Three-electron Systems Research Articles

Parameters for some resonances in photoionization for the lithium isoelectronic series

Expressions, derived from time dependent Hartree-Fock theory, for the locations, widths and profile indices of some resonances in the photoionization cross sections of three-electron systems are evaluated for members of the lithium series. The resonances are those which arise from the (1s2)1S → (1snp)1P excitations of the core. The dependence of these resonance parameters on nuclear charge, Z, is examined and it is shown that asymptotically they are independent of Z. Expressions for their Z dependences are presented.

Triply excited states of three-electron atomic systems

A formalism is discussed whereby many-electron systems may be described by a great number of product functions of different configurations in a manageable way. Calculations are made to find triply excited energy levels of three-electron atomic systems, using products of hydrogenic functions with a $Z$ equal to that of the nucleus. Results for the lowest $^{2}P^{o}$ and $^{2}S^{e}$ levels in the Li isoelectronic sequence, $Z=1 \mathrm{to} 10$, are presented. The results of calculations for the energies of $^{2}D^{e}$ triply excited states in ${\mathrm{He}}^{\ensuremath{-}}$ are also reported. The calculated energies are compared with available experimental and other theoretical values. Finally, the lowest $^{2}S^{e}$ and two lowest $^{2}P^{o}$ energy levels in the isoelectronic sequence are fitted by a $\frac{1}{Z}$ perturbation expansion. It is seen that this expansion can be used to estimate the energy levels of such states for higher nuclear charges.

Projected general spin-orbital wavefunctions for three-electron systems

Spin-projected Hartree-Fock calculations using general spin orbitals are performed on some S, P, D and F states of Li and on the ground states of its isoelectric ions with Z=4-10. The energy -7.447596 au is obtained for Li 22S, compared to -7.447565 au with the spin-optimized SCF method and -7.432727 au with the ordinary restricted Hartree-Fock method. Full configuration-interaction results for the same basis sets are also reported.

Electronic correlation studies. III. Self-correlated field method. Application to2S ground state and2P excited state of three-electron atomic systems

The self-correlated field method is based on the insertion in the group product wave function of pair functions built upon a set of correlated ''local'' functions and of ''nonlocal'' functions. This work is an application to three- electron systems. The effects of the outer electron on the inner pair are studied. The total electronic energy and some intermediary results such as pair energies, Coulomb and exchange ''correlated'' integrals, are given. The results are always better than those given by conventional SCF computations and reach the same level of accuracy as those given by more laborious methods used in correlation studies. (auth)

Spin-coupled wavefunctions

A new method of formulating the construction of spin-coupled wave-functions, in which each electron is ascribed to a different spatial function, is described. The direct use of the matrix representatives, ∪(P), of the permutation operators P, normally required in constructing antisymmetric wavefunctions, is avoided. Instead, antisymmetrizing operators are introduced as matrices, in such a way as to considerably reduce the amount of data required to characterize the spin-state, thereby increasing the computational efficiency of the method. The theory is applied to the three-electron systems Li, HeH and LiH+ using a simple one-configuration model, in which the orbitals are written in elliptical coordinates, and the non-linear parameters are optimized numerically.

Radio-Frequency Spectrum ofHe−3andHe−4

All independent energy intervals have been measured in the metastable $1s2s2p^{4}P$ state of the helium negative ions $^{4}\mathrm{He}^{\ensuremath{-}}$ and $^{3}\mathrm{He}^{\ensuremath{-}}$. These results are of sufficient precision to provide a critical test of any proposed wave functions for this state and, when adequate wave functions become available, can be used to test relativistic and radiative corrections in three-electron atomic systems.

Applications of a simple molecular wavefunction. Part 5.—Floating Spherical Gaussian Orbital open-shell calculations: introduction

A procedure is described for applying the Floating Spherical Gaussian Orbital (FSGO) model to molecular systems in doublet ground states. Results are presented for several three-electron systems. For the transition state of the hydrogen atom/hydrogen molecule exchange reaction, a linear arrangement of the nuclei is preferred over a triangular one in agreement with other calculations. However, the activation energy is too large. The FSGO model, using two configurations, is able to produce a van der Waals type interaction both between H and H2 and between H and He. Although the energy of interaction is not as large as would be expected from their polarizabilities, the HeH system is predicted to be several times more stable than a possible H … H2 complex. A good internuclear distance and dissociation energy are calculated for He+2. A low dissociation energy is computed for LiH+. Various forms of H+4 have been studied. It is shown that it is possible for it to exist as a complex between H and H+3; this being preferred to one between H2 and H+2. Each three-electron FSGO wavefunction is fully optimised in 10–15 seconds CPU time (Cambridge Titan Computer; estimated time on the IBM 270/165 is less than ½ second).

Lifetimes of metastable X-ray emitters in helium- and lithium-like chlorine

We have used coincidence techniques to identify the charge states of metastable K X-ray emitters produced in foil excitation of energetic chlorine beams. At 55 MeV the emitting states are found to belong primarily to helium- and lithium-like chlorine ions. Charge-state separated decay curves for these ionic species are characterized by lifetimes of 2.2±0.3 ns and 0.95±0.2 ns for two- and three-electron systems respectively. The corresponding emitting states are thought to be the (1s2p) 3P 2 0 and ( s2 s2 p) 4 P 5 2 0 levels in helium- and lithium-like chlorine respectively, both of which may radiatively decay by M2 photon emission.

Variational calculations for the three-electron systems He- and Be+

A variational calculation involving configuration interaction has been carried out to obtain the eigenvalues of the symmetries 2S, 2Pe and 4S in He- and 2S in Be+. The eigenvalues of 1s2s2 2S, 1s2p2 2Pe and 1s2s3s 4S have been minimized by varying the exponential parameters present in the linear combinations of Slater-type orbitals used. In He- the level 1s2s2 2S has been minimized in matrices of order 14 and 19 (which includes 1s2sp configurations) resulting in values of -4.3756 and -4.3614 ryd, respectively. Neither the 2Pe nor the 4S levels were found to be bound. The results for the Be+ 2S multiplet, using matrices of similar order and content, are -20.2115 and -20.1736 ryd.

Magnetic-Dipole and Electric-Quadrupole Hyperfine Matrix Elements for the 1s22p2P States of Three-Electron Systems

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The dipole polarizabilities of three-electron systems

Configuration interaction wave functions are used to determine the dipole polarizabilities of normal Li, Be+,..., O5+. A value of α = 23.97 × 10-24 cm3 is obtained in the case of lithium, which is within the error range of the most recent experiments.

The fine structure of the2P terms of the lithium isoelectronic sequence

Ab initio calculations of the fine structure separation of the 1s2np 2P, n = 2, 3 and 4, states of the three-electron systems are made using wave functions of the open-shell type. Satisfactory agreement with experiment is obtained only in the case of the highly ionized systems.

Nuclear Magnetic Shielding Constants for Two-, Three-, and Four-Electron Atoms and Ions

Values of nuclear magnetic shielding constants for two-, three-, and four-electron atoms and ions are compared and discussed. It was concluded that the error sigma /sub NR/-- sigma /sub HF/ = delta sigma approximates 0 for two- and three-electron systems, but delta sigma > 0 or sigma /sub NR/ > sigma /sub HF/ for four-electron systems. (P.C.H.)

Correlation Effects in Two- and Three-Electron Systems

Correlation effects in the lithium isoelectronic series ${(1s)}^{2}2s^{2}S$ have been studied by computing the expectation values of a number of one- and two-electron operators. Calculations were performed using both Hartree-Fock and configuration interaction wave functions. The operators considered are $\ensuremath{\Sigma}{{r}_{i}}^{N}$ ($N=\ensuremath{-}2, \ensuremath{-}1, 2, 4$), ${\ensuremath{\delta}}^{3}({\mathrm{r}}_{i})$, ${{p}_{i}}^{2}$, $\ensuremath{\Sigma}{i>j}^{}{\ensuremath{\delta}}^{3}({\mathrm{r}}_{\mathrm{ij}})$, $\ensuremath{\Sigma}{i>j}^{}({\mathrm{r}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{r}}_{j})$, and $\ensuremath{\Sigma}{i>j}^{}({\mathrm{p}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{p}}_{j})$. Computations were performed for $Z=3, 4, 5, \mathrm{and} 8$. For comparison purposes the same expectation values were computed from configuration interaction wave functions of comparable accuracy for the two-electron systems ${(1s)}^{2}^{1}S$ with $Z=2, 3, 4, 5, \mathrm{and} 8$. The results of the two-electron computations show reasonable agreement with more accurate computations and with empirical estimates. For the three-electron systems the main conclusions are: (a) Correlation has little effect on the one-electron expectation values, (b) the expectation values of $\ensuremath{\Sigma}{i>j}^{}({\mathrm{r}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{r}}_{j})$ and $\ensuremath{\Sigma}{i>j}^{}({\mathrm{p}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{p}}_{j})$ are proportionately larger for three- than for two-electron systems, and (c) the configuration interaction approach probably gives poor estimates of the expectation value of $\ensuremath{\Sigma}{i>j}^{}{\ensuremath{\delta}}^{3}({\mathrm{r}}_{\mathrm{ij}})$ even though this value appears to converge as the number of terms in the wave function is increased.

Oscillator Strengths for Lithium and Other Three-Electron Systems

Analytical wave functions of the open shell type have been computed for the 1 s2np2 P, n = 2, 3, 4, states of the three-electron systems and oscillator strengths for the 2 s , 3 s , 4 s –2 p , 3 p , 4 p transitions in Li I , Be II ,. . ., F VII have been calculated using the dipole length and dipole velocity formulae. Satisfactory agreement is in general obtained with the values determined by the method of Bates & Damgaard (1949).

Hyperfine Splitting of the Ground State of Lithium-like Ions

The hyperfine splitting of the 1s2 2s 2S states of the three-electron systems is calculated using the simplest form of open-shell wave functions. The result for lithium is in good agreement with the very accurate calculation of Nesbet and with experiment. The values for the isoelectronic sequence can be closely represented by an expansion in powers of Z-1 of the form = a03{0.5Z3 -1.472Z2+0.838Z+0.616} where f is the operator introduced by Nesbet.

Open shell calculations for the two- and three-electron ions

Open shell calculations are made for all the three-electron atomic systems of the second period of the periodic table. These results are compared with corresponding two-electron and experimental energies. It is noted that the improvement of open shell over closed shell calculations becomes less with increasing atomic number. In parallel trend the disparity of both from the experimental results increases with Z. An electron affinity of zero is found for the helium atom.

The Approximations Involved in Calculations of Atomic Interaction and Activation Energies

The assumptions underlying the computation of interaction energy among several atoms, and especially of activation energies, have been analyzed and discussed for the case of small overlapping of the one-electron wave functions involved. Some formal justification has been found for several types of approximation which have hitherto been used without adequate examination. Computations of the energy of several three-electron systems are presented as material illustrating the applicability of some of the assumptions to typical cases. It is concluded that in general any modification of the complete Heitler-London treatment of the system may lead to errors comparable in magnitude to the quantity to be computed. The so-called ``semi-empirical method,'' which involves many such modifications, owes its success to a happy cancellation of errors, and, until some theoretical explanation of this cancellation is found, it must assume the status and responsibilities of a purely empirical method.