Ritz variational principle

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. 

The determinants Jj form now a very convenient set of trial functions. The Raleigh-Ritz variational principle, keeping the determineuits As fixed and varying only the coefficients aim -. -nipin, leads to the following matrix eigenvalue problem ... 

Variational methods [6] for the solution of either the Schrodinger equation or its perturbation expansion can be used to obtain approximate eigenvalues and eigenfunctions of this Hamiltonian. The Ritz variational principle,... 

For the inequality between ground state energies, required for the second step, a minimum principle for the ground state energy (2.40) is used. However, while the Ritz variational principle is well established in the non-relativistic... 

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... 

E. BESALU and J. MARTI, Exploring the Raylei -Ritz variational principle. J. Chem. Educ., 75, 105 (1998). 

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. 

Presently, the widely used post-Hartree-Fock approaches to the correlation problem in molecular electronic structure calculations are basically of two kinds, namely, those of variational and those of perturbative nature. The former are typified by various configuration interaction (Cl) or shell-model methods, and employ the linear Ansatz for the wave function in the spirit of Ritz variation principle (c/, e.g. Ref. [21]). However, since the dimension of the Cl problem rapidly increases with increasing size of the system and size of the atomic orbital (AO) basis set employed (see, e.g. the so-called Paldus-Weyl dimension formula [22,23]), one has to rely in actual applications on truncated Cl expansions (referred to as a limited Cl), despite the fact that these expansions are slowly convergent, even when based on the optimal natural orbitals (NOs). Unfortunately, such limited Cl expansions (usually truncated at the doubly excited level relative to the IPM reference, resulting in the CISD method) are unable to properly describe the so-called dynamic correlation, which requires that higher than doubly excited configurations be taken into account. Moreover, the energies obtained with the limited Cl method are not size-extensive. 

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... 

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. 

The static and dynamic mean-field equations are derived, for a given energy-density functional, by variation with respect to the single-particle wave functions Ritz variational principle of minimal energy yields the static mean-field equations,... 

Let us, therefore, examine what the direct minimization of E X) in Ai leads to. According to the Ritz variational principle, E X) is bounded from below by the exact, full Cl, energy, so that... 

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

The proof of the Rayleigh-Ritz variation principle (Section 6-12) involves essentially two ideas. The first is that any function can be expanded into a linear combination of other functions that span the same function space. Thus, for example, exp(/ x) can be expressed as cos(fo) + i sin(fo). An exponential can also be written as a linear combination of powers of the argument ... 

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