K E Banyard

Momentum distributions for two-electron systems: electron correlation and the Coulomb hole

Evaluation of the distribution function f(p12), where p12 is the momentum difference mod p1-p2 mod , enables the investigation of the concept of a Coulomb hole Delta f(p12) in momentum space. Results are presented for the isoelectronic systems H-, He and Li+. The electron correlation within each CI wavefunction was analysed into its radial and angular components so that the structure and composition of Delta f(p12) could be assessed. The two-particle momentum radial density distribution and several two-particle expectation quantities are also examined. The findings indicate that, in momentum space, the radial components of correlation produce effects characteristic of total correlation in position space whereas, by contrast, angular correlation creates an opposite effect. Thus the shape and formation of Delta f(p12) proves to be considerably more complex than that found for its counterpart in position space.

Electron correlation in momentum space: Z-dependent variations throughout a series of Li-like ions

Momentum space is used to analyse electron correlation effects for the individual intra- and intershell electron pairs in the ground and first excited states of a series of Li-like ions when 3<or=Z<or=8. Coulomb shifts Delta f(p12) and partial Coulomb shifts Delta g(p12;p1), where p12= mod p1-p2 mod and p1= mod p1 mod , are reported for the pair descriptions K alpha K beta , K alpha L alpha and K beta L alpha in each state. Total Coulomb shifts are also examined. Radial and angular correlations are found to have opposing influences on the interparticle momentum distributions for each electron pair in the ground states and also for the intrashell electron pairs in the first excited states. Such a feature permits the identification of the major component of correlation in different regions of momentum space. Comparisons are made with a previous Z-dependent analysis of these systems in position space.

Momentum space correlation effects in the 2 3S, 2 1P and 2 3P states of He

A Dirac-Fourier transformation of correlated and Hartree-Fock wavefunctions for some (1s, nl) states of He allowed the authors to analyse electron correlation effects in momentum space for excited states of differing symmetry and spin multiplicity. For the 2 3S, 2 1P and 2 3P states, Coulomb shifts and partial Coulomb shifts were determined, as well as certain radial and angular distribution functions. For 2 3S and 2 1P it was established that, like the ground state of He, angular and radial correlation have opposing effects on the momentum distributions. In contrast to this, for 2 3P these two components of correlation work in unison, as occurs in position space. A rationalisation of this behaviour is given, along with an interpretation of the complex nature of the corresponding partial Coulomb shifts. The latter revealed significant changes in the relative importance of angular and radial correlation as the momentum of a test electron was increased. The success of this analysis rested essentially on a detailed knowledge of the inter-vectorial angular shifts. For these excited states, the use of angular expectation values alone proved to be too insensitive to explain the quite marked differences in the correlation characteristics.

Electron correlation in momentum space: the ground state of Li(2S)

Electron correlation is examined in momentum space for the individual intra- and inter-shell electron pairs in the ground state of Li(2S). It is established that the radial and angular components of the correlation have opposing influences on the inter-particle momentum distributions f(p12) and g(p12;p1), where p12= mod p1-p2 mod and p1= mod p1 mod . By examining the Coulomb shifts Delta f(p12) and, in particular, the partial Coulomb shifts Delta g(p12;P1) for each electron pair, such a feature enables one to identify the major component of correlation in different regions of momentum space. This represents a distinct advantage over the more traditional position space approach. For K alpha K beta , not only was it found that intra-shell correlation effects were influenced by the occupation of the 2s orbital but, more interestingly, the nature of the influence was seen to change markedly throughout different regions of space. For the K alpha L alpha and K beta L alpha inter-shells, the importance of angular correlation was highlighted and, for K alpha L alpha , the consequence of the Fermi effect was observed. Comparisons were made with a previous, and complementary, position-based study of Li(2S) and with the correlation effects found in momentum space for closed shell Li+(1S) and Be(1S).

Angular and radial correlation effects in momentum space for H-, He and Li+

The effect of correlation on the one- and two-particle momentum distributions is analysed by using the natural expansion representation of a configuration-interaction space wavefunction. Several expectation values are reported with emphasis being given to two-electron properties. In particular, the angular and radial components of correlation are assessed by the determination of various correlation coefficients. The introduction of total angular correlation is found to increase the probability of the two momentum vectors being aligned-as is indicated by the positive values obtained for the angular correlation coefficients.

Momentum space: effects of correlation in the doubly excited state of He-like ions

Momentum properties are examined for the -like systems when . Each doubly excited state (DES) is represented in momentum space by the Dirac - Fourier transform of the Aspromallis configuration-interaction (A-CI) wavefunction. Various truncations of the natural expansions of the transformed A-CI functions are used to investigate characteristics of several probability distribution functions and momentum expectation values. The sensitivity of such properties with respect to the ordered introduction of Coulomb correlation effects is of particular interest. This is especially so in the region of low momenta since, in position (or real) space, it corresponds to the outer or valence regions of an electron density. Coulomb shifts, based on distribution functions for momentum differences, and the separate angular and radial correlation effects are analysed for each Z. Comparisons are made, in momentum space, with the doubly occupied ground states and, briefly for He, with the singly excited state . Shapes and magnitudes of the Coulomb shifts are also related to the Coulomb holes in position space. Changes in statistical correlation coefficients highlight variations in the radial and angular components of electron correlation for the and states of the He-like ions as the electronic momenta decrease in magnitude.

Intra- and inter-shell correlation effects in Li-like ions: Coulomb holes and their interpretation

Electron correlation has been examined in detail within the K alpha K beta , K alpha L alpha and K beta L alpha shells in position space for a series of Li-like systems in their ground state (1s22s alpha )2S. This necessitated the partitioning of the two-particle space-spin density and was straightforward for the Hartree-Fock description. For the correlated density, a previously devised partitioning technique based on a many-electron representation of a CI wavefunction was employed. Using existing wavefunctions, Coulomb holes, partial Coulomb holes and several one- and two-particle expectation values, and their percentage changes due to correlation, were evaluated for the intra- and inter-electronic shells. Since 3<or=Z<or=8, it was possible to examine the Z-dependent trends of the radial and angular components of electron correlation for each shell. The presence or otherwise of Fermi correlation in the inter-shell descriptions had a marked influence on their Coulomb correlation characteristics. The findings for the Li-like intra- and inter-shells were compared both with a previous analysis of the He-like and Be-like series of ions and also with a similar examination of the 21S and 23S excited states of He.

The doubly-excited state 2p2 3P for 1<or=Z<or=4: Coulomb holes derived from explicitly correlated wavefunctions

Doubly-excited states (DES) of simple atoms involve, by comparison with the ground state, relatively slow moving electrons which should therefore be more responsive to electron correlation. Hence, for the 2p2 3P state, correlation effects have been analyzed in detail in terms of Coulomb holes, partial Coulomb holes and (rn12) when 1<or=Z<or=4. Comparisons are made with the 1s2 1S ground state and with the singly-excited state 1s2p, 3P. As for the lower states, each DES was described by an accurate explicitly correlated wavefunction. For each Z, a similarity of characteristics, but not of scale, is found between the DES results and those for the ground state. The Coulomb holes for the 1s2p 3P states have a significantly different shape. The correlation effect for each DES system exceeds that for the corresponding lower states examined here.

Coulomb correlation in the 21S, 23S, 21P and 23P states of He

Electron correlation has been examined in detail for the 21S, 23S, 21P and 23P states of He and comparisons are made with the ground state. The correlated and Hartree-Fock descriptions of these (1s, nl) excited states are provided by the wavefunctions of Tweed (1973) and Davidson (1965), respectively. Coulomb holes, various partial Coulomb holes and several one- and two-particle expectation values are presented. The concept of partial Coulomb holes and their collective presentation has surfaces as been demonstrated to be particularly informative. These surfaces enabled us to interpret correlation effects in different regions of space and so provide a rationalisation of the correlation mechanisms in each state. For all four states, correlation caused a significant inward shift of density from the outer regions due to a reduction in nuclear shielding. However, in the inner regions around the nucleus, the influence of correlation was more complicated and depended not only on the multiplicity of the state but also on its symmetry.

Potential energy curves and spectroscopic constants for BeH+, BH and CH+

Potential energy curves are reported for the diatomic hydrides BeH+, BH and CH+ in their ground states and, in each instance, an allowance was made for all the pair-correlation energies. Spectroscopic constants were obtained by means of a Dunham analysis and comparisons were made with experimental values.