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 spaces

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(Ψα-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-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.     

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.     

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.     

Experimental and calculated momentum densities for the complete valence orbitals of the antimicrobial agent diacetyl

The first electronic structural study of the complete valence shell binding energy spectra of the antimicrobial agent diacetyl, encompassing both the outer and inner valence regions, is reported. The binding energy spectra as well as the individual orbital momentum profiles have been measured by using a high resolution （e, 2e） electron momentum spectrometer （EMS） at an impact energy of 1200eV plus the binding energy, and using symmetric noncoplanar kinematics. The experimental orbital electron momentum profiles are compared with self-consistent field （SCF） theoretical profiles calculated using the Hartree-Fock approximation and Density Functional theory predictions in the target Kohn-Sham approximation which includes some treatment of correlation via the exchange and correlation potentials with a range of basis sets. The pole strengths of the main ionization peaks from the inner valence orbitals are estimated.     

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.     

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.     

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.     

Orbital responses to methyl sites in CnH2n＋2 （n= 1-6）

Orbital responses to methyl sites in C_nH_2n+2 （n=1-6） are studied by B3LYP/TZVP based on the most stable geometries using the B3LYP/aug-cc-pVTZ method.Vertical ionization energies are produced using the SAOP/et-pVQZ model for the complete valence space.The highest occupied molecular orbital （HOMO） investigations indicate the pelectron profiles in methane,ethane,propane,and n-butane.By increasing the number of carbon-carbon bonds in lower momentum regions,the s,p-hybridized orbitals are built and display strong exchange and correlation interactions in lower momentum space （P<0.50 a.u.）.Meanwhile,the relative intensities of the isomers in lower momentum space show the strong bonding number dependence of the carbon-carbon bonds,meaning that more electrons have contributed to orbital construction.The study of representative valence orbital momentum distribution further confirms that the structural changes lead to evident electronic rearrangement over the whole valence space.An analysis based on the isomers reveals that the valence orbitals are isomer-dependent and the valence ionization energy experiences an apparent shift in the inner valence space.However,such shifts are greatly reduced in the outer valence space.Meanwhile,the opposite energy shift trend is found in the intermediate valence space.     

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 momentum spectroscopy of NF3

The electronic structure of nitrogen trifluoride was investigated by combining the high-resolution electron momentum spectroscopy with the high-level calculations. The experimental binding energy spectra and the momentum distributions of each orbital were compared with the results of Hartree-Fock, density functional theory （DFT）, and symmetry-adapted- cluster configuration-interaction （SAC-CI） methods. SAC-CI and DFT-B3LYP with the aug-cc-pVTZ basis set can well reproduce the binding energy spectra and the observed momentum distributions of the valence orbitals except 1 a2 and 4e orbitals. It was found that the calculated momentum distributions using DFT-B3LYP are even better than those using the high-level SAC-CI method.     

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.     

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.     

Ab initio investigation of the structural and unusual electronic properties of α-CuSe （klockmannite）

In this article, a computational analysis has been performed on the structural properties and predominantly on the electronic properties of the α-CuSe （klockmannite） using density functional theory. The studies in this work show that the best structural results, in comparison to the experimental values, belong to the PBEsol-GGA and WC-GGA functionals. However, the best results for the bulk modulus and density of states （DOSs） are related to the local density approximation （LDA） functional. Through utilized approaches, the LDA is chosen to investigate the electronic structure. The results of the electronic properties and geometric optimization of α-CuSe respectively show that this compound is conductive and non-magnetic. The curvatures of the energy bands crossing the Fermi level explicitly reveal that major charge carriers in CuSe are holes, whose density is estimated to be 0.86×1022 hole/cm3. In particular, the Fermi surfaces in the first Brillouin zone demonstrate interplane conductivity between （001） planes. Moreover, the charge carriers among them are electrons and holes simultaneously. The conductivity in CuSe is mainly due to the hybridization between the d orbitals of Cu atoms and the p orbitals of Se atoms. The former orbitals have the dual nature of localization and itinerancy.     

Exact Solution of D-Dimensional Klein-Gordon Oscillator with Minimal Length

Specific modifications of the usual canonical commutation relations between position and momentum operators have been proposed in the literature in order to implement the idea of the existence of a minimal observable length. Here, we study a consequence of this proposal in relativistic quantum mechanics by solving in the momentum space representation the Klein-Gordon oscillator in arbitrary dimensions. The exact bound states spectrum and the corresponding momentum space wave function are obtained. Following Chang et al, （Phys. Rev. D 65 （2002） 125027）, we discuss constraint that can be placed on the minimal length by measuring the energy levels of an electron in a penning trap.     

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.