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Evolution of the elastic sheet around the valence electron density of a bond in a Si crystal. Starting with a perfect sphere, the first snapshot shown ͑ left ͒ is taken after 200 iterations, the second after 1000 iterations, and the third ͑ right ͒ after convergence to the zero-flux surface after 10 000 iterations. The larger spheres indicate the position of the Si atoms. The integrated charge of the enclosed volume is 1.976 electrons. The calculation took 34 min on a 400 MHz Pentium computer.
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We have developed a new method for finding and representing dividing surfaces which can, for example, be used to identify "atoms" in molecules or condensed phases based on Bader's definition. Given the total electron density of the system, the dividing surface is taken to be the zero-flux surface, i.e., the surface on which the normal component of...
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... applied the above algorithm to partitioning of the valence electron densities obtained in DFT/PW91 pseudopotential 8 calculations of Si crystal, bulk ice, and wa- ter clusters containing from 2 to 6 water molecules. The Si structures studied include the bulk bond Fig. 2, the bond between a pair of atoms which have been rotated in the Si crystal Fig. 3, and the bond between the two atoms forming a dumbbell interstitial Fig. 4. This last structure is espe- cially complex, being composed of a total of five local maxima, leading to five different zero-flux surfaces to de- scribe the valence charge density in the bonding ...
Context 2
... the region enclosed by the converged elastic sheet using 2000 particles, which is shown in Fig. 2, the integrated charge density amounts to 1.977 electrons, within 1.1% of the expected value. Figure 3 shows the zero-flux surface of the bond be- tween two atoms that are rotated in the bulk. This configu- ration is metastable and is found along the minimum energy path of the concerted exchange process proposed by Pandey. ...
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Citations
... [44] Grid methods are also applied in the context of computed electron densities. [45][46][47] Two more algorithms are based on different principles, such as the "elastic sheet" method [48] and a finite element method. [49] In contrast, recent developments [50,51] turned back to the original triangulation method of Biegler-K€ onig. ...
Atomic multipole moments associated with a spherical volume fully residing within a topological atom (i.e., the β sphere) can be obtained analytically. Such an integration is thus free of quadrature grids. A general formula for an arbitrary rank spherical harmonic multipole moment is derived, for an electron density comprising Gaussian primitives of arbitrary angular momentum. The closed expressions derived here are also sufficient to calculate the electrostatic potential, the two types of kinetic energy, as well as the potential energy between atoms. Some integrals have not been solved explicitly before but through recursion and substitution are broken down to more elementary listed integrals. The proposed method is based on a central formula that shifts Gaussian primitives from one center to another, which can be derived from the well-known plane-wave expansion (or Rayleigh equation). © 2018 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.
... The analysis is restricted to the localization and characterization of the CPs and to the calculation of the Hessian matrix eigenvalues. Various algorithms provide accurate separatrices from numerical densities [235][236][237][238], enabling tthe calculation of basin properties. The InteGriTy program developed by C. Katan and coworkers [236] is available in open access at [239] performs the complete topological analysis of experimental charge densities based on the Hansen-Coppens multipole formalism [240]. ...
This contribution explains how the topological methods of analysis of the electron density and related functions such as the electron localization function (ELF) and the electron localizability indicator (ELI-D) enable the theoretical characterization of various metal-metal (M–M) bonds (multiple M–M bonds, dative M–M bonds). Examples are taken in both bulk metals, alloys and molecular complexes. Metallic bonds as well as weak partially covalent M–M interactions, are described and characterized unambiguously combining AIM (atoms in molecules) and ELF/ELI-D topological analysis.
... In both cases, the analysis was limited to the localization and characterization of the critical points and to the calculation of the Hessian matrix eigenvalues. Different algorithms enable accurate separatrices from numerical densities [272][273][274] and, therefore, the calculation of basin properties. The InteGriTy program developed by C. Katan and coworker 273 is available in open access at the URL: http://www.gmcm.univ-rennes1. ...
Accurate electron densities and related functions are sources of chemical information enabling the understanding of the structure and chemical properties of molecules and solids. The quantum mechanical backgrounds of the electron probability distribution functions (the electron density and the pair functions), of the electron localization function (ELF), of the source function, and of some other related local descriptors of the electron distribution and of its properties are presented in the first part of this chapter. The rough data provided by these functions do not always reveal all of the expected chemical information, such as the characterization of bonding properties, and, therefore, methodological bridges are necessary to recover the phenomenological chemical concepts. The chapter is focused on the topological approaches that provide a description in the position space consistent with the Lewis structures and with the valence shell electron pair repulsion model. The theory of dynamical systems, applied to the gradient field of the electron density and of the ELF, is the most used mathematical tool enabling a rigorous partition of the space into basins of attractors that are chemically significant and also to propose criteria for a qualitative characterization of the bonding interactions and of their evolution during a chemical reaction. A deep quantitative insight is further achieved by integrating the one particle and pair densities as well as other density of property functions over the basin volumes. The statistical analysis of the variance and covariance of the basin populations provide a measure of the electron delocalization. The use of these techniques of analysis of the electron density properties is illustrated by a series of examples belonging to the field of inorganic chemistry.
... Different approaches for condensed, periodic systems have relied on analytic expressions of the density 3,4 or discretizing the charge density trajectories [5][6][7][8][9] . Early algorithms were based on the electron density calculated from analytical wavefunctions of small molecules, and integration along the gradient paths. ...
We propose an efficient, accurate method to integrate the basins of attraction of a smooth function defined on a general discrete grid and apply it to the Bader charge partitioning for the electron charge density. Starting with the evolution of trajectories in space following the gradient of charge density, we derive an expression for the fraction of space neighboring each grid point that flows to its neighbors. This serves as the basis to compute the fraction of each grid volume that belongs to a basin (Bader volume) and as a weight for the discrete integration of functions over the Bader volume. Compared with other grid-based algorithms, our approach is robust, more computationally efficient with linear computational effort, accurate, and has quadratic convergence. Moreover, it is straightforward to extend to nonuniform grids, such as from a mesh-refinement approach, and can be used to both identify basins of attraction of fixed points and integrate functions over the basins.
... The proton is thus transfered from Uracil to its primary hydration shell (in about 7 fs, see Fig. 5). We have checked that the transferred particle is a proton through an analysis of the charge reorganisation within the framework of the atoms in molecules (AIM) approach [45] and using the algorithm introduced in references [46,47]. This is the fastest dissociation scheme observed over the 10 different initial configurations investigated in this study for core holes on either nitrogen atom (N1 and N3) of Uracil: NH bond dissociation occurs at t = 9 fs, when the bond distance becomes larger than 1.5Å. ...
We present a series of ab initio density functional based
calculations of the fragmentation dynamics of core-ionized
biomolecules. The computations are performed for pure liquid water,
aqueous and isolated Uracil. Core ionization is
described by replacing the 1s
2 pseudopotential of one atom of the target
molecule (C, N or O) with a pseudopotential for a 1s
1 core-hole state.
Our results predict that the dissociation of core-ionized water
molecules may be reached during the lifetime of inner-shell vacancy
(less than 10fs), leading to OH bond breakage as a primary outcome.
We also observe a second fragmentation channel in which total Coulomb explosion
of the ionized water molecule occurs. Fragmentation pathways are found
similar for pure water or when the water molecule is in the primary hydration
shell of the uracil molecule. In the latter case, the proton may be transferred
towards the uracil oxygen atoms. When the core hole is located on the uracil
molecule, ultrafast dissociation is only observed in the aqueous environment
and for nitrogen-K vacancies, resulting in proton transfers towards
the hydrogen-bonded water molecule.
... While this approach works for small molecules, a high density of descent trajectories is needed to accurately represent the surface away from the critical points, and the method has been criticized as being computationally expensive for large systems [8, 5]. It was also found to fail for complex bonding geometries and when the radial integration rays have multiple intersection points with the dividing surface91011. Several improvements to this original approach have been proposed. ...
A computational method for partitioning a charge density grid into Bader volumes is presented which is efficient, robust, and scales linearly with the number of grid points. The partitioning algorithm follows the steepest ascent paths along the charge density gradient from grid point to grid point until a charge density maximum is reached. In this paper, we describe how accurate off-lattice ascent paths can be represented with respect to the grid points. This improvement maintains the efficient linear scaling of an earlier version of the algorithm, and eliminates a tendency for the Bader surfaces to be aligned along the grid directions. As the algorithm assigns grid points to charge density maxima, subsequent paths are terminated when they reach previously assigned grid points. It is this grid-based approach which gives the algorithm its efficiency, and allows for the analysis of the large grids generated from plane-wave-based density functional theory calculations.
Local density of states and electric charge in regions defined for individual atoms and molecules using grid based Bader analysis is presented for N2 and CO2 adsorbed on a platinum electrode in the presence of an applied electric field. When the density of states is projected onto Bader regions, the partial density of states for the various subregions correctly sums up to the total density of states for the whole system, unlike the commonly used projection onto spheres which results in missing contributions from some regions while others are over counted, depending on the radius chosen. The electrode is represented by a slab with a missing row reconstructed Pt(110)-(1 × 2) surface to model an edge between micro-facets on the surface of a nano-particle catalyst. For both N2 and CO2, a certain electric field window leads to adsorption. The binding of N2 to the electrode is mainly due to polarization of the molecule but for CO2 hybridization occurs between the molecular states and the states of the Pt electrode.
In a previous publication we derived a criterion for defining interatomic surfaces in condensed systems based on the zero flux surface (ZFS) of the electronic wavefunction's gradient which holds the condition of flux zero at every point. Here this criterion is examined and physical as well as mathematical properties are illustrated and discussed; in the light of this discussion we conclude that the definition illustrated in this work represents a natural extension of Bader's definition of atoms in molecules.
An algorithm is presented for carrying out decomposition of electronic charge density into atomic contributions. As suggested by Bader [R. Bader, Atoms in Molecules: A Quantum Theory, Oxford University Press, New York, 1990], space is divided up into atomic regions where the dividing surfaces are at a minimum in the charge density, i.e. the gradient of the charge density is zero along the surface normal. Instead of explicitly finding and representing the dividing surfaces, which is a challenging task, our algorithm assigns each point on a regular (x,y,z) grid to one of the regions by following a steepest ascent path on the grid. The computational work required to analyze a given charge density grid is approximately 50 arithmetic operations per grid point. The work scales linearly with the number of grid points and is essentially independent of the number of atoms in the system. The algorithm is robust and insensitive to the topology of molecular bonding. In addition to two test problems involving a water molecule and NaCl crystal, the algorithm has been used to estimate the electrical activity of a cluster of boron atoms in a silicon crystal. The highly stable three-atom boron cluster, B3I is found to have a charge of −1.5e, which suggests approximately 50% reduction in electrical activity as compared with three substitutional boron atoms.
We have studied the electric field near water clusters and in ice Ih using first principles calculations. We employed Møller-Plesset perturbation theory (MP2) for the calculations of the clusters up to and including the hexamer, and density functional theory (DFT) with a gradient dependent functional [Perdew-Wang (PW91)] for ice Ih as well as the clusters. The electric field obtained from the first principles calculations was used to test the predictions of an induction model based on single center multipole moments and polarizabilities of an isolated water molecule. We found that the fields obtained from the induction model agree well with the first principles results when the multipole expansion is carried out up to and including the hexadecapole moment, and when polarizable dipole and quadrupole moments are included. This implies that accurate empirical water interaction potential functions transferable to various environments such as water clusters and ice surfaces could be based on a single center multipole expansion carried out up to the hexadecapole. Since point charges are not included, the computationally intensive Ewald summations can be avoided. Molecular multipole moments were also extracted from the first principles charge density using zero flux dividing surfaces as proposed by Bader. Although the values of the various molecular multipoles obtained with this method are quite different from the ones resulting from the induction model, the rate of convergence of the electric field is, nevertheless, quite similar.
