A recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, <i>J. Chem. Theory Comput.</i> <b>2021</b>, <i>17</i>, 1535) is further developed and applied to the highly accurate calculation of the ground-state energy of two- (He, Li⁺, and H^-) and three- (Li) electron atoms. The method has been implemented with explicitly correlated many-particle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilli-Hartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained. Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory.

We show that the numerical results contained in a recent paper are affected by a non optimal implementation of the methods which have been used to obtain these results. A careful analysis done using the Rayleigh-Ritz method provides a rigorous upper bound for the energy of the ground state of an electron in a two dimensional potential generated by the edge dislocation, as well as precise values for the excited states. The extrapolation of the results corresponding to different subspaces is used to obtain a precise estimate of the fundamental energy of the model. The energies of the first 500 states that we have calculated are in perfect agreement with the expected asymptotic behavior.

We show that the numerical results contained in a recent paper are affected by a non optimal implementation of the methods which have been used to obtain these results. A careful analysis done using the Rayleigh-Ritz method provides a rigorous upper bound for the energy of the ground state of an electron in a two dimensional potential generated by the edge dislocation, as well as precise values for the excited states. The extrapolation of the results corresponding to different subspaces is used to obtain a precise estimate of the fundamental energy of the model. The energies of the first 500 states that we have calculated are in perfect agreement with the expected asymptotic behavior.

The stability of the random field Ising model (RFIM) against spin glass (SG) fluctuations, as investigated by Mézard and Young, is naturally expressed via Legendre transforms, stability being then associated with the non-negativeness of eigenvalues of the inverse of a generalized SG susceptibility matrix. It is found that the signal for the occurrence of the SG transition will manifest itself in free-energy {\sl fluctuations\/} only, and not in the free energy itself. Eigenvalues of the inverse SG susceptibility matrix is then approached by the Rayleigh Ritz method which provides an upper bound. Coming from the paramagnetic phase {\sl on the Curie line,\/} one is able to use a virial-like relationship generated by scaling the {\sl single\/} unit length $ (D<6; $ in higher dimension a new length sets in, the inverse momentum cut off). Instability towards a SG phase being probed on pairs of {\sl distinct\/} replicas, it follows that, despite the repulsive coupling of the RFIM the effective pair coupling is {\sl attractive\/} (at least for small values of the parameter $ g\bar Δ, $ $ g $ the coupling and $ \bar Δ$ the effective random field fluctuation). As a result, \lq\lq bound states\rq\rq\ associated with replica pairs (negative eigenvalues) provide the instability signature. {\sl Away from the Curie line\/}, the attraction is damped out till the SG transition line is reached and paramagnetism restored. In $ D<6, $ the SG transition always precedes the ferromagnetic one, thu

The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, which originated in the context of solving physical boundary-value The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, which originated in the context of solving physical boundary-value problems. It is named after Lord Rayleigh and Walther Ritz. In this method, an infinite-dimensional linear operator is approximated by a finite-dimensional compression, enabling the use of a numerical eigenvalue algorithm. It is used in all applicat The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, which originated in the context of solving physical boundary-value problems. It is named after Lord Rayleigh and Walther Ritz. In this method, an infinite-dimensional linear operator is approximated by a finite-dimensional compression, enabling the use of a numerical eigenvalue algorithm. It is used in all applications that involve approximating eigenvalues and eigenvectors, often under different names. In quantum mechanics, where a system of particles is described using a Hamiltonian, it uses trial wave functions to approximate the ground-state eigenfunction. In the context of the finite-element method, it is mathematically the same as the Ritz-Galerkin method. In mechanical and structural engineering, it is used to approximate the eigenmodes and resonant frequencies of a structure. A related adaption of Rayleigh-Ritz known as Hamiltonian truncation can be used in quantum field theory. That is, the ground-state energy is less than this value. The trial wave-function will always give an expectation value larger than or equal to the ground-energy. If the trial wave function is known to be orthogonal to the ground state, then it will provide a boundary for the energy of some excited state. The Ritz ansatz function is a linear combination of N known basis functions { Ψ i } {\displaystyle \left\lbrace \Psi _{i}\right\rbrace } , parametrized by unknown coefficients:

Estimation of hematocrit profile symmetry recovery length downstream from a bifurcation. Downstream from a microvascular bifurcation the distribution of blood cells in the vessel lumen is not symmetric. A diffusion process is used to model the rearrangement of red cells as blood flows between junctions in the microcirculation. A Fourier series approach is used to solve the model diffusion convection equation in slit geometry. Both flat and parabolic velocity profiles are considered. The eigenvalues, found using the Rayleigh-Ritz method, are used to find an upper bound on distance required for a symmetric red cell distribution to be obtained. The method has also been applied to cylindrical geometry and the computed symmetry recovery lengths are compared to distances between bifurcations measured in vivo. These estimates indicate that red cell distributions are frequently asymmetric in the microcirculation. Such asymmetries can have a strong effect on plasma skimming and material balance calculations. Published in Biorheology (1989)

Upper and lower bounds of the first four natural frequencies of elastic clamped arcs, which vibrate in a plane perpendicular to that of the initial curvature of the arcs, are obtained by applying to curved beams a method, based on differential operator theory, originally proposed by Lehmann and Maehly. The principal advantage of the method is that it provides, at the same time, the set of upper and lower bounds of natural frequencies of vibrating systems. The center lines of the arcs are in the forms of circles, cycloids, catenaries, and parabolas. Numerical results are presented in tabular form. The upper-bound frequencies so obtained are compared with those obtained by the application of the Rayleigh-Ritz method.

Large deformation mechanics of the enucleated eyeball. Large deformation of enucleated pig eyeballs under rigid cylindrical indenters was studied analytically and experimentally. The analytic model for the eyeball consists of a fluid-filled spherical membrane composed of an incompressible, elastic material with an exponential strain energy function. The Rayleigh-Ritz technique provided an approximate solution via a potential energy formulation. Comparison with results from tests on eyeballs and a water-filled rubber (Mooney-Rivlin) shell shows good agreement at large deflection, where membrane action dominates. Due to the highly nonlinear stress-strain relations for the sclera, the load remains relatively small until the indenter displacement approaches 40-60 percent of the eyeball radius, and then the load increases rapidly. Depending on the indenter size, either a perforation or a rupture type of failure occurs. Published in Journal of biomechanical engineering (1984)