Coulomb potential, regularizing

In the high-field limit (F > 1 atomic unit meaning that it is greater than the binding potential) the smoothed Coulomb potential in Eq. (2) can be treated as a perturbation on the regular, classical motion of a free electron in an oscillating field. So, let us first consider the Hamiltonian for the one-dimensional motion of a free electron in the... 

This expansion is valid only when q(f>—E)<2mc. For the Coulomb potentials when r 0 this assumption breaks down and the expansion is unjustified. Clearly, the expansion above is valid only for regular potentials such that the classical velocity of the particle is everywhere small compared to the velocity of light. On the assumption that such an expansion is justified, substitution in equation (50) gives the ESC Hamiltonian as an expansion of o ... 

Equation (1) is obtained by using an expansion in E/ 2c - Vc) on the Dirac Fock equation. This expansion is valid even for a singular Coulombic potential near the nucleus, hence the name regular approximation. This is in contrast with the Pauli method, which uses an expansion in (E — V)I2(. Everything is written in terms of the two component ZORA orbitals, instead of using the large and small component Dirac spinors. This is an extra approximation with respect to the original formalism. 

Patkowski K, Jeziorski B, Szalewicz K (2001) Symmetry-adapted perturbation theory with regularized Coulomb potential. J Mol Struct (Theochem) 547 293-307... 

If the potential V(r) is a pure Coulomb potential the asymptotic partial wave is given by the regular Coulomb function (4.64), apart from a constant phase factor. We strictly have no incident plane wave since the Coulomb potential modifies the wave function everywhere. We make the normalisation of the Coulomb distorted wave t/j,j(k,r) analogous to that of (4.83) by choosing the phase factor to be the Coulomb phase shift [Pg.95]

Moreover, -> molecular interaction fields are calculated for each molecule in terms of similarity indices instead of the usual interaction potential functions, such as Len-nard-Jones and Coulomb potential functions. Similarity fields are calculated representing the similarity between molecules and different probe atoms. In particular, the similarity values at the intersections of the regularly spaced grid (1.1 and 2.0 A) relative to the yth physico-chemical property between the ith compound and a probe atom is calculated as ... 

Hydrogen [54] and helium [55] atoms are known to exhibit regular/chaotic dynamics in the presence of external field of different colors and intensities. Chaotic ionization from the Rydberg states of the atoms [54,55] has been very intriguing for the experimentalists. Both QFD [56,57] and quantum theory of motion (QTM) [58,59] have been able to explain the quantum domain behavior of the classically chaotic systems [60], In QFD, the quantum dynamics is mapped onto that of a probability fluid of density and current density p(r, t) and j(r, t) respectively obtainable as solutions to the QFD equations. The fluid moves under the influence of the classical Coulomb potential augmented by a quantum potential defined as... 

Introducing a cut-off of the Coulomb potential at short distances introduces some arbitrariness. A method of regularization that works with the unmodified Coulomb potential is to take into account the anomalous magnetic moment of the electron. The Dirac operator now reads (in dimensionless units)... 

For the Coulomb potential, the regular solution has the asymptotic form fk r) sin (kr-I-1 log2kr - -I-a ) and fulfils fhe condifion of analyfic-... 

For a Coulomb potential the cut-off function / varies from 0 to 1 as illustrated in Fig. 7. The iterative solution of (103) provides regular terms that finally lead to a regular expansion of the Bloch and des Cloizeaux effective Hamiltonians. The first terms are ... 

Fig. E.l. The dispersion curves of the four lowest bound states (solid and dashed) for a regularized Coulomb potential with a K = 0) = 1. Even (odd n) parity states (solid curves) and odd (even n) parity states (dashed curves). The particle-hole continuum is bounded by the dotted curves. The energies are in units of Ej.

We see then that the contributions from the regularized two-electron relativistic operators to the Fock operator are split between two terms. If we want the screening contribution in the denominators, we cannot write the Coulomb potential as an integral... 

A simple choice for a regularized Coulomb potential would be... 

The difference between the original and regularized Coulomb potentials. 

Abstract. Classical regular and chaotic dynamics of the particle bound in the Coulomb plus linear potential under the influence of time-periodical perturbations is treated using resonace analysis. Critical value of the external field at which chaotization will occur is evaluated analytically based on the Chirikov criterion of stochasticity. 

---