Rayleigh-Ritz Variational Method Research Articles

A Schrödinger Equation for Evolutionary Dynamics

We establish an analogy between the Fokker–Planck equation describing evolutionary landscape dynamics and the Schrödinger equation which characterizes quantum mechanical particles, showing that a population with multiple genetic traits evolves analogously to a wavefunction under a multi-dimensional energy potential in imaginary time. Furthermore, we discover within this analogy that the stationary population distribution on the landscape corresponds exactly to the ground-state wavefunction. This mathematical equivalence grants entry to a wide range of analytical tools developed by the quantum mechanics community, such as the Rayleigh–Ritz variational method and the Rayleigh–Schrödinger perturbation theory, allowing us not only the conduct of reasonable quantitative assessments but also exploration of fundamental biological inquiries. We demonstrate the effectiveness of these tools by estimating the population success on landscapes where precise answers are elusive, and unveiling the ecological consequences of stress-induced mutagenesis—a prevalent evolutionary mechanism in pathogenic and neoplastic systems. We show that, even in an unchanging environment, a sharp mutational burst resulting from stress can always be advantageous, while a gradual increase only enhances population size when the number of relevant evolving traits is limited. Our interdisciplinary approach offers novel insights, opening up new avenues for deeper understanding and predictive capability regarding the complex dynamics of evolving populations.

Sensitivity of Localized Surface Plasmon Resonance and Acoustic Vibrations to Edge Rounding in Silver Nanocubes.

Precise knowledge of the dependence of nano-object properties on their structural characteristics such as their size, shape, composition, or crystallinity, in turn, enables them to be finely characterized using appropriate techniques. Spectrophotometry and inelastic light scattering spectroscopy are noninvasive techniques that are proving highly robust and efficient for characterizing the optical response and vibrational properties of metal nano-objects. Here, we investigate the optical and vibrational properties of monodomain silver nanocubes synthesized by the chemical route, with edge length ranging from around 20 to 58 nm. The synthesized nanocrystals are not perfectly cubic and exhibit rounded edges and corners. This rounding was quantitatively taken into account by assimilating the shape of the nanocubes to superellipsoids. The effect of rounding on their optical response was clearly evidenced by localized surface plasmon resonance spectroscopy and supported by calculations based on the discrete dipole approximation method. The study of their acoustic vibrations by high-resolution low-frequency Raman scattering revealed a substructure of the T2g band, which was analyzed as a function of rounding. The measured frequencies are consistent with the existence of an anticrossing pattern of the two T2g branches. Such an avoided crossing in the T2g modes is clearly evidenced by calculating the vibrational frequencies of silver nanocubes using the Rayleigh-Ritz variational method that accounts for both their real size, shape, and cubic elasticity. These results show that it is possible to assess the rounding of nanocubes, including by means of ensemble spectroscopic measurements on well-calibrated particles.

Polar compactons and solitons in a two dimensional optical waveguide: Theory and simulations

The propagation of optical cylindrical compactons and solitons in a two dimensional nonlocal nonlinear waveguide is deeply investigated by carrying a particular emphasis on its nonlinear response function. Considering the weakly nonlocal medium, the two dimensional nonlocal nonlinear Schrödinger equation describing the propagation of signal light is derived. This equation contains the basic nonlinear Schrödinger equation with additional nonlinear terms proportional to the second order space derivative of coordinates in the transverse directions. By seeking the solution in polar form, this equation is separated into real and imaginary parts, followed by the analysis of the equilibrium points and their stability, predicting the type of solutions. These solutions are found both with linear and nonlinear phases through direct integration. In order to seek polar solutions, the nonlocal nonlinear Schrödinger equation is expressed in cylindrical coordinate and the resulting equation, containing additional terms inversely proportional to the radius r is not easily solvable. The type of solutions of this equation is predicted by the investigation of new equilibrium points and their stability. The analytical cylindrical compactons and pulse solitons as solutions of this equation are provided by using the Rayleigh–Ritz variational method, usually used for symmetric radial solitons to approximate pulse solitons in polar coordinate, and the stability of these solutions is checked through numerical investigations.

Nonlinear vibration characteristics of composite pyramidal truss sandwich cylindrical shell panels with amplitude dependence

Recently studies on the bending, compression, buckling, and other static issues of composite pyramidal truss sandwich cylindrical shell (CPTSCS) panels have been reported. However, rare literature has explored the nonlinear vibration characteristics of the CPTSCS panels. In this paper, their amplitude-dependent vibration properties are studied both experimentally and theoretically. First, a fabrication flowchart is proposed to obtain such panel specimens, and a pulse excitation method via a hammer and sweeping frequency and fixed excitation techniques provided by a base vibration shaker are employed to measure the nonlinear variation of vibration parameters. As a result, the amplitude-dependent vibration phenomenon is confirmed to exist in both the pyramidal truss core and composite skins. After the nonlinear assumptions of Young's moduli, shear moduli, and loss factors of the pyramidal truss core and composite skins are defined based on Jones-Nelson nonlinear material theory, complex modulus method, Gibson equivalent principle, etc., a novel theoretical model is developed to predict the nonlinear natural frequencies, damping ratios, and resonant responses under different base excitation energies, in which the first order shear deformation theory, Rayleigh Ritz variation method, modal strain energy approach, and modal superposition technique are adopted to solve key vibration parameters. The pre-defined amplitude-dependent fitting coefficients that are vital to the completeness of the current model are further identified using the measured frequency and damping data. Also, detailed tests are done to make sure that both the proposed model and the predictions are correct. Finally, the effects of critical geometric and material parameters of the CPTSCS panels on the amplitude-dependent vibration characteristics are evaluated based on the calculated dynamic results, with some practical suggestions being refined for improving the vibration suppression capability of such advanced composite structures.

Ground state energy of the hydrogen atom inside penetrable spherical cavities; variational approach

In this work we calculate the ground state energy of the hydrogen atom confined in a sphere of penetrable walls of radius Rc. Inside the sphere the system is subject to a Coulomb potential, whereas outside of it the potential is a finite constant V0. The energy is obtained as a function of Rc and V0 by means of the Rayleigh-Ritz variational method, in which, the trial function is proposed as a free particle wave function within a finite square well potential but including an exponential factor that takes into account the electron-nucleus Coulomb attraction. For an impenetrable sphere, , the energy grows fast as Rc approaches zero. On the other hand, when the height of the barrier V0 is finite, the energy increases slowly as Rc goes to zero. We also compute the Fermi contact term, nuclear magnetic screening, polarizability, pressure and tunneling as a function of Rc and V0. As expected, these physical quantities approach the corresponding values of the free hydrogen atom as Rc grows. We also discuss the pressure-induced ionization of the hydrogen atom. The present results are found in good agreement with those previously published in the literature.

Transition frequencies between 2S and 2P states of lithium-like ions

The Schrödinger equation for the 2S and 2P states of the lithium-like ions Z = 5–7, 9–10 is solved by using the Rayleigh-Ritz variational method in Hylleraas coordinates. The leading-order relativistic and QED corrections are calculated perturbatively and higher-order corrections are estimated. The transition frequencies between the 2S1/2 and 2P J (J = 1/2, 3/2) states are determined and compared with experimental and other theoretical results. Specifically, isotope shifts are also calculated for B2+.

Electron affinity of LiH-

In this work we present high-accuracy benchmark-quality calculations of the electron affinity (EA) of the LiH molecule in a framework that does not assume the Born–Oppenheimer (BO) approximation. The EA is calculated as a difference between the total energies of and LiH. The calculations of the energies are performed using the Rayleigh-Ritz variational method with large basis sets of all-particle explicitly correlated Gaussian functions (ECGs). Up to 14,000 ECGs are used in the calculations for each system. The nonlinear parameters of the ECGs are optimised by by the minimisation of the total non-relativistic energy of the system using an approach that employs the energy gradient determined with respect the parameters. The and LiH non-relativistic non-BO wave functions are subsequently used to calculate the leading relativistic corrections. The calculated EA is well converged in terms of the size of the basis sets and the obtained value falls within the uncertainty of the best available experimental result.

Variational approach to the Schrödinger equation with a delta-function potential

We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.

Study of the transition from resonance to bound states in quantum dots embedded on a nanowire using the [formula omitted] method

We study the band structure of semiconductor nanowires with quantum dots embedded in them. The band structure is calculated using the Rayleigh–Ritz variational method. We consider quantum dots of two different types, one type is defined by electrostatic potentials applied to the nanowire, while the other one is defined by adding materials with band offsets with respect to the band parameters of the nanowire. We are particularly interested in the appearance of discrete energy levels in the gap between the conduction band and the valence band of the nanostructure, and in the dependence of the energy of these levels with the intensity of a magnetic field applied along the wire. It is shown that several scenarios are possible, being of particular interest the possibility of transforming states of the discrete into resonances and vice versa.

Variational calculation of the lowest exciton states in phosphorene and transition metal dichalcogenides

Several transition metal dichalcogenides (TMDs) can be exfoliated to produce nearly two-dimensional (2D) semiconductor layers supporting robust excitons with non-hydrogenic Rydberg series of states. Black phosphorus (BP) can also be layered to create a nearly 2D material with interesting properties including its pronounced in-plane anisotropy that influences, in particular, exciton states making them different from those in other 2D semiconductors. We apply the Rayleigh–Ritz variational method to evaluate the energies and approximate the wavefunctions of the ground and lowest excited states of the exciton in a 2D semiconductor with anisotropic effective masses of electrons and holes. The electron–hole interaction is described by the Rytova–Keldysh potential, which is considered beyond the standard zero-thickness approximation. The exciton binding energies calculated for BP and TMD (molybdenum disulphide and tungsten disulphide) monolayers are compared with previously published data.

Negative ion of hydrogen in dense semi-classical plasmas: Stability and zero-energy resonances

The effects of dense semi-classical plasma (DSCP) on the ground state of the negative ion of hydrogen (H−) and on the dynamics of electron–hydrogen scattering have been investigated. DSCP is described by an effective potential which takes care of the collective effects of the plasma at large distances as well as the quantum mechanical effects of diffraction at small distances. An elaborative wave function is employed in the Rayleigh–Ritz variational method to compute the ground state energy of H− for various values of the plasma parameters. In particular, critical values of the plasma parameters are calculated accurately to make a detailed study on the stability of the ion embedded in DSCP. Furthermore, parameters related to the ground state of H− and H are used in the effective range theory to study the effects of DSCP on the dynamics of low-energy e − H(1s) scattering. Special emphasis is given to investigate the phenomenon of zero-energy resonances by computing the singlet scattering length near the critical values of the plasma parameters.

Free Vibrations of Anisotropic Nano-Objects with Rounded or Sharp Corners.

An extension of the Rayleigh–Ritz variational method to objects with superquadric and superellipsoid shapes and cylinders with cross-sections delimited by a superellipse is presented. It enables the quick calculation of the frequencies and displacements for shapes commonly observed in nano-objects. Original smooth shape variations between objects with plane, convex, and concave faces are presented. The validity of frequently used isotropic approximations for experimentally relevant vibrations is discussed. This extension is expected to facilitate the assignment of features observed with vibrational spectroscopies, in particular in the case of single-nanoparticle measurements.

Energy levels and radiative transitions of Mg VII and Si IX

Term energies, line strengths, transition probabilities, and transition wavelengths among the (1s2)2s22p2, 2s22p3p, 2s2p3, 2s22p3s and 2s22p3d 1,3,5L (L = S, P, D, F) configuration states in Mg VII and Si IX are calculated by using the multiconfiguration Rayleigh-Ritz variation method and restricted variation method. The transition line strengths and transition probabilities of the electric dipole transitions are both given in length and velocity gauges. Differences between these two gauge values are discussed. The calculated atomic parameters show good agreement with the observed experimental results and other theoretical data. Furthermore, the uncertainty of each electric dipole transition line strength is estimated. For several transition parameters, uncertainties are decreased compared to the values from the National Institute of Standards and Technology Atomic Spectra Database.

Spectrum of theq-Schrödinger equation by means of the variational method based on the discreteq-Hermite I polynomials

In this work, the [Formula: see text]-Schrödinger equations with symmetric polynomial potentials are considered. The spectrum of the model is obtained for several values of [Formula: see text], and the limiting case as [Formula: see text] is considered. The Rayleigh–Ritz variational method is adopted to the system. The discrete [Formula: see text]-Hermite I polynomials are handled as basis in this method. Furthermore, the following potentials with numerous results are presented as applications: [Formula: see text]-harmonic, purely [Formula: see text]-quartic and [Formula: see text]-quartic oscillators. It is also shown that the obtained results confirm the ones that exist in the literature for the continuous case.

Nonrelativistic energies and fine-structure splittings for the Rydberg P states of lithium

The Schr\odinger equation for the $nP$ $(4\ensuremath{\le}n\ensuremath{\le}10)$ states of lithium is solved using the Rayleigh-Ritz variational method in Hylleraas coordinates, where the asymptotic behavior of the wave function that is specific to a Rydberg state is built in explicitly to improve the accuracy of the solution. The nonrelativistic variational energies of the $nP$ $(4\ensuremath{\le}n\ensuremath{\le}10)$ states reach an accuracy of ${10}^{\ensuremath{-}13}$ to ${10}^{\ensuremath{-}14}$. In addition, the fine-structure splittings of these states are also calculated and compared with experimental results.

Energy levels and transition probabilities of N+, F3+, and Ne4+ ions

Abstract Term energies, oscillator strengths, transition probabilities, and transition wavelengths among the low-lying states of (1s2)2s22p2, 2s22p3p, 2s2p3, 2s22p3s, and 2s22p3d 1,3,5 L L = S, P, D, F in N+, F3+, and Ne4+ ions were calculated by using the multiconfiguration Rayleigh-Ritz variation method and restricted variation method. The transition oscillator strengths and transition probabilities for the electric dipole transitions are both given in length and velocity gauges. Deviations between these two gauge values are discussed. The calculated atomic parameters are in good agreement with the observed experimental results and other theoretical data. Furthermore, the uncertainty of each electric dipole transition is estimated. Several uncertainties of transition parameters are improved when comparing with values from national institute of standards and technology NIST database. Atomic parameters presented in this paper should be useful for identifying the levels as well as for precise spectral modeling in astrophysical and laboratory plasmas in the future work.

Energy levels of a coupled-rotors model

We calculate the energy levels of a coupled-rotors model that has proved useful for the analysis of the multiple tunnelling peaks in the inelastic neutron scattering from 4-methylpyridine crystals. We resort to the Rayleigh–Ritz variational method with a basis set for each irreducible representation. The energy levels in terms of the rotor coupling strength exhibit a rich pattern of crossings between states of different symmetry and avoided crossings between states of the same symmetry.

Energies, radial expectation values, hyperfine structures of the ground state and highly excited states for C and O2+

The Rayleigh–Ritz variational method with multiconfiguration interaction wave functions is used to calculate energies, radiative transitions and radial expectation values of the [Formula: see text] [Formula: see text] ground state and the [Formula: see text], [Formula: see text], [Formula: see text] highly excited states of C and [Formula: see text]. Hyperfine structure parameters and magnetic coupling constants of these states are also calculated in this work. The present calculations agree well with theoretical and experimental values available in the literature. Other data not reported in the literature are expected to offer valuable benchmarks for future research.

Properties of the Positronium Negative Ion Embedded in Non-ideal Classical Plasmas

Properties of the positronium negative ion embedded in non-ideal classical plasmas have been studied theoretically. A pseudopotential, derived from a solution of Bogolyubov’s hierarchy equations, is used to describe the interaction potentials of the charged particles in the ion. A large basis set is employed in Rayleigh–Ritz variational method to compute accurately various quantities, such as binding energy, cusp values, annihilation rate, associated with the ground state of the ion. A detailed study is made on the effects of non-ideality of plasma on those quantities. In particular, special emphasis is given to determine the ranges of plasma screening parameters within which the ion remains stable.

Stability of the negative ion of hydrogen in nonideal classical plasmas.

The stability of the negative ion of hydrogen (H^{-}) embedded in nonideal classical plasma has been studied by computing the ground state energy of the ion quite accurately. The interactions among the charged particles in plasma have been modelled by a pseudopotential, derived from a solution of Bogolyubov's hierarchy equations. An extensive basis set is employed in Rayleigh-Ritz variational method to compute the ground state energy of H^{-} for various values of plasma parameters. Effects of nonideality of plasma on the stability of the ion have been investigated in detail for a wide range of nonideality. Particular emphasis is made to compute accurately the critical values of the plasma screening parameters.