Full Wave Function Research Articles

AB initio calculation of the two-photon absorption cross section of the X 1 Σ g+ → (E,F) 1 Σ g+ IN H 2

The two-photon absorption cross section of X 1Σ g + → 1Σ g + of H 2 has been computed using near full Cl wavefunctions. The convergence of the two-photon transition amplitude is achieved by enforcing the variational wavefunctions to satisfy the off-diagonal S(O) and S(−1) sum rules. The theoretical cross section agrees well with experiment.

Cluster analysis of the full configuration interaction wave functions of cyclic polyene models

AbstractThe first two members of the cyclic polyene homologous series are studied over a wide range of the coupling constant using the Hubbard and Pariser–Parr–Pople model Hamiltonians. The full and various limited configuration interaction (CI) correlation energies and wave functions are calculated exploiting the unitary group approach. The formalism for the cluster analysis of the exact wave function expressed through the unitary group formalism electronic Gelfand states is developed and applied to the full CI wave functions of the cyclic polyene models studied. It is shown that the connected tetraexcited clusters become essential in the fully correlated limit and that their contribution also significantly increases with electron number even for the coupling constant corresponding to the spectroscopic parametrization of the model Hamiltonians used.

Theoretical study of the spin-orbit coupling constants of the c(2p)3Πu, d(3p)3Πu, k(4p)3Πu, i(3d)3Πg, r(4d)3Πg, j(3d)3Δg, and (4f)3Δu states of H2

Theoretical spin-orbit coupling constants as a function of internuclear distance A(R), are reported for the c(2p)3Πu, d(3p)3Πu, k(4p)3Πu, i(3d)3Πg, r(4d)3Πg, j(3d)3Δg, and (4f)3Δu states of H2. Full configuration-interaction wave functions and large Slater-type basis (STO) sets were used. A 14σ14π8δ2φ basis set optimized for the c3Πu state was used in all 3Π state calculations and a 14σ12π8δ2φ basis set optimized for the j 3Δg state was used in the 3Δ calculations. The A(R) were vibrationally averaged using our theoretical potentials. In addition, we used the more accurate c3Πu and i3Πg theoretical potentials of Kolos and Rychlewski [J. Mol. Spectrosc. 66, 428 (1977)], and for the d3Πu state a potential derived from the experimental data of Dieke [J. Mol. Spectrosc. 2, 494 (1958)]. The resulting theoretical Av values (MHz) for the v = 0, N = 1 rovibrational level compare with the experimental values (in parenthesis) as follows: c(2p)3Πu: −3887.22(−3740.987); d(3p)3Πu: −863.85 (−814.5); k(4p)3Πu: −398.8 (−306.7); i(3d)3Πg: −144.59 (−146); j(3d)3Δg: −400.82 (−409). The corresponding theoretical values for the r(4d)3Πg and (4f)3Δu states, for which no experimental data exist, are −47.1 and −114.78 MHz, respectively. These values, as well as the calculated Av for many of the higher vibrational levels, should be useful in future experimental work.

On the Ability of the Interacting Boson Model to Describe Nuclear Deformation Effects

The characterization of many-fermion wave functions as condensates of bosons built out of pairs of particles is further analyzed. It is emphasized that the properties of the many particle systems are extremely sensitive to small components in the structure of the boson. In particular a truncation of the boson to the dominant angular momentum components may lead to results that are qualitatively different from those obtained from the full wave function.

Charge and spin correlation structures of conjugated π systems: Analysis of full CI wave functions of the PPP hamiltonian

AbstractAn analysis of the electronic correlation structures by means of the charge and spin correlation functions is carried out for full CI wave functions of four, five, and six membered conjugated π systems described by the Pariser–Parr–Pople Hamiltonian. The low‐lying states of these systems are classified as covalent (CV) and ionic (IN) states depending on whether the probability of finding two electrons simultaneously at the same position is small or large. It is found that many of excited CV states, the typical ones of which are the 21Ag state of linear π systems, have stronger CV character than the ground CV state, and their spin coupling structures are different from each other as well as from that of the ground CV state. The spin coupling structure in the ground CV state has an “antiferromagnetic” spin arrangement in favor of antiparallel coupling between nearest neighbor spins while in excited CV states the extent of the antiparallel spin coupling between nearest neighbor sites is decreased. IN states, which are less common for low‐lying states than CV ones, are also found to have characteristic modulations in the charge correlation. In particular, the charge correlations in the lowest singlet IN states, 11Bu of linear π systems, 11B2g of cyclobutadiene and 11B1U of benzene, are alternating.

The ground state of deformed nuclei as a boson condensate

It is shown that the ground state of deformed nuclei can be considered as a condensate of bosons that do not have a well-defined angular momentum. Values for the quadrupole moment and the particle number that are very close to the values obtained using the full boson wave function are obtained by retaining only the s- and d-parts of the boson wave function. By comparing with the many-shell (realistic) situation we found the limitations of the single-shell calculations.

Analysis ofπ−He4scattering andHe4(π−,n)H3reactions at 290 MeV

This study evaluates the amplitude for the pion absorption reaction /sup 4/He(..pi../sup -/,n)/sup 3/H to first order in the pion absorption operator for a pion laboratory kinetic energy of 290 MeV and the associated angular distribution of emitted neutrons is compared with recent experimental results. The pion absorption amplitude requires the full elastic scattering wave functions for the ..pi../sup -/-/sup 4/He and n-/sup 3/H systems and the bulk of this study is the generation of the ..pi../sup -/-/sup 4/He full elastic scattering wave function in the effective channel approach. The effective channel approach expresses the ..pi../sup -/-/sup 4/He full elastic scattering state in terms of explicit elastic and inelastic channel wave functions. The input to the effective channel approach is a phenomenological ..pi..N separable interaction which satisfactorily accounts for the ..pi..N 33 resonance and a complex local Gaussian potential which, in conjunction with the separable interaction, produces the approximate spin and isospin averaged ..pi..N phase shifts and charge exchange cross section. A judicious choice of the form factor of the separable interaction allows an exact evaluation of the pion-nucleus potentials occurring in the application of the effective channel approach. These potentials are nonseparable and couple each partial wave to itselfmore » and to its two next neighbors as well as to the partial waves of the other channel. The complex local ..pi..N Gaussian potential also allows the analytic construction of the associated terms of the effective channel approach. The resulting two-channel system of integrodifferential equations is solved by iteration. The parameters of the ..pi..N potential are then varied to produce the observed total and inelastic ..pi../sup -/-/sup 4/He cross sections and an angular distribution in reasonable accord with experiment to study pion absorption.« less

Exact solution of the Faddeev equations for the harmonic oscillator ground state

The Faddeev equations in $N$ space dimensions for three identical spinless particles in their ground state interacting via harmonic oscillator, two-body potentials is solved analytically. Unlike the well-known Schr\"odinger solution, the individual Faddeev amplitudes may be negative and contain a long-range component of arbitrary strength. Upon symmetrizing to obtain the full wave function, this component vanishes and the Schr\"odinger solution results. Three-dimensional plots of the various wave functions are presented.NUCLEAR STRUCTURE Faddeev calc., configuration space, harmonic oscillator.

A variational approach to two-body correlations in atoms

A variational method for calculating the correlation structure of atoms is proposed. The full wavefunction is written in the form Psi = Omega Phi where Phi is a sum of determinantal functions. Omega is expanded in terms of one-, two-, three-, etc. body cluster operators. The following approximations are made: (i) the cluster expansion is truncated; (ii) the cluster operators are taken to be local or to involve bilinearly spin, orbital angular momentum and linear momentum operators; and (iii) the cluster operators are expanded in terms of separable operators. The resultant expressions involve at most two-dimensional integrals, and the sums are tractable. Applications are discussed.

On the description of direct nuclear rearrangement reactions using various coupled channel equations

The standard description of direct nuclear rearrangement reactions by means of the distorted wave Born approximation (DWBA) or the coupled reaction channel (CRC) method has provided little insight into the role of rearrangement coupling effects in these reactions. Essential to these calculational schemes is the bound state approximation (BSA), i.e., the replacement of the full wave function by product wave functions corresponding to relevant asymptotic channels. The role of the continuum in rearrangement effects is thus ignored. Many-body scattering theory provides a powerful, mathematically correct tool for assessing the validity of these approximate schemes and for improving the treatment of the coupling mechanisms. In this paper we use one of the many-body scattering theories, the channel coupling array (CCA) theory, to study rearrangement coupling effects in the BSA both theoretically and computationally. The results of these calculations are compared with those from the DWBA and the CRC scheme for the transfer reactions 16O(d, p) 17O(2s 1 2 ), 16O(α, 3He) 17O(2s 1 2 ), and 19F( 3He, d) 20He(0 +). The associated elastic cross sections are also determined. Coupling effects are found to be very small in the BSA. They are overestimated considerably in the common “post” form of the CRC equations. In general, the CCA and CRC results are similar in shape and in magnitude. We argue that the exact coupling effects can be much larger and indicate how one could accomodate these large effects in equations having the same structure as the CCA or CRC equations. In such equations the rearrangement effects will mainly reveal themselves either through modified optical potentials, in which case a DWBA procedure — which uses first-order distorted waves — is well justified, or through large coupling potentials in which case higher order CCA or CRC type equations must be used. In the second case, and possibly also in the first case, the effective coupling potentials will be quite different from the short range interaction occurring in the usual DWBA.

Test of bound-state approximations in a three-body model of rearrangement collisions

A common approximation in atomic, molecular, and nuclear rearrangement processes is to neglect $n$-body breakup contributions ($n\ensuremath{\ge}3$) by replacing the full wave function by (sums of) product wave functions, each of which is related to a specific asymptotic arrangement channel. We test this bound-state approximation in a simple three-body model of nuclear stripping by comparing bound-state approximation calculations with exact and distorted-wave Born-approximation-type calculations. Calculated are the stripping and elastic amplitudes and cross sections for deuteron energies ${E}_{d}=1.78, 6.7, 11.2, \mathrm{and} 15.12$ MeV. The bound-state approximation is best below the breakup threshold (${E}_{\mathrm{BU}}=2.225$ MeV), where it provides an excellent fit in the forward direction and a qualitative fit over the whole angular range. For ${E}_{d}=6.7$ MeV the breakup cross section is still very small; however, the intermediate continuum states already play an important role and the bound-state approximation becomes quite poor. At this energy the distorted-wave Born approximation-type calculation, which is based on exact elastic wave functions and therefore accounts partly for the continuum, is still quite good. For the two higher energies both continuum and multistep effects appear to be very important, rendering both distorted-wave Born approximation and bound-state approximation poor approximations outside the forward region. The qualitative features of the elastic cross sections are explained in terms of the momentum dependence of the bound-state wave functions and two-body $T$ matrices.NUCLEAR REACTIONS Quality of DWBA and bound-state approximations; Three-body methods; Model ($d$,$p$) calculations at ${E}_{d}=1.7\ensuremath{-}15$ MeV; Multistep effects

Three-body model of deuteron breakup and stripping, II

A previously investigated three-body model of the deuteron-nucleus system, limited to relative angular momentum l = 0 for the two active nucleons, is reevaluated. Full attention is given to self-consistency between elastic and breakup channels. Introduction of the reaction of breakup on the elastic channel now reduces the elastic reflection coefficients in low partial waves by nearly a factor of 2 and causes substantial shifts in phase. Breakup amplitudes in low partial waves are also greatly reduced. As before, the breakup part of the wavefunction contains a broad spectrum of n- p continuum states. The breakup part of the wavefunction at zero n- p separation is localized at small radii, within and just outside the target nucleus, where it is comparable in magnitude with the projected elastic channel wavefunction. As a result, the projected elastic channel wavefunction is a poor approximation to the full wavefunction at n- p coincidence. Deuteron stripping theories that use the projected elastic wavefunction in a truncated waves Born series must correspondingly be quite misleading. To investigate deuteron stripping further, the exact result of the coupled channels calculation is compared with several standard approximate models. Although there is a close qualitative resemblance among the results of all the approaches, the best single approximation to the coupled channels result is found from the familiar phenomenological approach, in which a local optical potential is fitted to the elastic scattering “observed” in the coupled channels calculation. The coupled channels results are also used to analyze the approximations in the Johnson-Soper method. Several formal aspects of the three-body model are discussed.

Empirical Renormalization of the One-Body Gamow-Tellerβ-Decay Matrix Elements in the1s−0dShell

..cap alpha..-decay data from nuclei with 17 < or = A < or = 39 are analyzed with full sd-space shell-model wave functions to determine the effective values of the single-particle matrix elements of the Gamow-Teller operator between all combinations of sd-shell orbitals.

The structure of the many-body wavefunction for scattering

We show that the scattered part of the many-body wavefunction initiated by two incoming clusters is given by a fully connected operator acting on the initial channel state. The structure of this operator suggests a division of the full wavefunction into two-cluster components. A set of coupled equations in both the differential and integral form is then derived for these components. These equations have structure and properties similar to the three-body equations of Faddeev. We demonstrate that each component has outgoing waves in a unique two-cluster partition. The transition amplitude for any final arrangement can therefore be extracted directly from the outgoing waves in the relevant components.

Influence of the effective charges on the five-fold differential cross section for ionization of hydrogen and helium

Five-fold differential cross sections for the ionization of hydrogen and helium have been calculated and in the case of helium compared with the experiments of Ehrhardt and co-workers (1972). It is shown that exchange and capture processes are relevant only for primary energies below 250 eV and 80 eV respectively. The two outgoing electrons are described by a product of Coulomb waves in which the charges are treated as parameters in order to simulate the screening of the residual ion. The different sets of effective charges yield results which greatly differ in the angular position of the binary encounter maximum and the ratio of intensities in the binary encounter and the recoil peak. The results indicate that it is more important to know the full wavefunction in the region where all particles are close together rather than to require fulfilment of the proper asymptotic conditions.

Electronic quadrupole moments of excited states and the charge wavefunction of the many-electron theory

It is shown that the electronic quadrupole moment as a measure of the charge distribution in non-closed shell systems is given not by the Hartree-Fock wavefunction, ΦRHF, but by the “charge distribution wavefunction” (the “charge wavefunction”) of NCMET, ϑc. The electronic quadrupole moments, Θ, of Be 1 1s22s2p3p0, B I 1s22s22p 2p0, B I 1s22s2p24P, B I 1s22s2p22D, and CI 1s22p32D0 are calculated by the full charge wavefunctions of NCMET and the Hartree-Fock wavefunctions. In C II, RHF gives Θ = 0.0 while the NCMET values are Θ = 0.07726 for J = 52 and Θ = 0.05408 for J = 32. Where several l ≠ 0 orbitals are involved in the excited state, the RHF results can differ sizably from the accurate NCMET charge wavefunction results. Recent beam measurements of electronic quadrupole moments are pertinent to these conclusions.

A theoretical study of core-polarisation effects on E2 transitions in mass-19 nuclei and in the 19F(p, p′) 19F reaction

First-order core-polarisation corrections are applied to full intermediate-coupling wave functions of the mass-19 system in a calculation of E2 transition rates. The theoretical E2 strengths are found to be smaller than those measured experimentally by a factor which varies between about 1 and 2 depending on the particular transition. The calculation is performed by first calculating a set of spectroscopic amplitudes which contain all the “one-body” information of the nuclear states. The same spectroscopic amplitudes are used in a DWBA calculation of the 19F(p, p') 19F reaction in which the effective density-dependent interaction of Pandharipande is used for the projectile target-nucleon interaction. The importance of using fully antisymmetrised DWBA matrix elements and also of including core-polarisation effects are clearly evident. Excellent theoretical fits for angular distributions of inelastically scattered protons from the lowest members of the ground-state rotational band are obtained. The theoretical cross sections must be normalised by similar factors to those required to bring the calculated E2 transition strengths into agreement with experiment, thus indicating the suitability of the interaction used.

Configuration interaction studies of NF and NF +

Full potential energy curves and wavefunctions are calculated for NF and NF + in the Born-Oppenheimer approximation with a configuration interaction method employing a minimal basis of Slater type orbitals. Resulting energy differences, vibrational and rotational constants are compared to experimental results and predictions are made of these quantities for states which have not yet been experimentally identified.

Energy-Optimized Spin Orbitals in Accurate Electronic Calculations

We consider the eigenfunctions of well-defined operator related to $\ensuremath{\int}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}},2,\dots{},N)\ifmmode\times\else\texttimes\fi{}H(1,2,\dots{},N)\ensuremath{\Psi}(1,2,\dots{},N)d(2,3,\dots{},N)$, where $\ensuremath{\Psi}$ is a configuration-interaction (CI) wave function which includes the Hartree-Fock configuration and all double excitations and $H$ is the $N$-electron Hamiltonian of an atomic or molecular system. They satisfy for the energy the same optimum convergence properties as L\"owdin's natural spin orbitals satisfy for the wave function. They are equal to the natural spin orbitals when $\ensuremath{\Psi}$ is the exact wave function or a full CI wave function; $\ensuremath{\Psi}$ may correspond to the ground state or to any excited state.

Integral equation method for the determination of the scattering amplitude

An integral equation, derived from the Lippmann-Schwinger equation for the full wave function, is solved by converting it to matrix form by use of numerical quadrature. The scattering amplitude is then determined. The procedure is applied to the screened Coulomb potential. Quite good agreement with exact results is achieved with not too many quadrature points. In particular, fairly accurate total cross sections are obtained by use of the optical theorem. The numerical procedure does not, however, appear to converge rapidly to the exact results.