Calculation of magnetic multipole moment integrals using translation formulas for Slater-type orbitals

Using translation formulas for Slater type orbitals (STO’s) the infinite series through the overlap integrals are derived for the magnetic multipole moment integrals. By the use of the derived expressions the magnetic multipole moment integrals, therefore, the magnetic properties of molecules can be evaluated most efficiently and accurately. The convergence of the series is tested by calculating concrete cases. An accuracy of 10⁻⁵ for the computer results is obtained in the case 2 ^p -pole magnetic moment integrals for 1 ≤ v ≤ 5, and for the arbitrary values of internuclear distances and screening constants of atomic orbitals.     

Construction of the two-electron contribution to the Fock matrix by numerical integration

A novel method to numerically calculate the Fock matrix is presented. The Coulomb operator is re-expressed as an integral identity, which is discretized. The discretization of the auxiliary t dimension separates the x, y, and z dependencies transforming the two-electron Coulomb integrals of Gaussian-type orbitals (GTO) to a linear sum of products of two-dimensional integrals. The s-type integrals are calculated analytically and integrals of the higher angular-momentum functions are obtained using recursion formulae. The contributions to the two-body Coulomb integrals obtained for each discrete t value can be evaluated independently. The two-body Fock matrix elements can be integrated numerically, using common sets of quadrature points and weights. The aim is to calculate Fock matrices of enough accuracy for electronic structure calculations. Preliminary calculations indicate that it is possible to achieve an overall accuracy of at least 10^?12 E _h using the numerical approach.     

Kirkwood–Buff integrals of finite systems: shape effects

The Kirkwood–Buff (KB) theory provides an important connection between microscopic density fluctuations in liquids and macroscopic properties. Recently, Krüger et al. derived equations for KB integrals for finite subvolumes embedded in a reservoir. Using molecular simulation of finite systems, KB integrals can be computed either from density fluctuations inside such subvolumes, or from integrals of radial distribution functions (RDFs). Here, based on the second approach, we establish a framework to compute KB integrals for subvolumes with arbitrary convex shapes. This requires a geometric function w(x) which depends on the shape of the subvolume, and the relative position inside the subvolume. We present a numerical method to compute w(x) based on Umbrella Sampling Monte Carlo (MC). We compute KB integrals of a liquid with a model RDF for subvolumes with different shapes. KB integrals approach the thermodynamic limit in the same way: for sufficiently large volumes, KB integrals are a linear function of area over volume, which is independent of the shape of the subvolume.     

Upon the utility of spherical harmonic expansions in the evaluation of cluster integrals

A novel procedure for the analytic evaluation of cluster integrals is given. By means of a result of Silverstone and Moats which transforms the spherical harmonic expansion of a function around a given point into a new spherical harmonic expansion around a displaced point, a 3N-dimensional cluster integral forN point particles (N > 2) may be reduced to 2N+1 trivial integrals andN– 1 interesting integrals, an improvement over the usual reduction to six trivial integrals and3N–6 nontrivial integrals. For hard spheres, theN–1 integrals involve only a series of simple polynomials taken between linear algebraic bounds.This work was supported in part by the National Science Foundation under Grant No. CHE79-20389.     

Determination of consistent semiempirical one-centre integrals based on coupled-cluster theory

ABSTRACT

We examine the one-centre integrals used in semiempirical molecular orbital theory, for the elements H–Ne. The currently used parameters do not provide good estimates for the relative energies of ionised states of atoms. Directly calculating the one-electron integrals U _ss and U _pp with coupled-cluster theory and fitting the two-electron repulsion integrals G _ss and G _pp to accurate coupled-cluster ionisation curves improves this behaviour. Since all the remaining parameters can be derived from these, the number of fitted variables is reduced from seven to two. The two-parameter model provides qualitative agreement with coupled-cluster theory for all ionisation potentials (IPs) and the principal electron affinity (EA). To obtain quantitative agreement for the principal IP and EA, U _ss and U _pp are included as variables in a four-parameter model. We discuss the new parameters and implications for the development of new, consistent semiempirical Hamiltonians.     

Convergence of spherical harmonic expansions for the evaluation of hard-sphere cluster integrals

ForN particles (N>2), by means of a spherical harmonic expansion of Silverstone and Moats, a 3N-dimensional cluster may be reduced to 2N+1 trivial integrals andN–1 interesting integrals. For hard spheres, theN–1 interesting integrals are products of polynomials integrated between binomial bounds. With simple clusters, closed forms are obtained; for more complex clusters, infinite series inl (ofY _lm ) appear. It is here shown for representative cases that these series converge exponentially rapidly, the leading pair of terms accounting for all but a few tenths of a percent of the total cluster integral.     

Double exponential transformation for computing three-center nuclear attraction integrals

ABSTRACT

Three-center nuclear attraction integrals, which arise in density functional and ab initio calculations, are one of the most time-consuming computations involved in molecular electronic structure calculations. Even for relatively small systems, millions of these laborious calculations need to be executed. Highly efficient and accurate methods for evaluating molecular integrals are therefore all the more vital in order to perform the calculations necessary for large systems. When using a basis set of B functions, an analytical expression for the three-center nuclear attraction integrals can be derived via the Fourier transform method. However, due to the presence of the highly oscillatory semi-infinite spherical Bessel integral, the analytical expression still remains problematic. By applying the S transformation, the spherical Bessel integral can be converted into a much more favorable sine integral. In the present work, we then apply two types of double exponential transformations to the resulting sine integral, which leads to a highly efficient and accurate quadrature formula. This method facilitates the approximation of the molecular integrals to a high predetermined accuracy, while still keeping the calculation times low. The fast convergence properties analyzed in the numerical section illustrate the advantages of the method.     

Analytic two-loop master integrals for tW production at hadron colliders: I

We present the analytic calculation of two-loop master integrals that are relevant for tW production at hadron colliders. We focus on the integral families with only one massive propagator. After selecting a canonical basis, the differential equations for the master integrals can be transformed into the d ln form. The boundaries are determined by simple direct integrations or regularity conditions at kinematic points without physical singularities. The analytical results in this work are expressed in terms of multiple polylogarithms, and have been checked via numerical computations.     

Identities for Hypergeometric Integrals of Different Dimensions

Given complex numbers m₁, I₁ and nonnegative integers m₂, I₂, such that m₁+m₂ = I₁+ I₂, we define I₂-dimensional hypergeometric integrals I_a,b(z; m₁, m₂, I₁, I₂), a,b = 0,. . . ,min)(m₂,I₂), depending on a complex parameter z. We show that I_a,b(z;m₁, m₂,I₁, I₂) = I_a,b(z;I₁, I₂,m₁,m₂), thus establishing an equality of I₂ and m₂-dimensional integrals. This identity allows us to study asymptotics of the integrals with respect to their dimension in some examples. The identity is based on the ( _k, _k,) duality for the KZ and dynamical differential equations.Mathematics Subject Classifications (2000). 33C70, 33C80, 81R10     

Vanishing of the Kontsevich Integrals of the Wheels

We prove that the Kontsevich integrals (in the sense of the formality theorem) of all even wheels are equal to zero. These integrals appear in the approach to the Duflo formula via the formality theorem. The result means that for any finite-dimensional Lie algebra g, and for invariant polynomials f, g [S ^·(g)]^g one has f · g = f * g, where * is the Kontsevich star product, corresponding to the Kirillov–Poisson structure on g^*. We deduce this theorem form the result contained in math.QA/0010321 on the deformation quantization with traces.     

Partially coherent vectorial cosh-Gaussian beams beyond the paraxial approximation

The concept of partially coherent vectorial nonparaxial cosh-Gaussian (ChG) beams is introduced, and their analytical propagation expressions for the cross-spectral density matrix in free space are derived by using the generalized vectorial Rayleigh diffraction integrals. Some interesting cases, in particular, the vectorial nonparaxial Gaussian-Schell-model (GSM) beams are discussed and treated as special cases of our general expressions. It is shown that the f and f_σ parameters play a crucial role in determining the vectorial property and nonparaxiality of partially coherent ChG beams, but the decentered parameter additionally affects their behavior.     

Onsager's reaction field for the Potts model from the path integral

A new path integral formulation for theq-state Potts model is proposed. This formulation reproduces known results for the Ising model (q=2) and naturally extends these results for arbitraryq. The mean field results for both the Ising and the Potts models are obtained as a leading saddle point contribution to the corresponding functional integrals, while the systematic computation of corrections to the saddle point contribution produces the Onsager reaction field terms, which forq=2 coincide with results already known for the Ising model.     

Velocity-space contours of collision integrals

Based on the recently found closed-form expressions of the Boltzmann collision integrals in a rigid-sphere gas for multi-Maxwellian distributions, a few typical sets of contour surfaces of the integrals in the space of molecular velocities are presented. These show graphically the tendency toward equilibrium under the influence of collisions. A brief preliminary comparison with Monte Carlo results is also given.     

Calculation of Electric Mult ipole Moment Integrals Using Translation Formulas for Slater-Type Orbitals

Using translation formulas for Slater-type orbitals the infinite series through the overlap integrals are derived for the electric multipole moment integrals. By the use of the derived expressions the electric multipole moment integrals, and therefore, the electric properties of molecules can be evaluated most efficiently and accurately. The convergence of the series is tested by calculating concrete cases. An accuracy of for 10^-5 the computer results is obtained for 1 ≤ v ≤ 5, and for the arbitrary values of internuclear distances and screening constants of atomic orbitals.     

Recurrence equations for the computation of correlations in the 1/r2 quantum many-body system

In a previous paper the two-particle distribution function and one-particle density matrix for the quantum many-body system with the 1/r ² pair potential have been expressed as limiting cases of Selberg correlation integrals. Recurrence equations are derived which allow rapid evaluation of these multidimensional integrals. The exact results for the two-particle distribution are compared with the harmonic approximation.     

Vectorial Hermite--Laguerre--Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh--Sommerfeld integrals, the free-space propagation expressions for vectorial Hermite--Laguerre--Gaussian (HLG) beams beyond the paraxial approximation are derived. The far-field expressions and the scalar paraxial results are given as special cases of our general expressions. The intensity distributions of vectorial nonparaxial HLG beams are studied and illustrated with numerical examples.     

On the Evaluation Overlap Integrals with the Same and Different Screening Parameters Over Slater Type Orbitals via the Fourier-Transform Method

In this paper, derivation of analytical expressions for overlap integrals with the same and different screening parameters of Slater type orbitals (STOs) via the Fourier-transform method is presented. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals with same and different screening parameters mentioned as finite sums of Gegenbauer, Gaunt, binomial coefficients, and STOs.     

Cartesian Expressions for Surface and Regular Solid Spherical Harmonics Using Binomial Coefficients and Its Use in the Evaluation of Multicenter Integrals

In this study, we present cartesian expressions for surface and regular solid spherical harmonics using binomial coefficients. The obtained results are useful for the calculation of multicenter molecular integrals over Slater type orbitals, especially for large values of quantum numbers. The main advantage of presented formulae in the evaluation of multicenter integrals is discussed.     

A recursive reduction of tensor Feynman integrals

We perform a new, recursive reduction of one-loop n-point rank R   tensor Feynman integrals [in short: (n,R)$(n,R)$-integrals] for n?6$n?6$ with R?n$R?n$ by representing (n,R)$(n,R)$-integrals in terms of (n,R−1)$(n,R-1)$- and (n−1,R−1)$(n-1,R-1)$-integrals. We use the known representation of tensor integrals in terms of scalar integrals in higher dimension, which are then reduced by recurrence relations to integrals in generic dimension. With a systematic application of metric tensor representations in terms of chords, and by decomposing and recombining these representations, a recursive reduction for the tensors is found. The procedure represents a compact, sequential algorithm for numerical evaluations of tensor Feynman integrals appearing in next-to-leading order contributions to massless and massive three- and four-particle production at LHC and ILC, as well as at meson factories.