Beryllium Isoelectronic Sequence Research Articles

Calculation of atomic polarisabilities by finite-difference methods. Part 2.—Study of the beryllium isoelectronic sequence

Finite–difference methods are used in a study of the polarisabilities of the beryllium isoelectronic sequence Li– to Ne6+. Two approximations to the Coupled Perturbed Hartree–Fock scheme are considered for the evaluation of static dipole and quadrupole polarisabilities, it being demonstrated that, provided the appropriate perturbation equations are solved accurately, both yield results which differ only marginally from those obtainable from the fully coupled method. Similarly encouraging calculations of the dynamic polarisability of Be are also recorded. The practicability of accurate polarisability calculations on medium and large systems is briefly discussed in the light of the present results.

Effect of the exchange integral K2s2p on the 2p‐orbitals of the 1P and 3P states of the beryllium isoelectronic sequence arising from the configuration (1s)2(2s)(2p)

AbstractThe 1P and 3P states arising from the configuration (1s)2(2s)(2p) of the Be isoelectronic sequence are investigated. In the single configuration approximation, the energies of the two states are expressed as E0 + K2s2p and E0 ‐ K2s2p, respectively. K2s2p is the exchange integral between the 2s and 2p electrons and E0 is the energy of a model in which K2s2p is deleted. First we calculate the 2s‐ and 2p‐orbitals in this model. Second, by taking account of K2s2p in this model, effects of this term on the 2p‐orbitals in the 1,3P states are investigated. In this manner, an explanation is given for the following facts which are obtained from a minimal Slater‐type orbital set; (1) for Be and B+, the 2p‐orbital of the 1P state is broader than that of the 3P state; (2) for C2+, the extension of the 2p‐orbital in the two states is almost the same; (3) for O4+ and Ne6+, in contrast to Be and B+, the 2p‐orbital of the 1P state is tighter than that of the 3P state.

First order pair functions for the beryllium isoelectronic sequence

The first order pair equations have been solved for the four electron sequence Li− to Ne6+. A variational perturbation procedure is blended with numerical methods to generate a method for computing pair functions which is both accurate and efficient with respect to time of computation.

Wavefunctions and oscillator strengths of the beryllium iso-electronic sequence

Configuration interaction wavefunctions for the 1s22s2 1S, 1s22s2p 3P and 1s22s2p 1P states are calculated for Be I, B II, C III, N IV, O V, F VI and Ne VII and are used to calculate oscillator strengths for the 1S to 1P transition.

Electron densities derived from line intensity ratios: Beryllium isoelectronic sequence

A direct method for determining electron densities from emission line intensities of ions in the beryllium isoelectronic sequence is described and then applied to the analysis of extreme ultraviolet Ciii and Ov spectra from both quiet and active areas in the solar transition region. The results are consistent with a value of NeTe = 6 × 1014 cm-3K for the quiet Sun at temperatures of 5 × 104 to 3 × 105K. Electron densities are approximately five times greater in active regions than in the quiet Sun.

On the energy difference between the lP and 3P states of the beryllium isoelectronic sequence

AbstractWave functions of the 1S (ground state), 3P and 1P states for the beryllium isoelectronic sequence have been obtained in various approximations.The HF 2p orbitals for the 1P and 3P states are similar except for Be, where the 2p orbital is quite diffuse for the 1P state.The difference between the experimental E(1P) – E(3P) and the HF E(1P) – E(3P) is 0.62 eV for Be and 1.17 ∼ 1.40 eV for B+ ∼ F5+. The disagreements are attributed to the correlation effects between the 2s and 2p electrons. This is confirmed by ci calculations.It is shown that a limited basis SCF calculation reproduces the above feature of the HF results if we treat the orbital exponents as the variational parameters. The use of the Slater values for the orbital exponents is shown to be inadequate especially for the Be 1P state.The conclusions of this paper will be useful for discussing the V–T separations of H2 and C2H4.

The [formula omitted] transition probability in the beryllium isoelectronic sequence

The nuclear charge expansion method is used to calculate the (2p 2) 1D−(2s2p) 1P o transition probabilities for the beryllium isoelectronic sequence correct to first order. Discrepancies between measurements and earlier calculations are resolved.

Coherent and Incoherent X-Ray Scattering by Bound Electrons. II. Three- and Four-Electron Atoms

Atomic form factors and incoherent-scattering functions for the lithium and beryllium isoelectronic sequences through $Z=8$ have been calculated with correlated ground-state wave functions. The form factors differ from Hartree-Fock values by 1-2% and differ from Hartree-Slater values by as much as 10%. The incoherent-scattering functions are much more dependent on correlation, with the values for neutral beryllium differing from the Hartree-Fock values by as much as 30% at very small values of the momentum transfer.

Dipole Hyperpolarizabilities of S-State Atoms and Ions in the First Two Rows

The Z expansion technique for calculating atomic properties has been extended to the calculation of dipole hyperpolarizabilities. The leading two terms are of order Z−10 and Z−11 in the closed-shell case and of order Z−7 and Z−8 in the open-shell case. The corresponding coefficients have been calculated exactly, and also within the uncoupled Hartree-Fock approximation. Numerical results have been obtained for the ground states of the helium, lithium, and beryllium isoelectronic sequences, and for the metastable S1 and S3 states of the helium sequence. Our Hartree-Fock values for the ground states of the helium sequence are in good agreement with those of Langhoff, Lyons, and Hurst. However, the value derived from the leading two terms of the exact Z expansion for neutral helium is about 40% lower, and even for N5+, the discrepancy is still 10%. For the other sequences, the discrepancies are much larger, extending even to the signs of the hyperpolarizabilities of the lithium and beryllium sequences. The polarizabilities of the above sequences are also included for comparison with other data to indicate the rate of convergence of the asymptotic Z expansions.

Intercombination-Line Oscillator Strengths in the Helium and Beryllium Isoelectronic Sequences

Intercombination-Line Oscillator Strengths in the Helium and Beryllium Isoelectronic Sequences

Atomic scattering factors for the lithium- and beryllium-isoelectronic sequences from accurate analytic Hartree-Fock wave functions*

The most widely used atomic scattering factors are based on numerical self-consistent field (SCF) wave functions, some of which are of low accuracy. Atomic scattering factors obtained from recent analytic Hartree-Fock wave functions of higher accuracy than the older numerical SCF functions are reported here for the lithium- and beryllium-isoelectronic sequences. In both sequences, the calculations are made for the nuclear charges 3 ≤ Z ≤ 10. These results represent previously unreported SCF factors for the two sequences except for Li, Be and C +2 . The present factors for Li, Be and C +2 are found to be in excellent agreement with the corresponding factors determined from recent, accurate numerical Hartree-Fock solutions. The use of such analytic functions permits high-precision evaluation of the form factors for arbitrary values of λ −1 sin θ , an operation which is less readily performed using the conventional numerical solutions. In addition, the influence of increasing nuclear charge on various approximate wave functions is examined by comparing the SCF factors for these isoelectronic sequences with the factors derived from other types of wave functions. In this connection, the relative reliability of the SCF factors is discussed. Finally, an explanation is given for the observation that for a given atomic or ionic species of the sequences, the various scattering curves obtained from different approximate wave functions tend to cross in such a manner so that the areas beneath them are approximately equal.

Atomic scattering factors for the lithium- and beryllium-isoelectronic sequences from accurate analytic Hartree-Fock wave functions

Atomic scattering factors for the lithium- and beryllium-isoelectronic sequences from accurate analytic Hartree-Fock wave functions

An expansion method for calculating atomic properties. II

The accuracy of the methods of calculating the expectation values of perturbed atomic systems is analyzed by means of an expansion in inverse powers of the nuclear charge. Results are presented for the quadrupole polarizabilities and nuclear-shielding factors of all members of the helium and beryllium isoelectronic sequences and it is shown that the loss of accuracy entailed in replacing the full self-consistent method by a simpler one is small. Anestim ate is made of the importance of configuration interaction between the (1s 2 2s 2 ) 1 S and (1s 2 2p 2 ) 1 S states of the beryllium sequence.

An expansion method for calculating atomic properties

An expansion method is described which provides a simple and rapid means of calculating for any atom the expectation values of sums of one-electron operators. All the equations which arise can be solved analytically and the results are obtained as functions of the nuclear charge. For inner shells the accuracy is comparable with that of the Hartree—Fock approxi­mation. The method gives a quantitative description of the effects of direct and exchange interactions between electron shells. Results are given for all members of the helium and beryllium iso-electronic sequences.