S-wave helium

The quantum mechanics of the s-wave model of the helium atom was investigated by Draeger et al. (1994). It turns out that for certain classes of states the energy levels of s-wave helium are very close to the energy levels of the real helium atom. 

Handke, G., Draeger, M. and Friedrich, H. (1993). Classical dynamics of s-wave helium, Physica A197, 113-129. 

Fig. 12, Liapunov exponents of the unstable periodic orbits labelled — + + +, where each period contains one slow Coulombic oscillation of the farther electron and n Coulombic oscillations of the closer electron in between two crossings of the line ri = T2 in coordinate space. The ordinate measures the products n2 on a linear scale and the abscissa measures n on a logarithmic scale. The solid dots show the results for s-wave helium (Eq. (16)) [55], the open circles show the corresponding results for collinear helium (Eq. (15)) [52]. The linear behaviour for large n illustrates the proportionality of /l to (log n)/n. (Results are for charge Z = 2 in the Hamiltonian, Eq. (14))...

The reason for the large difference between the values of A for positrons and electrons at an energy of 2 eV is that for positrons the s-wave phase shift passes through zero at the Ramsauer minimum and the dominant contribution to the cross section therefore comes from the p-wave, which is quite strongly peaked in the forward and backward directions. In contrast, there is no Ramsauer minimum in electron-helium scattering, and the isotropic s-wave contribution to aT is dominant at this energy. 

Fig. 3.6. Positron-helium s-wave phase shifts for three helium models -,...

Several other calculations of the first few partial-wave phase shifts for positron-helium scattering have been carried out using a variety of approximation methods in all cases, however, rather simple uncorrelated helium wave functions have been used. Drachman (1966a, 1968) and McEachran et al. (1977) used the polarized-orbital method, whereas Ho and Fraser (1976) used a formulation based on the static approximation, with the addition of several short-range correlation terms, to determine the s-wave phase shifts only. The only other elaborate variational calculations of the s-wave phase shift were made by Houston and Drachman (1971), who employed the Harris method with a trial wave function similar to that used by Humberston (1973, 1974), see equation (3.77), and with the same helium model HI. Their results were slightly less positive than Humberston s HI values, and are therefore probably less... 

Campeanu, R.I., and Humberston, J.W. (1977a). The scattering of s-wave positrons by helium. J. Phys. B At. Mol. Phys. 10 L153-L158. 

D.A. Homer, C.W. McCurdy, T.N. Rescigno, Electron-helium scattering in the S-wave model using exterior complex scaling, Phys. Rev. A 71 (2005) 012701. 

All beryllium nuclei contain four protons and therefore +4 electronic units, so that four electrons orbit the nucleus of the neutral atom. Its electronic configuration is is2 2s2. This can be abbreviated as an inner core of inert helium (a noble gas) plus two s-wave electrons in the second radial s state (He)2S2. This locates Be at the top of Group IIA (Mg, Ca, Sr, Ba) of the periodic table. Beryllium therefore has valence +2. 

The collinear model (Eq. (15)) has been successfully used in the semiclassical description of many bound and resonant states in the quantum mechanical spectrum of real helium [49-52] and plays an important role for the study of states of real helium in which both electrons are close to the continuum threshold [53, 54]. The quantum mechanical version of the spherical or s-wave model (Eq. (16)) describes the Isns bound states of real helium quite well [55]. The energy dependence of experimental total cross sections for electron impact ionization is reproduced qualitatively in the classical version of the s-wave model [56] and surprisingly well quantitatively in a quantum mechanical calculation [57]. The s-wave model is less realistic close to the break-up threshold = 0, where motion along the Wannier ridge, = T2, is important. 

All of our orbitals have disappeared. How do we escape this terrible dilemma We insist that no two elections may have the same wave function. In the case of elections in spatially different orbitals, say. Is and 2s orbitals, there is no problem, but for the two elechons in the 1 s orbital of the helium atom, the space orbital is the same for both. Here we must recognize an extr a dimension of relativistic space-time... 

Mulliken, R. S., Proc. Natl. Acad. Sci. U.S. 38, 160, "A comparative survey of approximate ground sate wave functions of helium atom and hydrogen molecule/ ... 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

Just before returning to Europe in 1929, Slater generalized into an N-electron system the wave function used by Pauling in the treatment of helium in the 1928 Chemical Reviews essay. The title of Slater s paper, "The Self-Consistent Field and the Structure of Atoms," shows his debt to Hartree, although Slater s method turned out to be a great deal more practical than Hartree s, as well as consistent with the methods of Heitler, London, and Pauling.70... 

In 1881 L. Palmieri thought he detected helium in a yellow amorphous sublimation product from Vesuvius. When he heated it in the Bunsen flame, he was able to observe the D3 spectroscopic line with a wave length of 5875 Angstrom units (69, TO). Although R. Nasini and F. Anderlini were unable in 1906 to produce this line by similarly heating minerals known to contain helium, they believed that, if the helium in Palmieri s mineral was bound endothermally, he might possibly have observed its spectrum in this manner (69, 71). 

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