Partial-wave singlet

Figure 4.19 The partial-wave singlet (full curves) and triplet (broken curves) absorption cross sections in e+ + H(1s) collisions, plotted versus the incident positron energy measured from the threshold energy for positronium formation. Results of hyperspherical closecoupling calculations including the absorption potential —iVabs in the Hamiltonian. Note that the thresholds Etu for the full and broken curves are different by 0.841 meV, the hyperfme splitting. Figure from Ref. [16].

Several applications, such as predictions of narrow scattering resonances for heavy alkali-metal atomic species or spin relaxation in collisions of atoms in excited Zee-man states, may require the inclusion of higher-order corrections (e.g.. Refs. [29-32]) in Equation 11.5. Whereas the singlet and triplet potentials depend only on the inter-atomic distance, r = r, these additional interactions are generally directional, that is, they couple different partial waves in the relative motion of the atoms. All examples of bound-state and scattering phenomena throughout this chapter might be described, however, at least qualitatively, on the basis of the isotropic Born-Oppenheimer potentials only. 

The CMU group is part of the CERN collaboration, PS 185, which studies hyperon-antihyperon (AA, AS + XA,XL) production at LEAR through the reaction pp- YY. The total cross sections, differential cross sections, final state polarizations, and spin correlation coefficients are studied near threshold where the number of partial waves contributing to the production mechanism is limited, simplifying calculations. In addition, from the spin correlation coefficients, Cij, it is possible to extract the singlet fraction, the probability that the two hyperons are produced with their spins coupled to zero, as S=(l+Cxx""Cyy+Czz)/4 (where Cij is a measure of the probability that one of the produced particles will have a positive spin component in direction f i if the second particle has a positive spin component in direction j). We have also extracted the decay asymmetry parameters of the A and A as a test of a possible technique for investigating CP violation in a system other than the neutral kaons. 

Various schemes exist to try to reduce the number of CSFs in the expansion in a rational way. Symmetry can reduce the scope of the problem enormously. In die TMM problem, many of die CSFs having partially occupied orbitals correspond to an electronic state symmetiy other than that of the totally symmetric irreducible representation, and dius make no contribution to the closed-shell singlet wave function (if symmetry is not used before the fact, die calculation itself will determine the coefdcients of non-contributing CSFs to be zero, but no advantage in efdciency will have been gained). Since this application of group dieoiy involves no approximations, it is one of the best ways to speed up a CAS calculation. 

Of course, with HF wave functions in hand, it is possible to carry out post-HF calculations to partially correct for electron correlation effects. The poor quality of the HF wave functions, however, militate against any treatment much less sophisticated than coupled-cluster. At the CCSD(T)/cc-pVDZ level, the predicted energy of the lowest closed-shell singlet is in fair agreement with experiment (other data in the table suggest that use of a triple- basis set would improve the CCSD(T) estimate). The energy of the second closed-shell singlet state... 

Figure 3 Schematic depiction of the wave function for the lowest singlet state of methylene, showing how the presence of the. .. tt ) configuration results in rabbit ear GVB orbitals that partially confine the two electrons of the lone pair to different regions of space...

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