Rayleigh Ritz variational method

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

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 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. 

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

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, < >, ,... 

An alternative way, based on a pure formulation, of computing degenerate continumm orbitals at a given energy E for atomic and molecular systems, was proposed by Froese Fischer and Idrees [188], based on an extension of the Rayleigh-Ritz-Galerkin method for bound states. This is a variational approach... 

The available methods in molecular electronic structure theory are illustrated in Figure 1 with a family tree of quantum chemistry labeled with the acronyms of some of the most often used methods. The variety is a bit daunting to newcomers, who might be cautioned by a comment by Levine If you learn enough abbreviations you can convince some people that you know quantum chemistry. Flowever, as for most areas of science, electronic structure theory looks much worse from the outside than from the inside. The tree has three main branches density functional theory (DFT), quantum Monte Carlo (QMC), and Rayleigh-Ritz variational theory (RRV). Each of these leads to additional branches. In addition there are a number of interbranch connections indicated by dotted lines. We give a brief description below of the DFT and RRV branches and their relation to QMC, which is described in sections follow-... 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

The Rayleigh-Ritz variational theory is the basis for so-called variational methods in which an estimate of the energy of a system is calculated for an approximate trial wavefunction usually assembled from combinations of atomic orbitals. Expectation values of the energy may be calculated accurately for many trial wavefunctions and are upper bounds to the true energy. If the parameters of the trial wavefunctions are varied systematically, the lowest upper bound to the energy for a particular form of trial wavefunction may be determined (thus the term variational ). The trial functions must satisfy certain restrictions such... 

The variational approach received a major boost also when it was realised [79] that the simplest variational method - the Kohn variational principle, which is essentially the Rayleigh-Ritz variational principle for eigenvalues modified to incorporate scattering boundary conditions - is free of anomalous (i.e., spurious, unphysical) singularities if it is formulated with S-matrix type boundary conditions rather than standing wave boundary conditions as had been typically used previously. It is useful first to state the Kohn variational approach for the general inelastic scattering. Thus the variational expression for the S-matrix is... 

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... 

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. 

The variational quantum Monte Carlo method (VMC) is both simpler and more efficient than the DMC method, but also usually less accurate. In this method the Rayleigh-Ritz quotient for a trial function 0 is evaluated with Monte Carlo integration. The Metropolis-Hastings algorithm " is used to sample the distribution... 

Variational principles play an important role in the solution of the Schrbdinger equation, e.g. the Ritz-method [5] for the solution of partial differential equations and the Rayleigh-Ritz method [6] for the calculation of bound states of atoms and molecules. The oldest variational method for scattering problems was introduced 1944 by Hulthen [7]. Four years later Hulthen and Kohn [8, 9, 10] independently developed what is now known as the Hulthen-Kohn... 

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