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.     

Sur les intégrales de mouvement du système de sinus-Gordon discret – On Integrals of Motion of the discrete sine-Gordon System

We give explicit formulas for some densities of integrals of motion for the discrete sine-Gordon system (quantum or not). The generating function for the densities of integrals of motion may be seen as the expansion of the logarithm of a certain continued fraction (possibly quantum). In the case of q root of the unity, we show that these integrals of motion can be identified to the classical integrals of motion.     

Quantum corrections in <Emphasis Type="Italic">N</Emphasis> = 1 supersymmetric theories with cubic superpotential,regularized by higher covariant derivatives

Using the higher covariant derivative regularization we calculate a two-loop β-function for the N = 1 supersymmetric Yang-Mills theory with the matter superfields, containing the cubic superpotential. It is found that all integrals, defining this function, are integrals of total derivatives.     

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.     

Quantum algebra structure of certain Jackson integrals

Theq-difference system satisfied by Jackson integrals with a configuration ofA-type root system is studied. We explicitly construct some linear combination of Jackson integrals, which satisfies the quantum Knizhnik-Zamolodchikov equation for the 2-point correlation function ofq-vertex operators, introduced by Frenkel and Reshetikhin, for the quantum affine algebra . The expression of integrands for then-point case is conjectured, and a set of linear relations for the corresponding Jackson integrals is proved.     

关于线性阻尼振子第一积分的研究

通过引入一维线性阻尼振子基本积分来构造其他第一积分, 包括不含时的积分. 将这种方法推广到多维情形, 构造二维和n维线性阻尼振子不同形式的第一积分; 证明不同类型的二维线性阻尼振子都存在三个独立的不含时的第一积分, n维线性阻尼振子存在2n-1个独立的不含时的第一积分. 利用变量变换将线性阻尼振子的第一积分变换成简谐振子形式的第一积分. 关键词： 线性阻尼振子 第一积分 基本积分 简谐振子     

On the origin of integrability in matrix models

The matrix integrals involved in 2d lattice gravity are studied at finiteN. The integrable systems that arise in the continuum theory are shown to result directly from the formulation of the matrix integrals in terms of orthogonal polynomials. The partition function proves to be a tau function of the Toda lattice hierarchy. The associated linear problem is equivalent to finding the polynomial basis which diagonalizes the partition function. The cases of one Hermitian matrix, one unitary matrix, and Hermitian matrix chains all fall within the Toda framework.Research supported in part by DOE contract DE-FG02-90ER-40560, an NSF Presidential Young Investigator Award, and the Alfred P. Sloan Foundation     

关于谐振子第一积分的研究

提出一维谐振子基本积分的概念, 并利用它们构造其他积分. 将上述概念和方法推广到多维谐振子, 利用直接构造法证明不同类型的二维谐振子都存在三个与时间无关的独立的第一积分, n维谐振子系统存在2n-1个与时间无关的独立的第一积分. 讨论了分振动频率比为有理数和无理数的二维非对称谐振子的特征. 关键词： 谐振子 第一积分 基本积分 不含时积分     

Spin-orbit interactions from self consistent field wavefunctions

Effects of the spin-orbit Hamiltonian H_LS , including both the spin-same orbit and spin-other orbit terms, are studied at the self consistent field (SCF) level of theory. Separate calculations are carried out for each state considered, and biorthogonal orbitals are constructed for the evaluation of matrix elements. Doublet-doublet and singlet-triplet interactions are discussed. The evaluation of the Gaussian basis function integrals is described in a Cartesian component representation, these integrals being directly related to one and two electron second derivative integrals. This new SCF spin-orbit code is used to (i) determine the spatial dependence of the spin-orbit parameters for the Renner-Teller ²B₁, ²A₁ states of H₂O⁺, (ii) determine the spin-orbit splitting of the ²Π states of OH as a function of bond-length and (iii) calculate the radiative lifetime of the a ³Σ⁺ state of NO⁺. In each case these calculations are compared with more sophisticated configuration interaction studies, but it is found that the evaluation of these effects near equilibrium geometries at the SCF level is sufficiently accurate, provided that a large basis set is used.     

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.     

Calculation of absorption and resonance Raman cross sections of ClO2 in the gas‐phase and in solution: including Duschinsky effects

The time‐correlation function formalism has been used to calculate resonance Raman cross sections, excitation profiles, and electronic absorption spectra of the OClO molecule in the gas‐phase and in different solvents like cyclohexane, chloroform, and water. The multidimensional time domain integrals that arise in these calculations have been evaluated for the case in which an X²B₁ òA₁ electronic transition takes place between displaced‐distorted‐rotated harmonic potential energy surfaces. Ab initio calculations have been performed to provide the spectroscopic constants required for the evaluation of these integrals. The calculated absorption spectra and resonance Raman cross sections have been compared with the experimental results. Copyright © 2009 John Wiley & Sons, Ltd.     

Generating function and integral representation of hermite polynomials in physical problems

Using the generating function for Hermite polynomialsH _n two integrals developed in computation of the bond energy of atomic nucleus have been solved. Furthermore the probability distribution of the coordinate of quantum harmonic oscillator has been derived with the aid of suitable integral representation ofH _n(x). Presented computations are simpler then usually exploited recurrence equation technique.     

Power counting and regularization through loop integrations for multiple Feynman integrals in Minkowski space

A regularization procedure with a regularization parameter is developed which may be applied to multiple Feynman integrals in Minkowski space. The regularization is carried out inmomentum space and provides a rigorous method for studying Feynman integrals as multiple integrals in real variable theory. The regularized integrals are defined by changing the measure of integration _i dx _i to _i (1+x _i ² )^–/2 dx _i , >0, with a corresponding change defined inMinkowski space. We then develop a power counting convergence criterion for the absolute convergence of the integrals in terms of the parameter as a function of the so-called power asymptotic coefficients of Feynman integrands. An application to quantum electrodynamics is carried out.Work supported by the Department of National Defence Award under CRAD No. 3610-637:F4122.     

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.     

Evaluation of one-centre electron interaction integrals over slater type atomic orbitals

Simplified expressions for one-centre electron interaction integrals in general and Slater-Condon parameters as well as repulsion integrals of the type (aa|aa) in particular have been obtained over Slater-type atomic orbitals. Results calculated using these expressions are in agreement with those due to other authors.     

Feynman-Jackson integrals

Abstract

We introduce perturbative Feynman integrals in the context of q-calculus generalizing the Gaussian q-integrals introduced by Díaz and Teruel. We provide analytic as well as combinatorial interpretations for the Feynman-Jackson integrals.     

Asymptotic Approximations in Quantum Calculus

Abstract

This paper aims to study the asymptotic approximation of some functions defined by the q-Jackson integrals, for a fix q ∈]0, 1[. For this purpose, we shall attempt to extend the classical methods by giving their q-analogues. In particular, a q-analogue of the Watson’s lemma is discussed and new asymptotic expansions of the q?j _α Bessel function and of the q-complementary error function are established.     

Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

A Feynman formula is a representation of the semigroup, generated by an initial-boundary value problem for some evolutionary equation, by a limit of integrals over Cartesian powers of some space E, the integrands being some elementary functions. The multiple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real interval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space.