Showing papers on "Random phase approximation published in 1999"
TL;DR: In this paper, the effects of short-range correlation are investigated in the local spin density (LSD) approximation and the generalized gradient approximation (GGA) for atoms, molecules, and jellium surfaces.
Abstract: Since long-range electron-electron correlation is treated properly in the random phase approximation (RPA), we define short-range correlation as the correction to the RPA. The effects of short-range correlation are investigated here in the local spin density (LSD) approximation and the generalized gradient approximation (GGA). Results are presented for atoms, molecules, and jellium surfaces. It is found that (1) short-range correlation energies are less sensitive to the inclusion of density gradients than are full correlation energies, and (2) short-range correlation makes a surprisingly small contribution to surface and molecular atomization energies. In order to improve the accuracy of electronic-structure calculations, we therefore combine a GGA treatment of short-range correlation with a full RPA treatment of the exchange-correlation energy. This approach leads to jellium surface energies close to those of the LSD approximation for exchange and correlation together (but not for each separately).
112 citations
TL;DR: In this paper, a generalized form of proton-neutron quasi-particle RPA model with separable Gamow-Teller forces was used to calculate the capture and decay rates for sd-shell nuclei in stellar environment.
102 citations
TL;DR: In this article, the renormalization of the spin-orbit-coupling constant of two-dimensional electrons by electron-electron interactions was studied, and it was shown that, similarly to the g factor, renormalisation corresponds to the enhancement, although the magnitude of the enhancement is weaker than that for the G factor.
Abstract: We study theoretically the renormalization of the spin-orbit-coupling constant of two-dimensional electrons by electron-electron interactions. We demonstrate that, similarly to the g factor, the renormalization corresponds to the enhancement, although the magnitude of the enhancement is weaker than that for the g factor. For high-electron concentrations (small interaction parameter ${r}_{s})$ the enhancement factor is evaluated analytically within the static random phase approximation. For large ${r}_{s}\ensuremath{\sim}10,$ we use an approximate expression for effective electron-electron interaction, which takes into account the local field factor, and calculate the enhancement numerically. We also study the interplay between the interaction-enhanced Zeeman splitting and interaction-enhanced spin-orbit coupling.
82 citations
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TL;DR: In this article, a relativistic RPA calculation for the monopole and dipole compression modes in nuclei is performed, where the main contributions from the pairs of Fermi to Dirac sea states are through the exchange of the scalar meson, while the vector mesons play a negligible role.
Abstract: A fully consistent relativistic RPA calculation is performed for the monopole and dipole compression modes in nuclei. The emphasis is put on the effects of Dirac sea states which are generally neglected in relativistic RPA calculations. It is found that these effects can be quite important for the isoscalar monopole mode. The main contributions from the pairs of Fermi to Dirac sea states are through the exchange of the scalar meson, while the vector mesons play a negligible role. Numerical results of relativistic RPA are checked with the constrained relativistic mean field model in the monopole case. A good agreement beteween monopole energies calculated in RRPA and in time-dependent relativistic mean field approach is achieved. For the monopole compression mode, a comparison of experimental and calculated energies gives a value of 250 $\sim$ 270 MeV for the nuclear matter incompressibility. A large discrepancy remains between theory and experiment in the case of the dipole compression mode.
56 citations
TL;DR: In this article, the reorientation of the magnetization of a ferromagnetic monolayer is calculated with the help of many-body Green's function theory, which allows, in contrast to other spin wave theories, a satisfactory calculation of magnetic properties over the entire temperature range of interest since interactions between spin waves are taken into account.
Abstract: The reorientation of the magnetization of a ferromagnetic monolayer is calculated with the help of many-body Green's function theory. This allows, in contrast to other spin wave theories, a satisfactory calculation of magnetic properties over the entire temperature range of interest since interactions between spin waves are taken into account. A Heisenberg Hamiltonian plus a second-order uniaxial single-ion anisotropy and an external magnetic field is treated by the Tyablikov (Random Phase Approximation: RPA) decoupling of the exchange interaction term and the Anderson-Callen decoupling of the anisotropy term. The orientation of the magnetization is determined by the spin components $\la S^\alpha\ra$ ($\alpha=x,y,z$), which are calculated with the help of the spectral theorem. The knowledge of the orientation angle $\Theta_0$ allows a non-perturbative determination of the temperature dependence of the effective second-order anisotropy coefficient. Results for the Green's function theory are compared with those obtained with mean-field theory (MFT). We find significant differences between these approaches.
50 citations
TL;DR: In this paper, the indirect nuclear spin-spin coupling constants of PbH4 were calculated using a basis set which was specially optimized for correlated calculations of spin coupling constants and the effects of nuclear motion were investigated by calculating the coupling constants as a function of the totally symmetric stretching coordinate.
Abstract: We report ab initio calculations of the indirect nuclear spin–spin coupling constants of PbH4 using a basis set which was specially optimized for correlated calculations of spin–spin coupling constants. All nonrelativistic contributions and the most important part of the spin–orbit correction were evaluated at the level of the random phase approximation. Electron correlation corrections to the coupling constants were calculated with the multiconfigurational linear-response method using extended complete and restricted active space wavefunctions as well as with the second-order polarization propagator approximation and the second-order polarization propagator approximation with coupled-cluster singles and doubles amplitudes. The effects of nuclear motion were investigated by calculating the coupling constants as a function of the totally symmetric stretching coordinate. We find that the Fermi contact term dominates the Pb‐H coupling, whereas for the H‐H coupling it is not more important than the orbital paramagnetic and diamagnetic contributions. Correlation affects mainly the Fermi contact term. Its contribution to the one-bond coupling constant is reduced by correlation, independent of the method used; however, the different correlated methods give ambiguous results for the Fermi contact contribution to the H‐H couplings. The dependence of both coupling constants on the Pb‐H bond length is dominated by the change in the Fermi contact term. The geometry dependence is, however, overestimated in the random phase approximation.
48 citations
TL;DR: In this paper, the authors present microscopic calculations of neutrino propagation in hot neutron matter above the nuclear density within the framework of the random phase approximation, and they find that spin zero sound is present at zero temperature for all Skyrme forces considered.
Abstract: We present microscopic calculations of neutrino propagation in hot neutron matter above the nuclear density within the framework of the random phase approximation. Calculations are performed for nondegenerate neutrinos using various Skyrme effective interactions. We find that for densities just above the nuclear density, spin zero sound is present at zero temperature for all Skyrme forces considered. However, it disappears rapidly with increasing temperature due to a strong Landau damping. As a result the mean free path is given, to a good approximation, by the mean field value. Because of the renormalization of the bare mass in the mean field, the medium is more transparent as compared to the free case. We find, in contrast, that at several times the nuclear density, a new type of behavior sets in due to the vicinity of a magnetic instability. It produces a strong reduction of the mean free path. The corresponding transition density, however, occurs in a region where inputs from more realistic calculations are necessary for the construction of a reliable Skyrme-type parametrization.
40 citations
TL;DR: In this article, the properties of the low-lying, collective 2+ states in neutron-rich oxygen isotopes are investigated in the framework of self-consistent microscopic models with effective Skyrme interactions.
Abstract: The properties of the low-lying, collective 2+ states in neutron-rich oxygen isotopes are investigated in the framework of self-consistent microscopic models with effective Skyrme interactions. In RPA the excitation energies E2+ can be well described but the transition probabilities are much too small as compared to experiment. Pairing correlations are then accounted for by performing quasiparticle RPA calculations. This improves considerably the predictions of B(E2) values and it enables one to calculate more reliably the ratios Mn/Mp of neutron-to-proton transition amplitudes. A satisfactory agreement with the existing experimental values of Mn/Mp is obtained.
31 citations
TL;DR: The use of moment densities for understanding phase transition kinetics in polydisperse systems is discussed in this paper, where the role of size partitioning and the distinction between quenched and annealed phase diagrams is emphasised.
Abstract: The use of moment densities for understanding phase transition kinetics in polydisperse systems is discussed. The role of size partitioning and the distinction between quenched and annealed phase diagrams is emphasised. Polydisperse Flory–Huggins theory is used as an example, and Cahn–Hilliard type equations for the local moment densities are constructed for this model via a random phase approximation. They are used to examine the initial growth rate of density fluctuations which are found to reflect the annealed and quenched spinodal lines in the phase diagram.
29 citations
TL;DR: In this article, a variational approach to collective excitations in a boson formalism based on quasiparticles is discussed, where the variables entering into the definition of these phonons are fixed simultaneously by minimizing the expectation value of the boson Hamiltonian in the vacuum of the phonons.
Abstract: We discuss a variational approach to collective excitations in a boson formalism based on quasiparticles. Bosons are defined in correspondence with pairs of quasiparticles and boson images of fermion operators are constructed by means of a mapping procedure of Marumori-type. Phonons of the type used within the random phase approximation ~RPA! are introduced as Bogoliubov transformations of these bosons. The variables entering into the definition of these phonons as well as of the quasiparticle operators are fixed simultaneously by minimizing the expectation value of the boson Hamiltonian in the vacuum of the phonons. The approach is tested within an exactly solvable two-level model which is characterized by a pairing Hamiltonian. A quite good agreement is found for the energies of the ground state and of the first 0 excited state. The comparison with the Bardeen-Cooper-Schrieffer method and the quasiparticle RPA as well as with some recent selfconsistent RPA-type approaches is discussed. @S0556-2813~99!01103-6#
18 citations
TL;DR: It was found that the static upper critical dimension, duc=2D/(2−D), discriminates between Gaussian (or screened) and non-Gaussian regimes, whereas its dynamical counterpart, duca=2duc, distinguishes between the simple Rouse and the renormalized Rouse behavior.
Abstract: The Martin–Siggia–Rose functional integral technique is applied to the dynamics of a D-dimensional manifold in a melt of similar manifolds. The integration over the collective variables of the melt can be simply implemented in the framework of the dynamical random phase approximation. The resulting effective action functional of the test manifold is treated by making use of the self-consistent Hartree approximation. As an outcome the generalized Rouse equation of the test manifold is derived and its static and dynamic properties are studied. It was found that the static upper critical dimension, duc=2D/(2−D), discriminates between Gaussian (or screened) and non-Gaussian regimes, whereas its dynamical counterpart, duc=2duc, distinguishes between the simple Rouse and the renormalized Rouse behavior. We have argued that the Rouse mode correlation function has a stretched exponential form. The subdiffusional exponents for this regime are calculated explicitly. The special case of linear chains, D=1, shows go...
TL;DR: In this paper, small angle neutron scattering and neutron spin echo experiments on polyethylene-block-polyethylethylene (PE-PEE) copolymers with molecular weights of 16.500 and 68.000 g/mol were performed.
Abstract: Diblock copolymers in the melt exhibit order–disorder phase transitions (ODT), which are accompanied by strong concentration fluctuations. These transitions are generally described in terms of the random phase approximation (RPA) of Leibler and Fredrickson, which is able to explain small angle scattering results in the neighborhood of the ODT, in particular around the correlation peak at q*. The RPA theory has been extended to include dynamical phenomena, predicting the short time relaxation of the dynamic structure factor in polymeric multicomponent systems. We report small angle neutron scattering and neutron spin echo experiments on polyethylene–block-polyethylethylene (PE-PEE) and poly(ethylene-propylene)–block-polyethylethylene (PEP-PEE) copolymers with molecular weights of 16.500 and 68.000 g/mol, which explore the structure and dynamics of these block copolymers. Studying melts with different hydrogen/deuterium labeling it was possible to observe experimentally the different relaxation modes of suc...
TL;DR: In this paper, angle-resolved Raman scattering observations of the temperature dependent Landau damping of the acoustic plasmon in an electron bilayer system realised in a GaAs double quantum well structure are presented.
Abstract: We report angle-resolved Raman scattering observations of the temperature dependent Landau damping of the acoustic plasmon in an electron bilayer system realised in a GaAs double quantum well structure. Corresponding calculations of the charge-density excitation spectrum of the electron bilayer using forms of the random phase approximation (RPA), and the static local field formalism of Singwi, Tosi, Land and Sj\"{o}lander (STLS) extended to incorporate non-zero electron temperature $T_{\rm e}$ and phenomenological damping, are also presented. The STLS calculations include details of the temperature dependence of the intra- and inter-layer local field factors and pair-correlation functions. Good agreement between experiment and the various theories is obtained for the acoustic plasmon energy and damping for $T_{\rm e} \lesssim T_{\rm F}/2$, the Fermi temperature. However, contrary to current expectations, all of the calculations show significant departures from our experimental data for $T_{\rm e} \gtrsim T_{\rm F}/2$. From this, we go on to demonstrate unambiguously that real local field factors fail to provide a physically accurate description of exchange correlation behaviour in low dimensional electron gases. Our results suggest instead that one must resort to a {\em{dynamical}} local field theory, characterised by a {\em{complex}} field factor to provide a more accurate description.
TL;DR: In this paper, the normal state conductivity of a system of interacting large polarons is calculated within the Random Phase approximation and some numerical results are presented, where the behavior of the optical absorption as a function of the charge carrier density and of the temperature is analyzed for different values of the electron-phonon coupling constant.
Abstract: The normal state conductivity, σ(ω), of a system of interacting large polarons is calculated within the Random Phase approximation and some numerical results are presented. The behaviour of the optical absorption as a function of the charge carrier density and of the temperature is analyzed for different values of the electron-phonon coupling constant. It is shown that σ(ω)exhibits features similar to those observed in the infrared spectra of the cuprates.
TL;DR: In this paper, the relativistic mean-field (RMF) wave function of nucleus and the particle-hole residual interactions in the RRPA are calculated from the same effective Lagrangian.
Abstract: A fully consistent relativistic random-phase approximation (RRPA) is studied in the sense that the relativistic mean-field (RMF) wavefunction of nucleus and the particle-hole residual interactions in the RRPA are calculated from the same effective Lagrangian. A consistent treatment of RRPA within the RMF approximation, i.e., no sea approximation, has to include also the pairs formed from the Dirac states and occupied Fermi states, which is essential for satisfying the current conservation. The inverse energy-weighted sum rule for the isoscalar giant monopole mode is investigated in the constrained RMF. It is found that the sum rule is fulfilled only in the case where the Dirac state contributions are included.
TL;DR: In this article, a method to do ab initio molecular dynamics in electronically excited systems within the random phase approximation (RPA) was proposed using a dynamical variational treatment of the RPA frequency, which corresponds to the electronic excitation energy of the system.
Abstract: This paper describes a method to do ab initio molecular dynamics in electronically excited systems within the random phase approximation (RPA). Using a dynamical variational treatment of the RPA frequency, which corresponds to the electronic excitation energy of the system, we derive coupled equations of motion for the RPA amplitudes, the single particle orbitals, and the nuclear coordinates. These equations scale linearly with basis size and can be implemented with only a single holonomic constraint. Test calculations on a model two level system give exact agreement with analytical results. Furthermore, we examined the computational efficiency of the method by modeling the excited state dynamics of a one-dimensional polyene lattice. Our results indicate that the present method offers a considerable decrease in computational effort over a straight-forward configuration interaction (singles) plus gradient calculation performed at each nuclear configuration.
TL;DR: Using the spectral function F'(z)/F(z) the RPA correlation energy and other properties of a finite system can be written as a contour integral in a compact way.
Abstract: Using the spectral function F'(z)/F(z) the RPA correlation energy and other properties of a finite system can be written as a contour integral in a compact way. This yields a transparent expression and reduces drastically the numerical efforts for obtaining reliable values. The method applied to pairing vibrations in rotating nuclei as an illustrative example.
01 Aug 1999-Nuclear Instruments & Methods in Physics Research Section B-beam Interactions With Materials and Atoms
TL;DR: In this article, the wavevector and frequency-dependent dielectric matrix of bulk crystals were calculated by using first-principles, all-electron Kohn-Sham states in the integral of the irreducible polarizability in the random phase approximation.
Abstract: We calculate the wave-vector and frequency-dependent dielectric matrix of bulk crystals by use of first-principles, all-electron Kohn–Sham states in the integral of the irreducible polarizability in the random phase approximation. From this we determine the macroscopic “head” element, the ( 0 , 0 ) -element, of the inverse matrix (the energy loss function), and integrate over energy and momentum transfers to obtain the electronic energy loss of protons at low velocities. Numerical results are given for hexagonal close packed lithium.
TL;DR: In this article, the breakdown of the isospin symmetry, at the level of the quasiparticle mean field approximation, and its partial restoration by effective interactions at the quasi-particle random phase approximation ~QRPA! level of approximation, are studied.
Abstract: The breakdown of the isospin symmetry, at the level of the quasiparticle mean field approximation, and its partial restoration by effective interactions, at the quasiparticle random phase approximation ~QRPA! level of approximation, are studied. The method used to define effective symmetry breaking two-body interactions has been applied previously to particle-number and rotational symmetry violations. The connection between the present approach and the proton-neutron QRPA method with renormalized two-particle interactions is discussed. The formalism is applied to calculate nuclear matrix elements for Fermi double beta decay transitions. @S0556-2813 ~99!03101-5#
TL;DR: In this article, the role of the deformation, pairing and Coulomb interaction in double-beta decay was investigated and it was found that the inclusion of both deformation and pairing influences the 2-decay matrix element significantly.
Abstract: The two-neutrino double-beta decay (2-decay) for 76Ge 76Se is studied within the framework of the deformed Hartree-Fock formalism. An interaction given by Kuo and modified by Ahalpara et al is used. The results of our model calculation are compared with those obtained using the shell model and the quasiparticle random phase approximation. The role of the deformation, pairing and Coulomb interaction in double-beta decay is investigated. We have found that the inclusion of both deformation and pairing influences the 2-decay matrix element significantly.
TL;DR: In this article, the wave functions of the ground and excited states of an exactly solvable model are expanded in the basis of the exact solutions, and it is found that the RPA wave functions are orthogonal to the exact solution while the RRPA and SCRPA ones have small but finite overlaps with the exact results.
Abstract: The random phase approximation (RPA), the renormalized RPA (RRPA), and the self-consistent RPA (SCRPA) methods are applied to calculate the wave functions of the ground and excited states of an exactly solvable model. The approximated wave functions are expanded in the basis of the exact solutions. It is found that, when the RPA collapses, the RPA wave functions are orthogonal to the exact solutions while the RRPA and SCRPA ones have small but finite overlaps with the exact results. In spite of the apparently good agreement between the results of the RRPA, the SCRPA, and the exact solution, for the energy of the first excited state beyond the point of collapse, it is found that these approximations do not correctly describe the exact wave functions.
TL;DR: In this paper, a possible generation scheme for growing coherent plasma oscillations (plasma instability) is presented, where a sufficient population inversion can be maintained in the carrier distribution by appropriate injection-extraction configurations, form the basis of this scheme.
Abstract: A possible generation scheme for growing coherent plasma oscillations (plasma instability) is presented. Specific quantum well structures, where a sufficient population inversion can be maintained in the carrier distribution by appropriate injection–extraction configurations, form the basis of this scheme. Self-consistent random phase approximation calculations show that a population inversion, leading to a plasma instability, can occur in such structures. A comparison between the calculated and the observed differential conductance curves suggests that such quantum well structures could be designed as active regions for the generation of terahertz frequency radiation sources.
TL;DR: In this paper, the pn-QRPA equation of motion is solved by using a boson expansion technique adapted to the treatment of proton-neutron correlations, and the resulting wave functions are used to calculate the matrix elements of double-Fermi transitions.
Abstract: The proton-neutron quasiparticle random phase approximation (pn-QRPA) is extended to include next to leading order terms of the QRPA harmonic expansion. The procedure is tested for the case of a separable Hamiltonian in the SO(5) symmetry representation. The pn-QRPA equation of motion is solved by using a boson expansion technique adapted to the treatment of proton-neutron correlations. The resulting wave functions are used to calculate the matrix elements of double-Fermi transitions.
TL;DR: In this paper, the effects of thermal and quantal gap fluctuations in small superconductors are investigated by means of the static path plus random phase approximation to the partition function, which is implemented in an ensemble with fixed particle number parity (NP).
TL;DR: In this paper, the mean field phase diagram for charge density waves in quasi-one-dimensional electronic systems is computed taking into account both Pauli and orbital effects of external magnetic field.
Abstract: Taking into account both Pauli and orbital effects of external magnetic field we compute the mean field phase diagram for charge density waves in quasi-one-dimensional electronic systems. The magnetic field can cause transitions to CDW states with two types of the shifts of wave vector from its zero-field value. It can also stabilize the field-induced charge density wave. Furthermore, the critical temperature shows peaks at a new kind of magic angles.
TL;DR: In this paper, the static path plus random phase approximation (SPA1RPA) was formulated exactly in a canonical ensemble, for the case of a density decomposition of an arbitrary two-body Hamiltonian.
Abstract: The static path plus random phase approximation ~SPA1RPA! to the partition function is formulated exactly in a canonical ensemble, for the case of a density decomposition of an arbitrary two-body Hamiltonian. The canonical mean field plus RPA approach is also discussed. Numerical results are shown for a quadrupole interaction, where excellent agreement with the exact canonical results for a light nucleus is obtained. An effective canonical mean field plus RPA approach is in this case also developed, which avoids the sharp phase transition of the ordinary mean field and provides a reliable estimate of full SPA1RPA results. @S0556-2813~99!02901-5#
TL;DR: In this paper, the photo cross section for photoionization of alkali-metal clusters is calculated using the random phase approximation (RPA) on an equal footing for both, the low and the high energy regime.
Abstract: Photoionization of alkali-metal clusters is investigated theoretically for different photon energy regimes At low energies the photo cross section is characterized by the well-known plasmon peak resulting from collective electron dynamics For high energies the ionization cross section exhibits an oscillatory behavior, which can be explained by single-particle effects In this paper we use the random phase approximation (RPA) to calculate the photo cross section on an equal footing for both, the low- and the high-energy regime Thereby, we can show that the cross sections for photoionization calculated in the collective RPA and in the single-particle picture indeed merge for high photon energies Moreover, we demonstrate that the oscillatory behavior can already be identified at low photon energies where the cross section is not yet exponentially small Hence, it should be possible to identify the oscillations experimentally
TL;DR: In this article, a generalized form of proton-neutron quasiparticle RPA model with separable Gamow-Teller forces was used to calculate weak interaction rates for sd-shell nuclei in stellar environment.
Abstract: Allowed weak interaction rates for sd-shell nuclei in stellar environment are calculated using a generalized form of proton-neutron quasiparticle RPA model with separable Gamow-Teller forces. Twelve different weak rates are calculated for each nucleus as a function of temperature and density. This project consists of calculation of weak rates for a total of 709 nuclei with masses ranging from A = 18 to 100. This paper contains calculated weak rates for sd-shell nuclei. The calculated capture and decay rates take into consideration the latest experimental energy levels and ft value compilations. The results are also compared with earlier works. Particle emission processes from excited states, previously ignored, are taken into account, and are found to significantly affect some beta decay rates.
TL;DR: In this paper, the renormalized random phase approximation for hot finite Fermi systems is evaluated with the use of the thermo field dynamics formalism, which takes into account the Pauli principle in a more proper way than the usual thermal RPA.
Abstract: The renormalized random phase approximation for hot finite Fermi systems is evaluated with the use of the thermo field dynamics formalism. This approximation treats vibrations of a hot finite Fermi system as harmonic ones but takes into account the Pauli principle in a more proper way than the usual thermal RPA, thus incorporating a new type of correlations in a thermal ground state. To demonstrate advantages of the approximation and to analyze a range of its validity, it is applied to the exactly solvable Lipkin model. A comparison is made with the exact grand canonical ensemble calculations, results of the thermal Hartree – Fock approximation and the thermal random phase approximation. The intrinsic energy of the system, the heat capacity, the average value of the quasispin operator z-projection and the particle number variance are calculated as functions of temperature. On the whole, the thermal renormalized RPA appears to be a better approximation than the other two. Its advantage is especially evident in the vicinity of the phase transition point. It is found that within TRRPA the phase transition occurs at lower temperature than in THFA and TRPA.
TL;DR: In this article, a computational method based on continuous transformation of origin for the current density (CTOCD) induced within the electron cloud by an external homogeneous, static magnetic field has been employed to calculate atomic contributions to magnetic susceptibilities.
Abstract: The conventional random phase approximation (RPA) of the polarization propagator theory and a computational method based on continuous transformation of origin for the current density (CTOCD) induced within the electron cloud by an external homogeneous, static magnetic field has been employed to calculate atomic contributions to magnetic susceptibilities The diamagnetic part of the magnetic susceptibility is written in terms of the polarization propagator Since the paramagnetic term may also be obtained from the propagator it is thus possible to compute both contributions at the same level of approximation The evaluated average susceptibility is independent of the origin of the vector potential, but depends on the origin of the reference frame The atomic contributions to the diamagnetic and paramagnetic parts of the magnetic susceptibility are derived by applying off-diagonal hypervirial relations which are exactly fulfilled if the state functions are exact eigenfunctions of a model Hamiltonian The r