Zero-temperature second random phase approximation and its formal properties.

The random phase approximation (RPA) is fully renormalized to take into account the ground-state correlations (g.s.c.) beyond the RPA, which arise due to the fermion structure of the collective RPA operators. As compared to the conventional renormalized RPA (RRPA), the novelty of the present extended renormalized RPA (ERRPA) is the inclusion of the particle-particle $(\mathrm{pp})$ and hole-hole $(\mathrm{hh})$ g.s.c. beyond the RPA along with the $\mathrm{ph}$ ones in the complete set of ERRPA equations. The formalism is illustrated by numerical calculations within a schematic four-level model. The results show that the effects of g.s.c. beyond the RPA due to $\mathrm{pp}$ (and $hh)$ transitions can cause a partial or complete compensation of the ones due to $\mathrm{ph}$ transitions. This reduces the deviation from the RPA energy-weighted sum of strength within the ERRPA as compared to the one within the conventional RRPA.

Vibrational states of nuclei in the random phase approximation

Our understanding of nuclear structure is built upon mean-field theories such as Hartree-Fock and time-dependent Hartree-Fock. The small-amplitude limit of the latter is the random phase approximation (RPA), which is widely used to model giant resonances in nuclei. Despite this popularity, RPA has been mostly validated against toy models; tests against complex models are scarce in the literature. We perform a thorough test of the RPA against full 0hw shell model (SM) calculations, including in our investigation binding energies, scalar ground-state observables, for which we develop a new method, and transition strengths. We allow deformed Hartree-Fock solutions and compare results for spherical and deformed nuclei. We obtain reasonable agreement between RPA and SM, albeit with some significant failures. Particularly, we found that the low-lying collectivity is poorly described for deformed mean-field solutions, which we interpret as incomplete symmetry restoration in RPA. Results for observables, and in particular for J2, also point out toward the same conclusion regarding the symmetries of the ground state. We also prove, both analytically and numerically, that a long-standing theorem regarding RPA is violated in the case of deformation. The worse violation appears for low-lying transitions, such as isoscalar E2, which we consider as a third argument for an incomplete symmetry restoration.

The study is concerned with electromagnetic wave (EM) scattering by a random sea surface in the presence of coherent wave patterns. The coherent patterns are understood in a broad sense as the existence of certain dynamical coupling between linear Fourier components of the water wave field. We show that the presence of weakly nonlinear wave patterns can significantly change the EM scattering compared to the case of a completely random wave field. Generalizing the Random Phase Approximation (RPA) we suggest a new paradigm for EM scattering by a random sea surface. The specific analysis carried out in the paper synthesizes the small perturbation method for EM scattering and a weakly nonlinear approach for wind wave dynamics. By investigating, in detail, two examples of a random sea surface composed of either Stokes waves or horse-shoe (‘crescent-shaped’) patterns the mechanism of the pattern effect on scattering is revealed. Each Fourier harmonic of the scattered EM field is found to be a sum of contributions due to different combinations of wave field harmonics. Among these ‘partial scatterings’ there are phase-dependent ones and, therefore, the intensity of the resulting EM harmonic is sensitive to the phase relations between the wind wave harmonics. The effect can be interpreted as interference of partial scatterings due to the co-existence of several phase-related periodic scattering grids. A straightforward generalization of these results enables us to obtain, for a given wind wave field and an incident EM field, an a priori estimate of whether the effects due to the patterns are significant and the commonly used RPA is inapplicable. When the RPA is inapplicable, we suggest its natural generalization by re-defining the statistical ensemble for water surface. First, EM scattering by an ‘elementary’ constituent pattern should be considered. Each such scattering is affected by the interference because the harmonics comprising the pattern are dynamically linked. Then, ensemble averaging, which takes into account the distribution of the pattern parameters (based on the assumption that the phases between the patterns are random), should be carried out. It is shown that, generally, this interference does not vanish for any statistical ensemble due to dynamical coupling between water wave harmonics. The suggested RPA generalization takes into account weak non-Gaussianity of water wave field m contrast to the traditional RPA which ignores it.

We investigate the reliability of transition strengths computed in the random-phase approximation (RPA), comparing with exact results from diagonalization in full $0\hbar\omega$ shell-model spaces. The RPA and shell-model results are in reasonable agreement for most transitions; however some very low-lying collective transitions, such as isoscalar quadrupole, are in serious disagreement. We suggest the failure lies with incomplete restoration of broken symmetries in the RPA. Furthermore we prove, analytically and numerically, that standard statements regarding the energy-weighted sum rule in the RPA do not hold if an exact symmetry is broken.

A random phase approximation (RPA) calculation and a direct sum over states is used to calculate second−order optical properties and van der Waals coefficients. A basis set expansion technique is used and no continuumlike functions are included in the basis. However, unlike other methods we do not force the basis functions to satisfy any sum−rule constraints but rather the formalism (RPA) is such that the Thomas−Reiche−Kuhn sum rule is satisfied exactly. Central attention is paid to the dynamic polarizability from which most of the other properties are derived. Application is made to helium and molecular hydrogen. In addition to the polarizability and van der Waals coefficients, results are given for the molecular anisotropy of H2, Rayleigh scattering cross sections, and Verdet constants as a function of frequency. Agreement with experiment and other theories is good. Other energy weighted sum rules are calculated and compare very well with previous estimates. The practicality of our method suggests its applications to larger molecular systems and other properties.

The fulfilment of sum rules for the loss function which is determined using different models of the dielectric function (DF): the random phase approximation (RPA), Mermin, and extended models of RPA and Mermin is investigated1. In the extended models the dynamic collision frequency is used, which is calculated according to computer simulations and within the Born-Mermin model. It is shown that the DF obtained by the method of moments satisfy all sum rules. For other models of DF these equalities hold in part. Mermin and RPA models do not satisfy known, associated with convergent frequency zero and fourth moment, sum rules, but satisfy the second sum rule. These models take into account the electron-electron interaction only with the introduction of electron correction on a local-field, but do not take into account the electron-ion interaction.

In the present paper, we assess the performance of methods based on the random phase approximation (RPA) and on σ-functionals for predicting static optical properties, i.e., dipole moment, polarizability, and first and second hyperpolarizability, of small- and medium-sized molecules, including chain-like systems. First, we provide accurate reference data by coupled-cluster singles, doubles, with perturbative triples calculations with sufficiently large basis sets. The RPA and σ-functional calculations are carried out post-self-consistently using input orbitals and eigenvalues from the hybrid density-functional calculation. The optimal fraction of exact non-local exchange in these calculations is found to be quite high, around 0.5-0.6 in RPA and around 0.8-1.0 in σ-functional methods. σ-functional methods, however, proved to be less sensitive than RPA methods with respect to the amount of exact non-local exchange used in the generation of their input data. σ-functional methods are shown to outperform in accuracy RPA methods and various other considered density-functional theory methods for static optical properties and, thus, are well-suited for the calculation of linear and non-linear optical properties.

The hybrid QM:QM approach is extended to adsorption on transition metal surfaces. The random phase approximation (RPA) as the high-level method is applied to cluster models and, using the subtractive scheme, embedded in periodic models which are treated with density functional theory (DFT) that is the low-level method. The PBE functional, both without dispersion and augmented with the many-body dispersion (MBD), is employed. Adsorption of methane and ethane on the Pt(111) surface is studied. For methane in a 2 × 2 surface cell, the hybrid RPA:PBE and RPA:PBE+MBD results, -14.3 and -16.0 kJ mol-1, respectively, are in close agreement with the periodic RPA value of -13.8 kJ mol-1 at significantly reduced computational cost (factor of ∼50). For methane and ethane, the RPA:PBE results (-14.3 and -17.8 kJ mol-1, respectively) indicate underbinding relative to energies derived from experimental desorption barriers for relevant loadings (-15.6 ± 1.6 and -27.2 ± 2.9 kJ mol-1, respectively), whereas the hybrid RPA:PBE+MBD results (-16.0 and -24.9 kJ mol-1, respectively) agree with the experiment well within experimental uncertainty limits (deviation of -0.4 ± 1.5 and +2.3 ± 2.9 kJ mol-1, respectively). Finding a cluster that adequately and robustly represents the adsorbate at the bulk surface is important for the success of the RPA-based QM:QM scheme for metals.

The magnetic properties of solids are typically analyzed in terms of Heisenberg models where the electronic structure is approximated by interacting localized spins. However, even in such models the evaluation of thermodynamic properties constitutes a major challenge and is usually handled by a mean field decoupling scheme. The random phase approximation (RPA) comprises a common approach and is often applied to evaluate critical temperatures although it is well known that the method is only accurate wellbelowthe critical temperature. In the present work we compare the performance of the RPA with a different decoupling scheme proposed by Callen as well as the mean field decoupling of interacting Holstein-Primakoff (HP) magnons. We consider three-dimensional (3D) as well as two-dimensional (2D) model systems where the Curie temperature is governed by anisotropy. In 3D, the Callen method is the most accurate in the classical limit, and we show that the Callen decoupling (CD) produces the best agreement with experiments for bcc Fe, fcc Ni and fcc Co with exchange interactions obtained from first principles. In contrast, for low spin systems where a quantum mechanical treatment is pertinent, the HP and RPA methods are superior to the CD. In 2D systems with magnetic order driven by single-ion anisotropy, it is shown that HP fails rather dramatically and both RPA and Callen approaches severely overestimates Curie temperatures. The most accurate approach is then constructed by combining RPA with the CD of single-ion anisotropy, which yields the correct lack of order forS=1/2. We exemplify this by the case of monolayer CrI3using exchange constant extracted from experiments.

The collective excitation phenomena in atomic nuclei are studied in two different formulations of the Random Phase Approximation (RPA): (i) RPA based on correlated realistic nucleon-nucleon interactions constructed within the Unitary Correlation Operator Method (UCOM), and (ii) relativistic RPA (RRPA) derived from effective Lagrangians with density-dependent meson-exchange interactions. The former includes the dominant interaction-induced short-range central and tensor correlations by means of an unitary transformation. It is shown that UCOM-RPA correlations induced by collective nuclear vibrations recover a part of the residual long-range correlations that are not explicitly included in the UCOM Hartree-Fock ground state. Both RPA models are employed in studies of the isoscalar monopole resonance (ISGMR) in closed-shell nuclei across the nuclide chart, with an emphasis on the sensitivity of its properties on the constraints for the range of the UCOM correlation functions. Within the Relativistic Quasiparticle RPA (RQRPA) based on Relativistic Hartree-Bogoliubov model, the occurrence of pronounced low-lying dipole excitations is predicted in nuclei towards the proton drip-line. From the analysis of the transition densities and the structure of the RQRPA amplitudes, it is shown that these states correspond to the proton pygmy dipole resonance.

In the present paper a number conserving random phase approximation is derived as a special case of the recently developed random phase approximation in general symmetry projected quasiparticle mean fields. All the occurring integrals induced by the number projection are performed analytically after writing the various overlap and energy matrices in the random phase approximation equation as polynomials in the gauge angle. In the limit of a large number of particles the well-known pairing vibration matrix elements are recovered. We also present a new analytically number projected variational equation for the number conserving pairing problem.

The characteristic energy loss spectrum of solids is analyzed with the aid of the dielectric formulation of the many-body problem developed by the authors. It is shown that a measurement of the angular distribution of inelastically scattered fast electrons is a direct measurement of the imaginary part of the inverse dielectric constant of the solid, at the frequency and momentum of the energy transfer to the electrons in the solid. The calculations of the dielectric constant by means of first-order perturbation theory and within the random phase approximation are discussed with regard to their implications for the characteristic energy loss spectrum. It is shown that plasmons provide the dominant excitation mode at low momentum transfer for the free electron gas and for most solids; an explicit calculation is given of the role of screening in reducing the role of the independent electron excitations in this region. A comparison is made between characteristic energy loss experiments and other techniques for exploring the energy level structure of solids, including the non-relativistic Compton effect, optical transmission and reflection experiments, and x-ray emission and absorption studies. An analogous formulation of the optical properties of solids is given, which shows that in general the transverse and longitudinal dielectric constants will not be equal. They are equal within the random phase approximation, so that a careful comparison of optical data with characteristic energy loss data permits a direct verification of the validity of the random phase approximation. Such a comparison likewise yields information on the magnitude of the screening of individual electron transitions and permits the unambiguous identification of plasmon excitations. Finally it is shown that to the extent that the random phase approximation is valid, the results of a characteristic energy loss experiment may be used for a direct experimental determination of the ground-state energy of the electrons in a solid.

Random Phase Approximation (RPA) is one of the main approximation tools in studying nuclear structure. Here, we present a graphical user interface (GUI) Fortran code: Particle–Particle Random Phase Approximation (PPRPA) version 1, 2015. The code performs Tamm–Dancoff approximation (TDA) and Random Phase Approximation (RPA) calculations of nuclear structure of nuclei having A±2 nucleons in the total angular momenta and isospin (JT) scheme. The Hamiltonian is diagonalized with a given input model space, single-particle energies and interaction. Space function has been expanded to include orbits s, p, d, f, g and h. The current version of the code allows the user to test 20 orbits only. All possible eigenvalues and amplitudes within a model space are calculated. The single-particle density, charge distribution density and transition density are calculated in the basis of the harmonic oscillator potential. The primary utility of the PPRPA code is providing a visual tool to implementation and understanding of the collective excitation techniques TDA and RPA. Impact of the program includes all students, researchers and all those interested in knowing the facts about the structure of the atom nucleus and about the success of approximation methods in different branches of science.

We have assessed the performance of the non-self-consistent random phase approximation (RPA) on two pressure-induced structural phase transitions, diamond to \ensuremath{\beta}-Sn Si in Si and \ensuremath{\alpha}-quartz to stishovite in SiO${}_{2}$. The calculated equilibrium lattice properties of the four structures are in better agreement with experimental results than are those from several semilocal functionals. The energy differences between the high- and low-pressure phases are found to be 0.37 eV/Si and 0.39 eV/SiO${}_{2}$, respectively. The transition pressure obtained from our RPA calculations for diamond to \ensuremath{\beta}-Sn in Si is 12.2 GPa, in excellent agreement with the experimental value 11.3--12.6 GPa. However, the \ensuremath{\alpha}-quartz to stishovite phase-transition pressure in SiO${}_{2}$ is found to be 5.6 GPa, lower than the experimental 7.46 GPa; the Perdew-Burke-Ernzerhof (PBE) semilocal functional gives the transition pressure closest to experiment in this case. We conclude that the non-self-consistent, nonlocal RPA accurately describes the insulator-to-metal transition in Si, where semilocal density functionals tend to fail. But the RPA error cancellation that is nearly perfect in many solids, including Si, may be less perfect in solid SiO${}_{2}$, as it is in many molecules.