Full temporal density-matrix treatment of dual wavelength, arbitrary bandwidth, pulsed laser excitation of atoms with degenerate states in high collisional media

A full time-dependent theory, based on the Density-Matrix (D.M.) formalism, for two-step pulsed excitation of atoms in collision-dominated media by pulsed laser light of arbitrary bandwidth is presented. The atoms consist of three levels, of which each one, in turn, can consist of an arbitrary number of degenerate states. The atoms are exposed both to quenching collisions as well as elastic collisions. From a general set of density-matrix equations a more manageable, reduced set of fully time-dependent D.M. equations is formulated (the total number of equations is not more than 10, in contrast to the N² equations needed for a general set of D.M. equations, N being the total number of states within all three levels). The following approximations and assumptions have been made: the rotating-wave approximation; all individual transition probabilities between different states within a given pair of levels are the same (implying that the only input parameters for the transition probabilities are the spontaneous emission rates); all coherences between different states within each level are washed out by the high collisional rates; and the laser light is linearly polarized with an arbitrary bandwidth of Lorentzian shape. The full time-dependent equations are then solved in the steady-state limit of the non-diagonal elements, yielding time-dependent rate-equation-like population transfer equations. A few fully time-dependent simulations of some typical cases are given.     

A BASIC program for least-squares estimation of the parameters influencing line shapes in multi-site chemical exchange in nuclear magnetic resonance spectrometry

The OPTAO program in BASIC solves modified Bloch equations and calculates the line shape for a given set of parameters, which are then optimized by a Gauss-Newton non-linear least-squares method to provide the best fit to experimental data. Up to five sites can be handled. The program is suitable for personal computers.     

The ab initio calculation of molecular electric, magnetic and geometric properties

We give an account of some recent advances in the development of ab initio methods for the calculation of molecular response properties, involving electric, magnetic, and geometric perturbations. Particular attention is given to properties in which the basis functions depend explicitly both on time and on the applied perturbations such as perturbations involving nuclear displacements or external magnetic fields when London atomic orbitals are used. We summarize a general framework based on the quasienergy for the calculation of arbitrary-order molecular properties using the elements of the density matrix in the atomic-orbital basis as the basic variables. We demonstrate that the necessary perturbed density matrices of arbitrary order can be determined from a set of linear equations that have the same formal structure as the set of linear equations encountered when determining the linear response equations (or time-dependent self-consistent-field equations). Additional components needed to calculate properties involving perturbation-dependent basis sets are flexible one- and two-electron integral techniques for geometric or magnetic-field differentiated integrals; in Kohn-Sham density-functional theory (KS-DFT), we also need to calculate derivatives of the exchange-correlation functional. We describe a recent proposal for evaluating these contributions based on automatic differentiation. Within this framework, it is now possible to calculate any molecular property for an arbitrary self-consistent-field reference state, including two- and four-component relativistic self-consistent-field wave functions. Examples of calculations that can be performed with this formulation are presented.     

The efficiency of open tubular gas chromatographic columns part II. A comparison of theoretical equations for the efficiency of support coated open tubular columns

Summary A detailed comparison is given of the various equations derived by Golay and that derived by Hawkes for assessing the theoretical efficiency of SCOT columns.The terms in these equations which describe the resistance to mass transfer in both the gas phase and in the liquid phase are given in a form which permits the equations to be compared directly.A numerical comparison is then made of the potential efficiency predicted by the different equations for five partition ratio values for a set of nine idealized SCOT columns with a wide range of porous layer thickness and liquid loading. The potential column efficiencies are expressed in terms of the maximum theoretical mean specific plate number.It is shown that appreciable differences occur between the theoretical efficiencies given by the different equations when the porous layer thickness and liquid loading are high, but these differences are small for most of the columns used in practice.     

Multivariate data modelling through Piecewise generalized HDMR method

This work aims to develop a new High Dimensional Model Representation (HDMR) based method which can construct an analytical structure for a given multivariate data modelling problem. Modelling multivariate data through a divide-and-conquer method stands for multivariate data partitioning process in which we deal with a number of less variate data sets instead of a single N dimensional problem. Generalized HDMR is one of these methods used to model a multivariate data set which has a number of scattered nodes with associated function values. However, Generalized HDMR includes a linear equation system with huge number of unknowns and equations to be solved. This equation sometimes has linearly dependent equations in it and this is an undesirable situation. This work offers a new method named Piecewise Generalized HDMR method which bypasses this disadvantage as well as reducing the mathematical complexity and CPU time needed to complete the algorithm of the previous method. Our new method splits the given problem domain into subdomains, applies the Generalized HDMR philosophy to each subdomain and superpositions the information coming from these subdomains. The algorithm of this new method and a number of numerical implementations are given in this paper.     

MATHEMATICAL ASPECTS OF THE ULTRAVIOLET PHOTOCHEMISTRY OF PYRIMIDINE DINUCLEOSIDE PHOSPHATES

Abstract A mathematical formulation of the Kinetics of dimner and hydrate formation in pyrimidine dinucleosides upon ultraviolet (u.v.) radiatin is presented. The formulation allows for the possibilities of forming a number of types of dimers each with different cross sections for formtion and reversal, and of forming photo-irreversible products such as hydrates. The general differential equations descrbinjg the kinetics are solved in terms of a series of exponential terms which are functions of all the cross sections. The way in which any given species of hotoproduct is dependent upon the cross sections is illustrted in several examples. The results have been used to predict the maximum amount of dirner which may be produced for a given set of cross sections and the exposure at which this maximum is expected. The conditions under which dimers may be detected in a mixture of parent and dimer upon re-irradiation at short wavelengths are examined in terms of the differential equations.     

A proposed definition of resonant states

The widely cited definition of quantization in terms of square-integrable wave functions does not apply to continuum wave functions, to such phenomena as metastable states, or many-body resonances. A better philosophical foundation for quantum mechanics separates the probabilistic aspects based on square integrable Hilbert space functions from the dynamical aspects based upon the solutions of Shroedinger's (or Dirac's) equation. A Hilbert space may have a non-Hilbert space basis, which may be described by Stieltjes integrals and a spectrum measure. This viewpoint is expounded by reference to a very detailed analysis of a simple model, through which a precise definition of a Bohr–Feshbach resonance can be given. We propose a definition of a “metastable state,” showing that it is consistent with accepted usage, and that it overcomes a series of objections which have been catalogued by Simon. Its rate of decay is given by the Fourier–Stieltjes transform of the spectral density function; it is moreover the longest-lived initially localized state which can be formed from a small span of energy eigenfunctions near its mean energy.     

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.     

Classical limit of quantum mechanics in the large

Hilbert space may be regarded as a convenient standard approximation by interpolation and extrapolation to a unitary space, in general decomposable into a sum of semisimple spaces. In the limit we expect a particle theory expanded in powers of h according to the number of interacting particles. In the classical limit h tending to zero, phase space, with equations of motion reducible to Hamiltonian form, replaces Hilbert space. The modification of states by observing them is taken care of by considering probability distributions of petty ensembles. If the equations for any single observer can be made autonomous by replacing empirical time by universal time with an arrow we have a causal system. We then obtain the relation between probability and negentropy required for the second law of thermodynamics. An approximate Newtonian theory provides proximate particles with internal and external variables and admits the Poincaré group. For the internal variables we have approximately the Breit interaction. For the external variables we have equivalence for observers of the homogeneous Lorentz group of relativity. We introduce grand ensembles of ultimate particles, and nebulae as proximate particles in the large. We assume the Einstein principle of equivalence for the ten parameter set of observers suggested by relativity and suppose the second law of thermodynamics holds for each observer. The Einstein law of gravitation follows in classical theory to the order of the reciprocal of the large constant, in general with positive natural curvature as well as that corresponding to mass. Replacing particle interactions by fields we include them in classical theory.     

Numerical solution of ordinary differential equations by Fluctuationlessness theorem

This paper presents a numerical method based on Fluctuationlessness Theorem for the solution of Ordinary Differential Equations over appropriately defined Hilbert Spaces. We focus on the linear differential equations in this work. The approximated solution is written in the form of an nth degree polynomial of the independent variable. The unknown coefficients are obtained by setting up a system of linear equations which satisfy the initial or boundary conditions and the differential equation at the grid points, which are constructed as the independent variable’s matrix representation restricted to an n dimensional subspace of the Hilbert Space. An error comparison of the numerical solution and the MacLaurin series with the analytical solution is performed. The results show that the numerical solution obtained here converges to the analytical solution without using too many mesh points.     

On identifiability for chemical systems from measurable variables

The dynamics of the composition of chemical species in reacting systems can be characterized by a set of autonomous differential equations derived from mass conservation principles and some elementary hypothesis related to chemical reactivity. These sets of ordinary differential equations (ODEs) are basically non-linear, their complexity grows as much increases the number of substances present in the reacting media and can be characterized by a set of phenomenological constants (kinetic rate constants) which contains all the relevant information about the physical system. The determination of these kinetic constants is critical for the design or control of chemical systems from a technological point of view but the non-linear nature of the ODEs implies that there are hidden correlations between the parameters which maybe can be revealed with a identifiability analysis.     

Evaluation of multicenter integrals over Slater-type atomic orbitals by expansion in terms of complete sets

Multicenter integrals over Slater-type orbitals can be expanded in series with known coefficients if they are considered as elements of appropriate Hilbert spaces. Some test calculations for the three-center nuclear attraction integrals are given.     

Extending the Gibbs tangent plane semicontinuous mixtures

The computation of phase equilibrium for semicontinuous mixtures results in a set of algebraic-functional equations which are commonly solved using the pseudocomponent approach or reducing the order of the functional equations to a set of algebraic equations by means of a Petrov–Galerkin approach. Both approaches can be taken as particular cases of the method of weighed residuals. In order to perform stability analysis of phase equilibrium, the classical tangent plane criterion algorithms can be used as the first approach. There is a lack of functional formulation for the stability criterion in the literature. In some special cases, such as mixtures containing continuous families of heavy-components, a new formulation of the tangent plane criterion is required. In the present work, functional extensions of the Gibbs tangent plane criterion are described. The use of the proposed criteria was exemplified by calculating solid–liquid phase equilibrium of a crude oil containing asphaltenes on titration with a precipitating agent, n-pentane.     

A heuristic model of damped quantum rotation effects in nuclear magnetic resonance spectra

The damped quantum rotation (DQR) theory describes temperature effects in NMR spectra of hindered molecular rotators composed of identical atoms arranged in regular N-gons. In the standard approach, the relevant coherent dynamics are described quantum mechanically and the stochastic, thermally activated motions classically. The DQR theory is consistent. In place of random jumps over one, two, etc., maxima of the hindering potential, here one has damping processes of certain long-lived coherences between spin-space correlated eigenstates of the rotator. The damping-rate constants outnumber the classical jump-rate constants. The jump picture is recovered when the former cluster appropriately around only as many values as the number of the latter. The DQR theory was confirmed experimentally for hindered methyl groups in solids and even in liquids above 170 K. In this paper it is shown that for three-, four-, and sixfold rotators, the Liouville space equations of NMR line shapes, derived previously with the use of the quantum mechanical reduced density matrix approach, can be be given a heuristic justification. It is based on an equation of motion for the effective spin density matrix, where the relevant spin hamiltonian contains randomly fluctuating terms. The occurrence of the latter can be rationalized in terms of fluctuations of the tunneling splittings between the torsional sublevels of the rotator, including momentary liftings of the Kramers degeneracies. The question whether such degeneracy liftings are physical or virtual is discussed. The random terms in the effective hamiltonian can be Monte Carlo modeled as piecewise constant in time, which affords the stochastic equation of motion to be solved numerically in the Hilbert spin space. For sixfold rotators, this way of calculating the spectra can be useful in the instances where the Liouville space formalism of the original DQR theory is numerically unstable.     

The Sherman equations as a nonlinear Perron eigenvalue problem

When a chemical sample composed of N elements is analyzed using sequential selective excitation by a tunable polyenergetic X-ray beam and selective measurement of the characteristic X-rays, the production of secondary fluorescence does not interfere with the measurements. This experimental situation leads to a particular case of the Sherman equations which can be written as a set of non-linear equations. The same kind of equations are also obtained when we excite a chemical sample with a polyenergetic X-ray beam and neglect the production of secondary fluorescence. This set of equations can be regarded as a non-linear eigenvalue problem. A non-linear extension of the Perron Frobenious theorem ensures that there is one and only one physically acceptable solution, and also leads to a method to obtaining it. The propagation off measurements errors of sample fluorescence to errors in the calculated sample concentrations, has been simulated, and the results show that the solution is well conditioned. The case of production of secondary fluorescence can not be treated, in general, as a nonlinear Perron eigenvalue problem, but it has been shown that it is rather plausible that Sherman equations corresponding to the actual chemical elements and that include the production of secondary fluorescence have one and only one physically acceptable solution. An exhaustive search could elucitate the existence and unicity of solutions for the equations corresponding to the actual chemical elements.     

Diagrammatic kinetic theory for a lattice model of a liquid. I. Theory

We present a diagrammatic formalism for the time correlation functions of density fluctuations for an excluded volume lattice gas on a simple d-dimensional hypercubic lattice. We consider a multicomponent system in which particles of different species can have different transition rates. Our theoretical approach uses a Hilbert space formalism for the time dependent dynamical variables of a stochastic process that satisfies the detailed balance condition. We construct a Liouville matrix consistent with the dynamics of the model to calculate both the equation of motion for multipoint densities in configuration space and the interactions in the diagrammatic theory. A Boley basis of fluctuation vectors for the Hilbert space is used to develop two formally exact diagrammatic series for the time correlation functions. These theoretical techniques are generalizations of methods previously used for spin systems and atomic liquids, and they are generalizable to more complex lattice models of liquids such as a lattice gas with attractive interactions or polymer models. We use our formalism to construct approximate kinetic theories for the van Hove correlation and self-correlation function. The most simple approximation is the mean field approximation, which is exact for the van Hove correlation function of a one component system but an approximation for the self-correlation function. We use our first diagrammatic series to derive a two site multiple scattering approximation that gives a simple analytic expression for the spatial Fourier transform of the self-correlation function. We employ our second diagrammatic series to derive a simple mode coupling type approximation that provides a system of equations that can be solved for the self-correlation function.     

Dynamic behavior of interfaces: Modeling with nonequilibrium thermodynamics

In multiphase systems the transfer of mass, heat, and momentum, both along and across phase interfaces, has an important impact on the overall dynamics of the system. Familiar examples are the effects of surface diffusion on foam drainage (Marangoni effect), or the effect of surface elasticities on the deformation of vesicles or red blood cells in an arterial flow. In this paper we will review recent work on modeling transfer processes associated with interfaces in the context of nonequilibrium thermodynamics (NET). The focus will be on NET frameworks employing the Gibbs dividing surface model, in which the interface is modeled as a two-dimensional plane. This plane has excess variables associated with it, such as a surface mass density, a surface momentum density, a surface energy density, and a surface entropy density. We will review a number of NET frameworks which can be used to derive balance equations and constitutive models for the time rate of change of these excess variables, as a result of in-plane (tangential) transfer processes, and exchange with the adjoining bulk phases. These balance equations must be solved together with mass, momentum, and energy balances for the bulk phases, and a set of boundary conditions coupling the set of bulk and interface equations. This entire set of equations constitutes a comprehensive continuum model for a multiphase system, and allows us to examine the role of the interfacial dynamics on the overall dynamics of the system. With respect to the constitutive equations we will focus primarily on equations for the surface extra stress tensor.     

Average band structure of smectic C* type random materials

We consider a Smectic C* type structure which exhibits alignment fluctuations which can be described by noise associated with both director angles. We take the propagation of axially incident electromagnetic waves, and from Maxwell equations, we establish the stochastic governing set of equations corresponding to optical phenomena. We note that this set of equations can be expressed as a linear vector stochastic system of differential equations with multiplicative noise. We use a procedure to calculate from the stochastic differential set of equations, the governing equations for the expected value of the electromagnetic transverse magnetic and electric fields for a certain autocorrelation function, and calculate explicitly their corresponding band structure for a particular spectral noise density. We have shown that the average resulting electromagnetic fields exhibit a biased decaying exponential dependence which impedes to propagate the waves in one sense while it permits them in the other sense. We have also found a remarkable widening of the band gap and the appearance of new local maxima for the modes without band gap.     

Spatial and temporal progressions of spatial statistical moments in linear chromatography

Generalizations of existing models of chromatography allow the spatial and temporal progressions of all spatial statistical moments in linear chromatography to be given as the solution to a set of ordinary differential equations. Basic strategies of simplifying these equations are described.     

The construction of the Gilmore–Perelomov coherent states for the Kratzer–Fues anharmonic oscillator with the use of the algebraic approach

By applying the algebraic approach and the displacement operator to the ground state, the unknown Gilmore–Perelomov coherent states for the rotating anharmonic Kratzer–Fues oscillator are constructed. In order to obtain the displacement operator the ladder operators have been applied. The deduced SU(1, 1) dynamical symmetry group associated with these operators enables us to construct this important class of the coherent states. Several important properties of these states are discussed. It is shown that the coherent states introduced are not orthogonal and form complete basis set in the Hilbert space. We have found that any vector of Hilbert space of the oscillator studied can be expressed in the coherent states basis set. It has been established that the coherent states satisfy the completeness relation. Also, we have proved that these coherent states do not possess temporal stability. The approach presented can be used to construct the coherent states for other anharmonic oscillators. The coherent states proposed can find applications in laser-matter interactions, in particular with regards to laser chemical processing, laser techniques, in micro-machinning and the patterning, coating and modification of chemical material surfaces.