Remarks on the second order error in the variational calculation of expectation values

1977 ◽  
Vol 18 (2) ◽  
pp. 342 ◽  
Author(s):  
Saul T. Epstein
Keyword(s):  
Variational Calculation ◽  
Second Order ◽  
Expectation Values ◽  
Order Error

scholarly journals Origin of the residual line width under frequency-switched Lee–Goldburg decoupling in MAS solid-state NMR

2020 ◽  
Vol 1 (1) ◽  
pp. 13-25 ◽  
Author(s):  
Johannes Hellwagner ◽  
Liam Grunwald ◽  
Manuel Ochsner ◽  
Daniel Zindel ◽  
Beat H. Meier ◽  
...  
Keyword(s):  
Solid State ◽  
Line Width ◽  
Floquet Theory ◽  
Pulse Sequence ◽  
Second Order ◽  
Magic Angle ◽  
Line Broadening ◽  
Third Order ◽  
Order Error ◽  
Error Terms

Abstract. Homonuclear decoupling sequences in solid-state nuclear magnetic resonance (NMR) under magic-angle spinning (MAS) show experimentally significantly larger residual line width than expected from Floquet theory to second order. We present an in-depth theoretical and experimental analysis of the origin of the residual line width under decoupling based on frequency-switched Lee–Goldburg (FSLG) sequences. We analyze the effect of experimental pulse-shape errors (e.g., pulse transients and B1-field inhomogeneities) and use a Floquet-theory-based description of higher-order error terms that arise from the interference between the MAS rotation and the pulse sequence. It is shown that the magnitude of the third-order auto term of a single homo- or heteronuclear coupled spin pair is important and leads to significant line broadening under FSLG decoupling. Furthermore, we show the dependence of these third-order error terms on the angle of the effective field with the B0 field. An analysis of second-order cross terms is presented that shows that the influence of three-spin terms is small since they are averaged by the pulse sequence. The importance of the inhomogeneity of the radio-frequency (rf) field is discussed and shown to be the main source of residual line broadening while pulse transients do not seem to play an important role. Experimentally, the influence of the combination of these error terms is shown by using restricted samples and pulse-transient compensation. The results show that all terms are additive but the major contribution to the residual line width comes from the rf-field inhomogeneity for the standard implementation of FSLG sequences, which is significant even for samples with a restricted volume.

scholarly journals Direct Solution of Second Order Ordinary Differential Equations Using a Class of Hybrid Block Methods

2020 ◽  
Vol 5 (2) ◽  
Author(s):  
Emmanuel A Areo ◽  
Nosimot O Adeyanju ◽  
Sunday J Kayode
Keyword(s):  
Initial Value Problems ◽  
Hybrid Methods ◽  
Multistep Method ◽  
Second Order ◽  
Block Method ◽  
Initial Value ◽  
Order Error ◽  
Block Methods ◽  
Grid Points ◽  
Step Number

This research proposes the derivation of a class of hybrid methods for solution of second order initial value problems (IVPs) in block mode. Continuous linear multistep method of two cases with step number k = 4 is developed by interpolating the basis function at certain grid points and collocating the differential system at both grid and off-grid points. The basic properties of the method including order, error constant, zero stability, consistency and convergence were investigated. In order to examine the accuracy of the methods, some differential problems of order two were solved and results generated show a better performance when comparison is made with some current methods.Keywords- Block Method, Hybrid Points, Initial Value Problems, Power Series, Second Order 

scholarly journals First- and second-order error estimates in Monte Carlo integration

2016 ◽  
Vol 208 ◽  
pp. 29-34 ◽  
Author(s):  
R. Bakx ◽  
R.H.P. Kleiss ◽  
F. Versteegen
Keyword(s):  
Monte Carlo ◽  
Error Estimates ◽  
Second Order ◽  
Monte Carlo Integration ◽  
Order Error

Perturbation treatment of the Hartree–Fock equations for the ground states of the boron-like ions

1969 ◽  
Vol 47 (7) ◽  
pp. 699-705 ◽  
Author(s):  
C. S. Sharma ◽  
R. G. Wilson
Keyword(s):  
Ground State ◽  
Atomic System ◽  
Ground States ◽  
Great Accuracy ◽  
Second Order ◽  
Expectation Values ◽  
Third Order ◽  
First Order ◽  
Hartree Fock ◽  
The Third

The first-order Hartree–Fock and unrestricted Hartree–Fock equations for the ground state of a five electron atomic system are solved exactly. The solutions are used to evaluate the corresponding second-order energies exactly and the third-order energies with great accuracy. The first-order terms in the expectation values of 1/r, r, r2, and δ(r) are also calculated.

Second-order error; error that illuminates context

Kybernetes ◽  
2014 ◽  
Vol 43 (9/10) ◽  
pp. 1354-1361
Author(s):  
Ted Krueger
Keyword(s):  
Design Methodology ◽  
Second Order ◽  
Order Error ◽  
Content Type

Purpose – The purpose of this paper is to propose a novel classification of errors. Design/methodology/approach – A review of the classification of errors in several disciplines is undertaken. Findings – The role of errors in the delineation of the frameworks in which they occur is suggested.

scholarly journals Theory of Second Order Numerical Simulation Method of Enhanced Oil Production

2019 ◽  
Vol 11 (2) ◽  
pp. 73
Author(s):  
Yirang Yuan ◽  
Aijie Cheng ◽  
Danping Yang ◽  
Changfeng Li
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Oil Production ◽  
Coupled System ◽  
Simulation Method ◽  
Second Order ◽  
Optimal Order ◽  
Difference Operators ◽  
Order Error ◽  
L2 Norm

A kind of second-order implicit upwind fractional steps finite difference method is presented in this paper to numerically simulate the coupled system of enhanced (chemical) oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and optimal order error estimates in l2 norm are derived.

scholarly journals First and second order error estimates for the Upwind Source at Interface method

2004 ◽  
Vol 74 (249) ◽  
pp. 103-123 ◽  
Author(s):  
Theodoros Katsaounis ◽  
Chiara Simeoni
Keyword(s):  
Error Estimates ◽  
Second Order ◽  
Order Error
Sign in / Sign up

Export Citation Format

Share Document