Potenciais singulares na equação de Schrodinger unidimensional via transformada de Laplace

The present work has as its goal to treat well known and interesting unidimensional cases from quantum mechanics through an unusual approach within this eld of physics. The operational method of Laplace transform, in spite of its use by Erwin Schrödinger in 1926 when treating the radial equation for the hydrogen atom, turned out to be forgotten for decades. However, the method has gained attention again for its use as a powerful tool from mathematical physics applied to the quantum mechanics, appearing in recent works. The method is specially suitable to the approach of cases where we have potential functions with even parity, because this implies in eigenfunctions with de ned parity, and since the domain of this transform ranges from 0 to ∞, it su ces that we nd the eigenfunction in the positive semi axis and, with the boundary conditions imposed over the eigenfunction at the origin plus the continuity (discontinuity) of the eigenfunction and its derivative, we make the odd, even or both parity extensions so we can get the eigenfunction along all the axis. Factoring the eigenfunction behavior at in nity and origin, we take the due care with the points that might bring us problems in the later steps of the solving process, thus we can manipulate the Schrödinger's Equation regardless of time, so that way we make it convenient to the application of Laplace transform. The Chapter 3 shows the methodology that must be followed in order to search for the solutions to each problem

Synthesis of lanthanide-alkylaluminium bimetallic complexes and studies on their catalytic activities for polymerization of some polar monomers

Three new lanthanide (Ln)-alkylaluminium (Al) bimetallic complexes with the formula [(mu-CF3CO2)(2)Ln(mu-CF3CHO2)AIR(2) . 2THF](2) (Ln = Nd, Y, R=i-C4H9 (i-Bu); Ln=Eu, R=C2H5(Et); THF=tetrahydrofuran) were synthesized by the reaction of Ln(CF,CO,), (Ln=Nd, Y) with HAI (i-Bu)(2) and of Eu(CF3CO2)(3) with AlEt(3), respectively. Their crystal structures were determined by X-ray diffraction at 233 K. [(mu-CF3CO2)(2)Nd (mu-CF3CHO2)Al(i-Bu)(2) . 2THF](2) (Nd-Al) and [(mu-CF3CO2)(2)Y(mu-CF3CHO2)Al(i-Bu)(2) . 2THF](2) (Y-Al) are isomorphous and crystallize in space group with a=12.441(3) Angstrom [12.347(5) Angstrom for Y-Al], b=12.832(3) Angstrom [12.832(4) Angstrom], c=11.334(3) Angstrom [11.292(8) Angstrom], alpha=104.93 (2)degrees [104.45(4)degrees], beta=98.47(2)degrees [98.81(4)degrees], gamma=64.60(2)degrees [64.30(3)degrees], R=0.519 [0.113], R(w)=0.0532 [0.110], Z=1 and [(mu-CF3CO2)(2)Eu(CF3CHO2)AlEt(2) . 2THF](2)(Eu-Al) in space group P2(1)/n with a=11.913(6) Angstrom, b=14.051(9) Angstrom, c=17.920(9) Angstrom, alpha=101.88(11)degrees, beta=gamma=90 degrees, R=0.0509, R(w)=0.0471 and Z=2. The six CF3CO2- (including CF3CHO2-) of each complex, among which pairs are equivalent, coordinated to Ln and Al in three patterns: (A) the two oxygen atoms in one of the three CF3CO2- type coordinated to two different Ln; (B) the two oxygen atoms in the second of CF3CO2- type coordinated to Ln and Al, respectively; (C) one of the two oxygen atoms in the third CF3CO2- type bidentately coordinated to two Ln and another oxygen coordinated to Al and one of the two Ln, respectively. Unlike types A and B, in type C the carboxyl carbon with a hydrogen atom bonded to it was found to appear as an sp(3)-hybridized configuration rather than an sp(2)-one. 1D and 2D NMR results further confirmed the existence of such a disproportionated CF3CHO2- ligand. Methyl methacrylate (MMA) and epichlorohydrin (ECH) could be polymerized by Y-Al or Eu-Al as a single-component catalyst and highly syndiotactic poly(MMA) was obtained. THF could also be polymerized by Y-Al in the presence of a small amount of ECH.

Combined R-matrix eigenstate basis set and finite-difference propagation method for the time-dependent Schrodinger equation: The one-electron case

In this work we present the theoretical framework for the solution of the time-dependent Schrödinger equation (TDSE) of atomic and molecular systems under strong electromagnetic fields with the configuration space of the electron’s coordinates separated over two regions; that is, regions I and II. In region I the solution of the TDSE is obtained by an R-matrix basis set representation of the time-dependent wave function. In region II a grid representation of the wave function is considered and propagation in space and time is obtained through the finite-difference method. With this, a combination of basis set and grid methods is put forward for tackling multiregion time-dependent problems. In both regions, a high-order explicit scheme is employed for the time propagation. While, in a purely hydrogenic system no approximation is involved due to this separation, in multielectron systems the validity and the usefulness of the present method relies on the basic assumption of R-matrix theory, namely, that beyond a certain distance (encompassing region I) a single ejected electron is distinguishable from the other electrons of the multielectron system and evolves there (region II) effectively as a one-electron system. The method is developed in detail for single active electron systems and applied to the exemplar case of the hydrogen atom in an intense laser field.

Calculating the tetrahedral bond angle using spherical polars and the dot product

A proof using the methane tetrahedroid bond angle can be obtained by using spherical polar coordinates to calculate the Cartesian coordinates of the hydrogen atoms, then using the dot product of vectors.

Spherical, cylindrical and tetrahedral symmetries; hydrogenic states at high magnetic field in Si:P

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Finite-difference calculation of donor energy levels in a spherical quantum dot subject to a magnetic field

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On the electronic nature of low-barrier hydrogen bonds in enzymatic reactions

The electronic nature of low-barrier hydrogen bonds (LBHBs) in enzymatic reactions is discussed based on combined low temperature neutron and x-ray diffraction experiments and on high level ab initio calculations by using the model substrate benzoylacetone. This molecule has a LBHB, as the intramolecular hydrogen bond is described by a double-well potential with a small barrier for hydrogen transfer. From an “atoms in molecules” analysis of the electron density, it is found that the hydrogen atom is stabilized by covalent bonds to both oxygens. Large atomic partial charges on the hydrogen-bonded atoms are found experimentally and theoretically. Therefore, the hydrogen bond gains stabilization from both covalency and from the normal electrostatic interactions found for long, weak hydrogen bonds. Based on comparisons with other systems having short-strong hydrogen bonds or LBHBs, it is proposed that all short-strong and LBHB systems possess similar electronic features of the hydrogen-bonded region, namely polar covalent bonds between the hydrogen atom and both heteroatoms in question.

Metal-free graphitic carbon nitride as mechano-catalyst for hydrogen evolution reaction

Graphitic carbon nitride (g-C3N4), as a promising metal-free catalyst for photo-catalytic and electrochemical water splitting, has recently attracted tremendous research interest. However, the underlying catalytic mechanism for the hydrogen evolution reaction (HER) is not fully understood. By using density functional theory calculations, here we have established that the binding free energy of hydrogen atom (ΔGH∗0) on g-C3N4 is very sensitive to mechanical strain, leading to substantial tuning of the HER performance of g-C3N4 at different coverages. The experimentally-observed high HER activity in N-doped graphene supported g-C3N4 (Zheng et al., 2014) is actually attributed to electron-transfer induced strain. A more practical strategy to induce mechanical strain in g-C3N4 is also proposed by doping a bridge carbon atom in g-C3N4 with an isoelectronic silicon atom. The calculated ΔGH∗0 on the Si-doped g-C3N4 is ideal for HER. Our results indicate that g-C3N4 would be an excellent metal-free mechano-catalyst for HER and this finding is expected to guide future experiments to efficiently split water into hydrogen based on the g-C3N4 materials.

The perturbational treatment of the process of hydrogen abstraction by ketones

The change in energy during hydrogen abstraction by ketones is estimated for different electronic states as a function of the intermolecular orbital overlap employing perturbation theory. The results suggest that ketones preferentially undergo the in-plane reaction and abstract a hydrogen atom in their triplet nπ* state. For ketones where the triplet ππ* state lies below the triplet nπ* state, hydrogen abstraction can take place in the ππ* state owing to the crossing of the zero order reaction surfaces of the nπ* and ππ* states.

Completeness of the Coulomb wave functions in quantum mechanics

We give an explicit, direct, and fairly elementary proof that the radial energy eigenfunctions for the hydrogen atom in quantum mechanics, bound and scattering states included, form a complete set. The proof uses only some properties of the confluent hypergeometric functions and the Cauchy residue theorem from analytic function theory; therefore it would form useful supplementary reading for a graduate course on quantum mechanics.

Role of free valences and spin densities on the migrating atom in location of saddle points in radical-exchange reactions

The principle of the conservation of bond orders during radical-exchange reactions is examined using Mayer's definition of bond orders. This simple intuitive approximation is not valid in a quantitative sense. Ab initio results reveal that free valences (or spin densities) develop on the migrating atom during reactions. For several examples of hydrogen-transfer reactions, the sum of the reaction coordinate bond orders in the transition state was found to be 0.92 +/- 0.04 instead of the theoretical 1.00 because free valences (or spin densities) develop on the migrating atom during reactions. It is shown that free valence is almost equal to the square of the spin density on the migrating hydrogen atom and the maxima in the free valence (or spin density) profiles coincide (or nearly coincide) with the saddle points in the corresponding energy profiles.

Halogen Bonds in Crystal Engineering: Like Hydrogen Bonds yet Different

CONSPECTUS: The halogen bond is an attractive interaction in which an electrophilic halogen atom approaches a negatively polarized species. Short halogen atom contacts in crystals have been known for around 50 years. Such contacts are found in two varieties: type I, which is symmetrical, and type II, which is bent. Both are influenced by geometric and chemical considerations. Our research group has been using halogen atom interactions as design elements in crystal engineering, for nearly 30 years. These interactions include halogen center dot center dot center dot halogen interactions (X center dot center dot center dot X) and halogen center dot center dot center dot heteroatom interactions (X center dot center dot center dot B). Many X center dot center dot center dot X and almost all X center dot center dot center dot B contacts can be classified as halogen bonds. In this Account, we illustrate examples of crystal engineering where one can build up from previous knowledge with a focus that is provided by the modern definition of the halogen bond. We also comment on the similarities and differences between halogen bonds and hydrogen bonds. These interactions are similar because the protagonist atoms halogen and hydrogen are both electrophilic in nature. The interactions are distinctive because the size of a halogen atom is of consequence when compared with the atomic sizes of, for example, C, N, and O, unlike that of a hydrogen atom. Conclusions may be drawn pertaining to the nature of X center dot center dot center dot X interactions from the Cambridge Structural Database (CSD). There is a clear geometric and chemical distinction between type I and type II, with only type II being halogen bonds. Cl/Br isostructurality is explained based on a geometric model. In parallel, experimental studies on 3,4-dichlorophenol and its congeners shed light on the nature of halogen center dot center dot center dot halogen interactions and reveal the chemical difference between Cl and Br. Variable temperature studies also show differences between type I and type II contacts. In terms of crystal design, halogen bonds offer a unique opportunity in the strength, atom size and interaction gradation; this may be used in the design of ternary cocrystals. Structural modularity in which an entire crystal structure is defined as a combination of modules is rationalized on the basis of the intermediate strength of a halogen bond. The specific directionality of the halogen bond makes it a good tool to achieve orthogonality in molecular crystals. Mechanical properties can be tuned systematically by varying these orthogonally oriented halogen center dot center dot center dot halogen interactions. In a further development, halogen bonds are shown to play a systematic role in organization of LSAMs (long range synthon aufbau module), which are bigger structural units containing multiple synthons. With a formal definition in place, this may be the right time to look at differences between halogen bonds and hydrogen bonds and exploit them in more subtle ways in crystal engineering.

Photoelectron angular distributions from above threshold ionization of hydrogen atoms in strong laser fields

We apply a scattering theory of nonperturbative quantum electrodynamics to study the photoelectron angular distributions (PADs) of a hydrogen atom irradiated by linearly polarized laser light. The calculated PADs show main lobes and jetlike structure. Previous experimental studies reveal that in a set of above-threshold-ionization peaks when the absorbed-photon number increases by one, the jet number also increases by one. Our study confirms this experimental observation. Our calculations further predict that in some cases three more jets may appear with just one-more-photon absorption. With consideration of laser-frequency change, one less jet may also appear with one-more-photon absorption. The jetlike structure of PADs is due to the maxima of generalized phased Bessel functions, not an indication of the quantum number of photoelectron angular momentum states.

Átomos hidrogenoides em espaços com dimensionalidade D6=3: os casos não relativístico e relativístico

A possibilidade da existência de átomos de hidrogênio estáveis em dimensões superiores a três é abordada. O problema da dimensionalidade é visto como um problema de Física, no qual relacionam-se algumas leis físicas com a dimensão espacial. A base da análise deste trabalho faz uso das equações de Schrödinger (não relativística) e de Dirac (relativística). Nos dois casos, utiliza-se a generalização tanto do setor cinemático bem como o setor de interação coulombiana para variar o parâmetro topológico dimensão. Para o caso não relativístico, os auto-valores de energia e as auto-funções são obtidas através do método numérico de Numerov. Embora existam soluções em espaços com dimensões superiores, os resultados obtidos no presente trabalho indicam que a natureza deve, de alguma maneira, se manifestar em um espaço tridimensional.

Description of two new myxosporean species parasitic in freshwater fishes from the Yangtze River in China

Two new species of myxosporeans (Myxosporea: Myxidiidae), Myxidium tuanfengensis sp. n. and Zschokkella saurogobionis sp. n., Parasitic in freshwater fishes collected from the Yangtze River of China are described in this paper. M. tuanfengensis was found in the liver parenchyma and intestine lumen of Leptobotia taeniops Sauvage, 1878, while Z. saurogobionis was found in the gall bladder of Saurogobio dumerili Bleeker, 1871. The diagnostic characters of M. tuanfengensis are: round or elliptical polysporous plasmodia averaging 118 mum in size; spore oval in frontal view with smooth surface and nearly spindle-shape in sutural view with slightly sinuous sutural ridge, averaging 19.5 x 9.75 x 8.9 mum in size; two large spherical polar capsules 6.8 mum in diameter, with polar filament wound in 4 to 5 coils. The diagnostic characters of Z. saurogobionis are: spore elliptical in both frontal and sutural view measuring 18.3 x 9.8 x 10.8 mum in size; fine sutural ridge in S-form, spore shell marked with 10 to 12 distinct lines paralleled with the sutural line; two spherical polar capsules, 6.7 mum in diameter, with polar filament in 5 coils.