One-range addition theorems for generalized integer and noninteger μ Coulomb,and exponential type correlated interaction potentials with hyperbolic cosine in position,momentum,and four-dimensional

The formulae are established in position,momentum,and four-dimensional spaces for the one-range addition theorems of generalized integer and noninteger μ Coulomb,and exponential type correlated interaction potentials with hyperbolic cosine（GCTCP and GETCP HC）.These formulae are expressed in terms of one-range addition theorems of complete orthonormal sets of Ψα-exponential type orbitals（Ψ α-ETO）,α-momentum space orbitals（α-MSO）,and zα-hyperspherical harmonics（zα-HSH） introduced.The one-range addition theorems obtained can be useful in the electronic structure calculations of atoms and molecules when the GCTCP and GETCP HC in position,momentum,and four-dimensional spaces are employed.     

Unified treatment of one-range addition theorems for integer and non-integer n-STO, -GTO and -generalized exponential type orbitals with hyperbolic cosine in position, momentum and four-dimensional spaces

Simpler formulas are derived for one-range addition theorems for the integer and noninteger n generalized exponential type orbitals, momentum space orbitals, and hyperspherical harmonics with hyperbolic cosine (GETO HC, GMSO HC, and GHSH HC) in position, momentum and four-dimensional spaces, respectively. The final results are expressed in terms of one-range addition theorems of complete orthonormal sets of ψα -exponential type orbitals, α - momentum space orbitals and z α -hyperspherical harmonics. We notice that the one-range addition theorems for integer and noninteger n-Slater type orbitals and Gaussian type orbitals in position, momentum and four dimensional spaces are special cases of GETO HC, GMSO HC, and GHSH HC. The theorems presented can be useful in the accurate study of the electronic structure of atomic and molecular systems.     

One-Range Addition Theorems in Terms of ψ^α -ETOs for STOs and Coulomb-Yukawa Like Correlated Interaction Potentials of Integer and Noninteger Indices

The expansion formulas in terms of complete orthonormal sets of ψ^α -exponential type orbitals are established for the Slater type orbitals and Coulomb-Yukawa-like correlated interaction potentials of integer and noninteger indices. These relations are used in obtaining their unsymmetrical and symmetrical one-range addition theorems. The final results are especially useful in the calculations of multicentre multielectron integrals occurring when Hartree-Fock-Roothaan and explicitly correlated methods are employed.     

Calculation of multicentre nuclear attraction integrals over Slater-type orbitals using unsymmetrical one-range addition theorems

Using the unsymmetrical one-range addition theorems introduced by one of the authors with the help of complete orthonormal sets of Ψα-exponential type orbitals(α = 1,0,1,2,...),this paper presents the sets of series expansion relations for multicentre nuclear attraction integrals over Slater-type orbitals arising in Hartree-Fock-Roothaan equations for molecules.The final results are expressed through multicentre charge density expansion coefficients and basic integrals.The convergence of the series is tested by calculating concrete cases for arbitrary values of parameters of orbitals.     

Methyl orbital signatures in 2-amino-1-propanol

Electron density distributions of 2-aminoethanol (2AE) and 2-amino-1-propanol (2AP) are calculated in both the coordinate and the momentum spaces using the B3LYP/TZVP method. Using the dual space analysis, molecular orbital signatures of the methyl substituent in 2AP are identified with respect to 2AE. Relaxations of the geometry and the valence orbital in 2AP are found to be due to the insertion of the methyl group. Five orbitals, not four orbitals, are identified as the methyl signatures. They are orbital 5a in the core shell, orbitals 9a and 10a in the inner valence shell, and orbitals 15a and 16a in the outer valence. In the inner valence shell, the attachment of methyl to 2AE causes a splitting of its orbital 8a into orbitals 9a and 10a of 2AP, whereas in the outer valence shell, the methyl group results in the insertion of an additional orbital pair of 15a and 16a. The frontier molecular orbitals 21a, 20a, and 19a are found to have no significant role in the methylation of 2AE.     

Fisher and Shannon information entropies for a noncentral inversely quadratic plus exponential Mie-type potential

In this work, we determine the Fisher and Shannon entropies, the expectation values and the squeeze state for a noncentral inversely quadratic plus exponential Mie-type potential analytically.The proposed potential is solved under the Schr?dinger equation using a special Greene Aldrich approximation to the centrifugal term to obtain a normalised wave function within the framework of the Nikiforov–Uvarov method. Numerical results are obtained for different screening parameters:α?=?0.1, 0.12 and 0.13 for varying real constant parameter(B). The numerical solutions are obtained only for ground state. The numerical results of Fisher entropy both for position and momentum spaces are in good agreement with existing literature. The normalisation constant, wave function, and probability density plots are carried out using a well designed Mathematica algorithm.The Fourier transform of position space entropy gives the momentum space entropy.     

Methyl orbital signatures in 2-amino-l-propanol

Electron density distributions of 2-aminoethanol （2AE） and 2-amino-l-propanol （2AP） are calculated in both the coordinate and the momentum spaces using the B3LYP/TZVP method. Using the dual space analysis, molecular orbital signatures of the methyl substituent in 2AP are identified with respect to 2AE. Relaxations of the geometry and the valence orbital in 2AP are found to be due to the insertion of the methyl group. Five orbitals, not four orbitals, are identified as the methyl signatures. They are orbital 5a in the core shell, orbitals 9a and 10a in the inner valence shell, and orbitals 15a and 16a in the outer valence. In the inner valence shell, the attachment of methyl to 2AE causes a splitting of its orbital 8a into orbitals 9a and 10a of 2AP, whereas in the outer valence shell, the methyl group results in the insertion of an additional orbital pair of 15a and 16a. The frontier molecular orbitals 21a, 20a, and 19a are found to have no significant role in the methylation of 2AE.     

Simulating electron momentum spectra of iso-C2H2Cl2： A study of the core electronic structure

Electron momentum spectroscopy(EMS) has been used for the first time to study core electronic structure of isoC2H2Cl2. In the present work, the pronounced difference between ionization energies of two C1 score orbitals(2A1 and 3A1) is seen as a chemical shift of 3 eV, which is due to different chemical environments of the related carbon atoms. Both the calculated spherically averaged core electron momentum distributions(MDs) and three-dimensional electron momentum density maps show that these core molecular orbitals(MOs) 2A1and 3A1 exhibit strong atomic orbital characteristics in real and momentum space. However, the core states 2B2 and 4A1, which are almost degenerate and related to two equivalent atoms, exhibit notable differences between the momentum and position depictions. In contrast to the position space, the momentum density maps of these two core MOs highlight the interference effects which are due to the nuclear positions. The 2B2 orbital of iso-C2H2Cl2 is the antisymmetric counterpart of the 4A1 core orbital in real space. However, it relates to the 4A1 orbital by an exchange of maxima and minima in momentum space. Due to interference effects between electrons scattered from different atomic centers, modulations with a periodicity of 1.12 a.u. can be seen in the computed momentum densities, which tend to decay with increasing electron momenta. Accordingly, the EMS can not only effectively image the electronic structure of compounds by studying valence orbitals, but also provides direct information on the nature of the nuclear geometry by investigating the core states.     

Exact solutions for nonlinear Schr?dinger equation in phase space: applications to Bose－Einstein condensate

The stationary-state nonlinear Schr?dinger equation, which models the dilute-gas Bose－Einstein condensate, is introduced within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. The exact solutions of equation are obtained in the phase space, by means of the wave-mechanics method. The eigenfunctions in position and momentum spaces are obtained through the ‘Fourier-like' projection transformation from the phase space eigenfunctions. The eigenfunction with a hypersecant part is discussed as an example.     

Electron Momentum Spectroscopy of the Frontier Orbitals of Chlorodifluoromethane

We report on the first measurement of the electron momentum distributions of the three outermost valence orbitals for chorodifluoromethane(CHF2Cl)by binary(e,2e) electron momentum spectroscopy.The experimental data are compared with Hartree-Fock and density functional theory(DFT) calculations employing 6-31G,6-311 G^** and AUG-cc-pVQZ basis sets.For the summed momentum distribution of 8α′ 5α″ 7α′ orbitals,the DFT/.AUG-cc-pVQZ calculation gives the best fit.A very large and diffuse basis set,AUG-cc-pVQZ,is employed in the calculations to approach the Hartree-Fock limit of the basis set,but the improvement of the calculation quality is little in comparison with that calculated with the 6-311 G^** basis set,This indicates that the 6-311 G^** basis set is nearly saturated for the calculations of these three orbitals of CHF2Cl,and it is unnecessary to employ a larger basis set in the calculations.     

Conformation effects on the molecular orbitals of serine

This paper calculates the five most stable conformers of serine with Hartree--Fock theory, density functional theory (B3LYP), Moller--Plesset perturbation theory (MP4(SDQ)) and electron propagation theory with the 6-311++G(2d,2p) basis set. The calculated vertical ionization energies for the valence molecular orbitals of each conformer are in agreement with the experimental data, indicating that a range of molecular conformations would coexist in an equilibrium sample. Information of the five outer valence molecular orbitals for each conformer is explored in coordinate and momentum spaces using dual space analysis to investigate the conformational processes, which are generated from the global minimum conformer Ser1 by rotation of C₂--C₃ (Ser4), C₁--C₂ (Ser5) and C₁--O₂ (Ser2 and Ser3). Orbitals 28a, 27a and 26a are identified as the fingerprint orbitals for all the conformational processes.     

Construction of Different Kinds of Atomic and Molecular Orbitals Using Complete Orthonormal Sets of ψ^α-ETO in Single Exponent Approximation

Using complete orthonormal sets of ψ^α-exponential type orbitals in single exponent approximation the new approach has been suggested for construction of different kinds of functions which can be useful in the theory of linear combination of atomic orbitals. These functions can be chosen properly according to the nature of the problems under consideration. This is rather important because the choice of the basis set may be play a crucial role in applications to atomic and molecular problems. As an example of application, different atomic orbitals for the ground states of the neutral and the first ten cationic members of the isoelectronic series of He atom are constructed by the solution of Hartree-Fock-Roothaan equations using ψ^1, ψ^0 and ψ^-1 basis sets. The cMculated results are close to the numerical Hartree-Fock values. The total energy, expansion coefficients, orbital exponents and virial ratio for each atom are presented.     

Electron impact ionization of neon and neonic ions under distorted-wave Born approximation

The （e, 2e） triple differential cross sections of 2s orbitals of neon and neonic ions （Z = 11-14） are calculated using a distorted-wave Born approximation under coplanar asymmetric geometry. The calculated results show that, with the increase in the nuclear charge number Z, the amplitude of triple differential cross sections decreases. The angle difference between the binary peak position and the direction of momentum transfer gradually increases with the increase in the nuclear charge Z, and a new structure appears at an ejected angle 90° 〈 θ2 〈 120°. Three kinds of collision processes are proposed to illustrate the formation mechanism of such collision peaks.     

Nucleon internal structure: a new set of quark, gluon momentum, angular momentum operators and parton distribution functions

It is unavoidable to deal with the quark and gluon momentum and angular momentum contributions to the nucleon momentum and spin in the study of nucleon internal structure. However we never have the quark and gluon momentum, orbital angular momentum and gluon spin operators which satisfy both the gauge invariance and the canonical momentum and angular momentum commutation relation. The conflicts between the gauge invariance and canonical quantization requirement of these operators are discussed. A new set of quark and gluon momentum, orbital angular momentum and spin operators, which satisfy both the gauge invariance and canonical momentum and angular momentum commutation relation, are proposed. The key point to achieve such a proper decomposition is to separate the gauge field into the pure gauge and the gauge covariant parts. The same conflicts also exist in QED and quantum mechanics and have been solved in the same manner. The impacts of this new decomposition to the nucleon internal structure are discussed.     

Addition Formulae of Discrete KP,q-KP and Two-Component BKP Systems

In this paper,we construct the addition formulae for several integrable hierarchies,including the discrete KP,the q-deformed KP,the two-component BKP and the D type Drinfeld–Sokolov hierarchies.With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies,we prove that these addition formulae are equivalent to these hierarchies.These studies show that the addition formula in the research of the integrable systems has good universality.     

Geodesics of Spherical Dilaton Spacetimes

The properties of spherical dilaton black hole spacetimes are investigated through a study of their geodesics. The closed and non-closed orbits of test particles are analysed using the effective potential and phase-plane method. The stability and types of orbits are determined in terms of the energy and angular momentum of the test particles. The conditions of the existence of circular orbits for a spherical dilaton spacetime with an arbitrary dilaton coupling constant α are obtained. The properties of the orbits and in particular the position of the innermost stable circular orbit are compared to those of the Reissner-Nordstrom spacetime. The circumferential radius of innermost stable circular orbit and the corresponding angular momentum of the test particles increase for α≠ 0.     

Exact solution of the one-dimensional Klein-Gordon equation with scalar and vector linear potentials in the presence of a minimal length

Using the momentum space representation, we solve the Klein-Gordon equation in one spatial dimension for the case of mixed scalar and vector linear potentials in the context of deformed quantum mechanics characterized by a finite minimal uncertainty in position. The expressions of bound state energies and the associated wave functions are exactly obtained.     

Five-Dimensional Cosmological Model with Variable G and A

Einstein＇s field equations with C and A both varying with time are considered in the presence of a perfect fluid for five-dimensional cosmological model in a way which conserves the energy momentum tensor of the matter content. Several sets of explicit solutions in the five-dimensional Kaluza-Klein type cosmological models with variable G and A are obtained. The diminishment of extra dimension with the evolution of the universe for the five-dimensional model is exhibited. The physical properties of the models are examined.     

Hamiltonian analysis of 4-dimensional spacetime in Bondi-like coordinates

We discuss the Hamiltonian formulation of gravity in four-dimensional spacetime under Bondi-like coordinates {v,r,x~a,a=2,3}. In Bondi-like coordinates, the three-dimensional hypersurface is a null hypersurface, and the evolution direction is the advanced time v. The internal symmetry group SO(1,3) of the four-dimensional spacetime is decomposed into SO(1,1), SO(2), and T~±(2), whose Lie algebra so(1,3) is decomposed into so(1,1), so(2), and t~±(2) correspondingly. The SO(1,1) symmetry is very obvious in this type of decomposition, which is very useful in so(1,1) BF theory. General relativity can be reformulated as the four-dimensional coframe(e_μ~I) and connection(ω_μ~(IJ))dynamics of gravity based on this type of decomposition in the Bondi-like coordinate system. The coframe consists of two null 1-forms e~-, e~+and two spacelike 1-forms e~2, e~3. The Palatini action is used. The Hamiltonian analysis is conducted by Dirac's methods. The consistency analysis of constraints has been done completely. Among the constraints, there are two scalar constraints and one two-dimensional vector constraint. The torsion-free conditions are acquired from the consistency conditions of the primary constraints about π_(IJ)~μ. The consistency conditions of the primary constraints π_(IJ)~0=0 can be reformulated as Gauss constraints. The conditions of the Lagrange multipliers have been acquired. The Poisson brackets among the constraints have been calculated. There are 46 constraints including 6 first-class constraints π_(IJ)~0=0 and 40 second-class constraints. The local physical degrees of freedom is 2.The integrability conditions of Lagrange multipliers n_0, l_0, and e_0~A are Ricci identities. The equations of motion of the canonical variables have also been shown.     

Experimental and Theoretical Investigation of Valence Orbitals in 1,4-Dioxane by Electron momentum Spectroscopy

The binding energy spectrum of all valence orbitals and the momentum distributions of highest occupied molecular orbital （HOMO： 8ag）, 7bu＋7ag, 4bu, 2bg ＋4ag and 2au in 1, 4-dioxane are investigated by electron momentum spectroscopy （EMS） with 600 eV impact energy. The experimental results are consistent with theoretical calcula- tions of C2h chair conformation using the Hartree-Fock method and density functional theory with 6-311＋＋G^＊＊ and AUG-CC-PVTZ basis sets.