Coulomb problem

Strain M C, Scuseria G E and Frisch M J 1996 Linear scaling for the electronic quantum coulomb problem Science 271 51-3... 

However, unlike point charges, the continuous charge distributions that occur in quantum chemistry have varying extents and the applicability of the multipole approximation is not only limited by the distance but also by the extent or diffuseness of the charge distribution. This additional complexity makes a transfer of the concepts of the fast multipole method to applications in quantum chemistry less straightforward. Therefore it should come as no surprise that several adaptations to extend the applicability of the FMM to the Coulomb problem with continuous charge distributions have been suggested. These lead to... 

Strain, M. C., Scuseria, G. E., Frisch, M. J., 1996, Achieving Linear Scaling for the Electronic Quantum Coulomb Problem , Science, 271, 51. 

Modern aspects of the three-body Coulomb problems... 

For this reason, we will restrict our subsequent approach to planar configurations of the two electrons and of the nucleus, with the polarization axis within this plane. This presents the most accurate quantum treatment of the driven three body Coulomb problem to date, valid in the entire nonrelativistic parameter range, without any adjustable parameter, and with no further approximation beyond the confinement of the accessible configuration space to two dimensions. Whilst this latter approximation certainly does restrict the generality of our model, semiclassical scaling arguments suggest that the unperturbed three... 

Another well-defined configuration of the classical three body Coulomb problem with unambiguous quantum correspondence is the collinear antisymmetric stretch configuration, where the electrons are located on opposite sides of the nucleus. In contrast to the frozen planet orbit, the antisymmetric stretch is unstable in the axial direction (G.S. Ezra et.al., 1991 P. Schlagheck et.al., 2003), with the two electrons colliding with the nucleus in a perfectly alternating way (Fig. 3 (left)). Hence, already the one dimensional treatment accounts for the dominant classical decay channel of this configuration. As for the frozen planet, there are doubly excited states of helium associated to the periodic orbit of the ASC as illustrated in Fig. 3 (left). 

The relationship between alternative separable solutions of the Coulomb problem in momentum space is exploited in order to obtain hydrogenic orbitals which are of interest for Sturmian expansions of use in atomic and molecular structure calculations and for the description of atoms in fields. In view of their usefulness in problems where a direction in space is privileged, as when atoms are in an electric or magnetic field, we refer to these sets as to the Stark and Zeeman bases, as an alternative to the usual spherical basis, set. Fock s projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. Its generalization to higher spaces permits to build up multielectronic and multicenter orbitals. 

The hyperspherical method, from a formal viewpoint, is general and thus can be applied to any N-body Coulomb problem. Our analysis of the three body Coulomb problem exploits considerations on the symmetry of the seven-dimensional rotational group. The matrix elements which have to be calculated to set up the secular equation can be very compactly formulated. All intervals can be written in closed form as matrix elements corresponding to coupling, recoupling or transformation coefficients of hyper-angular momenta algebra. 

Only the first two terms in the series give contributions larger than 1 Hz to the ground state splitting in muonium. As usual, in the Coulomb problem, expansion in the series for the binding corrections goes over the parameter... 

Za) without any factors of tt in the denominator. This is characteristic for the Coulomb problem and emphasizes the nonradiative nature of the relativistic corrections. 

We now convert the radial equation to the standard form of the coulomb problem by introducing p(r), which is defined by3 R(r) = p(r)/r. With this substitution the radial equation of Eq. (2.9) may be written as... 

M. J. Englefield, Group Theory and the Coulomb Problem (Wiley Interscience, New York, 1972). 

Early applications of WKB approximations to the Coulomb problem in Schrodinger theory demonstrated the necessity and expediency of the Kramers modification ) ... 

Expression (3.113) must be applied in the collision integral (3.71). An important example is the Coulomb problem. In this case the bound states must be interpreted as atoms. For systems with Coulomb interaction we get from (3.113) an expression that has a clear physical meaning ... 

In this section we shall apply the realizations of so(2, 1) to physical systems, such as the nonrelativistic Coulomb problem, the three-dimensional isotropic harmonic oscillator, Schrodinger s relativistic equation (Klein-Gordon... 

In atomic units the Hamiltonian for the nonrelativistic Coulomb problem with nuclear charge Z is given by... 

We are interested in the actual two-electron atom and ions.Thus we need to investigate the -dependence of the flow on the TCM. Thanks to the similarity between celestial problem and Coulomb problem, for our Coulomb systems, the same argument is easily shown following the discussion of Ref. [22]. 

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