Bounds to expectation values and exponentially decreasing upper bounds to the one-electron density of atoms

1978 ◽  
Vol 11 (1) ◽  
pp. 17-24 ◽  
Author(s):  
T Hoffmann-Ostenhof ◽  
M Hoffmann-Ostenhof
Keyword(s):  
Electron Density ◽  
Upper Bounds ◽  
Expectation Values ◽  
The One

Developing orbital-free quantum crystallography: the local potentials and associated partial charge densities

2021 ◽  
Vol 77 (4) ◽  
Author(s):  
Vladimir Tsirelson ◽  
Adam Stash
Keyword(s):  
Density Functional Theory ◽  
Electron Density ◽  
Density Functional ◽  
Electronic Energy ◽  
Physical Content ◽  
Functional Theory ◽  
Orbital Free ◽  
The One ◽  
Physical And Chemical ◽  
Quantum Crystallography

This work extends the orbital-free density functional theory to the field of quantum crystallography. The total electronic energy is decomposed into electrostatic, exchange, Weizsacker and Pauli components on the basis of physically grounded arguments. Then, the one-electron Euler equation is re-written through corresponding potentials, which have clear physical and chemical meaning. Partial electron densities related with these potentials by the Poisson equation are also defined. All these functions were analyzed from viewpoint of their physical content and limits of applicability. Then, they were expressed in terms of experimental electron density and its derivatives using the orbital-free density functional theory approximations, and applied to the study of chemical bonding in a heteromolecular crystal of ammonium hydrooxalate oxalic acid dihydrate. It is demonstrated that this approach allows the electron density to be decomposed into physically meaningful components associated with electrostatics, exchange, and spin-independent wave properties of electrons or with their combinations in a crystal. Therefore, the bonding information about a crystal that was previously unavailable for X-ray diffraction analysis can be now obtained.

scholarly journals Evaluation of the one electron expectation values for different wave function of Be atom

2007 ◽  
Vol 4 (3) ◽  
pp. 393-396
Author(s):  
Baghdad Science Journal
Keyword(s):  
Wave Function ◽  
Slater Determinant ◽  
Open Shell ◽  
Expectation Values ◽  
Electronic Density ◽  
Expectation Value ◽  
Hartree Fock ◽  
Shell System ◽  
Electronic Wave Function ◽  
The One

The aim of this work is to evaluate the one- electron expectation value from the radial electronic density function D(r1) for different wave function for the 2S state of Be atom . The wave function used were published in 1960,1974and 1993, respectavily. Using Hartree-Fock wave function as a Slater determinant has used the partitioning technique for the analysis open shell system of Be (1s22s2) state, the analyze Be atom for six-pairs electronic wave function , tow of these are for intra-shells (K,L) and the rest for inter-shells(KL) . The results are obtained numerically by using computer programs (Mathcad).

Weak interactions in chain polymers [M(μ-X)2 L 2]∞ (M = Zn, Cd; X = Cl, Br; L = substituted pyridine) – an electron density study

2009 ◽  
Vol 65 (5) ◽  
pp. 600-611 ◽  
Author(s):  
Ruimin Wang ◽  
Christian W. Lehmann ◽  
Ulli Englert
Keyword(s):  
Hydrogen Bonds ◽  
Electron Density ◽  
Critical Points ◽  
Weak Interactions ◽  
Data Sets ◽  
X Ray Diffraction ◽  
Density Distributions ◽  
The One ◽  
Experimental Electron Density ◽  
Density Study

The experimental electron-density distributions in crystals of five chain polymers [M(μ-X)2(py)2] (M = Zn, Cd; X = Cl, Br; py = 3,5-substituted pyridine) have been obtained from high-resolution X-ray diffraction data sets (sin θ/λ > 1.1 Å−1) at 100 K. Topological analyses following Bader's `Atoms in Molecules' approach not only confirmed the existence of (3, −1) critical points for the chemically reasonable and presumably strong covalent and coordinative bonds, but also for four different secondary interactions which are expected to play a role in stabilizing the polymeric structures which are unusual for Zn as the metal center. These weaker contacts comprise intra- and inter-strand C—H...X—M hydrogen bonds on the one hand and C—X...X—C interhalogen contacts on the other hand. According to the experimental electron-density studies, the non-classical hydrogen bonds are associated with higher electron density in the (3, −1) critical points than the halogen bonds and hence are the dominant interactions both with respect to intra- and inter-chain contacts.

Exploring the electron density localization in MoS2 nanoparticles using a localized-electron detector: Unraveling the origin of the one-dimensional metallic sites on MoS2 catalysts

2018 ◽  
Vol 20 (31) ◽  
pp. 20417-20426 ◽  
Author(s):  
Yosslen Aray ◽  
Antonio Díaz Barrios
Keyword(s):  
Electron Density ◽  
The Other ◽  
Local Moment ◽  
Electron Detector ◽  
Moment Representation ◽  
One Dimensional ◽  
Mos2 Nanoparticles ◽  
Mos2 Catalysts ◽  
The One

The nature of the electron density localization in two MoS2 nanoclusters containing eight rows of Mo atoms, one with 100% sulphur coverage at the Mo edges (n8_100S) and the other with 50% coverage (n8_50S) was studied using a localized-electron detector function defined in the local moment representation.

Properties of Aromaticity Indices Based on the One-Electron Density Matrix†

2007 ◽  
Vol 111 (28) ◽  
pp. 6521-6525 ◽  
Author(s):  
Jerzy Cioslowski ◽  
Eduard Matito ◽  
Miquel Solà
Keyword(s):  
Density Matrix ◽  
Electron Density ◽  
Electron Density Matrix ◽  
The One

Atomic interactions in ethylenebis(1-indenyl)zirconium dichloride as derived by experimental electron density analysis

2005 ◽  
Vol 61 (4) ◽  
pp. 418-428 ◽  
Author(s):  
Adam I. Stash ◽  
Kiyoaki Tanaka ◽  
Kazunari Shiozawa ◽  
Hitoshi Makino ◽  
Vladimir G. Tsirelson
Keyword(s):  
Electron Density ◽  
Topological Analysis ◽  
Electronic Energy ◽  
Ligand Interaction ◽  
Atomic Interactions ◽  
Particle Potential ◽  
Density Analysis ◽  
The One ◽  
Electron Density Analysis ◽  
Experimental Electron Density

A topological analysis of the experimental electron density in racemic ethylenebis(1-indenyl)zirconium dichloride, C20H16Cl2Zr, measured at 100 (1) K, has been performed. The atomic charges calculated by the numerical integration of the electron density over the zero-flux atomic basins demonstrate the charge transfer of 2.25 e from the Zr atom to the two indenyl ligands (0.19 e to each) and two Cl atoms (0.93 e to each). All the atomic interactions were quantitatively characterized in terms of the electron density and the electronic energy-density features at the bond critical points. The Zr—C2 bond paths significantly curved towards the C1—C2 bond were found; no other bond paths connecting the Zr atom and indenyl ligand were located. At the same time, the π-electrons of the C1—C2 bond are significantly involved in the metal–ligand interaction. The electron density features indicate that the indenyl coordination can be approximately described as η1 with slippage towards η2. The `ligand-opposed' charge concentrations around the Zr atom were revealed using the Laplacian of the electron density and the one-particle potential; they were linked to the orbital representations. Bonds in the indenyl ligand were characterized using the Cioslowski–Mixon bond-order indices calculated directly from the experimental electron density.

Sharp upper bounds on perfect retrieval in the Hopfield model

1999 ◽  
Vol 36 (03) ◽  
pp. 941-950 ◽  
Author(s):  
Anton Bovier
Keyword(s):  
Fixed Points ◽  
Lower Bounds ◽  
Upper Bound ◽  
Random Variables ◽  
Upper Bounds ◽  
Negative Association ◽  
Hopfield Model ◽  
Gradient Dynamics ◽  
One Step ◽  
The One

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

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