Factorization of certain evolution operators using lie algebra: Convergence theorems

This work concerns convergence theorems with a sequence of -approximants for exponential evolution operators with Lie operator arguments. A companion paper presents the formulation of the -approximants. The theorems presented in this paper give the conditions which are sufficient for convergence of the sequences. Although the main emphasis will be on convergence properties of the one-dimensional case, the generalization to multidimensional cases is quite straightforward.Supported by the Department of Energy.     

构造SO~2分子势能面的李代数方法

把李代数方法得到的SO~2分子的代数Hamiltonian,利用相干态基经典化并找到一个新的变换,将分子的键角引入,而得到SO~2分子的势能面。由该势能面计算的解离能,所给出的势能面的立体图和相应的等高线以及力常数与其他方法给出的相一致。该方法可以推广到多原子分子及反应体系。     

“Basis” lie algebra of electronic fock space: Application to evaluation of matrix elements of spin tensor operators

A new set of generators of the operator algebra over the electronic Fock space is introduced. It is shown that with this set of generators the “basis” Lie algebra can be associated and that the operator algebra of the Fock space is the homomorphic image of the corresponding universal enveloping algebra. The algebraic structure revealed is used for deriving the reduction formulas for the elements of the simplest spin tensor operators between the Gelfand states.     

A new general algorithmic method in reaction syntheses using linear algebra

We discuss here a new general linear algebraic method (both model and algorithm) for describing and generating (among others) minimal reactions and also minimal mechanisms in stoichiometry, or dimensionless groups in physics as well. (Further applications in process network syntheses will be discussed in .) With some minor modifications of the input this method can be extended for several related questions: for generating direct and overall reactions, direct (steady state) mechanisms, for finding the possible resulting (overall) reactions among all possible mechanisms, etc.Computational results in section 4 show the speed of our algorithm.We give also mathematical background and results in sections 3, 5 and 6. However, we do not restrict ourselves to mathematics only, we also talk on the language of chemistry, too.The theoretical results in sections 3.2, 3.3, 5 and the computational examples in section 4 are completely new, further theoretical results will appear in and in .     

A computational method for symmetry factorization without using symmetry operators

A computational method is suggested for symmetry factorization of symmetric matrices according to the symmetry properties of their basis. The method avoids the use of symmetry operators and is convenient for computer programming.     

Effective operators and the Z-expansion method

The application of effective operators in the Z-expansion method is explored using hydrogenic functions. The relevant perturbation summations are evaluated using unit tensor operators and the results applied to the 1s²2s²2p² + 1s²2p⁴ complex of the carbon isoelectronic series.     

Rotationally inelastic molecule–surface scattering: dynamical lie algebraic method

The rotationally inelastic molecule–surface scattering is analyzed using dynamical Lie algebraic method. We treat, by example, the simple model of the scattering of NO from a rigid, flat Ag(111) surface. The explicit expressions of transition probability and the probability current density are obtained. It is proved that dynamical Lie algebraic method can be useful for describing the scattering problems. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 500–510, 2000     

Analysis of the interaction between molecules and IR-laser radiation using one-dimensional lie groups

The effect of IR-laser radiation on molecules has been examined by using one-dimensional Lie groups in a triangular matrix to solve the equation for a nonuniform harmonic oscillator. The red shift effect is explained.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 31, No. 4, pp. 232–237, July–August, 1995.     

Determination of oxygen in certain gases : Improved winkler method

An improved method is described for the determination of micro-amounts of oxygen gas in argon, carbon dioxide, helium, hydrogen and nitrogen by the Winkler. procedure. The technique is such that a mean standard deviation of ± 0.33 ppm oxygen with a reproducibility of ± 0.67 is obtained for gases varying in oxygen content from 1 to 30 ppm. The results are reproducible on a day-to-day basis. The reagent blank is equivalent to about 2.5 ppm with a deviation of 0.2 ppm. The method can be used for oxygen gas determination from 1 to 1.50 ppm. The accuracy of the method is ± 0.1 ppm for the lower range of oxygen concentrations.The method may also be extended to higher oxygen-containing gases, except that the oxygen determination is made by titration and smaller gas samples are used.     

A rapid and precise method for the determination of oxygen in certain gases: Modified brady method

The modified brady method is a rapid one, requiring only about one hour for the analysis, but requires large samples of gas, 5 to 20 liters. The precision of the method described here-in is ± 0.004 to ± 0.4 in the low range of oxygen-concentration (0.01 to 0. 10 part per million oxygen),± 0.01 to ± 0.07 in the medium range of oxygen-concentration (0.1 to l .0 part per million oxygen)and ± 0.0 to ± 0.2 in the upper range of oxygen-concentration (1 to 15 parts per million oxygen). The liest precision (winkler method) previously reported was ± 0.3 part per million oxygen. The range of the method is 0.01 to 15 parts per million oxygen. The minimum amount of oxygen detectible is 0.005 part per million.     

On the evolution equations for reduced density operators in quantum open systems

Evolution equations for transition probabilities of reduced density operators in quantum open systems are derived. Information contained in such equations is obtained from spectral resolutions and the role of memory kernels is elucidated within the scenario of many body theory as a function of the self-energy fields. As an analytical example of this formulation relaxation times for dissipative systems are evaluated in terms of the interaction between subsystems.     

Formulation of the LCAS MS SCF method within the Gaussian basis set

Within the presented LCAS MS (linear combination of atomic spinors–molecular spinors) SCF formalism both large and small components of the spinor radial parts have been expanded within the Gaussian basis set. The respective expressions for matrix elements as well as for one- and two-electron integrals are given.     

Matrix elements for the Morse potential using ladder operators

The algebra of the two-dimensional harmonic oscillator is exploited to obtain matrix elements between eigenstates of the Morse potential. This follows after mapping the latter into the radial equation of the former problem by means of a change of variable and the use of the angular variable as a dummy variable.     

Numerical calculation of diatomic transition operators: The variable-phase and -amplitude method

The variable-phase method is extended to calculate two-atom off-the-energy shell t-matrix elements. A study is presented of the off-shell wavefunctions and t-matrix elements of the HH pair, for both the lowest ¹Σ and ³Σ interaction potentials.     

Molecular structural formulas as one-electron density and hamiltonian operators: the VIF method extended

The valency interaction formula (VIF) method is given a broader and more general interpretation in which these simple molecular structural formulas implicitly include all overlaps between valence atomic orbitals even for interactions not drawn in the VIF picture. This applies for VIF pictures as one-electron Hamiltonian operators as well as VIF pictures as one-electron density operators that constitute a new implementation of the VIF method simpler in its application and more accurate in its results than previous approaches. A procedure for estimating elements of the effective charge density-bond order matrix, Pmunu, from electron configurations in atoms is presented, and it is shown how these lead to loop and line constants in the VIF picture. From these structural formulas, one finds the number of singly, doubly, and unoccupied molecular orbitals, as well as the number of molecular orbitals with energy lower, equal, and higher than -1/2Eh, the negative of the hydrogen atom's ionization energy. The VIF results for water are in qualitative agreement with MP2/6311++G3df3pd, MO energy levels where the simple VIF for water presented in the earlier literature does not agree with computed energy levels. The method presented here gives the simplest accurate VIF pictures for hydrocarbons. It is shown how VIF can be used to predict thermal barriers to chemical reactions. Insertion of singlet carbene into H2 is given as an example. VIF pictures as one-electron density operators describe the ground-state multiplicities of B2, N2, and O2 molecules and as one-electron Hamiltonian operators give the correct electronegativity trend across period two. Previous implementations of VIF do not indicate singly occupied molecular orbitals directly from the pictorial VIF rules for these examples. The direct comparison between structural formulas that represent electron density and those that represent energy is supported by comparison of a simple electronegativity scale, chiD=N/n2, with well-known electronegativity scales of Pauling, Mulliken, and Allen. This scale comes from the method used to calculate Pmumu for sp3 hybridized period-two elements and is comparable to electronegativity because it has the same form as <1/r> for hydrogenic orbitals. It therefore provides a physical basis for the representation of one electron density and Hamiltonian operators by the same VIF picture.