Rayleigh variational 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... 

It will be recalled that the Variation Principle asks us to choose those values of cr which make the Rayleigh Ratio, y, stationary. We thus seek a situation in which a change in cr (r = 1,2,..., , in turn) produces no change in y. In other words, we shall require n conditions of the sort... 

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. 

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... 

As noted above, these authors also proved the general validity, for the Rayleigh problem, of the principle of exchange of stabilities. Further, by formulating the problem in terms of a variational principle, Pellew and South-well devised a technique which led to a very rapid and accurate approximation for the critical Rayleigh number. Later, a second variational principle was presented by Chandrasekhar (C3). A review by Reid and Harris (R2) also includes other approximate methods for handling the Benard problem with solid boundaries. 

However, there is a stationary variation principle of precisely the type employed in the quantum chemical linear variation method. In the derivation of the Roothaan equations based on finite basis set expansions of Schrodinger wavefimctions, one insists only that the Rayleigh quotient be stationary with respect to the variational parameters, and then assumes that the variational principle guarantees an absolute minimum. In the corresponding linear equations based on the Dirac equation, the stationary condition is imposed, but no further assumption is made about the nature of the stationary point. 

The principal computational steps are first to orthogonalize the Xijk basis set, and then to diagonalize the Hamiltonian matrix H in the orthogonalized basis set so as to satisfy the Rayleigh-Schrbdinger variational principle... 

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, < >, ,... 

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