The general setting for the zero‐flux condition: The lagrangian and zero‐flux conditions that give the heisenberg equation of motion

Generalizing our recent work on relativistic generalizations of the quantum theory of atoms in molecules, we present the general setting under which the principle of stationary action for a region leads to open quantum subsystems. The approach presented here is general and works for any Hamiltonian, and when a reasonable Lagrangian is selected, it often leads to the integral of the Laplacian of the electron density on the region vanishing as a necessary condition for the zero‐flux surface. Alternatively, with this method, one can design a Lagrangian that leads to a surface of interest (though this Lagrangian may not be, and indeed probably will not be, “reasonable”). For any reasonable Lagrangian for the electronic wave function and any two‐component method (related by integration by parts to the Hamiltonian) considered, the Bader definition of an atom is recaptured. © 2018 Wiley Periodicals, Inc.     

Calculation of atomic integration data

The calculation of average properties of atoms in molecules and interatomic surfaces is a difficult problem that requires the evaluation of two‐ and three‐dimensional integrals over regions with nontrivial borders. A mathematical formalism is presented that maps these regions onto the whole of ℝ² and/or ℝ³ and allows the construction of efficient and reliable numerical methods for the calculation of these integrals. These methods, which will be part of a forthcoming program package, are described and examples are given. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 1040–1048, 2000     

An efficient grid-based scheme to compute QTAIM atomic properties without explicit calculation of zero-flux surfaces

We introduce a method to compute atomic properties according to the "quantum theory of atoms in molecules." An integration grid in real space is partitioned into subsets, omega(i). The subset, omega(i), is composed of all grid points contained in the atomic basin, Omega(i), so that integration over Omega(i) is reduced to simple quadrature over the points in omega(i). The partition is constructed from deMon2k's atomic center grids by following the steepest ascent path of the density starting from each point in the grid. We also introduce a technique that exploits the cellular nature of the grid to make the algorithm faster. The performance of the method is tested by computing properties of atoms and nonnuclear attractors (energies, charges, dipole, and quadrupole moments) for a set of representative molecules.     

A fast and accurate algorithm for QTAIM integration in solids

A new algorithm is presented for the calculation of atomic properties, in the sense of the quantum theory of atoms in molecules. This new method, named QTREE , applies to solid‐state densities and allows the computation of the atomic properties of all the atoms in the crystal in seconds to minutes. The basis of the method is the recursive subdivision of a symmetry‐reduced wedge of the Wigner‐Seitz cell, which in turn is expressed as a union of tetrahedra, plus the use of β‐spheres to improve the performance. A considerable speedup is thus achieved compared with traditional quadrature‐based schemes, justified by the poor performance of the latter because of the particular features of atomic basins in solids. QTREE can use both analytical or interpolated densities, calculates all the atomic properties available, and converges to the correct values in the limit of infinite precision. Several gradient path tracing and integration techniques are tested. Basin volumes and charges for a selected set of 11 crystals are determined as a test of the new method. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

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

Several definitions of an atom in a molecule (AIM) in three‐dimensional (3D) space, including both fuzzy and disjoint domains, are used to calculate electron sharing indices (ESI) and related electronic aromaticity measures, namely, I_ring and multicenter indices (MCI), for a wide set of cyclic planar aromatic and nonaromatic molecules of different ring size. The results obtained using the recent iterative Hirshfeld scheme are compared with those derived from the classical Hirshfeld method and from Bader's quantum theory of atoms in molecules. For bonded atoms, all methods yield ESI values in very good agreement, especially for C–C interactions. In the case of nonbonded interactions, there are relevant deviations, particularly between fuzzy and QTAIM schemes. These discrepancies directly translate into significant differences in the values and the trends of the aromaticity indices. In particular, the chemically expected trends are more consistently found when using disjoint domains. Careful examination of the underlying effects reveals the different reasons why the aromaticity indices investigated give the expected results for binary divisions of 3D space. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011.     

Estimations of energy of noncovalent bonding from integrals over interatomic zero‐flux surfaces: Correlation trends and beyond

Bonding energies of 50 associates composed by neutral molecules (atoms) and bounded by various weak noncovalent interactions are calculated within the DFT framework using the PBE0/aug‐cc‐pVTZ combination. The electronic virial and electron density values at bond critical points together with their integrals over interatomic surfaces are tested to check their ability to estimate bonding energies. Two correlations schemes dealing with integrals over interatomic surface are suggested to estimate bonding energy of any noncovalent interaction. The physical meaning of explored and several known correlations is discussed. Methods to estimate interatomic surface integrals of electronic virial and electron density are proposed. © 2018 Wiley Periodicals, Inc.     

Stability of atomic oxygen chemisorption on Pt‐alloy surfaces

A density functional theory calculation is used to investigate the atomic oxygen (O) stability over platinum (Pt) and Pt‐based alloy surfaces. Here, the stability is connected with the preferential adsorption sites for O chemisorptions and the adsorption energy. Thus, the interaction mechanism between atomic O and metal surfaces is studied by using charge transfer analysis. In this present paper, atomic structure and binding energy of oxygen adsorption on the Pt(111) are in a very good agreement with experiment and previous density functional theory calculations. Furthermore, we obtained that the addition of ruthenium (Ru) and molybdenum (Mo) on the pure Pt surface enhances the adsorption energy. Our charge transfer analysis shows that the largest charge transfer contributing to the metal‐O bonding formation is observed in the case of O/PtRuMo surface followed by O/PtRu surface. This is in consistency with metal d‐orbital characteristic, where Mo has much more empty d‐orbital than Ru in correspondence to accept electrons from atomic oxygen. Copyright © 2016 John Wiley & Sons, Ltd.     

Occupation numbers for atomic shells in direct space bounded by the maxima of the one‐electron potential

Atomic shells defined as wells of the one‐electron potential $\nabla^{2}\sqrt{\rho}/2\sqrt{\rho}$ bounded by successive maxima of this electron density function give reasonable electron numbers for the occupation of shells with empty d orbitals. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 324–331, 2001     

Visualization and integration of quantum topological atoms by spatial discretization into finite elements

We present a novel algorithm to integrate property densities over the volume of a quantum topological atom. Atoms are grown outward, starting from a sphere centered on the nucleus, by means of a finite element meshing algorithm. Bond critical points and ring critical points require special treatment. The overall philosophy as well as intricate features of this meshing algorithm are given, followed by details of the quadrature over the finite elements. An effort has been made to design a streamlined and compact algorithm, focusing on the core of challenges arising in tracing the electron density's gradient vector field. The current algorithm also generates a new type of pictures that can be a Graphical User Interface. Excellent integration errors, L(Omega), are obtained, even for atoms with (narrow) tails or sharp corners.     

Zero‐valent bimetallic iron/copper catalyzed SET‐LRP: A dual activation by zero‐valent iron

In this work, bimetallic zero‐valent metal (Fe(0) powder and Cu(0) powder) was used to mediate the single electron transfer‐living radical polymerization (SET‐LRP) of methyl methacrylate at 25 °C in dimethyl sulfoxide. Different feed ratios of [Fe(0)]₀/[Cu(0)]₀ (0/1.5, 0.5/1, 0.75/0.75, 1/0.5, and 1.3/0.2) were explored. With the increase of Fe(0) feed, the polymerization rate was mildly depressed with a prolonged induction period. While, the control over the molecular weights was improved upon the increase of Fe(0). A best control (initiation efficiency = 91%) was achieved at [Fe(0)]₀/[Cu(0)]₀ = 1/0.5. A further increase of Fe(0) to the feed ratio of [Fe(0)]₀:[Cu(0)]₀ = 1.3: 0.2 led to a uncontrolled polymerization. Explorations of available solvents and ligands for this polymerization confirmed the SET‐LRP mechanism. It was suggested that Fe(0) might act as a dual role in this process: one was the activation agent for Cu(0), which favored a better control over the molecular weights; The other was an alternative catalyst for the activation of R‐X or P_n‐X to generate radicals, which assured a comparable polymerization rate as that of Cu(0). This work provided an alternative and economical catalyst for SET‐LRP, and would eventually reinforce the SET‐LRP technique. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Evaluation of atomic pressure in the multiple time‐step integration algorithm

In molecular dynamics (MD) calculations, reduction in calculation time per MD loop is essential. A multiple time‐step (MTS) integration algorithm, the RESPA (Tuckerman and Berne, J. Chem. Phys. 1992, 97, 1990–2001), enables reductions in calculation time by decreasing the frequency of time‐consuming long‐range interaction calculations. However, the RESPA MTS algorithm involves uncertainties in evaluating the atomic interaction‐based pressure (i.e., atomic pressure) of systems with and without holonomic constraints. It is not clear which intermediate forces and constraint forces in the MTS integration procedure should be used to calculate the atomic pressure. In this article, we propose a series of equations to evaluate the atomic pressure in the RESPA MTS integration procedure on the basis of its equivalence to the Velocity‐Verlet integration procedure with a single time step (STS). The equations guarantee time‐reversibility even for the system with holonomic constrants. Furthermore, we generalize the equations to both (i) arbitrary number of inner time steps and (ii) arbitrary number of force components (RESPA levels). The atomic pressure calculated by our equations with the MTS integration shows excellent agreement with the reference value with the STS, whereas pressures calculated using the conventional ad hoc equations deviated from it. Our equations can be extended straightforwardly to the MTS integration algorithm for the isothermal NVT and isothermal–isobaric NPT ensembles. © 2017 Wiley Periodicals, Inc.     

Structural properties and the effect of 2,6‐diaminoanthraquinone on G‐tetrad,non‐G‐tetrads,and mixed tetrads—A density functional theory study

Telomerase inhibitor causes the attrition of telomere length and consequently leading to senescence which require a lag period for cancer cells to stop proliferating. Telomeric sequences form quadruplex structures stabilized by tetrads. The structural and electronic properties related with interaction of 2,6‐diaminoanthraquinone and tetrads are the key step to elucidate the anticancer activity. The present study has been focused on the stability of the isolated tetrads and the effect of interaction of 2,6‐diaminoanthraquinone with G‐tetrad, non‐G‐tetrads, and mixed tetrads using density functional theory method in both gas and aqueous phases. The solvent interaction with the molecular systems has increased the stability of the isolated tetrads and complexes. The sharing of electron density between the interacting molecules is shown through electron density difference maps. The atoms in molecules theory and natural bond orbital analysis have been performed to study the nature of hydrogen bonds in the inhibitor interacting complexes. The linear correlation is shown between electron density [ρ(r)], and its Laplacian [(²ρ(r)] at the bond critical points. The strong binding nature of 2,6‐diaminoanthraquinone with studied tetrads reveals that this inhibitor is suitable to stabilize the above tetrads and inhibit the telomerase activity. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Quasi‐random integration in quantum chemistry: Efficiency,stability, and application to the study of small atoms and molecules constrained in spherical boxes

This project consists of two parts. In the first part, a series of test calculations is performed to verify that the integrals involved in the determination of atomic and molecular properties by standard self‐consistent field (SCF) methods can be obtained through Halton, Korobov, or Hammersley quasi‐random integration procedures. Through these calculations, we confirm that all three methods lead to results that meet the levels of precision required for their use in the calculation of properties of small atoms or molecules at least at a Hartree–Fock level. Moreover, we have ensured that the efficiency of quasi‐random integration methods that we have tested is Halton=Korobov>Hammersley?pseudo‐random. We also find that these results are comparable to those yielded by ordinary Monte Carlo (pseudo‐random) integration, with a calculation effort of two orders of smaller magnitude. The second part, which would not have been possible without the integration method previously analyzed, contains a first study of atoms constrained in spherical boxes through SCF calculations with basis functions adapted to the features of the problem: Slater‐type orbitals (STOs) trimmed by multiplying them by a function that yields 1 for 0 < r < (R‐δ), polynomial values for (R‐δ) < r < R and null for r > R, R being the radius of the box and δ a variationally determined interval. As a result, we obtain a equation of state for electrons of small systems, valid just in the limit of low temperatures, but fairly simple. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Density functional studies on hydrogen‐bonded clusters of hydrogen halides and the interaction on halide anions

Density functional theory (DFT) calculations have been performed to study the structures and stability of X^?·(HX)_n=2–5 clusters where X = F, Cl, Br at B3LYP/6‐311++G** level of theory. The presence of halide ions in these clusters disintegrates the hydrogen halide clusters. All the hydrogen halides are then hydrogen bonded to the centrally placed halide ions, thereby forming multiple hydrogen bonds. The interaction energies have been corrected for the basis set superposition error (BSSE) using Boy's counterpoise correction method. Evidence for the destruction of hydrogen bonds in hydrogen halide clusters due to the presence of halide ions is further obtained from topological analysis and natural bond orbital analysis. The chemical hardness and chemical potential have been calculated for all the anion clusters. The above analysis reveals that hydrogen bonding in these systems is not an essentially electrostatic interaction. The nature of the stabilization interactions operative in these multiple hydrogen‐bonded clusters has been explained in terms of many‐body contribution to interaction energies. From these studies, an attempt has been made to understand the nature of the molecular properties resulting from different electronegativities of the halogens. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Reply to ‘comment on “extending Hirshfeld‐I to bulk and periodic materials”’

The issues raised in the comment by Manz are addressed through the presentation of calculated atomic charges for NaF, NaCl, MgO, SrTiO $$_3$$ , and La $$_2$$ Ce $$_2$$ O $$_7$$ , using our previously presented method for calculating Hirshfeld‐I charges in solids (Vanpoucke et al., J. Comput. Chem. doi: 10.1002/jcc.23088). It is shown that the use of pseudovalence charges is sufficient to retrieve the full all‐electron Hirshfeld‐I charges to good accuracy. Furthermore, we present timing results of different systems, containing up to over 200 atoms, underlining the relatively low cost for large systems. A number of theoretical issues are formulated, pointing out mainly that care must be taken when deriving new atoms in molecules methods based on “expectations” for atomic charges. © 2012 Wiley Periodicals, Inc.     

Revisiting the foundations of the quantum theory of atoms in molecules: Toward a rigorous definition of topological atoms

An explicit classification of consistent variational constraints within the context of the “quantum theory of proper open subsystems” as well as the “quantum theory of atoms in molecules” (QTAIM) it presented. It is demonstrated that the general variational procedure is not sensitive enough to discriminate between different mathematically consistent variational conditions. The uniqueness of the regional kinetic energy is employed to derive the net zero‐flux condition and the regions satisfying this condition are named as quantum divided basins. A modified form of the local zero‐flux is proposed in order to define topological atoms within the context of the orthodox QTAIM. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Description of correlated densities for few‐electron atoms by simple functional forms

Simple analytical functional forms for the electron density of two‐ and three‐electron atoms which reproduce fairly the correlated (exact) values are presented. The procedure is based on the fitting of an auxiliary f(r) function which has adequate properties for this purpose and can be extended to more complex atoms. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 443–454, 1999