A geometric analysis of the Schrödinger equation

Vega

Int J of Quantum Chemistry

2005

Self Cite

ABSTRACT:The Schrö dinger-Riccati equation has been applied to the study of some systems of the He-isoelectronic series. The statistical analysis of the local ratios of the predicted function over an accurate function yields, with a sample mean of 1.000, the following confidence intervals (at a 99% level of confidence) for the true population means: He I, 0.0013 (for 24,896 points); Li II, 0.0015 (for 29,354 points); Be III, 0.0018 (for 31,290 points); C V, 0.0024 (for 28,639 points); and O VII, 0.0058 (for 23,784 points).

Section: Theoretical Detailsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Schrödinger–Riccati equation: Feasibility study for the He‐isoelectronic series

Vega

Int J of Quantum Chemistry

2005

Self Cite

“…where H is the Hamiltonian operator (consisting of the kinetic energy and potential-energy operators, T and V , respectively) and W and E denote one of its eigenfunctions and the corresponding eigenvalue, the associated local SRE [1,2] is…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…The details of the derivation of the SRE may be found in previous work [1,2] and here we just summarize the actual procedure for numerical calculations.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

See 1 more Smart Citation

Molecular Schr�dinger-Riccati calculations. Test for the hydrogen-ion molecule

Vega

Theoretical Chemistry Accounts: Theory, Computation, and Modeli

2001

The hydrogen-ion molecule has been used as a test system for the application of the SchroÈ dinger± Riccati formulation to molecular calculations. Some of the points discussed are the characteristics (quasiconstancy of the local energies, size, number of points) of the sampling region to be chosen, the dependence on the starting function, the precision of the calculations, and the excellent behaviour of the predicted function (by comparison to an accurate function).

Int J of Quantum Chemistry

Few sketches on connections between the Riccati and Ermakov–Milne–Pinney equations

Kryachko

2009

ABSTRACT:In this work, we focus on some connections that exist between the Riccati and Ermakov-Milne-Pinney equations, particularly those which are established via the Schrödinger equation. Some formulas that express the general solution of the Riccati and Schrödinger equations in terms of a particular solution of the Riccati equation are obtained. It is shown that the difference between the general and particular solutions of the normal Riccati equation satisfies the Ermakov-Milne-Pinney equation with λ = −1/4. The formulas derived in this work are discussed in view of their quantum chemical applications, especially related to the WKB approach that is applied to describe the semiclassical behavior of molecules. This discussion is illustrated by a simple model of quantum harmonic oscillator.