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 complete orthonormal sets for wave functions, and Slater orbitals of particles with arbitrary spin in coordinate, momentum and four-dimensional spaces

The new analytical relations of complete orthonormal sets for the tensor wave functions and the tensor Slater orbitals of particles with arbitrary spin in coordinate, momentum and four-dimensional spaces are derived using the properties of tensor spherical harmonics and complete orthonormal scalar basis sets of ψα-exponential type orbitals, ?α-momentum space orbitals and zα-hyperspherical harmonics introduced by the author for particles with spin s=0, where the . All of the tensor wave functions obtained are complete without the inclusion of the continuum and, therefore, their group of transformations is the four-dimensional rotation group O(4). The analytical formulas in coordinate space are also derived for the overlap integrals over tensor Slater orbitals with the same screening constant. We notice that the new idea presented in this work is the combination of tensor spherical harmonics of rank s with complete orthonormal scalar sets for radial parts of ψα-, ?α- and zα-orbitals, where .     

One-range Addition Theorems for Noninteger n Slater Functions using Complete Orthonormal Sets of Exponential Type Orbitals in Standard Convention

By the use of complete orthonormal sets of ${\psi ^{(\alpha^{\ast})}}$ -exponential type orbitals ( ${\psi ^{(\alpha^{\ast})}}$ -ETOs) with integer (for α ^* = α) and noninteger self-frictional quantum number α ^*(for α ^* ≠ α) in standard convention introduced by the author, the one-range addition theorems for ${\chi }$ -noninteger n Slater type orbitals ${(\chi}$ -NISTOs) are established. These orbitals are defined as follows $$\begin{array}{ll}\psi _{nlm}^{(\alpha^*)} (\zeta ,\vec {r}) = \frac{(2\zeta )^{3/2}}{\Gamma (p_l ^* + 1)} \left[{\frac{\Gamma (q_l ^* + )}{(2n)^{\alpha ^*}(n - l - 1)!}} \right]^{1/2}e^{-\frac{x}{2}}x^{l}_1 F_1 ({-[ {n - l - 1}]; p_l ^* + 1; x})S_{lm} (\theta ,\varphi )\\ \chi _{n^*lm} (\zeta ,\vec {r}) = (2\zeta )^{3/2}\left[ {\Gamma(2n^* + 1)}\right]^{{-1}/2}x^{n^*-1}e^{-\frac{x}{2}}S_{lm}(\theta ,\varphi ),\end{array}$$ where ${x=2\zeta r, 0<\zeta <\infty , p_l ^{\ast}=2l+2-\alpha ^{\ast}, q_l ^{\ast}=n+l+1-\alpha ^{\ast}, -\infty <\alpha ^{\ast} <3 , -\infty <\alpha \leq 2,_1 F_1 }$ is the confluent hypergeometric function and ${S_{lm} (\theta ,\varphi )}$ are the complex or real spherical harmonics. The origin of the ${\psi ^{(\alpha ^{\ast})} }$ -ETOs, therefore, of the one-range addition theorems obtained in this work for ${\chi}$ -NISTOs is the self-frictional potential of the field produced by the particle itself. The obtained formulas can be useful especially in the electronic structure calculations of atoms, molecules and solids when Hartree–Fock–Roothan approximation is employed.     

Theory of complete orthonormal sets of relativistic tensor wave functions and Slater tensor orbitals of particles with arbitrary spin in position, momentum and four-dimensional spaces

Using the complete orthonormal basis sets of nonrelativistic and quasirelativistic orbitals introduced by the author in previous papers for particles with arbitrary spin the new analytical relations for the 2(2s+1)-component relativistic tensor wave functions and Slater tensor orbitals in position, momentum and four-dimensional spaces are derived, where s=1/2,1,3/2,2,… . The relativistic tensor function sets are expressed through the corresponding nonrelativistic and quasirelativistic orbitals. The analytical formulas for overlap integrals over relativistic Slater tensor orbitals in position space are also derived.     

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.     

Use of combined Hartree–Fock–Roothaan theory in evaluation of lowest states of K[Ar]4s03d1 and Cr?+?[Ar]4s03d5 isoelectronic series over noninteger n-Slater type orbitals

By using noninteger n-Slater type orbitals in combined Hartree–Fock–Roothaan method, self-consistent field calculations of orbital and lowest states energies have been performed for the isoelectronic series of open shell systems K[Ar]4s ⁰3d ¹ (² D) (Z = 19–30) and Cr⁺[Ar]4s ⁰3d ⁵ (⁶ S) (Z = 24–30). The results of the calculations for the orbital and total energies obtained by using minimal basis-sets of noninteger n-Slater type orbitals are given in the tables. The results are compared with the extended-basis Hartree–Fock computations. The orbital and total energies are in good agreement with those presented in the literature. The results can be useful in the study of various properties of heavy atomic systems when the combined Hartree–Fock–Roothaan approach is employed.     

On Evaluation of Overlap Integrals with Noninteger Principal Quantum Numbers

By use of complete orthonormal sets of ψ^α exponential-type orbitals (ψ^α-ETOs,α=1,0,-1,-2,...) the series expansion formulas for the noninteger n Slater-type orbitals (NISTOs) in terms of integer n Slater-type orbitals (ISTOs) are derived. These formulas enable us to express the overlap integrals with NISTOs through the overlap integrals over ISTOs with the same and different screening constants. By calculating concrete cases the convergence of the series for arbitrary values of noninteger principal quantum numbers and screening constants of NISTOs and internuclear distances is tested. The accuracy of the results is quite high for quantum numbers, screening constants and location of STOs.     

New Quantization Method for Evaluation of Eigenenergies

A quantum dynamical equation is constructed as the limit of a sequence of functions (called Semiquantum momentum functions or SQMF). The quantum action variable J is defined as the limit of the sequence of contour integrals of SQMFs such that the quantization condition is J = n, where n is a nonnegative integer for eigenvalues and a noninteger for off eigenvalues. This quantization condition is exact and J is an analytic function of energy. Based on new definitions, an accurate numerical method is developed for obtaining eigenenergies. The method can be applied to both real and PT symmetric complex potentials. The validity and the accuracy of this new method is demonstrated with three illustrations.     

增加光子位移福克态的量子统计性质

引入了增加光子位移福克态 a^+m| α,n〉,其中 | α,n〉是位移福克态,m是整数.用数值计算的方法研究了这种态的量子统计性质.结果表明,在位移福克态上增加光子后可以使原有的光场呈现较强的压缩,并能使光场的亚泊松特性得到明显增强.     

Statistical complexity and Fisher-Shannon information in the H-atom

The Fisher-Shannon information and a statistical measure of complexity are calculated in the position and momentum spaces for the wave functions of the H-atom. For each level of energy, it is found that these two indicators take their minimum values on the orbitals that correspond to the highest orbital angular momentum.     

Spectral Gaps of Quantum Hall Systems with Interactions

A two-dimensional quantum Hall system without disorder for a wide class of interactions including any two-body interaction with finite range is studied by using the Lieb–Schultz–Mattis method [Ann. Phys. (N.Y.) 16:407 (1961)]. The model is defined on an infinitely long strip with a fixed, large width, and the Hilbert space is restricted to the lowest (n _max+1) Landau levels with a large integer n _max. We prove that, for a noninteger filling of the Landau levels, either (i) there is a symmetry breaking at zero temperature or (ii) there is only one infinite-volume ground state with a gapless excitation. We also prove the following two theorems: (a) If a pure infinite-volume ground state has a nonzero excitation gap for a noninteger filling , then a translational symmetry breaking occurs at zero temperature. (b) Suppose that there is no non-translationally invariant infinite-volume ground state. Then, if a pure infinite-volume ground state has a nonzero excitation gap, the filling factor must be equal to a rational number. Here the ground state is allowed to have a periodic structure which is a consequence of the translational symmetry breaking. We also discuss the relation between our results and the quantized Hall conductance, and phenomenologically explain why odd denominators of filling fractions giving the quantized Hall conductance are favored exclusively.     

Remarks on the Blow-up of the Euler Equations and the Related Equations

We consider the Euler system for inviscid incompressible fluid flows, and its perturbations in ⁿ, n2. We prove global well-posedness of this perturbed Euler system in the Triebel-Lizorkin spaces for initial vorticity which is small in the critical Besov norms. Comparison type theorems about the blow-up of solutions are proved between the Euler system and its perturbations. We also study the possiblity of the self-similar type of blow-up of solutions to the equations.     

探测二茂铁外价轨道(e,2e)反应中的扭曲波效应

利用第三代高效率电子动量谱仪,分别在600和1500 eV两种不同入射电子能量下获得了二茂铁(ferrocene)分子外价轨道的电离能谱和电子动量谱的相关实验结果.并利用非相对论与标量相对论密度泛函方法计算出了二茂铁的重叠型和交错型两种不同构象的理论动量谱.两种构象的外价轨道一一对应,理论电子动量谱基本一样.对二茂铁的外价轨道,在低动量区观测到了强烈的扭曲波效应,这与这些轨道主要由铁原子的3d轨道构成有关.通过相对论和非相对论计算结果的比较,表明相对论效应对于二茂铁的外轨道动量分布几乎没有影响. 关键词： 二茂铁分子 电子动量谱 相对论效应 扭曲波效应     

On the Evaluation of Two-Center Overlap Integrals over Integer and Noninteger n-Slater-Type Orbitals

In this study, two-center overlap integrals over Slater-type orbitals (STOs) with integer and noninteger principal quantum numbers in unaligned coordinate systems have been calculated using formulas for overlap integrals in aligned coordinate systems obtained by the author's previous work (T. Özdogan and M. Orbay, Int. J. Quant. Chem. 87 (2002) 15). The obtained results for integer case have been found to be in excellent agreement with the prior literature. On the other hand, to the best of authors knowledge, because of the scarcity of the literatures on the use of noninteger n-STOs in unaligned coordinate systems, the results for noninteger case have been tested with the limit of integer case, and good agreement has been obtained too.     

Remarks on potential spaces and Besov spaces in a Lipschitz domain and on Whitney arrays on its boundary

In this note, we propose to remove some small gaps in the theory of potential spaces H _p ^s (Ω) and Besov spaces B _p ^s (Ω), 1 < p < ∞, s ∈ ℝ, for a bounded Lipschitz domain Ω ⊂ ℝ ⁿ , n ⩾ 2. Namely, we discuss 1) the unified definitions of these spaces with s of any sign, the unified duality theorems and interpolation relations, 2) the possibility of constructing a function in these spaces with given array of traces of its derivatives on the boundary. To the memory of Leonid Romanovich Volevich The work was partially supported by the RFBR grant no. 07-01-00287.     

The all configuration mean energy multiconfiguration self-consistent-field method. I. Equal configuration weights

ABSTRACT

The All Configuration Mean Energy (ACME) conditions are a special case of state averaging for Multiconfigurational Self-Consistent-Field (MCSCF) orbital optimisation. The method is formulated using the Graphical Unitary Group Approach (GUGA) in which the Configuration State Function (CSF) basis is represented as walks within a Shavitt graph. This graphical formulation leads to efficient recursive algorithms for the energy and reduced density matrices (RDM) that are independent of the CSF dimension and that scale only as O(n²) where n is the number of occupied orbitals. The Hamiltonian matrix diagonalization step is obviated and the CSF expansion coefficients are neither referenced nor required during the orbital optimisation. This allows MCSCF orbital optimisation to be performed for essentially unlimited numbers of active orbitals and arbitrarily large CSF expansions. The discussion includes various types of CSF expansion spaces, the partitioning of the essential and redundant orbital optimisation parameters, the computation of the spin-density, and the formulation of state-specific analytic gradients and nonadiabatic coupling for high-level electronic structure methods that use the ACME MCSCF orbitals.     

Einstein metrics onS 3,R 3 andR 4 bundles

Starting from a 4n-dimensional quaternionic Kähler base space, we construct metrics of cohomogeneity one in (4n+3) dimensions whose level surfaces are theS ² bundle space of almost complex structures on the base manifold. We derive the conditions on the metric functions that follow from imposing the Einstein equation, and obtain solutions both for compact and non-compact (4n+3)-dimensional spaces. Included in the non-compact solutions are two Ricci-flat 7-dimensional metrics withG ₂ holonomy. We also discuss two other Ricci-flat solutions, one on theR ⁴ bundle overS ³ and the other on anR ⁴ bundle overS ⁴. These haveG ₂ and Spin(7) holonomy respectively.     

Adiabatic Times for Markov Chains and Applications

We state and prove a generalized adiabatic theorem for Markov chains and provide examples and applications related to Glauber dynamics of the Ising model over ℤ ^d /nℤ ^d . The theorems derived in this paper describe a type of adiabatic dynamics for l¹(\mathbbRⁿ₊)\ell^{1}(\mathbb{R}^{n}_{+}) norm preserving, time inhomogeneous Markov transformations, while quantum adiabatic theorems deal with ℓ ²(ℂ ⁿ ) norm preserving ones, i.e. gradually changing unitary dynamics in ℂ ⁿ .     

Conformally compactified homogeneous spaces. Possible observable consequences

Some arguments, based on the possible spontaneous violation of the cosmological principle (represented by the observed large-scale structures of galaxies), on the Cartan geometry of simple spinors, and on the Fock formulation of hydrogen atom wave equation in momentum space, are presented in favor of the hypothesis that space-time and momentum space should be both conformally compactified and should both originate from the two four-dimensional homogeneous spaces of the conformai group, both isomorphic (S ³ ×S ¹)/Z ₂ and correlated by conformal inversion, but should not necessarily be identified with them. Within this framework, the possible common origin for the SO(4) symmetry underlying the geometrical structure of the Universe, of Kepler orbits, and of the H atom is discussed. One of the consequences of the proposed hypothesis could be that any quantum field theory should be naturally free from both infrared and ultraviolet divergences. But then physical spaces defined as those where physical phenomena may be best described through some set of fields, could be different from those homogeneous spaces. A simple, exactly soluble, toy model, valid for a two-dimensional spacetime, is presented where the conjectured conformally compactified homogeneous spaces are both isomorphic to (S ¹ ×S ¹)/Z ₂,while the possible physical spaces could be two finite lattices which are dual since Fourier transforms, represented by finite, discrete, sums, may be well defined on them.     

Fluctuations of the Nodal Length of Random Spherical Harmonics

Using the multiplicities of the Laplace eigenspace on the sphere (the space of spherical harmonics) we endow the space with Gaussian probability measure. This induces a notion of random Gaussian spherical harmonics of degree n having Laplace eigenvalue E = n(n + 1). We study the length distribution of the nodal lines of random spherical harmonics.