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 $／varPsi ^／alpha $-exponential type orbitals （$／alpha = 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.

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（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 ex- ponential 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 Ca-exponential type orbitals, Ca- momentum space orbitals and za-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.

Analytical Study on Wave Diffraction from a Vertical Circular Cylinder in Front of Orthogonal Vertical Walls

In this paper, the principle of mirror image is used to transform the problem of wave diffraction from a circular cylinder in front of orthogonal vertical walls into the problem of diffraction of four symmetric incident waves from four symmetrically arranged circular cylinders, and then the eigenfunction expansion of velocity potential and Grafs addition theorem are used to give the analytical solution to the wave diffraction problem. The relation of the total wave force on cylinder to the distance between the cylinder and orthogonal vertical walls and the incidence angle of wave is also studied by numerical computation.

Some Remarks for the Relationships between the Generalized Bernoulli and Euler Polynomials

In this paper,we prove the Srivastava-Pint'er's addition theorems(see Applied Mathematic Lett.17(2004),375-380) by applying the another methods.We also provide some analoges of these addition theorems and deduce the corresponding special cases.

A Low Frequency Vector Fast Multipole Algorithm with Vector Addition Theorem

In the low-frequency fast multipole algorithm(LF-FMA)[19,20],scalar addition theorem has been used to factorize the scalar Green’s function.Instead of this traditional factorization of the scalar Green’s function by using scalar addition theorem,we adopt the vector addition theorem for the factorization of the dyadic Green’s function to realize memory savings.We are to validate this factorization and use it to develop a low-frequency vector fast multipole algorithm(LF-VFMA)for lowfrequency problems.In the calculation of non-near neighbor interactions,the storage of translators in the method is larger than that in the LF-FMA with scalar addition theorem.Fortunately it is independent of the number of unknowns.Meanwhile,the storage of radiation and receiving patterns is linearly dependent on the number of unknowns.Therefore it is worthwhile for large scale problems to reduce the storage of this part.In this method,the storage of radiation and receiving patterns can be reduced by 25 percent compared with the LF-FMA.

Diagonalizations of Vector and Tensor Addition Theorems

Based on the generalizations of the Funk-Hecke formula and the Rayleigh plan-wave expansion formula,an alternative and succinct derivation of the addition theorem for general tensor field is obtained.This new derivation facilitates the diagonalization of the tensor addition theorem.In order to complete this derivation,we have carried out the evaluation of the generalization of the Gaunt coefficient for tensor fields.Since vector fields(special case of tensor fields)are very useful in practice,we discuss vector multipole fields and vector addition theorem in details.The work is important in multiple scattering and fast algorithms in wave physics.

Vector Addition Theorem and Its Diagonalization

The conventional vector addition theorem is written in a compact notation.Then a new and succinct derivation of the vector addition theorem is presented thatis as close to the derivation of the scalar addition theorem. Newly derived expressionsin this new derivation are used to diagonalize the vector addition theorem. Thediagonal form of the vector addition theorem is important in the design of fast algorithmsfor computational wave physics such as computational electromagnetics andcomputational acoustics.

L^2-Decay rate for non-ergodic Jackson network

We establish the additive theorem of L^2-decay rate for multi- dimensional Markov process with independent marginal processes. Using this and the decomposition method, we obtain explicit upper and lower bounds for decay rate of non-ergodic Jackson network. In some cases, we get the exact decay rate.

Enhanced solution to the surface-volume-surface EFIE for arbitrarymetal-dielectric composite objects

The surface–volume–surface electric field integral equation(SVS-EFIE)can lead to complex equations,laborious implementation,and unacceptable computational complexity in the method of moments(MoM).Therefore,a general matrix equation(GME)is proposed for electromagnetic scattering from arbitrary metal–dielectric composite objects,and its enhanced solution is presented in this paper.In previous works,MoM solution formulation of SVSEFIE considering only three-region metal–dielectric composite scatters was presented,and the two-stage process resulted in two integral operators in SVS-EFIE,which is arduous to implement and is incapable of reducing computational complexity.To address these difficulties,GME,which is versatile for homogeneous objects and composite objects consisting of more than three sub-regions,is proposed for the first time.Accelerated solving policies are proposed for GME based on coupling degree concerning the spacing between sub-regions,and the coupling degree standard can be adaptively set to balance the accuracy and efficiency.In this paper,the reformed addition theorem is applied for the strong coupling case,and the iterative method is presented for the weak coupling case.Parallelism can be easily applied in the enhanced solution.Numerical results demonstrate that the proposed method requires only 11.6%memory and 11.8%CPU time on average compared to the previous direct solution.