Rayleigh-Ritz principle

Now we turn to the topic of interest, that of the ground state of a many -Fermion system represented by a grand ensemble at chemical potential /i. The appropriate Rayleigh-Ritz principle for such a system is... 

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 ... 

For the ground state, variational calculations are based on the Rayleigh-Ritz principle... 

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 ... 

Gross, E. K. U., Oliveira, L. N., Kohn, W., 1988a, Rayleigh-Ritz Variational Principle for Ensembles of Fractionally Occupied States , Phys. Rev. A, 37, 2805. 

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... 

Eq. (22) shows that the Coulomb fitting variation principle is a boimd fixim below, so improving the fitting basis raises the total energy. This behavior is the reverse of what h pens to the total energy as the orbital basis is augmented (because of the Rayleigh-Ritz variational principle). The difference can seem counter-intuitive to new users of the method. 

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... 

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. 

Applying the Rayleigh-Ritz variational principle, one obtains 4 H, ) < ( P //i )... 

For many years configuration interaction was regarded as the method of choice in describing electron correlation effeets in atoms and moleeules. The method is robust and systematic being firmly based on the Rayleigh-Ritz variational principle. The total electronie wavefimetion, is written as a linear eombination of A/ -electron determinantal functions, < >, ,... 

Komi, D. 1., T. Markovich, N. Maxwell, and E. R. Bittner. 2009. Supersymmetric quantum mechanics, excited state energies and wave functions, and the Rayleigh-Ritz variational principle a proof of principle study. Journal of Physical Chemistry A 113 (52) 15257. 

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