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In this proof‐of‐concept paper, we show how exchange‐correlation effects can be simply recovered for interatomic energies within the interacting quantum atoms decomposition when local, gradient generalized, or meta‐gradient generalized approximations are used in density functional theory (DFT) calculations. We also demonstrate how inhomogeneity and non‐local effects can be introduced even from a pure local scheme, without resorting to any orbital information. Finally, we provide numerical evidence on a database of selected energetic molecules that this decomposition scheme can be efficiently used to build accurate models for the prediction of molecular energies from an initial "cheap" DFT calculation. [ABSTRACT FROM AUTHOR]

Previously, numerical simulations of laser ablation of materials occurring under the action of ultrashort laser pulses in semiconfined samples and samples of finite thickness were carried out. Its thermal mechanism was described in terms of a one-dimensional unsteady heat equation in a coordinate system associated with a moving evaporation front. The action of the laser was taken into account through the source functions in the thermal conductivity equation, specifying the coordinate and time dependences of the laser source. In this work, similar simulations were carried out for semiconfined samples within the framework of a two-temperature thermal spike model, which consisted of two interrelated thermal conductivity equations for the electron gas and the crystal lattice. For the convenience of numerical simulation, in the equations of the thermal spike model, a transition was made to the coordinate system associated with the moving evaporation front of the material. Using numerical simulation, temperature profiles of the electron gas and crystal lattice at different times were obtained, and the dynamics of the temperatures of the electron gas and crystal lattice on the surface of the sample were calculated within the thermal spike model, taking into account the evaporation of the crystal lattice and the emission of electron gas from the surface of the sample. A comparative analysis of the numerical results obtained within both models was carried out. [ABSTRACT FROM AUTHOR]

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QTAIM and source function analysis were used to explore the non-covalent bonding in twelve different water clusters (H2O)n obtained by considering n = 2–7 and various geometrical arrangements. A total of seventy-seven O−H⋯O hydrogen bonds (HBs) were identified in the systems under consideration, and the examination of the electron density at the bond critical point (BCP) of these HBs revealed the existence of a great diversity of O−H⋯O interactions. Furthermore, the analysis of quantities, such as | V (r) | / G (r) and H (r) , allowed a further description of the nature of analogous O−H⋯O interactions within each cluster. In the case of 2-D cyclic clusters, the HBs are nearly equivalent between them. However, significant differences among the O−H⋯O interactions were observed in 3-D clusters. The assessment of the source function (SF) confirmed these findings. Finally, the ability of SF to decompose the electron density (ρ) into atomic contributions allowed the evaluation of the localized or delocalized character of these contributions to ρ at the BCP associated to the different HBs, revealing that weak O−H⋯O interactions have a significant spread of the atomic contributions, whereas strong interactions have more localized atomic contributions. These observations suggest that the nature of the O−H⋯O hydrogen bond in water clusters is determined by the inductive effects originated by the different spatial arrangements of the water molecules in the studied clusters. [ABSTRACT FROM AUTHOR]

In order to determine the effect of reservoir stress sensitivity on the inflow performance of multistage fractured horizontal wells during the development of low permeability reservoirs, a semi-analytical method was used to derive the point source function of infinite reservoirs considering stress sensitivity. Through perturbation transformation, Laplace transformation, superposition principle and other methods, the mathematical model of unstable seepage flow in multistage fractured horizontal wells during depletion development of low permeability reservoirs was established. Finally, the calculation accuracy of the model was compared with KAPPA well testing software. The results show that with the increase of production time, the stronger the stress sensitivity of the reservoir, the more upwarped the well test curve appears in the late flow period, and the upwarping time is advanced. Moreover, the sensitivity analysis of fracture geometric parameters and reservoir boundary size was carried out. [ABSTRACT FROM AUTHOR]

The C–X bond cleavage in different methyl halides (CH3X; X = Cl, Br, I) mediated by 5,6-dimethylbenzimidazole-bis(dimethylglyoximate)cobalt(II) (CoIICbx) was theoretically investigated in the present work. An SN2-like mechanism was considered to simulate the chemical process where the cobalt atom acts as the nucleophile and the halogen as the leaving group. The reaction path was computed by means of the intrinsic reaction coordinate method and analyzed in detail through the reaction force formalism, the quantum theory of atoms in molecules (QTAIM), and the calculation of one-electron density derived quantities, such as the source function (SF) and the spin density. A thorough comparison of the results with those obtained in the same reaction occurring in presence of 5,6-dimethylbenzimidazole-bis(dimethylglyoximate)cobalt(I) (CoICbx) was conducted to reveal the main differences between the two cases. The reactions mediated by CoIICbx were observed to be endothermic and possess higher activation energies in contrast to the reactions where the CoICbx complex is present. The latter was supported by the reaction force results, which suggest a relationship between the activation energy and the ionization potentials of the different nucleophiles present in the cleavage reaction. Moreover, the SF results indicates that the lower axial ligand (i.e., 5,6-dimethylbenzimidazole) exclusively participates on the first stage of the reaction mediated by the CoIICbx complex, while for the CoICbx case, it appears to have an important role along the whole process. Finally, the QTAIM charge analysis indicates that oxidation of the cobalt atom occurs in both cases; at the same time, it suggests the formation of an uncommon two-center one-electron bond in the CoIICbx case. The latter was confirmed by means of electron localization calculations, which resulted in a larger electron count at the Co–C interatomic region for the CoICbx case upon comparison with its CoIICbx counterpart. [ABSTRACT FROM AUTHOR]

Often with abundant of natural fracture, Carbonate reservoirs are characterized with the stress sensitive and dual media. Mostly, its flow model solved by numerical method. In this paper, the semi-analytical solution for this problem is presented: firstly, the point source function considering stress sensitivity in infinite dual medium reservoir is obtained in Laplace space by Perturbation Transformation and Laplace Transformation; Secondly, the Laplace space solution of multi-stage fracturing horizontal well in infinite plate reservoir is obtained by Image Principle and Superposition Principle; Finally, the spacial solution of multi-stage fracturing horizontal well is obtained by Stehfest numerical inversion and perturbation inverse transformation. The calculation results show that the flow regime of multi-stage fracturing horizontal well can be divided into six stages: Ⅰ- linear flow, Ⅱ-first radial flow, Ⅲ-double radial flow, Ⅳ-radial flow of natural fracture system, Ⅴ-channeling flow regime and Ⅵ-radial flow of the whole system. The impact of stress sensitivity of formation permeability on linear flow is lower, and mainly affecting the last five flow regimes, and the dimensionless pressure drop derivative curve tends to rise in the later stage of development, showing the characteristics of closed boundary. In this paper, when the reservoir stress sensitivity is not considered, the calculation results will produce a large error, and the wrong well test interpretation will be obtained.

Population balance equations are essential to scientists and engineers of widely varying discipline, applications have covered a wide range of processes involving bioreactors, catalytic reaction, and cell populations, but the lack of exact solutions necessitates the use of numerical methods. Group classification of Smoluchowski equation with a source is provided by use of the method of preliminary group classification. The determining equation is successfully solved by the method of developed Lie group analysis. All source functions, reduced equations, invariant solutions, explicit exact solutions and dynamic behavior analysis of solutions are presented in this paper. • Population balance equations for aggregation and breakage processes. • Explicit exact solutions of population balance equations with a source term. • Dynamic behavior analysis of solutions. • Preliminary group classification. [ABSTRACT FROM AUTHOR]

Abstract This article is devoted to the study of the source function for the Caputo–Fabrizio time fractional diffusion equation. This new definition of the fractional derivative has no singularity. In other words, the new derivative has a smooth kernel. Here, we investigate the existence of the source term. Through an example, we show that this problem is ill-posed (in the sense of Hadamard), and the fractional Landweber method and the modified quasi-boundary value method are used to deal with this inverse problem and the regularized solution is also obtained. The convergence estimates are addressed for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. In addition, we give a numerical example to illustrate the proposed method.

A QTAIM and IQA investigation on model compounds of two isostructural AgI and CuI coordination polymers (CPs) based on cyclic triimidazole (L), i.e. the [MIL]n 1D double‐stranded stair chain and the [MClL]n 3D network (M = Cu, Ag), has allowed light to be shed on the different emissive behaviour associated with the two metal ions. According to a previously reported investigation [Malpicci et al. (2021). Inorg. Chem. Front.8, 1312–1323], AgI CPs showed both fluorescence and multiple ligand‐centred room‐temperature phosphorescences, whereas CuI CPs displayed non‐thermally equilibrated halogen and metal‐to‐ligand charge transfer and two ligand‐centred phosphorescences, the latter observed only by their selective activation. Analysis of both local and integral QTAIM descriptors, including delocalization indices and source function, of the Ag—N and Cu—N bonds reveals a higher covalent and local character for the latter, explaining the greater metal–ligand electronic communication observed for the Cu compounds. Moreover, IQA investigation shows that the Cu—N bond is characterized by higher interaction energy, due to both higher electrostatic and exchange‐correlation contributions. Analysis on the M—X (M = Ag, Cu; X = I, Cl) bonds, also present in these structures, highlights a much higher covalent and local character with respect to the M—N bonds. [ABSTRACT FROM AUTHOR]

The C–X bond cleavage in different methyl halides (CH3X; X = Cl, Br, I) mediated by 5,6-dimethylbenzimidazole-bis(dimethylglyoximate)cobalt(II) (CoIICbx) was theoretically investigated in the present work. An SN2-like mechanism was considered to simulate the chemical process where the cobalt atom acts as the nucleophile and the halogen as the leaving group. The reaction path was computed by means of the intrinsic reaction coordinate method and analyzed in detail through the reaction force formalism, the quantum theory of atoms in molecules (QTAIM), and the calculation of one-electron density derived quantities, such as the source function (SF) and the spin density. A thorough comparison of the results with those obtained in the same reaction occurring in presence of 5,6-dimethylbenzimidazole-bis(dimethylglyoximate)cobalt(I) (CoICbx) was conducted to reveal the main differences between the two cases. The reactions mediated by CoIICbx were observed to be endothermic and possess higher activation energies in contrast to the reactions where the CoICbx complex is present. The latter was supported by the reaction force results, which suggest a relationship between the activation energy and the ionization potentials of the different nucleophiles present in the cleavage reaction. Moreover, the SF results indicates that the lower axial ligand (i.e., 5,6-dimethylbenzimidazole) exclusively participates on the first stage of the reaction mediated by the CoIICbx complex, while for the CoICbx case, it appears to have an important role along the whole process. Finally, the QTAIM charge analysis indicates that oxidation of the cobalt atom occurs in both cases; at the same time, it suggests the formation of an uncommon two-center one-electron bond in the CoIICbx case. The latter was confirmed by means of electron localization calculations, which resulted in a larger electron count at the Co–C interatomic region for the CoICbx case upon comparison with its CoIICbx counterpart.

We applied a set of advanced bonding descriptors to establish the hidden electron density features and binding energy characteristics of intermolecular DH∙∙∙A hydrogen bonds (OH∙∙∙O, NH∙∙∙O and SH∙∙∙O) in 150 isolated and solvated molecular complexes. The exchange‐correlation and Pauli potentials as well as corresponding local one‐electron forces allowed us to explicitly ascertain how electron exchange defines the bonding picture in the proximity of the H‐bond critical point. The electron density features of DH∙∙∙A interaction are governed by alterations in the electron localization in the H‐bond region displaying itself in the exchange hole. At that, they do not depend on the variations in the exchange hole mobility. The electrostatic interaction mainly defines the energy of H‐bonds of different types, whereas the strengthening/weakening of H‐bonds in complexes with varying substituents depends on the barrier height of the exchange potential near the bond critical point. Energy variations between H‐bonds in isolated and solvated systems are also caused the electron exchange peculiarities as follows from the corresponding potential and the interacting quantum atom analyses complemented by electron delocalization index calculations. Our approach is based on the bonding descriptors associated with the characteristics of the observable electron density and can be recommended for in‐depth studies of non‐covalent bonding. [ABSTRACT FROM AUTHOR]

We analytically obtained the frequency dependences of the absorption coefficient profile, taking into account the combined influence of the macroscopic motion of matter, Doppler and Voigt mechanisms of spectral line broadening. The effect of rotation of a spherical plasma on the formation of resonant fluorescence by barium ions upon absorption of laser radiation at a wavelength of 455.4 nm was studied. The deformation of the frequency shape of the laser radiation passing through the plasma and the emission spectrum is due to a shift in the absorption coefficient profile and its broadening caused by the rotational motion of the plasma. An increase in the resonant photons escape from the medium stipulated by the broadening of the emission spectrum of the resonance line is predicted. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of the source function for the Caputo–Fabrizio time fractional diffusion equation. This new definition of the fractional derivative has no singularity. In other words, the new derivative has a smooth kernel. Here, we investigate the existence of the source term. Through an example, we show that this problem is ill-posed (in the sense of Hadamard), and the fractional Landweber method and the modified quasi-boundary value method are used to deal with this inverse problem and the regularized solution is also obtained. The convergence estimates are addressed for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. In addition, we give a numerical example to illustrate the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we report on the results of experimental and theoretical (DFT calculation of an isolated molecule) investigation of electron density in a triphenylantimony disorbate complex (triphenylantimony bis(hexa-2,4-dienoate), Ph3Sb(O2CCH=CH–CH=CH–CH3)2 (1)). A topological analysis of the electron density was carried out in the framework of the quantum theory of atoms in molecules (QTAIM), which allows to study the nature of chemical bonds and the molecular graph in Ph3Sb(O2CCH=CH–CH=CH–CH3)2 complex. The molecular graph is an important tool for determining the interacting atoms. However, the molecular graph of the triphenylantimony disorbate complex did not show the presence of the "expected" intramolecular interactions between the antimony atom and the carbonyl oxygen one. Such a situation can be caused by electron density low curvature between these atoms. It is extremely difficult case, and sometimes it is not possible to find all the "expected" bond paths and critical points (3,−1). Thus, the molecular graph for this class of compounds does not provide a definitive picture of the chemical bonding and should be complemented with other descriptors, such as а source function (SF) and non-covalent interaction (NCI) index. It was found that in some cases using the SF on the NCI isosurface allows to interpret intramolecular interactions in the absence of a bond critical point more correctly. In this article, presence of intramolecular interaction in the absence of a bond critical point between the antimony atom and the carboxylate oxygen one was shown. The carboxylate fragment always acts as a source of the electron density for the Sb...O(carbonyl) interactions, whereas the antimony atom can be both a source and a sink for it. [ABSTRACT FROM AUTHOR]

In the article, we examine well-posedness questions in the Sobolev spaces of the inverse source problem in the case of a quasilinear parabolic system of the second order. The main part of the operator is linear. The overdetermination conditions are values of a solution at some collection of interior points. It is demonstrated that, in the case of at most linear growth of the nonlinearity, there exists a unique global (in time) solution and the problem is well-posed in the Sobolev classes. The conditions on the data are minimal and the results are sharp. [ABSTRACT FROM AUTHOR]

In oil and gas operation, horizontal well technology has become an important technique because of its numerous advantages. Despite these advantages, determination of pressure distribution expression is a problem because it involve 3D transient-flow that need to be solved. In this work, ten (10) models for pressure distribution for horizontal well under different boundary variation were derived following these steps for each of the models:(i) choosing a boundary condition for each axis (ii) selecting the appropriate source function for each axis and (iii)applying Newman product rule to arrive at the pressure expression. In conclusion, the following were observed.(a) along the xaxis, seven of the models have infinite-slab sources in infinite-slab sources(b) five of the models derived, along the yaxis they have infinite-plane sources in infinite-slab sources while for the remaining five along the y- axis they have infinite sources(c) along z-axis, six of the models have infinite-plane sources in infinite-slab sources while four of the remaining one have infinite sources(d) Three types of boundaries are observed:(i) Completely bounded boundaries (ii) Mixed boundaries (iii) infinite-acting boundaries(e)Three instantaneous source functions are involved in the pressure distribution expression. NOMENCLATURE ye= distance to the external boundary in y-direction hD= dimensionless height xe= distance to the external boundary in x-direction xD= arbitrary dimensionless distance along the x axis ze= distance to the external boundary in z-direction yD = arbitrary dimensionless distance along the y axis zw = Well coordinate in z-direction zD = arbitrary dimensionless distance along the z axis yw = Well coordinate in y-direction τ= dummy integration dimensionless time variable xw = Well coordinate in x-direction ɳ= diffusivity constant1 t = Time Keywords: Source function, Horizontal well, Reservoir, Boundary

In this paper we present the results of a high–resolution single crystal X–ray diffraction experiment of η6–[1,4–dihydrospiro(2H–3,1–benzoxazine–2,1′–cyclohexane)]tricarbonylchromium(0) and a subsequent charge density study based on a topological analysis according to quantum theory of atoms in molecules. The key point of this investigation is a comparison of the experimental topological characteristics of the electron density in the chromium atom coordination sphere with the experimentally–theoretical ones obtained on the basis of "whole–molecule" aspherical scattering factors (molecular invariom). We have shown that molecular invarioms adequately reproduce the deformation electron density in the coordination sphere of the chromium atom, molecular graphs, noncovalent interaction (NCI) index isorsurfaces as well as the population of d–orbitals of the Cr atom. In addition, we found that a comparative analysis of atomic charges and volumes obtained experimentally and experimentally–theoretically allows "to find" a reasonable value of the atomic charge. [ABSTRACT FROM AUTHOR]

In oil and gas operation, horizontal well technology has become an important technique because of its numerous advantages. Despite these advantages, determination of pressure distribution expression is a problem because it involve 3D transient-flow that need to be solved. In this work, ten (10) models for pressure distribution for horizontal well under different boundary variation were derived following these steps for each of the models: (i) choosing a boundary condition for each axis (ii) selecting the appropriate source function for each axis and (iii) applying Newman product rule to arrive at the pressure expression. In conclusion, the following were observed. (a) along the x-axis, seven of the models have infinite-slab sources in infinite-slab sources (b) five of the models derived, along the y-axis they have infinite-plane sources in infinite-slab sources while for the remaining five along the y-axis they have infinite sources (c) along z-axis, six of the models have infinite-plane sources in infinite-slab sources while four of the remaining one have infinite sources (d) Three types of boundaries are observed: (i) Completely bounded boundaries (ii) Mixed boundaries (iii) infinite-acting boundaries(e) Three instantaneous source functions are involved in the pressure distribution expression. [ABSTRACT FROM AUTHOR]

The hydrogen bond (H‐bond) is among the most important noncovalent interaction (NCI) for bioorganic compounds. However, no "energy border" has yet been identified to distinguish it from van der Waals (vdW) interaction. Thus, classifying NCIs and interpreting their physical and chemical importance remain open to great subjectivity. In this work, the "energy border" between vdW and H‐bonding interactions was identified using a dimer of water, as well as for a series of classical and nonclassical H‐bonding systems. Through means of the quantum theory of atoms in molecules and in particular the source function, it was possible to clearly identify the transition from H‐bonding to vdW bonding via analysis of the electronic structure. This "energy border" was identified both on elongating the interatomic interaction and by varying the contact angle. Hence, this study also redefines the "critic angle" previously proposed by Galvão et al. (J. Phys. Chem. A 2013, 117, 12668). Consequently, such "energy border" through an analysis of atomic basins volume variation was possible to identify the end of long‐range interactions. © 2019 Wiley Periodicals, Inc. The topological tool source function calculated at the bond critical point of CS dimer of water subjected to variation of angle with the fulcra in the hydrogen (H) between H‐donor (D) and the H‐acceptor (A) oxygen atoms θ(D − H⋯A) and a variation of A⋯D distance (RA⋯D) allowed to define the border between hydrogen bond (H‐bond) and van der Waals. [ABSTRACT FROM AUTHOR]

A chalcogen bond (ChB) is a σ-hole-based noncovalent interaction between a Lewis base and an electrophilic element of Group VI (O, S, Se, Te), which behaves as a Lewis acid. Recently, we demonstrated that halogen bond, the more familiar σ-hole-based interaction, is able to promote the enantioseparation of chiral compounds in HPLC environment. On this basis, an investigation to detect ChBs, functioning as stereoselective secondary interactions for HPLC enantioseparations, was started off and the results of this study are described herein. Our investigation also focused on the impact of the perfluorinated aromatic ring as a π-hole donor recognition site. For these purposes, seven atropisomeric fluorinated 3-arylthio-4,4’-bipyridines were designed, synthesized and used as potential ChB donors (ChBDs) with two cellulose-based chiral stationary phases (CSPs) containing carbonyl groups as ChB acceptors (ChBAs). In addition, one and two analogues lacking fluorine and sulphur, respectively, were prepared as terms of comparison. The design of the test analytes was computationally guided. In this regard, electrostatic potentials (EPs) associated with σ- and π-holes were computed and the atomic contributions to the sulphur EP maxima were derived using a molecular space partitioning in terms of Bader’s atomic basins. This procedure is akin to the Bader-Gatti electron density source function (SF) decomposition, yet suitably extended to the EP field. For five 3-substituted-4,4’-bipyridines, thermodynamic parameters were derived from van’t Hoff plots. Finally, the use of molecular dynamic (MD) simulation to model ChB in cellulose-analyte complexes was explored. Evidences that σ-hole and π-hole interactions can jointly drive HPLC enantioseparations through recognition sites generated by electronic charge depletion emerged from both experimental results and theoretical data. [ABSTRACT FROM AUTHOR]

The source function (SF) decomposes the electron density at any point into contributions from all other points in the molecule, complex, or crystal. The SF “illuminates” those regions in a molecule that most contribute to the electron density at a point of reference. When this point of reference is the bond critical point (BCP), a commonly used surrogate of chemical bonding, then the SF analysis at an atomic resolution within the framework of Bader's Quantum Theory of Atoms in Molecules returns the contribution of each atom in the system to the electron density at that BCP. The SF is used to locate the important regions that control the hydrogen bonds in both Watson–Crick (WC) DNA dimers (adenine:thymine (AT) and guanine:cytosine (GC)) which are studied in their neutral and their singly ionized (radical cationic and anionic) ground states. The atomic contributions to the electron density at the BCPs of the hydrogen bonds in the two dimers are found to be delocalized to various extents. Surprisingly, gaining or loosing an electron has similar net effects on some hydrogen bonds concealing subtle compensations traced to atomic sources contributions. Coarser levels of resolutions (groups, rings, and/or monomers‐in‐dimers) reveal that distant groups and rings often have non‐negligible effects especially on the weaker hydrogen bonds such as the third weak CH⋅⋅⋅O hydrogen bond in AT. Interestingly, neither the purine nor the pyrimidine in the neutral or ionized forms dominate any given hydrogen bond despite that the former has more atoms that can act as source or sink for the density at its BCP. © 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

The source function (SF) decomposes the electron density at any point into contributions from all other points in the molecule, complex, or crystal. The SF “illuminates” those regions in a molecule that most contribute to the electron density at a point of reference. When this point of reference is the bond critical point (BCP), a commonly used surrogate of chemical bonding, then the SF analysis at an atomic resolution within the framework of Bader's Quantum Theory of Atoms in Molecules returns the contribution of each atom in the system to the electron density at that BCP. The SF is used to locate the important regions that control the hydrogen bonds in both Watson–Crick (WC) DNA dimers (adenine:thymine (AT) and guanine:cytosine (GC)) which are studied in their neutral and their singly ionized (radical cationic and anionic) ground states. The atomic contributions to the electron density at the BCPs of the hydrogen bonds in the two dimers are found to be delocalized to various extents. Surprisingly, gaining or loosing an electron has similar net effects on some hydrogen bonds concealing subtle compensations traced to atomic sources contributions. Coarser levels of resolutions (groups, rings, and/or monomers‐in‐dimers) reveal that distant groups and rings often have non‐negligible effects especially on the weaker hydrogen bonds such as the third weak CH⋅⋅⋅O hydrogen bond in AT. Interestingly, neither the purine nor the pyrimidine in the neutral or ionized forms dominate any given hydrogen bond despite that the former has more atoms that can act as source or sink for the density at its BCP. © 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

Magnetic properties of open‐shell systems depend on their unpaired electron density distribution. Accurate spin density (SD) is difficult to retrieve, both from polarized neutron diffraction (PND) data and from quantum approaches, and its interpretation is not trivial. The Source Function is a useful tool to interpret SD distributions and their accuracy. It is here applied to analyze and compare the theoretical SD in a weakly ferromagnetically coupled end‐end azido dicopper complex with that in a strongly‐coupled end‐on complex. The Source Function enables to highlight the origin of the SD differences between the two dicopper complexes and among adopted computational approaches (CASSCF, DFT, UHF). Further insight is provided by partial Source Function SD reconstructions using given subsets of atoms. DFT methods exaggerate electron sharing between copper and the ligands, causing spin delocalization toward them and overestimating metal‐ligand spin polarization, while underestimating CASSCF spin information transmission between atoms. CAS(10,10) SD is closer to the PND SD than other adopted methods © 2018 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

Abstract: The extent of cyclic π‐electron delocalization in the N‐heterocyclic ring of an imidazol‐2‐ylidene (i.e., 1,3,4,5‐tetramethylimidazol‐2‐ylidene) and its corresponding imidazolium salt (i.e., 1,3,4,5‐tetramethylimidazolium chloride) has been investigated theoretically by using Bader's quantum theory of atoms in molecules (QTAIM) descriptors, delocalization indices, electron localizability indicators (ELI‐Ds), and the source function tool. In addition, the experimental electron density distribution for the imidazolium salt has been obtained and analyzed from 100 K X‐ray diffraction data. A significant drop is found in the ellipticity of the electron density along the Ccarbene−N bond path in the imidazol‐2‐ylidene. This is shown to be a natural consequence of the σ lone pair of the Ccarbene atom, which overwhelms the π‐electron density, rather than a sign of a significantly diminished degree of π‐electron delocalization in the imidazol‐2‐ylidene compared to its imidazolium salt. In fact, the source functions, the ELI‐Ds, and the delocalization indices all probe a quite similar extent of cyclic π‐electron delocalization in the N‐heterocyclic rings of the two compounds. [ABSTRACT FROM AUTHOR]

Exact solutions of a nonlinear Boltzmann kinetic equation with a source are constructed in the case of an isotropic distribution function and Maxwell model of isotropic scattering. These solutions are constructed with the use of an equivalence group such that one of its transformations uniquely identifies the class of the source functions that are linear in terms of the distribution function; moreover, the transformed equation has a zero right side. As a result, invariant solutions of the type of the Bobylev-Krook-Wu solutions can be explicitly found, in particular, those that admit physical interpretation. [ABSTRACT FROM AUTHOR]