Lower Bounds to Expectation Values: Two-Electron Atoms

1973 ◽  
Vol 51 (2) ◽  
pp. 260-264 ◽  
Author(s):  
D. P. Chong ◽  
F. Weinhold
Keyword(s):  
Perturbation Theory ◽  
Lower Bound ◽  
Excited States ◽  
Lower Bounds ◽  
Order Perturbation ◽  
Expectation Values ◽  
High Quality ◽  
Variational Calculations ◽  
Very High ◽  
Rule Out

Weinhold's lower bound formulas are used on rather accurate wavefunctions for the 11S ground states of H−, He, and Li+, and the 21S and 23S excited states of He. The results are found to be of very high quality, as judged by comparison with the best available variational calculations, and can be used to rule out many of the values calculated by Scherr and Knight from high-order perturbation theory.

"Zero" and "Off-Zero" Critical Concentrations in Solutions of Polydisperse Polymers with Very High Molar Masses

1995 ◽  
Vol 60 (10) ◽  
pp. 1661-1688 ◽  
Author(s):  
Karel Šolc ◽  
Karel Dušek ◽  
Ronald Koningsveld ◽  
Hugo Berghmans
Keyword(s):  
Perturbation Theory ◽  
Half Plane ◽  
Diphenyl Ether ◽  
Order Perturbation ◽  
Order Perturbation Theory ◽  
Critical Concentrations ◽  
Quadratic Interaction ◽  
Region Ii ◽  
Very High

Polymer solutions with a concentration-dependent interaction parameter g(ϕ) are known to have sometimes critical polymer concentrations ϕc converging to a non-zero value (a so-called "off-zero" limiting critical point (CP)), as the chain length, m, grows to infinity, rather than to zero as usual (a "zero" limiting CP). In this report the criteria for the existence of both types, known for binary solutions with a linear g = g0 + g1ϕ, are extended to cover polydisperse polymers with a quadratic interaction function g(ϕ). Its coefficients g2 and ∆g2 = g2 - g1 determine the number and type of limiting CPs. Accordingly, the plane g2, ∆g2 is divided into the regions I (a zero CP), II (an off-zero CP), and III (a zero + two off-zero CPs). The region II is restricted to the half-plane with ∆g2 < -1/6, whereas the other half-plane with ∆g2 > -1/6 is shared by I and III. By varying the interactions, two limiting CPs may be brought together and merged in a heterogeneous double limiting CP. Such instances define the boundaries between the regions: at the I/III line, two off-zero CPs merge, whereas at the II/III line an off-zero CP coincides with a zero CP. A first-order perturbation theory of the latter double CPs, and a second-order perturbation theory of single "zero" CPs are developed, enabling meaningful extrapolations of data on polymers with high but finite molar masses. The latter theory yields extrapolation formulas for determination of Η-temperature, taking into account the polymer polydispersity and the concentration dependence of g. Solutions of polyisobutene in diphenyl ether and, possibly, in benzene appear to present experimental examples of off-zero limiting critical concentrations.

Complementary variational principles in perturbation theory

1968 ◽  
Vol 303 (1475) ◽  
pp. 503-509 ◽  
Keyword(s):  
Perturbation Theory ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Lower Bounds ◽  
Upper Bound ◽  
Variational Principles ◽  
Upper And Lower Bounds ◽  
Quantum Mechanical ◽  
Simple Application ◽  
Mechanical Perturbation

Complementary upper and lower bounds are derived for second-order quantum-mechanical perturbation energies. The upper bound is equivalent to that of Hylleraas. The lower bound appears to be new, but reduces to that of Prager & Hirschfelder if a certain constraint is applied. A simple application to a perturbed harmonic oscillator is presented.

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