Rayleigh-Ritz variational calculation

As regards the ground state of a given symmetry, the fundamental theoretical tool is the existence of the energy lower bound. This fact allows Rayleigh-Ritz variational calculations to converge monotonically upon variation of the size of the basis set and/or of linear or nonlinear variational parameters. 

Rates of Convergence of Rayleigh-Ritz Variational Calculations on Atoms and Molecules... 

In this article I shall discuss how spatial dimension D impacts the rates of convergence of Rayleigh-Ritz variational calculations on atoms and molecules, in which the nuclei and electrons are assumed to interact by 1/r potentials (rather than Coulombic potentials, which are proportional to and hence overwhelm the kinetic energy... 

Powers of 3 appear in several contexts in the analysis of rates of convergence of Rayleigh-Ritz variational calculations ... 

In this section we shall examine the competition between singularity and localisation effects in Rayleigh-Ritz variational calculations performed by John Loeser and Dudley Herschbach [22] on heliumlike ions for a wide range of D and Z, using a Pekeris-type basis of products of generalised Laguerre functions... 

I should like to conclude by taking from my discussion in [5] a summary of the key issues, in order of importance, for performing highly accurate Rayleigh-Ritz variational calculations on atoms and molecules ... 

In quantum calculations, the Rayleigh-Ritz variational method is widely used to approximate the solution of the Schrodinger equation [86], To obtain exact results, one should expand the exact wave function in a complete basis set... 

Many of the calculations of quantum chemistry are based on the Rayleigh-Ritz variation principle which states For any normalized, acceptable function 4>,... 

The Rayleigh-Ritz variational theory is the basis for so-called variational methods in which an estimate of the energy of a system is calculated for an approximate trial wavefunction usually assembled from combinations of atomic orbitals. Expectation values of the energy may be calculated accurately for many trial wavefunctions and are upper bounds to the true energy. If the parameters of the trial wavefunctions are varied systematically, the lowest upper bound to the energy for a particular form of trial wavefunction may be determined (thus the term variational ). The trial functions must satisfy certain restrictions such... 

In bound-state calculations, the Rayleigh-Ritz or Schrodinger variational principle provides both an upper bound to an exact energy and a stationary property that determines free parameters in the wave function. In scattering theory, the energy is specified in advance. Variational principles are used to determine the wave function but do not generally provide variational bounds. A variational functional is made stationary by choice of variational parameters, but the sign of the residual error is not determined. Because there is no well-defined bounded quantity, there is no simple absolute standard of comparison between different variational trial functions. The present discussion will develop a stationary estimate of the multichannel A -matrix. Because this matrix is real and symmetric for open channels, it provides the most... 

In 1997, Pakiari and Mohammadi used the FSGO basis set for a perturbation variation Rayleigh Ritz (PV = RR) calculation. We used a matrix representation Schrodinger equation for the configuration interaction calculation. 

Variational principles play an important role in the solution of the Schrbdinger equation, e.g. the Ritz-method [5] for the solution of partial differential equations and the Rayleigh-Ritz method [6] for the calculation of bound states of atoms and molecules. The oldest variational method for scattering problems was introduced 1944 by Hulthen [7]. Four years later Hulthen and Kohn [8, 9, 10] independently developed what is now known as the Hulthen-Kohn... 

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