Evaluation of molecular spin-orbit integrals by a gaussian expansion method

A method is presented for the evaluation of molecular multi-centre spin-orbit integrals of Coulomb, hybrid and exchange types using cartesian gaussian functions. It is shown that if gaussian bases are employed the spin-same-orbit integrals reduce to one-electron field integrals and the spin-other-orbit integrals to two-electron field integrals. The final formulae for these integrals are found to be convenient for computer programming. Some relationships among the nuclear attraction, the field and the field-gradient integrals as well as among the electronic repulsion, the two-electron field, and the zero-field splitting integrals are discussed in connection with the spin-orbit integrals. Some calculations are carried out for the radicals CH₂ and NH.     

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     

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 surface integrals for triple collision integrals in kinetic theory of gases

The triple collision integrals in kinetic theory of gases are re-examined in terms of mass-normalized coordinate systems and new surface integrals are obtained from them. The five-dimensional surface integrals obtained are more symmetric than those known up to know and appear suggestive of further generalization to more complex problems of many-body systems.     

Definition of path integrals and rules for non-linear transformations

Functional integrals are defined as the limit of multidimensional integrals based on fundamental generating distributions. The “lattice choice” is put into a suitable functional form. The independence of the particular choice and the necessity of this fact are shown. Various forms of the path integrals are derived and discussed. The relation to the traditional ordering problem is pointed out. The mechanism of non-linear transformations of variables is investigated and rules are given. In the case of fields it turns out that the path integrals can also be considered for space translations.     

Integrals in the theory of four-particle atomic-molecular systems

Explicit expressions for four-particle Coulomb interaction integrals and overlap integrals for exponential basis functions dependent on five interparticle distances have been obtained. These formulas significantly simplify calculations in comparison with the general algorithm and are free from uncertainties arising in its application. The values of the integrals under consideration are calculated for a number of points forbidden in the general algorithm. The results obtained are applicable to analysis of four-particle integrals of the general form and calculation of four-particle atomic-molecular systems with partial allowance for the correlation of particle motion.     

Infinite-dimensional evolution equations and the representation of their solutions in the form of feynman integrals

In the paper, existence conditions for Feynman integrals in the sense of analytic continuation of Gaussian integrals with respect to operator arguments are found. A representation of Feynman integrals in the form of Gaussian integrals is also constructed and, finally, the class of evolution equations having solutions representable by Feynman integrals is described.     

Methods of reduction for Lagrange systems on time scales with nabla derivatives

The Routh and Whittaker methods of reduction for Lagrange system on time scales with nabla derivatives are studied.The equations of motion for Lagrange system on time scales are established, and their cyclic integrals and generalized energy integrals are given. The Routh functions and Whittaker functions of Lagrange system are constructed, and the order of differential equations of motion for the system are reduced by using the cyclic integrals or the generalized energy integrals with nabla derivatives. The results show that the reduced Routh equations and Whittaker equations hold the form of Lagrnage equations with nabla derivatives. Finally, two examples are given to illustrate the application of the results.     

三维静电场线性插值边界元中的解析积分方法

提出求解三维静电场的三角形线性插值边界元解析积分方法.针对含1/R和1/R²的积分项,将单元形状函数分解为常数项、含x的线性项和含y的线性项,从而将边界单元积分简化为6个基本积分组合,并导出其解析计算公式,避免了因形状函数改变而导致的重复计算.该方法不仅可以准确计算远离奇异情况下的边界元积分,而且可以准确计算一阶和二阶接近奇异积分以及一阶奇异积分.计算结果表明,在接近奇异积分和奇异积分比较突出的问题中,当数值积分方法不能给出正确结果时,用同样的边界元网格,解析积分方法可以给出正确的结果,提高了三维静电场线性插值边界元法的计算精度.     

The Number of Master Integrals is Finite

For a fixed Feynman graph one can consider Feynman integrals with all possible powers of propagators and try to reduce them, by linear relations, to a finite subset of integrals, the so-called master integrals. Up to now, there are numerous examples of reduction procedures resulting in a finite number of master integrals for various families of Feynman integrals. However, up to now it was just an empirical fact that the reduction procedure results in a finite number of irreducible integrals. It this paper we prove that the number of master integrals is always finite.     

Reduction of Chaplygin equations for nonholonomic systems on time scales by means of the cyclic integral and energy integral

In this paper, the reduction of the Chaplygin equations on time scales by means of the cyclic integral and energy integral are proposed and studied. The cyclic integrals and generalized energy integrals for nonholonomic systems on time scales are given. The Chaplygin equations on time scales are reduced by means of the cyclic integrals and the generalized energy integrals. The results show that if the systems have a cyclic integral and an energy integral, then the Chaplygin equations on time scales can be reduced by at least one degree and the form of the Chaplygin equations holds.     

Feynman formulas and path integrals for some evolution semigroups related to τ-quantization

This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n-fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.     

Extraordinary properties of functional integrals and groups of diffeomorphisms

A review of the work of the authors is presented, in which corollaries of the quasi-invariance of functional integrals on the Wiener measure with respect to the action of a group of diffeomorphisms are studied, and the behavior of functional integrals with nonlinear nonlocal change of variables of integration is investigated as well. Using these substitutions, the functional integrals over discontinuous paths can be determined. The simplest models of the (Euclidean) quantum field theory are offered, in which the presence of hidden internal symmetries or the allowance for discontinuous paths in functional integrals leads to a number of paradoxical properties contradicting the conventional view.     

Evaluation of the integrals of target/seabed scattering using the method of complex images

In this paper the integrals which arise in target/seabed scattering problems are examined. These are the integrals which express the propagation of a spherical harmonic term in a fluid above or below a half-space and the integrals which compute the conversion coefficients of outgoing spherical harmonics into incident spherical harmonics after a seabed reflection. A very efficient and accurate method of computing these integrals, using the method of complex images, is derived. Numerical comparisons with the exact integral expressions show the accuracy of this approach.     

Fields of stress and electric displacement produced by dislocations and disclinations in three-dimensional,anisotropic, elastic and piezoelectric media: elliptical Frank disclination

A statement is made on the theory of continuous distributions of dislocations and disclinations in anisotropic elastopiezoelectric media. The basic field equations governing the fields of stress functions, electric vector potential and incompatibility are presented and solved to give the fields of stress and electric displacement caused by a distribution of dislocations and disclinations. They are expressed in terms of the dislocation- and disclination-density tensors by means of the convolution integrals, extended throughout the medium, and the Fourier integrals. To treat the fields around discrete defects, that is dislocation and/or Frank disclination, the convolution integrals are replaced by the line integrals belonging to the loop of the defect. The fields of stress and electric displacement are given in terms of three quadruple integrals, which are converted into single integrals of explicitly given functions, in the case where the loop of the defect is elliptical. Numerical computations are carried out to estimate the fields in gallium arsenide. The values of those fields at a certain point of the body are presented. The contours and zero lines of the fields of dilatational stress and electric displacement in the plane placed parallel to and at a certain distance from the loop are illustrated.     

New relationships between Feynman integrals

New types of relationships between Feynman integrals are presented. It is shown that Feynman integrals satisfy functional equations connecting integrals with different values of scalar invariants and masses. A method is proposed for obtaining such relations. The derivation of functional equations for one-loop propagator- and vertex-type integrals is given. It is shown that a propagator-type integral can be written as a sum of two integrals with modified scalar invariants and one propagator massless. The vertex-type integral can be written as a sum over vertex integrals with all but one propagator massless and one external momenta squared equal to zero. It is demonstrated that the functional equations can be used for the analytic continuation of Feynman integrals to different kinematic domains.     

Covariant description of the spin states of a fermi particle moving in external fields

Covariant operators for the spin of a Fermi particle are considered. The choice of spin integrals of motion is determined for motion in steady isotropie electromagnetic fields. The spin integrals in the presence of an anomalous magnetic moment are examined.     

Feynman path integrals

Path integrals techniques are derived from a new definition [1] of Feynman path integrals. These techniques are used to establish that Feynman-Green functions for a given physical system are covariances of pseudomeasures suitable for its path integrals. The variance of a pseudomeasure is a more versatile tool than the Feynman-Green function it defines.     

The asymptotic behavior of massless Feynman integrals in the limit of several large scales

A general method is developed which permits the ealculation of the asymptotic behavior of Feynman integrals in the limit of several large seales, All those contributions which are power law suppressed are neglected. The parametric representation of Feynman integrals is employed, and is reviewed concisely. The main tool is the multiple Mellin transform of Feynman integrals with respect to their external momenta. A transformation of the Feynman parameters is represented which allows the systematic isolation of the leading poles in the multiple Mellin plane.