Eigenvalues of the boltzmann collision operator for binary gases: Relaxation of anisotropic distributions

The time eigenvalues of the Boltzmann collision operator for the hard-sphere cross section are calculated with a discrete ordinate method. The anisotropic part of the collision operator is considered in the present work. The approach to equilibrium of a gas initially non-maxwellian and anisotropic is considered.     

Eigenvalues of the Boltzmann collision operator for binary gases: Mass dependence

Eigenvalues of the Boltzmann collision operator are calculated versus mass ratio with two different methods. One method involves the expansion of the eigenfunctions in speed polynomials, whereas with the second method the eigenfunctions are evaluated at discrete points based on a particular gaussian quadrature rule. The discrete ordinate method proved to be superior provided the mass ratio was neither too large nor too small. The approach of the eigenvalues to the continuum boundary was also studied for several mass ratios.     

Analogy between real irreducible tensor operator and molecular two-particle operator

. Molecular matrix elements of a physical operator are expanded in terms of polycentric matrix elements in the atomic basis by multiplying each by a geometrical factor. The number of terms in the expansion can be minimized by using molecular symmetry. We have shown that irreducible tensor operators can be used to imitate the actual physical operators. The matrix elements of irreducible tensor operators are easily computed by choosing rational irreducible tensor operators and irreducible bases. A set of geometrical factors generated from the expansion of the matrix elements of irreducible tensor operator can be transferred to the expansion of the matrix elements of the physical operator to compute the molecular matrix elements of the physical operator. Two scalar product operators are employed to simulate molecular two-particle operators. Thus two equivalent approaches to generating the geometrical factors are provided, where real irreducible tensor sets with real bases are used. Received: 3 September 1996 / Accepted: 19 December 1996     

Comparative study of the Hartree–Fock and the Hartree–Fock–Slater method

The Hartree–Fock method (standard Roothaan closed-shell HF –LCAO theory) and the Hartree–Fock–Slater method (restricted HFS –LCAO –DV method developed by Baerends and Ros) have been compared with emphasis on the respective one-electron equations and on the matrix elements of the respective Fock operators. Using the same STO basis in the two cases, the matrix elements of the Fock operators and of their separate one-electron, Coulomb, and exchange contributions have been calculated for the same orbitals and density of the ground state of the diatomic molecule ZnO. The effects of methodical (exchange potential) and numerical (DV method, density fit) differences between the HF and HFS methods on the various matrix elements have been analyzed. As expected the methodical effect prevails and is responsible for the higher (less negative) values of the matrix elements of the HFS Fock operator compared to those of the HF Fock operator. Numerical effects are observable also and are caused by the difference in integration procedures (DV method), not by the density fit.     

A numerical solution of the linear Boltzmann equation using cubic B-splines

A numerical method using cubic B-splines is presented for solving the linear Boltzmann equation. The collision kernel for the system is chosen as the Wigner-Wilkins kernel. A total of three different representations for the distribution function are presented. Eigenvalues and eigenfunctions of the collision matrix are obtained for various mass ratios and compared with known values. Distribution functions, along with first and second moments, are evaluated for different mass and temperature ratios. Overall it is shown that the method is accurate and well behaved. In particular, moments can be predicted with very few points if the representation is chosen well. This method produces sparse matrices, can be easily generalized to higher dimensions, and can be cast into efficient parallel algorithms.     

Applications of fourier transforms in molecular orbital theory. Calculation of optical properties and tables of two-center one-electron integrals

Fourier transform methods initiated by Geller and Harris are applied to the calculation of optical properties of molecules. Tables of one-electron two-center integrals needed for the accurate computation of molecular absorption and optical activity are calculated by the Fourier transform method. A general theorem is derived which allows the angular part of the integrals to be treated by means of projection operators. The radial parts of the integrals are treated by the methods of Harris. The results are obtained in a simple closed form which avoids the usual transformation to local coordinates. The two-center integrals evaluated include matrix elements of the momentum operator, the dipole moment operator, the tensor operator , the quadrupole moment operator, and the angular momentum operator. These are evaluated between 1s, 2s, and 2p Slater-type atomic orbitals located on different atoms. The results are expressed as functions of the Slater exponents and of the relative coordinates of the two atoms.     

Addition theorem and matrix elements of radiative multipole operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole, or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of μth component of the angular momentum operator, L _μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), . The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator     

Addition Theorem and Matrix Elements of Radiative Multipole Operators

The addition theorem for radiative multipole operators, i.e., electric-dipole, electric-quadropole or magnetic-dipole, etc., is derived through a translational transformation. The addition theorem of the μth component of the angular momentum operator, L_μ (r), is also derived as a simple expression that represents a general translation of the angular momentum operator along an arbitrary orientation of a displacement vector and when this displacement is along the Z-axis. The addition theorem of the multipole operators is then used to analytically evaluate the matrix elements of the electric and magnetic multipole operators over the basis functions, the spherical Laguerre Gaussian-type function (LGTF), . The explicit and simple formulas obtained for the matrix elements of these operators are in terms of vector-coupling coefficients and LGTFs of the internuclear coordinates. The matrix element of the magnetic multipole operator is shown to be a linear combination of the matrix element of the electric multipole operator.     

Theory of two shells of atomic electrons using non-orthogonal radial orbitals

A theory for handling non-orthogonal radial orbitals of two shells of atomic electrons based on the mathematical apparatus of irreducible tensor operators is presented. The general expressions for one- and two-electron operator matrix elements are given.     

The algebraic approach for the derivation of ladder operators and coherent states for the Goldman and Krivchenkov oscillator by the use of supersymmetric quantum mechanics

The ladder operators for the Goldman and Krivchenkov anharmonic potential have been derived within the algebraic approach. The method is extended to include the rotating oscillator. The coherent states for the Goldman and Krivchenkov oscillator, which are the eigenstates of the annihilation operator and minimize the generalized position-momentum uncertainty relation, are constructed within the framework of supersymmetric quantum mechanics. The constructed ladder operators can be a useful tool in quantum chemistry computations of non-trivial matrix elements. In particular, they can be employed in molecular vibrational–rotational spectroscopy of diatomic molecules to compute transition energies and dipole matrix elements.     

Inclusion of inversion symmetry in centroid molecular dynamics: a possible avenue to recover quantum coherence

Inversion symmetry is included in the operator formulation of the centroid molecular dynamics (CMD). This work involves the development of a symmetry-adapted CMD (SA-CMD), here particularly for symmetrization and antisymmetrization projections. A symmetry-adapted quasidensity operator, as defined by Blinov and Roy [J. Chem. Phys. 115, 7822 (2001)], is employed to obtain the centroid representation of quantum mechanical operators. Numerical examples are given for a single particle confined to one-dimensional symmetric quartic and symmetric double-well potentials. Two SA-CMD simulations are performed separately for both projections, and centroid position autocorrelation functions are obtained. For each projection, the quality of the approximation as well as the accuracy are similar to those of regular CMD. It is shown that individual trajectories from two separate SA-CMD simulations can be properly combined to recover trajectories for Boltzmann statistics. Position autocorrelation functions are compared to the exact quantum mechanical ones. This explicit account of inversion symmetry provides a qualitative improvement on the conventional CMD approach and allows the recovery of some quantum coherence.     

No fluctuation approximation in any desired precision for univariate function matrix representations

The operator involving problems are mostly handled by using the matrix representations of the operators over a finite set of appropriately chosen basis functions in a Hilbert space as long as the related problem permits. The algebraic operator which multiplies its operand by a function is the focus of this work. We deal with the univariate case for simplicity. We show that a rapidly converging scheme can be constructed by defining an appropriate fluctuation operator which projects, in fact, to the complement of the space spanned by appropriately chosen finite number of basis functions. What we obtain here can be used in efficient numerical integration also.     

Influence of the Excited States of Atomic Nitrogen N(<Superscript>2</Superscript>D) and N(<Superscript>2</Superscript>P) on the Transport Properties of Nitrogen. Part I: Atomic Nitrogen Properties

In this paper, calculated values of the viscosity and thermal conductivity of atomic nitrogen, taking into account three species (the ground and two excited states), are presented. The calculations, which assume that the temperature dependent probability of occupation of the states is given by the Boltzmann factor, are performed for atmospheric-pressure in the temperature range from 1,000 to 20,000 K. Six collision integrals are used in calculating the transport coefficients and we have introduced new averaged collision integrals where the weight associated at each interacting species pair is the probable collision frequency. The influence of the collision integral values and energy transfer between two different species is studied. These results are compared which those of published theoretical studies.     

三重四极杆电感耦合等离子体质谱(ICP-MS)法测定氧化铕中13种稀土杂质元素

稀土杂质元素直接影响高纯单金属稀土材料的整体性能,是高科技领域许多材料的重要组成部分。通过考察最佳的消解酸量、温度、时间、氧气反应气流量、稀释气流量,建立了基于三重四极杆电感耦合等离子体质谱仪(ICP-MS/MS)直接测定氧化铕中13种稀土杂质元素分析方法。该方法采用0.1%基体直接进样,可以很大程度提高前处理分析效率。利用碰撞模式测定氧化铕稀土中的Y、La、Pr、Nd、Sm、Gd、Tb、Dy、Ho、Er、Yb、Lu元素,氧气质量转移模式测定氧化铕中的Tm,两种模式结合可以有效去除多原子干扰,实现氧化铕的稳定测试分析。通过对氧化铕标准物质（GBW02902）直接测定分析,结果表明,在碰撞和氧气质量转移模式下,各元素线性相关系数（r）均大于 0.9999,方法检出限为0.001~0.023 mg/kg,测试精密度优于1.99%,13种元素的测试值都在认定值的不确定度范围之内。该分析方法操作简单,测试稳定,效率高,为实验室进行氧化铕材料中REE杂质的准确测试分析提供思路和借鉴。     

Symmetrization of operator matrix elements

The method of Dupuis and King for generating matrix elements of a totally symmetric one-electron operator in terms of symmetry-distinct integrals only is generalized to the case of nontotally symmetric operators. For operators constructed from two-electron integrals, explicit reduction of integral processing to permutationally inequivalent symmetry-distinct integrals only is described, while for one-electron operators further reductions are derived using double coset decompositions. Finally, some computational consequences of this approach are briefly discussed.     

Exact quantum statistics for electronically nonadiabatic systems using continuous path variables

We derive an exact, continuous-variable path integral (PI) representation of the canonical partition function for electronically nonadiabatic systems. Utilizing the Stock-Thoss (ST) mapping for an N-level system, matrix elements of the Boltzmann operator are expressed in Cartesian coordinates for both the nuclear and electronic degrees of freedom. The PI discretization presented here properly constrains the electronic Cartesian coordinates to the physical subspace of the mapping. We numerically demonstrate that the resulting PI-ST representation is exact for the calculation of equilibrium properties of systems with coupled electronic and nuclear degrees of freedom. We further show that the PI-ST formulation provides a natural means to initialize semiclassical trajectories for the calculation of real-time thermal correlation functions, which is numerically demonstrated in applications to a series of nonadiabatic model systems.     

Quantum-instanton evaluation of the kinetic isotope effects

A general quantum-mechanical method for computing kinetic isotope effects is presented. The method is based on the quantum-instanton approximation for the rate constant and on the path-integral Metropolis-Monte Carlo evaluation of the Boltzmann operator matrix elements. It computes the kinetic isotope effect directly, using a thermodynamic integration with respect to the mass of the isotope, thus avoiding the more computationally expensive process of computing the individual rate constants. The method should be more accurate than variational transition-state theories or the semiclassical instanton method since it does not assume a single tunneling path and does not use a semiclassical approximation of the Boltzmann operator. While the general Monte Carlo implementation makes the method accessible to systems with a large number of atoms, we present numerical results for the Eckart barrier and for the collinear and full three-dimensional isotope variants of the hydrogen exchange reaction H + H2 --> H2 + H. In all seven test cases, for temperatures between 250 and 600 K, the error of the quantum instanton approximation for the kinetic isotope effects is less than approximately 10%.     

On the long way from the Coulombic Hamiltonian to the intermolecular energy surfaces: Concluding remarks at the tromsø conference,June 20–22, 1989

After a brief review of the main results given at the conference, the general properties of the Coulombic Hamiltonian for a system of electrons moving in a framework of moving atomic nuclei—considered as point charges—are discussed. Since this Hamiltonian is invariant under translations and rotations, the total momentum and the total angular momentum are constants of motion, which means that it is possible to separate the motion of the center of mass and the rotation of the system as a whole. Even if these separations are simple in principle, they lead to a mixing of the electronic and nuclear coordinates that complicates the transformed Hamiltonian. The general features of this Hamiltonian are discussed both in pure quantum mechanics and general quantum theory dealing with wave functions Ψ respective density matrices ρ or system operators T. The principles of the latter are derived from five simple axioms, and it is shown that pure quantum mechanics is a special case of the general theory and that the analogy between these two approaches is essential for the “economy of thinking.” It is indicated that the general theory of the shape and topology of the energy surface 〈H〉 = TrHΓ and its critical points, as a function of the system operator Γ involving both electronic and nuclear coordinates, is a very difficult mathematical problem and that calculation of this surface even for simple molecular systems represents a formidable computational problem, which has to be solved in order to be able to understand the nature of chemical reactions from first principles.     

Study of simulation method of time evolution of atomic and molecular systems by quantum electrodynamics

We discuss a method to follow step‐by‐step time evolution of atomic and molecular systems based on quantum electrodynamics. Our strategy includes expanding the electron field operator by localized wavepackets to define creation and annihilation operators and following the time evolution using the equations of motion of the field operator in the Heisenberg picture. We first derive a time evolution equation for the excitation operator, the product of two creation or annihilation operators, which is necessary for constructing operators of physical quantities such as the electronic charge density operator. We, then, describe our approximation methods to obtain time differential equations of the electronic density matrix, which is defined as the expectation value of the excitation operator. By solving the equations numerically, we show “electron‐positron oscillations,” the fluctuations originated from virtual electron‐positron pair creations and annihilations, appear in the charge density of a hydrogen atom and molecule. We also show that the period of the electron‐positron oscillations becomes shorter by including the self‐energy process, in which the electron emits a photon and then absorbs it again, and it can be interpreted as the increase in the electron mass due to the self‐energy. © 2014 Wiley Periodicals, Inc.