Correlated complex exponential wave functions for atomic-molecular systems

The application of flexible correlated complex exponential basis functions depending on all inter-particle distances in precision variational computations of atomic-molecular systems is considered. In contrast to three-particle systems, the application of such functions for four-particle systems is hindered by the cumber-some nature of the multistep algorithm of evaluation of the integrals determining the matrix elements of the energy operator. Transformations reducing the number of integrals to be evaluated are carried out. As a result, the computation volume is reduced by a factor of six: instead of 43 integrals, it is sufficient to take 6 integrals of the Coulomb interaction of pairs of particles and the overlap integral. The method of direct construction of formulas for many-particle integrals is discussed.     

Finite crossing sum rules for partial-wave helicity amplitudes

A method is given for deriving finite crossing some rules for partial wavesof helicity amplitudes of four-particle reactions with general spins and masses. The method constitutes an algorithm for constructing a complete set of sum rules and yields explicit formulas for them.     

关于F19,Ne20相干能级的结构

在前一篇文章中提出了二个核子空间结构对称的波函数。本文又提出了三个核子和四个核子空间结构全对称的波函数。将本文的波函数与Flowers和Redlich用壳模型计算的相应的波函数作了比较,结果发现这些波函数很相一致,波函数的重迭积分的平方都在80％以上。     

Coupled-cluster calculation for nuclear matter and comparison with the hole-line expansion

The Bochum truncation of the coupled-cluster equations is applied to the nuclear matter problem. Numerical solutions are obtained on the two- and three-particle levels. Approximate treatments for the four-particle level including the four-particle cluster term result in only small contributions, demonstrating the fast convergence of the coupled-cluster scheme. The results are in agreement with variational and hole-line expansion calculations.     

Probabilistic Remote Preparation of a Four-Particle Entangled W State for the General Case and for All Kinds of the Special Cases

We present a protocol for probabilistic remote preparation of a four-particle entangled W state. The quantum channel is composed of two partial entangled four-particle cluster states. We calculate the total successful probability and total classical communication cost required for the general case and for all kinds of the special cases, respectively. It is shown that for two maximally entangled four-particle cluster states, such a scheme for the general case has the total successful probability of 25% and only consumes the total classical communication of 1 bit, while this scheme for the special cases under certain conditions can possess successful probability of 50% or 100%, the required classical communication will only be 2 bits or 4 bits. Meantime, we give in detail all unitary transformations for the general case and for all kinds of the special cases, respectively.     

Probabilistic Remote Preparation of a Four-Particle Entangled W State for the General Case and for All Kinds of the Special Cases

We present a protocol for probabilistic remote preparation of a four-particle entangled W state.The quantum channel is composed of two partial entangled four-particle cluster states.We calculate the total successful probability and total classical communication cost required for the general case and for all kinds of the special cases,respectively.It is shown that for two maximally entangled four-particle cluster states,such a scheme for the general case has the total successful probability of 25%and only consumes the total classical communication of 1 bit,while this scheme for the special cases under certain conditions can possess successful probability of 50%or 100%,the required classical communication will only be 2 bits or 4 bits.Meantime,we give in detail all unitary transformations for the general case and for all kinds of the special cases,respectively.     

Efficient algorithm for two-center Coulomb and exchange integrals of electronic prolate spheroidal orbitals

We present a fast algorithm to calculate Coulomb/exchange integrals of prolate spheroidal electronic orbitals, which are the exact solutions of the single-electron, two-center Schrödinger equation for diatomic molecules. Our approach employs Neumann’s expansion of the Coulomb repulsion 1/∣x ? y∣, solves the resulting integrals symbolically in closed form and subsequently performs a numeric Taylor expansion for efficiency. Thanks to the general form of the integrals, the obtained coefficients are independent of the particular wavefunctions and can thus be reused later.Key features of our algorithm include complete avoidance of numeric integration, drafting of the individual steps as fast matrix operations and high accuracy due to the exponential convergence of the expansions.Application to the diatomic molecules O₂ and CO exemplifies the developed methods, which can be relevant for a quantitative understanding of chemical bonds in general.     

Angular momentum reduction in the four-body problem

A complete angular momentum analysis of the integral equations of motion for four identical spinless particles is given. After separation of the variables associated with rotation, the equations of motion take the form of an infinite set of coupled integral equations in three continuous variables. In general, the four-particle kinematic factors in the integral kernels are expressed as bilinear combinations of factors of the three-particle kinematic factors type. For the case of a separable two-particle interaction the equations obtained are simplified so that they are reduced to coupled integral equations in two continuous variables.     

Field theory of collective excitations III condensation of four-particle clusters

For many fermions with arbitrary two-body forces we investigate the effective action Г[?, α] where ? is the density matrix including pair correlations and α is the four-particle vertex. The physical state including all radiative corrections is obtained by extremizing Г[?, α] with respect to ? and α. Just as ? ≠ 0 corresponds to non-zero density and condensation of pairs, α ≠ 0 signals the presence of a condensate of four-particle clusters. A simple loop expansion is given for Г[?, α]. To lowest order, the extremum amounts to the Hartree-Fock-Bogoliubov equations. The next two steps bring in the four-particle condensate.     

Two-particle resonance poles and the energy spectra of three- and four-particle systems

The position of the two-particle resonance poles is shown to be the dominant factor in a number of interesting phenomena appearing in the energy spectra of bound three- and four-particle systems with separable two-particle interactions. The validity of the bound-state pole dominance assumption in few-particle systems is shown to depend on the location of the two-particle resonance poles. These results generalize the ground state collapse obtained earlier for the Tabakin rank-one separable interactions in the triton. However, our results also indicate that the collapse does not necessarily occur for more general nonlocal interactions.     

周期轨迹与不可积体系的量子化：Henon-Heiles体系的一个研究例子

研究了两个振子耦合的Henon-Heiles体系的周期轨迹与量子化问题.结果表明，周期轨迹的 作用量积分与体系的能量有着简单的线性关系.可以利用那些是整数值的周期轨迹的作用量 积分对不可积体系进行半经典量子化.由周期轨迹的物理内涵出发，揭示混沌体系的残余周 期轨迹具有与量子化有关的性质.这对于认识和理解经典力学与量子体系的联系关系及其物 理内涵有着深刻而重要的意义. 关键词： 周期轨迹 半经典量子化 混沌     

水平层状介质中电磁场并矢Green函数的一种快速新算法

本文利用提取直射波并结合自适应数字滤波等技术提出一种计算水平层状介质中电磁场并矢Green函数的快速算法. 首先将谱域Green函数中表征均匀介质作用的直射波提取出来并对其积分进行解析计算,这种处理降低了谱域Green函数的奇异性,可在很大程度上缩短其积分收敛区间. 然后在将谱域Green函数剩余部分对应积分转化为三个快速下降积分的基础上,引入一种自适应数字滤波算法对其进行快速求解. 最后通过具体算例验证了本文所述算法的有效性. 关键词： 并矢Green函数 快速算法 水平层状介质     

PV-reduction of sunset topology with auxiliary vector

The Passarino–Veltman (PV) reduction method has proven to be very useful for the computation of general one-loop integrals. However, not much progress has been made when it is applied to higher loops. Recently, we have improved the PV-reduction method by introducing an auxiliary vector. In this paper, we apply our new method to the simplest two-loop integrals, i.e., the sunset topology. We show how to use differential operators to establish algebraic recursion relations for reduction coefficients. Our algorithm can be easily applied to the reduction of integrals with arbitrary high-rank tensor structures. We demonstrate the efficiency of our algorithm by computing the reduction with the total tensor rank up to four.     

One-particle and few-particle billiards

We study the dynamics of one-particle and few-particle billiard systems in containers of various shapes. In few-particle systems, the particles collide elastically both against the boundary and against each other. In the one-particle case, we investigate the formation and destruction of resonance islands in (generalized) mushroom billiards, which are a recently discovered class of Hamiltonian systems with mixed regular-chaotic dynamics. In the few-particle case, we compare the dynamics in container geometries whose counterpart one-particle billiards are integrable, chaotic, and mixed. One of our findings is that two-, three-, and four-particle billiards confined to containers with integrable one-particle counterparts inherit some integrals of motion and exhibit a regular partition of phase space into ergodic components of positive measure. Therefore, the shape of a container matters not only for noninteracting particles but also for interacting particles.     

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.     

Equilibrium configurations for a system of N particles in the plane

We bring together work on central configurations of the classical N-body problem and on asymptotic properties of a system of repelling particles. By introducing a parameter β, we consider configurations that occur in these problems to be special cases of a more general system. We describe an asymmetric configuration obtained in the eight-particle system and investigate a bifurcation occurring in the four-particle system.     

Quartet states in the sd and fp shells

Two-proton-two-neutron correlations are studied in psd and fp shell nuclei in the frame of the aligned scheme approximation. Four-particle rotational states are obtained in terms of linear combinations of projected aligned Slater determinants and the resulting excitation energies and wave functions are compared with those of elaborate shell-model calculations. The close agreement obtained both for single j-shell and complete shell configuration spaces shows that rotational or quasi-rotational states are almost entirely generated by aligned configurations. The lowest four-particle states obtained in one j-shell can be explained as the rotational spectrum of an oblate intrinsic state. The lowest four-particle states of ¹⁶O, ²⁰Ne, ⁴⁴Ti and ⁶⁰Zn, calculated in a major shell, correspond to the rotational spectrum of a prolate intrinsic state and fit the existing data. Higher states are the result of a mixed deformation.     

Matrix elements of vibration kinetic energy operator of tetrahedral molecules in non-orthogonal-dependent coordinates

In this work, we propose an algorithm for calculating the matrix elements of the kinetic energy operator for tetrahedral molecules. This algorithm uses the dependent six-angle coordinates (6A) and takes into account the full symmetry of molecules. Unlike A.V. Nikitin, M. Rey, and Vl. G. Tyuterev who operate with the kinetic energy operator only in Radau orthogonal coordinates, we consider a general case. The matrix elements are shown to be a sum of products of one-dimensional integrals.     

Low-rank canonical-tensor decomposition of potential energy surfaces: application to grid-based diagrammatic vibrational Green's function theory

ABSTRACT

A new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm^?1 or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.     

Scattering amplitude of four non-relativistic particles

The exact integral equations for the non-relativistic four-particle scattering amplitudes are obtained, assuming only two-body forces. These equations can be solved when in the final state there are two pairs of particles with small relative energy of particles in the c.m. system of each pair. As an example, the reaction d+d→n+p+d is considered.