Pseudopotential Approaches to Localized Orbitals for Polyatomic Systems

First-order perturbation theory is employed to examine sound propagation in flowing media confined by a cylindrical waveguide. The use of perturbation theory allows examination of mode phase-speed changes due to any radially dependent flow w(r) as long as the flow magnitude is sufficiently small. The condition to be fulfilled is satisfied in the flow range: 0-0.3 m/s for the specific values of cylinder radius, ultrasound frequency, and sound speed analyzed in the present work [in the general case, however, the condition in Eq. (1) of the present work must be fulfilled]. This freedom of choice, i.e., the possibility to handle any radial flow profile, is used to analyze two flow profile cases: (1) where w(r) is a linear combination of a laminar flow profile and a flat profile corresponding to turbulent flow, and (2) where w(r) is a linear combination of a laminar flow profile and a more realistic logarithmic-dependent turbulent flow profile. In both cases, it is shown that large errors may result in ultrasound flow measurements if several modes are excited by the transmitting transducer, and that a logarithmic flow profile in the turbulent regime leads to somewhat larger measurement errors at high flow values as compared to assuming a simple flat profile in the turbulent regime.

This paper considers the scattering of an electromagnetic wave from a simplistic model of a random rough surface. The model surface is an ensemble of systems each of which consists of two conducting, parallel, Rayleigh hemicylinders of infinite length and equal radius on an infinite conducting plane. This is the simplest multiple-scattering case of the N-hemicylinder problem, and is examined to gain insights into the approximation methods used to study the more general case. The separation distance between the protrusions varies randomly from one member of the ensemble to another except that the hemicylinders are not permitted to overlap. This surface is illuminated by a plane wave whose electric field vector is parallel to the hemicylinder axes, i.e., having TM polarization. The exact solution for the ensemble average of the square of the scattering amplitude shows that multiple scattering makes a significant contribution to the correction term introduced by the no-overlap condition. The correction term is about a factor of 2 greater than that calculated by first-order perturbation theory, which neglects multiple scattering. This contradicts the tenet that perturbation theory is correct in the limit of small (i.e., Rayleigh) scatterers. We also use a recently developed variational principle to study scattering from the random rough-surface model and find a correction term in essential agreement with the exact result.

Analysis of X-ray scattering from a rough multilayer mirror in the first-order perturbation theory

A two reference fluid equation of state is derived for mixtures consisting of chain molecules, using Wertheim's first-order thermodynamic perturbation theory (TPT1). This equation of state is based on the compressibility factors of two reference fluids of different chain lengths, and avoids the need to estimate the correlation functions of reference fluids. It is shown that the recently proposed pure fluid equation of state based on observations of Wertheim's firstorder perturbation theory and generalized Flory theory due to Escobedo, F. A., and de Pablo, J. J. (1995, J. chem. Phys., 103, 1946) and Sheng, Y.-J., Panagiotopoulos, A. Z., and Kumar, S. K. (1995, J. chem. Phys., 103, 10315) is a direct prediction of this approach. The present study involves a rigorous derivation of the equation of state, and provides the basis to extend the equation of state to mixtures. For particular chain lengths of the two reference fluids, the equivalence of the SAFT and SAFTD equations of state are recovered. The equation of state is shown to be in good agreement with recent molecular simulation results for pure homonuclear and alternating copolymer hard chain fluids.

The relativistic Dirac Hamiltonian that describes the motion of electrons in a magnetic field contains only paramagnetic terms (i.e., terms linear in the vector potential A) while the corresponding nonrelativistic Schrödinger Hamiltonian also contains diamagnetic terms (i.e., those from an A2 operator). We demonstrate that all diamagnetic terms relativistically arise from second-order perturbation theory and that they correspond to a “redressing” of the electrons by the magnetic field. If the nonrelativistic limit is taken with a fixed no-pair Hamiltonian (no redressing), the diamagnetic term is missing. The Schrödinger equation is normally obtained by taking the nonrelativistic limit of the Dirac one-electron equation, we show why nonrelativistic use of the A2 operator is also correct in the many-electron case. In nonrelativistic approaches, diamagnetic terms are usually considered in first-order perturbation theory because they can be evaluated as an expectation value over the ground state wave function. The possibility of also using an expectation value expression, instead of a second-order expression, in the relativistic case is investigated. We also introduce and discuss the concept of “magnetically balanced” basis sets in relativistic calculations.

We show that the Zwanzig first-order perturbation theory can be obtained directly from a truncated Taylor series expansion of a two-body perturbation theory and that such truncation provides a more accurate prediction of thermodynamic properties than the full two-body perturbation theory. This unexpected result is explained by the quality of the resulting approximation for the fluid radial distribution function. We prove that the first-order and the two-body perturbation theories are based on different approximations for the fluid radial distribution function. To illustrate the calculations, the square-well fluid is adopted. We develop an analytical expression for the two-body perturbed Helmholtz free energy for the square-well fluid. The equation of state obtained using such an expression is compared to the equation of state obtained from the first-order approximation. The vapor-liquid coexistence curve and the supercritical compressibility factor of a square-well fluid are calculated using both equations of state and compared to Monte Carlo simulation data. Finally, we show that the approximation for the fluid radial distribution function given by the first-order perturbation theory provides closer values to the ones calculated via Monte Carlo simulations. This explains why such theory gives a better description of the fluid thermodynamic behavior.

An atom is defined as a region of space bound by a surface of local zero flux in the gradient vector field of the electron density. The same boundary condition defines a proper open system, one whose observables and their equations of motion are defined by quantum mechanics. Applied to a crystal, this boundary condition coincides with the original definition of the atomic cell in metallic sodium given by Wigner & Seitz. It is proposed that it be used to generalize the concept of a Wigner–Seitz cell, defining it as the smallest connected region of space bounded by a `zero-flux surface' and exhibiting the translational invariance of the crystal. This definition, as well as removing the arbitrary nature of the original method of construction of the cell in the general case, maximizes the relation of the cell and the derived atomic form factors to the physical form exhibited by the charge distribution of its constituent atoms. The topology of the electron density, as summarized in terms of its critical points, also defines the atomic connectivity and structure within a cell. Attention is drawn to the correspondence of the symmetries of the structural elements determined by the critical points with the site symmetries tabulated in International Tables for Crystallography. The atomic scattering factor is defined for an atom in a crystal and determined in ab initio calculations for diamond and silicon. The transferable nature of atomic charge distributions is demonstrated. It enables one to estimate a structure factor and its phase in a crystal using the density of an atom or functional group obtained in a molecular calculation. Atoms in a crystal, along with defects and vacancies, are identifiable with bounded regions of real space. Their properties are additive and are defined by quantum mechanics.

A number of numerical experiments were performed to assess the validity of diffusion theory and various perturbation methods for calculating the reactivity state of a severely disrupted liquid metal cooled fast breeder reactor (LMFBR). The disrupted configurations correspond, in general, to phases through which an LMFBR core could pass during a core disruptive accident (CDA). Two-reactor models were chosen for this study, the two zone, homogeneous Clinch River Breeder Reactor and the Large Heterogeneous Reactor Design Study Core. The various phases were chosen to approximate the CDA results predicted by the safety analysis code SAS3D. The calculational methods investigated in this study include the eigenvalue difference technique based on both discrete ordinate transport theory and diffusion theory, first-order perturbation theory, exact perturbation theory, and a new hybrid perturbation theory. Selected cases were analyzed using Monte Carlo methods. It was found that in all cases, diffusion theory and perturbation theory yielded results for the change in reactivity that significantly disagreed with both the discrete ordinate and Monte Carlo results. These differences were, in most cases, in a nonconservative direction.

Abstract. Nonlinear effects at the bottom profile of convex shape (non-reflecting beach) are studied using asymptotic approach (nonlinear WKB approximation) and direct perturbation theory. In the asymptotic approach the nonlinearity leads to the generation of high-order harmonics in the propagating wave, which result in the wave breaking when the wave propagates shoreward, while within the perturbation theory besides wave deformation it leads to the variations in the mean sea level and wave reflection (waves do not reflect from "non-reflecting" beach in the linear theory). The nonlinear corrections (second harmonics) are calculated within both approaches and compared between each other. It is shown that for the wave propagating shoreward the nonlinear correction is smaller than the one predicted by the asymptotic approach, while for the offshore propagating wave they have a similar asymptotic. Nonlinear corrections for both waves propagating shoreward and seaward demonstrate the oscillatory character, caused by interference of the incident and reflected waves in the second-order perturbation theory, while there is no reflection in the linear approximation (first-order perturbation theory). Expressions for wave set-up and set-down along the non-reflecting beach are found and discussed.

We develop a physically based equation of state that describes Mie ν-6 fluids with an accuracy comparable to that of state-of-the-art empirical models. The equation of state is developed within the framework of the uv-theory [T. van Westen and J. Gross, J. Chem. Phys. 155, 244501 (2021)], which is modified by incorporating the third virial coefficient B3 in the low-density description of the model. The new model interpolates between a first-order Weeks-Chandler-Andersen (WCA) perturbation theory at high densities and a modified first-order WCA theory that recovers the virial expansion up to B3 at low densities. A new algebraic equation for the third virial coefficient of Mie ν-6 fluids is developed-other inputs are taken from previous work. Predicted thermodynamic properties and phase equilibria are compared to a comprehensive database of molecular simulation results from the literature, including Mie fluids of repulsive exponents 9 ≤ ν ≤ 48. The new equation of state is applicable to states with densities up to ρ*(T*)⪅1.1+0.12T* and temperatures T* > 0.3. For the Lennard-Jones fluid (ν = 12), the performance of the model is comparable to that of the best empirical equations of state available. As compared to empirical models, the physical basis of the new model provides several advantages, however: (1) the new model is applicable to Mie fluids of repulsive exponents 9 ≤ ν ≤ 48 instead of only ν = 12, (2) the model leads to a better description of the meta-stable and unstable region (which is important for describing interfacial properties by classical density functional theory), and (3) being a first-order perturbation theory, the new model (potentially) allows an easier and more rigorous extension to non-spherical (chain) fluids and mixtures.

In earlier work by Gelbard and Lell, arguments based on perturbation theory were used to obtain relations between mean-square chord lengths and lattice eigenvalues for given bucklings. We show here that first-order perturbation theory does not give the lattice eigenvalue correctly to order B/sup 2/. When all missing terms in B/sup 2/ are inserted, the eigenvalue buckling relations remain formally unchanged, but the mean-square chord lengths must be redefined. The original and redefined mean-square chord lengths differ only insofar as events in successive fission generations are correlated. A reexamination of work based on the original relations indicates that earlier numerical results were substantially correct.

We consider three-photon--one-photon phase control of resonance-enhanced photonioization with a phase-diffusion field. As is well known, control is achieved because excitation via the fundamental field interferes with excitation via the third harmonic field, and the form of the interference (i.e., constructive or destructive) depends on the relative phase difference between the two fields. In the problem, the stochastic nature of the field influences control because the propagation constant of a dispersive medium depends on the field's fluctuating frequency. Here, we approach the influence of laser phase noise on control via (i) a physically intuitive zeroth-order approximation, (ii) first-order perturbation theory, and (iii) numerical simulation. Our results show that first-order perturbation theory is not attractive for the study of this problem, since it requires the evaluation of very high order correlation functions associated with the fundamental field"s fluctuating frequency. More importantly, however, numerical simulation demonstrates that highly efficient control can be attained even in the presence of very large linewidth fields.

The thermodynamic and structural properties of Yukawa hard chains are considered in this work. We specifically examined the influence of the Yukawa potential range parameter λ on the thermodynamic properties, inter- and intrachain correlation functions of the chain fluid. The compressibility factor or pressure of Yukawa chain fluids was calculated via a first-order perturbation theory using hard-sphere chain fluid as a reference. Monte Carlo simulations were performed to calculate the thermodynamic properties and inter- and intrachain correlation functions of the fluid for λ=1.8 and 3.0. Simulation results for the compressibility were compared with calculations from the first-order perturbation theory and the Statistical Associating Fluid Theory-Variable Range (SAFT-VR) model. We found that both theories represent the simulated data very well. In particular, the SAFT-VR equation provides an excellent estimate of the properties of Yukawa hard chains over wide ranges of temperature and density. Simulated data for interchain and intrachain correlation functions reveal that the local structure of the Yukawa hard chains is strongly effected by the range parameter of the potential.

Autopoietic Systems in General Phenomenal Domains

Application of the x-ray scattering (XRS) technique for studying super-smooth surfaces such as Si wafers is discussed. The XRS method is demonstrated to enable quantitative evaluation of power spectrum density (PSD) functions and effective roughness of super-smooth surfaces. Within the calculation of PSD functions, comparative study between first-order vector perturbation theory (FOPT) and generalized Harvey-shack theory (H-S) is performed. First-order perturbation theory which is widely accepted and has been extensively validated even for large scattered and incident angles for "smooth" surfaces considers the scattering amplitude as a power series in the roughness height; its scattering diagram is related to the statistical parameters of surface roughness (PSD functions) in a very simple way (linear). Therefore, PSD functions can be uniquely and directly extracted from the measured data. However, generalized Harvey-shack theory considers that scattering behavior is characterized by a surface transfer function which relates the scattering behavior to the surface topography. With the grazing incident angle less than critical value about 0.22 degree, three Si wafers with rms roughness of 0.29nm, 0.46nm and 0.67nm are inspected by XRS (λ=0.154nm) method. The calculated values are all in a good agreement with the results obtained from Atomic force microscope (AFM). However, the difference resulting from the limits of applicability of the theories used in XRS data processing appears and be analyzed. Both of the theories are not only used for optical surface characterization, but also can allow accurate predictions of image degradation due to scattering effects in grazing X-ray telescopes. The experimental schemes are also analyzed.