe properties of metallic solids poorly due to their underestimation of screening in metals. In this work, the authors replace the kinetic energy dependence of a sophisticated meta-GGA with an approximation based on the electronic density gradient and Laplacian. This Laplacian-level meta-GGA is tested on a diverse set of solid-state properties: geometries, cohesive energies, bulk moduli, ferromagnetic moments, monovacancy formation energies, and formation enthalpies. Most deficiencies of the parent meta-GGA in describing metals are remedied with its Laplacian-level variant.

The Distance between Minima of Electron Density and Electrostatic Potential as a Measure of Halogen Bond Strength.

In this study, we present results of a detailed topological analysis of electron density (ED) of 145 halogen-bonded complexes formed by various fluorine-, chlorine-, bromine-, and iodine-containing compounds with trimethylphosphine oxide, Me3PO. To characterize the halogen bond (XB) strength, we used the complexation enthalpy, the interatomic distance between oxygen and halogen, as well as the typical set of electron density properties at the bond critical points calculated at B3LYP/jorge-ATZP level of theory. We show for the first time that it is possible to predict the XB strength based on the distance between the minima of ED and molecular electrostatic potential (ESP) along the XB path. The gap between ED and ESP minima exponentially depends on local electronic kinetic energy density at the bond critical point and tends to be a common limiting value for the strongest halogen bond.

Meta-Local Density Functionals: A New Rung on Jacob's Ladder.

The homogeneous electron gas (HEG) is a key ingredient in the construction of most exchange-correlation functionals of density-functional theory. Often, the energy of the HEG is parameterized as a function of its spin density nσ, leading to the local density approximation (LDA) for inhomogeneous systems. However, the connection between the electron density and kinetic energy density of the HEG can be used to generalize the LDA by evaluating it on a geometric average nσavg(r) = nσ1–x(r)ñσx(r) of the local spin density nσ(r) and the spin density ñσ(r) of a HEG that has the local kinetic energy density τσ(r) of the inhomogeneous system. This leads to a new family of functionals that we term meta-local density approximations (meta-LDAs), which are still exact for the HEG, which are derived only from properties of the HEG and which form a new rung of Jacob’s ladder of density functionals [AIP Conf. Proc.2001, 577, 1]. The first functional of this ladder, the local τ approximation (LTA) of Ernzerhof and Scuseria [J. Chem. Phys.1999, 111, 911] that corresponds to x = 1 is unfortunately not stable enough to be used in self-consistent field calculations because it leads to divergent potentials, as we show in this work. However, a geometric averaging of the LDA and LTA densities with smaller values of x not only leads to numerical stability of the resulting functional but also yields more accurate exchange energies in atomic calculations than the LDA, the LTA, or the tLDA functional (x = 1/4) of Eich and Hellgren [J. Chem. Phys.2014, 141, 22410725494732]. We choose x = 0.50, as it gives the best total energy in self-consistent exchange-only calculations for the argon atom. Atomization energy benchmarks confirm that the choice x = 0.50 also yields improved energetics in combination with correlation functionals in molecules, almost eliminating the well-known overbinding of the LDA and reducing its error by two thirds.

Halogen and Chalcogen Bond Energies Evaluated Using Electron Density Properties.

Halogen (X-bond) and chalcogen bond (Ch-bond) energies for 36 complexes have been obtained at the RI-MP2/def2-TZVP level of theory, involving the heavier halogen and chalcogen atoms (Br, I, Se, Te). We have explored the existence of linear relationships between the interaction energies and the local kinetic energy densities at the bond critical points that characterize the σ-hole interactions (both electronic G(r) and potential V(r) energy densities). Interestingly, we have found strong relationships for halogen and chalcogen bonding energies, especially for the V(r) energy density, thus allowing to estimate the interaction energy without computing the separate monomers. This is also useful to estimate the interaction in monomeric systems (intramolecular X/Ch-bonds), as illustrated using several examples. Remarkably, we have also found a good relationship when in the same representation both halogen and chalcogen atoms are included, thus allowing to use the same empirical correlation for both interactions.

Correlations of NHN hydrogen bond energy with geometry and 1H NMR chemical shift difference of NH protons for aniline complexes.

In this computational work, we propose to use the NMR chemical shift difference of NH2 protons for 1:1 complexes formed by aniline and nitrogen-containing proton acceptors for the estimation of the hydrogen bond energy and geometry (N⋯H and N⋯N distances). The proposed correlations could be applied to other aromatic amines as well, in a gas phase, a solution, or a solid state, for both inter- and intramolecular hydrogen bonds. We considered a set of 21 complexes with the NHN hydrogen bond without proton transfer, including hydrogen bonds from weak to medium strong ones (2-21 kcal/mol), with neutral or anionic bases and with sp3 and sp2 hybridized nitrogen proton acceptors. For each complex apart from direct hydrogen bond energy calculation, we have tested several other ways to estimate the energy: (a) using a correlation between NH stretching band intensity and hydrogen bond energy and (b) using correlations between electron density properties at (3, -1) bond critical point (quantum theory of atoms in molecules analysis) and hydrogen bond energy. Besides for the studied type of complexes, we obtained refined linear correlations linking the local electron kinetic (G) and potential (V) energy densities with the hydrogen bond energy.

Effect of high-rate dynamic comminution on penetration of projectiles of various velocities and impact angles into concrete

The dynamic ‘overstress’, i.e., the apparent increase of strength of concrete at very high strain rates (10– $$10^{6}$$ /s) experienced in projectile impact and penetration, has recently been explained by a new theory with partial analogy to turbulence. The increase is attributed to comminution of concrete driven by the release of local kinetic energy of the shear strain rate field of forming fragments which, in projectile impact problems, exceeds the strain energy of the fragments by orders of magnitude. This theory gives the particle size distribution and the additional local kinetic energy density, $$\Delta K$$ , proportional to the deviatoric strain rate square. To match test results, $$\Delta K$$ must be dissipated during finite element simulations of impact. In previous simulations, $$\Delta K$$ was, at first, dissipated by an artificial equivalent viscosity (not empirical but predicted by the theory). Later it was found that dissipation by upscaling material tensile strength is equally effective. This theoretically justified upscaling is adopted here since it is more realistic when microplane constitutive model M7 for fracturing damage in concrete is used. All artificial damping is eliminated from the computer program. While previous simulations with the comminution theory were limited to orthogonal impacts, and only the cases of penetration of slabs of various thickness by projectile of one velocity and penetration depths for different velocities, the present study also analyzes further test data on oblique impacts at various impact angles up to $$35^\circ $$ , and on the exit velocities and penetration depths of projectiles of different velocities. For each test series on one and the same concrete, the material parameters are calibrated on one test and then, using the same parameters, all the other tests are predicted. All the predictions of exit velocities and penetration depths of projectiles, as well as entry and exit craters, are quite accurate.

The QTAIM Approach to Chemical Bonding in Triruthenium Carbonyl Cluster: [Ru3 (μ-H)(μ3-κ2-Haminox-N,N)(CO)9

The bonding in the hydride triruthenium carbonyl cluster [Ru3 (μ-H)(μ3-κ2-Haminox-N,N)(CO)9 is investigated by using the Quantum Theory of Atoms-in-Molecules (QTAIM). Calculated metal–metal and metal–ligand bond critical points (bcp) properties, electron density ρ(b), Laplacian ∇2ρ(b), local energy density H(b), local kinetic (Lagrangian) energy density G(b), local potential energy density V(b) ellipticity ε(b), and bond delocalization indices δ(A, B) are consistent with the relevant transition metal clusters in the literature. The topological data recognizes bond paths (bp) and bond critical points for metal-metal interactions in the core of the hydride triruthenium cluster [Ru(1)-Ru(3) and Ru(2)-Ru(3)]. However, no direct bond path is found for the interaction between the hydride bridged Ru atoms, while a non-negligible delocalization index δ(Ru(1)…Ru(2)) is obtained for this non-bonding interaction. An interaction of 5-center 6-electron type is proposed for the core. The topological parameters of Ru-N oxazoline ring bond suggest a pure σ-bond.

The Relationship of the Energy of Interaction of Alkali Metal Cations with an Aprotic Solvent Molecule with Quantum Topological Electron Density Characteristics

The optimal geometry and wave functions of the complexes [M(Solv)]+ (M = Li, Na, K; Solv is an aprotic solvent molecule) were calculated and the topological characteristics of the electron density distribution at the (3,–1) critical points (CP) of ion–molecule bonds were analyzed by the density functional theory in the B3LYP/6-31+G(d, p) approximation. The parametric dependences for the energy of ion–molecule bonds in terms of the local kinetic and potential electron energy densities at the bond CTs were proposed.

Revised M06-L functional for improved accuracy on chemical reaction barrier heights, noncovalent interactions, and solid-state physics

We present the revM06-L functional, which we designed by optimizing against a larger database than had been used for Minnesota 2006 local functional (M06-L) and by using smoothness restraints. The optimization strategy reduced the number of parameters from 34 to 31 because we removed some large terms that increased the required size of the quadrature grid and the number of self-consistent-field iterations. The mean unsigned error (MUE) of revM06-L on 422 chemical energies is 3.07 kcal/mol, which is improved from 3.57 kcal/mol calculated by M06-L. The MUE of revM06-L for the chemical reaction barrier height database (BH76) is 1.98 kcal/mol, which is improved by more than a factor of 2 with respect to the M06-L functional. The revM06-L functional gives the best result among local functionals tested for the noncovalent interaction database (NC51), with an MUE of only 0.36 kcal/mol, and the MUE of revM06-L for the solid-state lattice constant database (LC17) is half that for M06-L. The revM06-L functional also yields smoother potential curves, and it predicts more-accurate results than M06-L for seven out of eight diversified test sets not used for parameterization. We conclude that the revM06-L functional is well suited for a broad range of applications in chemistry and condensed-matter physics.

Synthesis, structural characterization, crystal structure and theoretical study of a Pd(II)-salen complex with propylene linkage

A Pd(II)-salen complex derived from salen ligand is reported. The reported complex is investigated by microanalyses (C, H, N), ESI-MS spectrometry, FT-IR, 1H and 13C NMR and UV/Vis spectroscopic studies. In addition, crystal structure measurement study has also been carried out in order to confirm the structure of Pd(II)-salen complex.In order to explore the insights into the structural bonding of the studied complex, computational measurements has been carried out. Combined topology and NBO studies were made to explore the nature of PdO and PdN bonding in the complex. The natural charges showed that the transfers of the negative charge from the ligand to palladium atom is at 1.4157–1.4312 e. Atom in a molecule (AIM) analysis showed the electron density (ρ(r) > 0.1) and its Laplacian (∇2ρ(r) > 0). These topological parameters showed that covalent bonding interactions are dominant in PdN and PdO bonds. However, PdN bonds have more covalent characters than PdO bonds, which is further confirmed by the ratio of local electron potential energy density to the local electron kinetic energy density (|V(r)|/G(r)) found to be higher for PdN bonds (1.1683–1.1993) as compared to PdO bonds (1.0689–1.0926). AIM and NBO reveal that shorter PdN and PdO bonds have higher interaction energies (Eint) and hence higher bond covalence.

How strong are H-bonds at the fully hydroxylated silica surfaces? Insights from the B3LYP electron density topological analysis

The calculation through the supermolecular approach of the hydrogen bond strength EHB between silanol groups at the surface of an ample class of silica-based materials is hindered by the intrinsic difficulty to define the “H-bond free” reference system. We propose, for the first time, to evaluate EHB by adopting the literature empirical correlation relating the Bader local electronic kinetic energy density Gb computed at the H⋅⋅⋅O bond critical point with EHB. Remarkably, EHB for the hydroxylated surfaces of quartz polymorphs correlates with surface formation energy, showing that the surface EHB is responsible of the surface stability. A number of correlations between hydrogen bond features are established, with that between EHB and the enhanced infrared intensity associated to surface hydrogen bond formation, obeying the literature formula semi-quantitatively. The present results are quite general and can be extended to other inorganic surfaces where hydrogen bonds between surface sites are the dominant features.

Physical meaning of the QTAIM topological parameters in hydrogen bonding.

This work examined the local topological parameters of charge density at the hydrogen bond (H-bond) critical points of a set of substituted formamide cyclic dimers and enolic tautomers. The analysis was performed not only on the total electron density of the hydrogen bonded complexes but also on the intermediate electron density differences derived from the Morokuma energy decomposition scheme. Through the connection between these intermediate electron density differences and the corresponding differences in topological parameters, the meaning of topological parameters variation due to hydrogen bonding (H-bonding) becomes evident. Thus, for example, we show in a plausible way that the potential energy density differences at the H-bond critical point properly describe the electrostatics of H-bonding, and local kinetic energy density differences account for the localization/delocalization degree of the electrons at that point. The results also support the idea that the total electronic energy density differences at the H-bond critical point describe the strength of the interaction rather than its covalent character as is commonly considered.

Complexes of XeHXe⁺ with simple ligands: a theoretical investigation on (XeHXe⁺)L (L = N₂, CO, H₂O, NH₃).

The structure, stability, and harmonic frequencies of the (XeHXe(+))L complexes (L = N2, CO, H2O, NH3) were investigated by ab initio and density functional theory (DFT) calculations. Their bonding situation was also assayed by natural bond orbital (NBO), atoms-in-molecules (AIM), and energy decomposition (EDA) analyses. For any L, we located a linear and a T-shaped isomer, whose energy difference progressively increases in the order N2 < CO < H2O < NH3 and ranges from nearly 0 to 4.5 kcal mol(-1). The absolute complexation energies of both the linear and the T-shaped isomers also increase in the same order, and their EDA analysis revealed the prevailing contribution of electrostatic interactions. The noncovalent character of the bonding between XeHXe(+) and L was confirmed by the AIM analysis. In particular, we based our investigation on the joint use of numerical AIM indices and graphic examination of the local Hamiltonian kinetic energy density, K(r). Interestingly, this function visually identifies the "covalent" regions occupied by XeHXe(+) and L and the "noncovalent" zones existing between them, which include, in particular, the bond critical point located on the Xe-L bond paths. Only for the linear (XeHXe(+))NH3 did the AIM analysis suggest an onset of covalency in the xenon-nitrogen interaction. Further work is in progress to examine the effectiveness of K(r), and its plotted forms, as a function of the bonding situation of noble-gas compounds.

Cl···Cl Interactions in Molecular Crystals: Insights from the Theoretical Charge Density Analysis

The structure, IR harmonic frequencies and intensities of normal vibrations of 20 molecular crystals with the X-Cl···Cl-X contacts of different types, where X = C, Cl, and F and the Cl···Cl distance varying from ~3.0 to ~4.0 Å, are computed using the solid-state DFT method. The obtained crystalline wave functions have been further used to define and describe quantitatively the Cl···Cl interactions via the electron-density features at the Cl···Cl bond critical points. We found that the electron-density at the bond critical point is almost independent of the particular type of the contact or hybridization of the ipso carbon atom. The energy of Cl···Cl interactions, E(int), is evaluated from the linking E(int) and local electronic kinetic energy density at the Cl···Cl bond critical points. E(int) varies from 2 to 12 kJ/mol. The applicability of the geometrical criterion for the detection of the Cl···Cl interactions in crystals with two or more intermolecular Cl···Cl contacts for the unique chlorine atom is not straightforward. The detection of these interactions in such crystals may be done by the quantum-topological analysis of the periodic electron density.

Charge Density Analysis of Ferulic Acid: Robustness of a Trifurcated C–H···O Hydrogen Bond

The toplogical features of a sporadic trifurcated C–H···O interaction region, where an oxygen atom acts as an acceptor of three weak hydrogen bonds, has been investigated by experimental and theoretical charge density analysis of ferulic acid. The interaction energy of the asymmetric molecular dimer formed by the trifurcated C–H···O motif, based on the multipolar model, is shown to be greater than the corresponding asymmetric O–H···O dimer in this crystal structure. Further, the hydrogen bond energies associated with these interaction motifs have been estimated from the local kinetic and potential energy densities at the bond critical points. The trends suggest that the interaction energy of the trifurcated C–H···O region is comparable to that of a single O–H···O hydrogen bond.

Advances in Understanding of Chemical Bonding: Inputs from Experimental and Theoretical Charge Density Analysis

The development of charge density analysis has undergone a major renaissance in the last two decades. In recent years, the characterization of bonding features associated with atoms in molecules and in crystals has been explored using high-resolution X-ray diffraction data (laboratory or synchrotron) complemented by high level ab initio theoretical calculations. The extraction of one electron topological properties, namely, electrostatic charges, dipole moment and higher moments, electrostatic potential, electric field gradients, in addition to evaluation of the local kinetic and potential energy densities, have contributed toward an understanding of the electron density distributions in molecular solids. New topological descriptors, namely, the source function (SF) and electron localization function (ELF) provide additional information as regards characterization of the topology of the electron density. In addition, delocalization indices have also been developed to account for bonding features pertinent to M-M bonds. The evaluation of these properties have contributed significantly toward the understanding of intra- and intermolecular bonding features in organic, inorganic, and biomolecules in the crystalline phase, with concomitant applications in the understanding of chemical reactivity and material/biological properties. In recent years, the focus has strongly shifted toward the understanding of structure-property relationships in organometallic complexes containing labile M-C bonds in the crystal structure with subsequent implications in catalysis. This perspective aims to highlight the major developments in electron density measurements in the past few years and provides pointers directed toward the potential use of this technique in future applications for an improved understanding of chemical bonding in systems that have been unexplored.

Violation of the Spatial Quantum Inequality and the Quantum Interest Conjecture: A New Approach

It is well known that the local kinetic energy density can be negative in quantum field theory. This is a “quantum effect” that is in contrast to classical physics where the kinetic energy density is always non-negative. It has been proposed in the literature that there exist “quantum inequalities” which place limits on this negative energy density. In this paper we will examine a type of quantum inequality call the spatial quantum inequality. The spatial quantum inequality is a lower bound on the weighted average of the kinetic energy density. It will be shown that for a massless scalar field in 1-1 dimensional space-time we can formulate a quantum state which violates the quantum inequality. In addition, we will look into the quantum interest conjecture. The quantum interest conjecture states that a negative energy pulse must always be closely associated with a positive energy pulse of greater magnitude. It will be shown that the quantum interest conjecture is violated.