A perturbation calculation of properties of the 1 s σ and ­2 p σ states of HeH 2+

1956 ◽  
Vol 238 (1213) ◽  
pp. 276-285 ◽  
Keyword(s):  
Dipole Moments ◽  
High Accuracy ◽  
Wave Functions ◽  
Quadrupole Moments ◽  
Perturbation Calculation ◽  
Third Order ◽  
The Third ◽  
Wide Range ◽  
Fifth Order ◽  
Kinetic And Potential Energies

A perturbation calculation, valid in the limit of large separations, of various properties of the 1 s σ and 2 p σ states of HeH 2+ is carried out. The total energy and the kinetic and potential energies are calculated to the fifth order, the dipole moments to the third order and the quadrupole moments to the second order. The results are compared with those obtained using exact and variationally determined two-centre wave functions and also with those obtained from an approximate application of perturbation theory and it is shown that perturbation calculations of molecular properties are capable of high accuracy over a wide range of nuclear separations.

A perturbation calculation of properties of the 2 pπ state of HeH 2+

1957 ◽  
Vol 240 (1221) ◽  
pp. 274-283 ◽  
Keyword(s):  
Dipole Moment ◽  
Total Energy ◽  
Second Order ◽  
Wave Functions ◽  
Quadrupole Moments ◽  
Perturbation Calculation ◽  
Third Order ◽  
The Third ◽  
Fifth Order ◽  
Kinetic And Potential Energies

A perturbation calculation, valid in the limit of large separations, of various properties of the 2 pπ state of HeH 2+ is carried out. The total energy and the kinetic and potential energies are calculated to the fifth order, the dipole moment to the third order and the quadrupole moments to the second order and the results compared with those obtained using exact and variationally determined two-centre wave functions. Some results are also given for the 2 pπ u and 3 dπ g states of H + 2 and the influence of nuclear symmetry at large separations is briefly discussed.

Synthesis and third-order nonlinear optical properties of novel ethynyl-linked heteropentamer composed of four porphyrins and one pyrene

2009 ◽  
Vol 13 (02) ◽  
pp. 275-282 ◽  
Author(s):  
Ning Sheng ◽  
Jing Sun ◽  
Yongzhong Bian ◽  
Jianzhuang Jiang ◽  
Dong Xu
Keyword(s):  
Optical Properties ◽  
Nonlinear Optical ◽  
Three Dimensional ◽  
Spectroscopic Methods ◽  
Third Order ◽  
Zinc Compounds ◽  
Nlo Properties ◽  
Metal Free ◽  
The Third ◽  
Wide Range

Novel heteropentameric porphyrins-pyrene arrays, in which four meso-tetraphenyl porphyrins are linked to the center unit of pyrene by four acetylenyl bonds, were designed and synthesized. The newly synthesized heteropentameric compounds have been characterized by a wide range of spectroscopic methods. The third-order nonlinear optical (NLO) properties of both the metal-free and zinc compounds of the three-dimensional arrays were investigated by Z-scan experiments, showing enhanced NLO properties compared with that of the porphyrin and pyrene monomers.

scholarly journals Achieving Seventh-Order Amplitude Accuracy in Leapfrog Integrations

2013 ◽  
Vol 141 (9) ◽  
pp. 3037-3051 ◽  
Author(s):  
Paul D. Williams
Keyword(s):  
Nonlinear System ◽  
Third Order ◽  
First Order ◽  
Time Stepping ◽  
The Third ◽  
Numerical Integrations ◽  
Seventh Order ◽  
Ocean Models ◽  
Stability And Accuracy ◽  
Fifth Order

Abstract The leapfrog time-stepping scheme makes no amplitude errors when integrating linear oscillations. Unfortunately, the Robert–Asselin filter, which is used to damp the computational mode, introduces first-order amplitude errors. The Robert–Asselin–Williams (RAW) filter, which was recently proposed as an improvement, eliminates the first-order amplitude errors and yields third-order amplitude accuracy. However, it has not previously been shown how to further improve the accuracy by eliminating the third- and higher-order amplitude errors. Here, it is shown that leapfrogging over a suitably weighted blend of the filtered and unfiltered tendencies eliminates the third-order amplitude errors and yields fifth-order amplitude accuracy. It is further shown that the use of a more discriminating (1, −4, 6, −4, 1) filter instead of a (1, −2, 1) filter eliminates the fifth-order amplitude errors and yields seventh-order amplitude accuracy. Other related schemes are obtained by varying the values of the filter parameters, and it is found that several combinations offer an appealing compromise of stability and accuracy. The proposed new schemes are tested in numerical integrations of a simple nonlinear system. They appear to be attractive alternatives to the filtered leapfrog schemes currently used in many atmosphere and ocean models.

scholarly journals Comparison of 6 Diode and 6 Transistor Mixers Based on Analysis and Measurement

2016 ◽  
Vol 2016 ◽  
pp. 1-17 ◽  
Author(s):  
J. Ladvánszky ◽  
K. M. Osbáth
Keyword(s):  
Field Effect Transistors ◽  
Input Power ◽  
Bipolar Transistors ◽  
Large Signal ◽  
Design Environment ◽  
Third Order ◽  
The Third ◽  
Wide Range ◽  
Measurement Results ◽  
Versus Input

Our goal is to overview semiconductor mixers designed for good large signal performance. Twelve different mixers were compared utilizing pn diodes, bipolar transistors, and/or junction field effect transistors. The main aspect of comparison is the third-order intercept point (IP3), and both circuit analysis and measurement results have been considered. IP3 has been analyzed by the program AWR (NI AWR Design Environment) and measured by two-tone test (Keysight Technologies). We provide three ways of improvement of large signal performance: application of a diplexer at the RF port, reduction of DC currents, and exploiting a region of RF input power with infinite IP3. In addition to that, our contributions are several modifications of existing mixers and a new mixer circuit (as illustrated in the figures). It is widely believed that the slope of the third-order intermodulation product versus input power is always greater than that of the first-order product. However, measurement and analysis revealed (as illustrated in the figures) that the two lines may be parallel over a broad range of input power, thus resulting in infinite IP3. Mixer knowledge may be useful for a wide range of readers because almost every radio contains at least one mixer.

Applicability of Kolmogorov's and Monin's equations of turbulence

1997 ◽  
Vol 353 ◽  
pp. 67-81 ◽  
Author(s):  
REGINALD J. HILL
Keyword(s):  
Structure Function ◽  
Structure Functions ◽  
Inertial Range ◽  
Homogeneous Turbulence ◽  
Order Structure ◽  
Third Order ◽  
Local Isotropy ◽  
The Third ◽  
Moderate Reynolds Number ◽  
Wide Range

The equation relating second- and third-order velocity structure functions was presented by Kolmogorov; Monin attempted to derive that equation on the basis of local isotropy. Recently, concerns have been raised to the effect that Kolmogorov's equation and an ancillary incompressibility condition governing the third-order structure function were proven only on the restrictive basis of isotropy and that the statistic involving pressure that appears in the derivation of Kolmogorov's equation might not vanish on the basis of local isotropy. These concerns are resolved. In so doing, results are obtained for the second- and third-order statistics on the basis of local homogeneity without use of local isotropy. These results are applicable to future studies of the approach toward local isotropy. Accuracy of Kolmogorov's equation is shown to be more sensitive to anisotropy of the third-order structure function than to anisotropy of the second-order structure function. Kolmogorov's 4/5 law for the inertial range of the third-order structure function is obtained without use of the incompressibility conditions on the second- and third-order structure functions. A generalization of Kolmogorov's 4/5 law, which applies to the inertial range of locally homogeneous turbulence at very large Reynolds numbers, is shown to also apply to the energy-containing range for the more restrictive case of stationary, homogeneous turbulence. The variety of derivations of Kolmogorov's and Monin's equations leads to a wide range of applicability to experimental conditions, including, in some cases, turbulence of moderate Reynolds number.

The mechanism of the Cannizzaro reaction of Formaldehyde

1954 ◽  
Vol 7 (4) ◽  
pp. 335 ◽  
Author(s):  
RJL Martin
Keyword(s):  
Fourth Order ◽  
Kinetic Data ◽  
Order Reaction ◽  
Rate Equations ◽  
Third Order ◽  
The Third ◽  
Hydride Ion ◽  
Cannizzaro Reaction ◽  
Wide Range ◽  
Mechanism Of The Reaction

For a wide range of concentrations of formaldehyde and alkali, the Cannizzaro reaction of formaldehyde can be described as the sum of a third and a fourth order reaction. However, the concentrations which are used for the rate equations must be corrected for the amount of methylene glycol anion present. The dissociation constant of methylene glycol as determined from the kinetic data is the same magnitude as that derived electrometrically. The mechanism of the reaction is interpreted as a reaction between formaldehyde and the hydride ion donors CH2(O-)(OH) and CH2(O-)(O-) It is shown why the third order reaction proposed by previous workers is not always applicable.

Bilateral series and Ramanujan radial limits of mock (false) theta functions

2019 ◽  
Vol 30 (04) ◽  
pp. 1950023
Author(s):  
Bin Chen
Keyword(s):  
Modular Forms ◽  
Theta Functions ◽  
Mock Theta Functions ◽  
Third Order ◽  
Radial Limits ◽  
The Third ◽  
False Theta Functions ◽  
Even Order ◽  
Fifth Order ◽  
Vector Valued

Ramanujan gave a list of seventeen functions which he called mock theta functions. For one of the third-order mock theta functions [Formula: see text], he claimed that as [Formula: see text] approaches an even order [Formula: see text] root of unity [Formula: see text], then [Formula: see text] He also pointed at the existence of similar properties for other mock theta functions. Recently, [J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci, Bilateral series and Ramanujan’s radial limits, Proc. Amer. Math. Soc. 143(2) (2014) 479–492] presented some similar Ramanujan radial limits of the fifth-order mock theta functions and their associated bilateral series are modular forms. In this paper, by using the substitution [Formula: see text] in the Ramanujan’s mock theta functions, some associated false theta functions in the sense of Rogers are obtained. Such functions can be regarded as Eichler integral of the vector-valued modular forms of weight [Formula: see text]. We find two associated bilateral series of the false theta functions with respect to the fifth-order mock theta functions are special modular forms. Furthermore, we explore that the other two associated bilateral series of the false theta functions with respect to the third-order mock theta functions are mock modular forms. As an application, the associated Ramanujan radial limits of the false theta functions are constructed.

NONLINEAR APPROACH OF ALPHA AND CLUSTER DECAYS IN THE REACTION CHANNEL

1996 ◽  
Vol 05 (02) ◽  
pp. 329-344 ◽  
Author(s):  
V.G. KARTAVENKO ◽  
A. LUDU ◽  
A. SĂNDULESCU ◽  
W. GREINER
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Reaction Channel ◽  
Nuclear Surface ◽  
Third Order ◽  
Properties Of Nuclei ◽  
The Third ◽  
Nonlinear Approach ◽  
Cubic Schrödinger Equation ◽  
Fifth Order

In the framework of nonlinear nuclear hydrodynamics with realistic effective Skyrme contact δ-interactions, a Schrödinger equation with a fifth order nonlinearity is deduced. By using quasi soliton-like solutions, which describe well the general properties of nuclei, we studied the behavior of alpha and cluster decays in the reaction channel. Previously, we studied the formation of clusters as solitons on nuclear surface with the help of the third order (cubic) Schrödinger equation.

scholarly journals Bounds on the conductivity of a random array of cylinders

1988 ◽  
Vol 417 (1852) ◽  
pp. 59-80 ◽  
Keyword(s):  
Fourth Order ◽  
Volume Fraction ◽  
Effective Conductivity ◽  
Basis Set ◽  
Circular Cylinders ◽  
Third Order ◽  
Entire Volume ◽  
The Third ◽  
Wide Range ◽  
Order Bounds

We consider the problem of determining rigorous third-order and fourth-order bounds on the effective conductivity σ e of a composite material composed of aligned, infinitely long, equisized, rigid, circular cylinders of conductivity σ 2 randomly distributed throughout a matrix of conductivity σ 1 . Both bounds involve the microstructural parameter ξ 2 which is an integral that depends upon S 3 , the three-point probability function of the composite (G. W. Milton, J. Mech. Phys. Solids 30, 177-191 (1982)). The key multidimensional integral ξ 2 is greatly simplified by expanding the orientation-dependent terms of its integrand in Chebyshev polynomials and using the orthogonality properties of this basis set. The resulting simplified expression is computed for an equilibrium distribution of rigid cylinders at selected ϕ 2 (cylinder volume fraction) values in the range 0 ≼ ϕ 2 ≼ 0.65. The physical significance of the parameter ξ 2 for general microstructures is briefly discussed. For a wide range of ϕ 2 and α = σ 2 /σ 1 , the third-order bounds significantly improve upon second-order bounds which only incorporate volume fraction information; the fourth-order bounds, in turn, are always more restrictive than the third-order bounds. The fourth-order bounds on σ e are found to be sharp enough to yield good estimates of σ e for a wide range of ϕ 2 , even when the phase conductivities differ by as much as two orders of magnitude. When the cylinders are perfectly conducting ( α = ∞), moreover, the fourth-order lower bound on σ e provides an excellent estimate of this quantity for the entire volume-fraction range studied here, i. e. up to a volume fraction of 65%.

Dynamic Analysis of Non-Uniform Beams With Time Dependent Elastic Boundary Conditions

1995 ◽  
Author(s):  
Sen Yung Lee ◽  
Shueei Muh Lin
Keyword(s):  
Boundary Conditions ◽  
Time Dependent ◽  
Polynomial Functions ◽  
Third Order ◽  
The Third ◽  
Elastic Boundary ◽  
Order Degree ◽  
Elastically Restrained ◽  
Fifth Order ◽  
Elastic Boundary Conditions

Abstract The dynamic response of a non-uniform beam with time dependent elastic boundary conditions is studied by generalizing the method of Mindlin-Goodman and utilizing the exact solutions of general elastically restrained non-uniform beams given by Lee and Kuo. The time dependent elastic boundary conditions for the beam are formulated. A general form of change of dependent variable is introduced and the shifting polynomials of the third order degree, instead of the fifth order degree polynomials taken by Mindlin-Goodman, are selected. The physical meaning of these shifting polynomial functions are explored. Finally, the limiting cases are discussed and several examples are given to illustrate the analysis.

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