Rayleigh-Ritz functional

HF (HF) theory is based on the idea that one takes an antisymmetrized product wavefunction and uses the variational principle to obtain the best possible approximation to the A -particle wavefunction that cannot be represented by such a single determinant. Thus, one inserts the single determinant into the Rayleigh-Ritz functional and performs a constraint variation of the orbitals. The results of the variational process are the famous HF equations that are satisfied by each of the orbitals ... 

The deflection results for the three approaches are plotted in Figure 5-10 as a function of the ratio of the principal stiffnesses, D., /D22. which gets large as 0 decreases. Thus, the larger the D.,0 and 025. the smaller 0 becomes and hence the more inaccurate both the Rayleigh-Ritz approach and the specially orthotropic approximation become. 

The presence of D g 26 governing differential equation and the boundary conditions renders a closed-form solution impossible. That is, in analogy to both bending and buckling of a symmetric angle-ply (or anisotropic) plate, the variation in lateral displacement, 5vy, cannot be separated into a function of x alone times a function of y alone. Again, however, the Rayleigh-Ritz approach is quite useful. The expression... 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

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... 

The density functional theory for ensembles is based on the generalized Rayleigh-Ritz variational principle [7]. The eigenvalue problem of the Hamiltonian H is given by... 

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... 

B. Klahn, W.A. Bingel, The convergence of the Rayleigh-Ritz method in quantum chemistry. II. Investigation of the convergence for special systems of Slater, Gauss and two-electron functions, Theor. Chim. Acta 44 (1977) 27-43. 

On the other hand, Marin and Cruz [16-18] used the direct variational method or Rayleigh-Ritz method with a trial function of the same form used by Gorecki and Byers Brown (Equation (25)). Marin and Cruz chose hydrogenic functions with a variational parameter as the exponent for the functions

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Varshni [22] studied the CHA by means of the Rayleigh-Ritz method. He proposed a modification of wave function (26), introducing an additional variational parameter /9. In his approach, the Is, 2p and 3d CHA wave functions are written as... 

Simulation of confinement by penetrable boxes represents a more realistic physical model. A very simple approach was proposed by Marin and Cruz [18], where they used the Rayleigh-Ritz variational method via a trial wave function for the ground state, which consists of two piecewise functions, one for the inner region (r < ro), and the other for the outer region (r > ro). The trial wave function is defined as follows ... 

Both techniques are based on the Rayleigh-Ritz variational principle using energy-independent basis functions of the muffin-tin orbital type [1.21]. 

Instead of applying tail cancellation as in Sect.2.1 where we derived the KKR-ASA equations, one may use the linear combination of muffin-tin orbitals (5.27) directly in a variational procedure. This has the advantages that it leads to an eigenvalue problem and that it is possible to include non-muffin-tin perturbations to the potential. According to the Rayleigh-Ritz variational principle, one varies y to make the energy functional stationary, i.e. 

To do this, we use the Rayleigh-Ritz variational principle in connection with the radial trial function of arbitrary logarithmic derivative D at the sphere boundary defined by the linear combination... 

Vibration-rotation partition function for HC1 obtained via standard Rayleigh-Ritz variational (var) basis-set methods from Topper et al. [46]. 

The energy of a system is given in terms of its exact wave function F by Eq. (1). If we seek instead a reliable estimate of the wave function, it is common to rely on the Rayleigh-Ritz principle ... 

Wheeler and collaborators [3], in the context of nuclear physics, showed at that time that the limit in the variational procedure potential itself was not reached. Indeed, the Rayleigh-Ritz (RR) variational scheme teaches us how to obtain the best value for a parameter in a trial function, i.e., exponents of Slater (STO) or Gaussian (GTO) type orbital, Roothaan or linear combination of atomic orbitals (LCAO) expansion coefficients and Cl coefficients. Instead, the generator coordinate method (GCM) introduces the Hill-Wheeler (HW) equation, an integral transform algorithm capable, in principle, to find the best functional form for a given trial function. We present the GCM and the HW equation in Section 2. 

Compute the compliance of the CDCB specimen by the Rayleigh-Ritz method [8] as a function of crack length and specimen-slope C = C(a,p). 

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... 

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