Two-electron integrations in the Quantum Theory of Atoms in Molecules with correlated wave functions

2005 ◽  
Vol 26 (4) ◽  
pp. 344-351 ◽  
Author(s):  
A. Martín Pendás ◽  
E. Francisco ◽  
M. A. Blanco
Keyword(s):  
Quantum Theory ◽  
Wave Functions ◽  
Atoms In Molecules

Quantum Theory

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Uncertainty Principle ◽  
Quantum Spin ◽  
Quantum States ◽  
Wave Functions ◽  
Symbolic Representations ◽  
Quantum Numbers ◽  
The Subject ◽  
Heisenberg's Uncertainty Principle

The subject of Chapter 8 is the fundamental principles of quantum theory, the abstract extension of quantum mechanics. Two of the entities explored are kets and operators, with kets being representations of quantum states as well as a source of wave functions. The quantum box and quantum spin kets are specified, as are the quantum numbers that identify them. Operators are introduced and defined in part as the symbolic representations of observable quantities such as position, momentum and quantum spin. Eigenvalues and eigenkets are defined and discussed, with the former identified as the possible outcomes of a measurement. Bras, the counterpart to kets, are introduced as the means of forming probability amplitudes from kets. Products of operators are examined, as is their role underpinning Heisenberg’s Uncertainty Principle. A variety of symbol manipulations are presented. How measurements are believed to collapse linear superpositions to one term of the sum is explored.

scholarly journals Reducing a Class of Two-Dimensional Integrals to One-Dimension with an Application to Gaussian Transforms

Atoms ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 53
Author(s):  
Jack C. Straton
Keyword(s):  
Quantum Theory ◽  
Arbitrary Function ◽  
Plane Waves ◽  
Wave Functions ◽  
One Dimension ◽  
Two Dimensional ◽  
Transition Amplitudes ◽  
Multidimensional Integrals ◽  
Coulomb Potentials

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers multiplied by an arbitrary function of xy/(x+y) multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, and Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions.

Ab initio charge density analysis of (B6C)2– and B4C3 species — How to describe the bonding pattern?

2006 ◽  
Vol 84 (5) ◽  
pp. 771-781 ◽  
Author(s):  
Cina Foroutan-Nejad ◽  
Gholam Hossein Shafiee ◽  
Abdolreza Sadjadi ◽  
Shant Shahbazian
Keyword(s):  
Quantum Theory ◽  
Carbon Atom ◽  
Charge Density ◽  
Ab Initio ◽  
Atoms In Molecules ◽  
Central Carbon Atom ◽  
B3lyp Method ◽  
Central Carbon ◽  
Density Analysis ◽  
Concrete Ring

In this study, a detailed topological charge density analysis based on the quantum theory of atoms in molecules (QTAIM) developed by Bader and co-workers, has been accomplished (using the B3LYP method) on the CB62– anion and three planar isomers of the C3B4 species, which had been first proposed by Exner and Schleyer as examples of molecules containing hexacoordinate carbon atoms. The analysis uncovers the strong (covalent) interactions of boron atoms as well as the "nondirectional" interaction of central carbon atom with those peripheral atoms. On the other hand, instabilities have been found in the topological networks of (B6C)2– and B4C3(para) species. A detailed investigation of these instabilities demonstrates that the topology of charge density has a floppy nature near the equilibrium geometries of the species under study. Thus, these species seems to be best described as complexes of a relatively concrete ring containing boron or carbon atoms and a central carbon atom that is confined in the plane of the molecule, but with nondirectional interactions with the surrounding atoms.Key words: hypervalency, hexacoordinate carbon, quantum theory of atoms in molecules, charge density analysis, ab initio methods.

scholarly journals Decomposition of d- and f-Shell Contributions to Uranium Bonding from the Quantum Theory of Atoms in Molecules: Application to Uranium and Uranyl Halides

Inorganics ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 88 ◽  
Author(s):  
Jonathan Tanti ◽  
Meghan Lincoln ◽  
Andy Kerridge
Keyword(s):  
Quantum Theory ◽  
Electronic Structures ◽  
Density Functional ◽  
Covalent Bond ◽  
Vibrational Frequencies ◽  
Atoms In Molecules ◽  
High Symmetry ◽  
Strong Correlations ◽  
Bond Stability ◽  
Fluoride Ligands

The electronic structures of a series of uranium hexahalide and uranyl tetrahalide complexes were simulated at the density functional theoretical (DFT) level. The resulting electronic structures were analyzed using a novel application of the Quantum Theory of Atoms in Molecules (QTAIM) by exploiting the high symmetry of the complexes to determine 5f- and 6d-shell contributions to bonding via symmetry arguments. This analysis revealed fluoride ligation to result in strong bonds with a significant covalent character while ligation by chloride and bromide species resulted in more ionic interactions with little differentiation between the ligands. Fluoride ligands were also found to be most capable of perturbing an existing electronic structure. 5f contributions to overlap-driven covalency were found to be larger than 6d contributions for all interactions in all complexes studied while degeneracy-driven covalent contributions showed significantly greater variation. σ-contributions to degeneracy-driven covalency were found to be consistently larger than those of individual π-components while the total π-contribution was, in some cases, larger. Strong correlations were found between overlap-driven covalent bond contributions, U–O vibrational frequencies, and energetic stability, which indicates that overlap-driven covalency leads to bond stabilization in these complexes and that uranyl vibrational frequencies can be used to quantitatively probe equatorial bond covalency. For uranium hexahalides, degeneracy-driven covalency was found to anti-correlate with bond stability.

On the transferability of electron density in binary vanadium borides VB, V3B4 and VB2

2015 ◽  
Vol 71 (6) ◽  
pp. 777-787 ◽  
Author(s):  
Bürgehan Terlan ◽  
Lev Akselrud ◽  
Alexey I. Baranov ◽  
Horst Borrmann ◽  
Yuri Grin
Keyword(s):  
Quantum Theory ◽  
Crystal Structures ◽  
Electron Density ◽  
Critical Points ◽  
Atoms In Molecules ◽  
Model Systems ◽  
Suitable Model ◽  
Interatomic Distances ◽  
Systematic Analysis ◽  
A Charge

Binary vanadium borides are suitable model systems for a systematic analysis of the transferability concept in intermetallic compounds due to chemical intergrowth in their crystal structures. In order to underline this structural relationship, topological properties of the electron density in VB, V3B4 and VB2 reconstructed from high-resolution single-crystal X-ray diffraction data as well as derived from quantum chemical calculations, are analysed in terms of Bader's Quantum Theory of Atoms in Molecules [Bader (1990). Atoms in Molecules: A Quantum Theory, 1st ed. Oxford: Clarendon Press]. The compounds VB, V3B4 and VB2 are characterized by a charge transfer from the metal to boron together with two predominant atomic interactions, the shared covalent B—B interactions and the polar covalent B—M interactions. The resembling features of the crystal structures are well reflected by the respective B—B interatomic distances as well as by ρ(r) values at the B—B bond critical points. The latter decrease with an increase in the corresponding interatomic distances. The B—B bonds show transferable electron density properties at bond critical points depending on the respective bond distances.

scholarly journals ELECTRONIC STRUCTURE OF METHYLIDINESULFOXIDALCANES

2020 ◽  
pp. 106-113
Author(s):  
Наталья Петровна Русакова ◽  
Елена Михайловна Чернова ◽  
Владимир Владимирович Туровцев ◽  
Юрий Димитриевич Орлов
Keyword(s):  
Electronic Structure ◽  
Quantum Theory ◽  
Functional Groups ◽  
Atoms In Molecules

В рамках «квантовой теории атомов в молекулах» (QTAIM) определено электронное строение молекул ряда метилидинсульфоксидалканов n-C H-S(О)СН, где 1 ≤ n ≤ 10. Составлена качественная шкала электроотрицательностей функциональных групп. Within the framework of the «quantum theory of atoms in molecules» (QTAIM), the electronic structure of the molecules of a series of methylidinesulfoxidealkanes n-CH-S(О)СН, where 1 ≤ n ≤ 10, was determined. A qualitative scale of electronegativities of functional groups was compiled.

scholarly journals π-Hole Tetrel Bonds—Lewis Acid Properties of Metallylenes

Crystals ◽  
2022 ◽  
Vol 12 (1) ◽  
pp. 112
Author(s):  
Sławomir J. Grabowski
Keyword(s):  
Quantum Theory ◽  
Crystal Structures ◽  
Lewis Acid ◽  
Natural Bond Orbital ◽  
Database Search ◽  
Lewis Base ◽  
Atoms In Molecules ◽  
Acid Properties ◽  
Structural Database ◽  
Tetrel Bonds

The MP2/aug-cc-pVTZ calculations were performed on the dihalometallylenes to indicate their Lewis acid and Lewis base sites. The results of the Cambridge Structural Database search show corresponding and related crystal structures where the tetrel center often possesses the configuration of a trigonal bipyramid or octahedron. The calculations were also carried out on dimers of dichlorogermylene and dibromogermylene and on complexes of these germylenes with one and two 1,4-dioxide molecules. The Ge⋯Cl, Ge⋯Br, and Ge⋯O interactions are analyzed. The Ge⋯O interactions in the above mentioned germylene complexes may be classified as the π-hole tetrel bonds. The MP2 calculations are supported by the results of the Quantum Theory of Atoms in Molecules (QTAIM) and the Natural Bond Orbital (NBO) approaches.

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