Rayleigh-Ritz variation

MacDonald J K L 1933 Successive approximations by the Rayleigh-Ritz variation method Phys. Rev4Z 830-3... 

Selective Photodynamic Control of Chemical Reactions A Rayleigh-Ritz Variational Approach... 

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. 

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... 

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 ... 

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... 

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

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