Integral Hellmann-Feynman theorem as a criterion of the quality of wave functions of atomic-molecular systems

It is proposed to use the integral Hellmann-Feynman theorem for the quality control and refinement of atomic and molecular wave functions. Its validity is verified for variational wave functions of the positronium ion e ⁺ e ^? e ^?, the negative ion of the hydrogen atom p _∞ ⁺ e ^? e ^? (H^?), and mesomolecules μ^?π⁺π⁺ and μ^? p ⁺ p ⁺. The relative violation of this theorem (10^?2) is six orders of magnitude larger than the error of the energy calculation (10^?8), which demonstrates its high sensitivity to the quality of wave functions. A way of refining wave functions on the basis of combination of the integral Hellmann-Feynman theorem for exactly solvable model and real atomic-molecular systems is proposed. A rule for verification of the mutual consistency of the wave functions of any three quantum-mechanical systems is formulated.     

Separation of a Slater Determinant Wave Function with a Neck Structure into Spatially Localized Subsystems

A method of separating a Slater determinant wave function with a two-center neck structure into spatially localized subsystems is proposed, and it is applied to α + α Margenau–Brink cluster wave functions and antisymmetrized molecular dynamics wave functions of the ground state in ¹⁰Be and a negative-parity largely deformed state in ³⁵S.     

6He β-decay from a three-body model of the A=6 system

The rate of ⁶He β-decay is computed from ⁶He and ⁶Li separable-potential three-body wave functions. The value of ft predicted with the wave functions derived from the most complete set of αN and NN interactions is 807±12 s compared to the experimental value of 807±2 s.     

Asymptotic behaviour of the Fourier coefficients of MAPW wave functions

In contrast to the conventional APW method the MAPW method imposes continuity conditions on the trial wave functions which guarantee that the wave functions and their first derivatives are continuous throughout the whole atomic polyhedron. It is shown that as a consequence of this fact the Fourier coefficients of the MAPW wave functions approach zero as 1/p ⁴ whereas conventional APW functions decay as 1/p ² only. For the MAPW Fourier coefficients asymptotic formulas are derived that may be useful in numerical calculations. Numerical results are given for metallic Li and Cu.     

Analysis of elastic and inelastic hadron scattering on 6,7Li nuclei within diffraction theory

Within Glauber-Sitenko multiple-scattering theory, the differential cross sections for elastic and inelastic proton, positive-pion, and positive-kaon scattering on ^6,7Li nuclei are calculated at incident-hadron energies ranging between 0.143 and 1.0 GeV. The ⁶Li and ⁷Li wave functions are taken in, respectively, the α2N and the αt clustermodel. The resulting cross sections are investigated as functions of the scattered-particle energy, parameters of the model wave functions, and various scattering multiplicities. It is concluded that a partial filling of the diffraction minimum in the cross section is due to the D-wave contribution to the wave function for the ⁶Li target nucleus.     

Structure of 0+ states in 18O

Analysis of spectroscopic factors obtained in single-nucleon transfer reactions leading to and from ¹⁸O yields two different sets of wave functions for the first three O⁺ states. One set of wave functions is in agreement with ¹⁶O(t, p)¹⁸O data for the three states-the other set is not. The wave functions that agree with the experimental data have the majority of the $\text{(s}\text{1}\text{2}\text{)}{}^2$ strength in the third O⁺ state.     

Bestimmung der Konfigurationsmischungen im RaE aus den β−-Übergängen Pb210→Bi210→Po210

From the anomalies in theβ-decays of Pb²¹⁰ and Bi²¹⁰, relations between the shell model wave functions of the ground states of Pb²¹⁰, Bi²¹⁰ and Po²¹⁰ are derived. These relations are however not satisfied by wave functions calculated under the assumption of two-body central forces of zero range. We make use of the fact that such forces seem to be a good approximation for even-even-nuclei (in our case Pb and Po), but not for odd-odd-nuclei (Bi in our case), and derive the wave function for the 1^? level of Bi²¹⁰ from the wave functions of Pb²¹⁰ and Po²¹⁰ using the relations obtained from theβ-decay anomalies.     

The reaction γ4He→3Hn π+ in impulse approximation

The reactionγ+⁴He→³H+n+π ⁺ is calculated in the impulse approximation. The influence of nuclear wave functions on the differential cross section is discussed numerically using Gaussian and Irving-Gunn wave functions for the triton and the α-particle. The differences are large enough to be looked for experimentally.     

The Condition of Invariance of the Energy of an Atomic-Molecular System with Respect to Rotations of the Jacobi Coordinates

An additional condition for the wave functions of the ground and excited states of atomic-molecular systems, which follows from the energy invariance with respect to rotations of the Jacobi coordinates, is studied. This condition makes a substantial contribution to the conventional criterion of the quality of wave functions, based on the variational principle for the average energy value. The explicit form of this condition is found for an arbitrary interaction potential dependent on the particle-particle distances and the requirements for realization of this condition are ascertained. Variational calculations of ³He²⁺ μ^? e ^? and ⁴He²⁺ μ^? e ^? mesoatoms in the basis of correlated exponential functions dependent on the particle-particle distances are performed with a detailed optimization of all nonlinear parameters for each individual energy level. A high sensitivity of the condition under study to the quality of approximate wave functions and the efficiency of the detailed optimization of the nonlinear variational parameters in calculation of each individual energy level are demonstrated.     

Study of the structure of light,unstable nuclei and the mechanism of elastic proton scattering

The review presents calculations of elastic p ⁶He-, p ⁸Li-, p ⁹Li- and p ⁹C scattering in the framework of the Glauber theory of multiple diffraction scattering at intermediate energies of 70 and 700 MeV/nucleon. The most significant result of the calculations is that we have utilized realistic three-body wave functions obtained within modern nuclear models. The relation is found between differential cross sections and intercluster potentials, where the nuclear wave functions are calculated. Conclusions are made concerning the types of potentials which describe most realistically the available experimental data. The method for calculation of three-body wave functions in α-n-n-, α-t-n-, ⁷Be-p-p-, α-t-2n-, and ⁷Li-n-n models is described with discussion of inter-cluster potentials and the quantum-number configurations taken into consideration. It is revealed how the wave functions and the nuclear electromagnetic characteristics calculated using these wave functions depend on the choice of intercluster potentials. The derivation of matrix elements (amplitudes) of pA scattering in the framework of the Glauber approach with three-body wave functions is presented by an example of ⁶He nucleus. Discussing the results of calculation of differential cross sections and the analyzing power (A _y ), we established how the calculated characteristics depend on a wave-function structure and dynamics of the process determined by a Glauber operator of multiple scattering. The calculated differential cross sections and analyzing powers are compared with available experimental data and calculations by other authors performed for different formalisms, which allows us to make justified conclusions.     

Bound-pion absorption followed by emission of two nucleons

The bound-pion absorption reaction, viz. ¹²C(π^?, NN), is studied using Hartree-Fock (HF) wave functions obtained with the unitary-model-operator approach starting with the hard-core nucleon-nucleon (NN) interaction. The inequality of the energies of the two outgoing nucléons is treated exactly and calculations are done using the “1N model” for π-absorption. Other effects taken into account are: NN scattering in the final state, contributions of all excited states of ¹⁰B and ¹⁰Be with E < 5 MeV, and effects of the strong π-nucleus interaction and the finite nuclear size on the bound-π wave function. Branching ratios and angular distributions of absorption rates are in better agreement with experimental data. The correct order of magnitude of the total absorption rate is reproduced. Whatever the effects of short-range correlations present in the HF wave functions, they are not masked by the NN final-state interaction. The contribution of excited states in ¹⁰B and ¹⁰Be is found to be quite large. Absorption rates obtained with the HF and oscillator wave functions differ significantly both in size and shape.     

Study of relativistic bound state wave functions in quasielastic (e,e′ p) reactions

The unpolarized response functions of the quasielastic ¹⁶O(e,e′ p)¹⁵N reaction are calculated for three different types of relativistic bound-state wave functions. The wave functions are obtained from relativistic Hartree, relativistic Hartree-Fock and density-dependent relativistic Hartree calculations that reproduce the experimental rms charge radius of ¹⁶O. The sensitivity of the unpolarized response functions to the single-particle structure of the different models is investigated in the relativistic plane-wave impulse approximation. Redistributions of the momentum dependence in the longitudinal and transverse response function can be related to the binding energy of the single-particle states. The interference responses R_LT and R_TT reveal a strong sensitivity to the small component of the relativistic bound-state wave function.     

Comparison of the noncoplanar 6Li(p,pd)4He reactions at 120 and 200 MeV

The ⁶Li(p, pd)⁴He reaction was studied at 200.2 MeV, at the quasi-free angle pair (θ_p, θ_d) = (54°, ?48.9°), for noncoplanarity angles φ from 0° to 28°. ⁶Li αd spectroscopic factors of 0.84 and 0.76 are deduced from our coplanar data at this energy and 120 MeV, respectively, for ground-state 2S Woods-Saxon wave functions. A recent microscopic three-body calculation predicts spectroscopic factors from 0.70 to 0.75; using the ground-state wave functions from this study, we deduce a factor of 0.76 from the 200 MeV data. DWIA calculations fit the measured integrated cross sections versus φ for spectator momenta P_α ? 100 MeV/c at both bombarding energies, but underpredict them for larger P_α. Momentum form factors were better reproduced with 1S αd cluster wave functions for a soft-core bound-state potential than with the 2S Woods-Saxon wave functions, but the former wave functions generate unphysically large (～1.25) spectroscopic factors.     

Comments about the stark effect on the energy levels of f9-electrons in cubic crystal fields

Effects of free ion wave functions and J-admixture, on the Stark split energy levels of f⁹ electron systems in cubic environments are studied. This is done by working out the example of Dy³⁺ ion in the CaF₂ matrix. It is shown that while the effect of free ion wave functions is rather small, the J-admixture has a larger effect on the Stark split levels.     

Meson-exchange currents in thermal n-3He radiative capture

The cross section for radiative capture of thermal neutrons by ³He arising from 1π, 1ρ and 1ω meson-exchange processes is calculated using oscillator and exponential wave functions for ³He (including a D-wave component) and oscillator wave functions for ⁴He (also including a D-wave component). A correlation function with a range of 0.25 fm is incorporated in the wave function to describe the short-range behaviour. Results are sensitive to the assumed range of the correlation function and to the triplet scattering length in the neutron scattering wave function, which are not known accurately. For a reasonable set of parameter choices, a combination of one-body impulse approximation (between D-states) and two-body meson-exchange current processes can explain the entire experimental cross section.     

Multipole expansion of non-local coupling potentials for cluster transfer and application to 16O(16O, 12C)20Ne

When the transfer of clusters and the symmetrization (antisymmetrization) of scattering wave functions is described by cluster models within the coupled-channel formalism, non-local coupling potentials arise. We suggest a procedure to calculate these potentials by a multipole expansion of all potentials and wave functions which depend on sums of vectors. The expansion coefficients are found by least-squares fit. The method is applied to the ¹⁶O(¹⁶O, ¹²C)²⁰Ne reaction, which is treated in the cluster model with two inert ¹²C- and α-clusters as constituents.     

Optical oscillator strengths for Be II and B III

A simple two-parameter analytic potential adjusted so as to produce the experimental energy levels is used to generate wave functions for the ground and excited states of the ions Be II and B III. Using these wave functions we calculate optical oscillator strengths for various excitations from the 1s ²2s(² S) ground state. The results are compared to experiment and other calculations.     

On the degeneracy of the 2H and 2P states from the atomic configuration d3 in the non-relativistic Hartree-Fock approximation

A new relation between the wave functions of the atomic states ²H and ²P arising from the configuration d³ and the wave functions of two molecular states arising from the configuration h³ in icosahedral symmetry molecules is presented. This relation gives an insight into the unknown origin of the non-relativistic Hartree-Fock degeneracy of the atomic states ²H and ²P.     

(π +,p) Reaction in12C

Hartree-Fock wave functions obtained from realistic nucleon-nucleon interaction are employed to calculate cross-sections for the reaction¹² C(π ⁺,p)¹¹ C(g.s.). These wave functions take into account central correlations between nucleons inside the nucleus. This itself is found to change the cross-section by more than an order of magnitude. The incoming pion is represented by a plane wave while proton-distortion is taken into account in the high-energy or semi-classical approximation, thereby determining the proton optical well parameters. These values agree well with those obtained by more conventional methods. Variation of the cross-section with the oscillator well parameter is also studied. Calculations have been made using the one-nucleon mechanism for the pion-absorption process.