Vibro-rotational states of the two-electron atom. I. Euler angles coordinate basis

Absolute Differential Cross Sections for Photo Double Ionization of Helium from theAb InitioHypersphericalR-Matrix Method with Semiclassical Outgoing Waves

“…Our method uses hyperspherical coordinates, namely the hyperradius R p r 2 1 1 r 2 2 , the angle of radial correlation a arctanr 1 ͞r 2 , the angle of angular correlation u 12 ͑ r 1 , r 2 ͒, complemented by a set of three Euler angles denoted collectively by V. This coordinate system, well suited to the description of a correlated motion, is numerically very convenient as only one variable is of infinite range. Projecting the 1 P o wave function C onto the appropriate Wigner functions D 16 01 ͑V͒ [13][14][15] eliminates the Euler angles from the treatment at the cost of introducing the two components F 6 ͑R, a, u 12 ͒. In the following, we shall omit the 6 labels for simplicity, and treat these two components as a unique vector.…”

Collective motion of two-electron atom in hyperspherical adiabatic approximation

“…Define Euler angles that connect the rotating and the fixed -body frames as follows: the rotation about the Z axis by an angle α ( 0≤ α 0 <2 π ), followed by a rotation around the new axis Y by an angle β ( 0 ≤ β ≤ π ) and, finally , the rotation around the new axis Z angle γ ( 0 ≤γ <2π). To find the expression of the Hamiltonian operator in the variables r1, r2, θ, α 0 , β, γ, we use the results of Ostrovsky and Nikitin [6,7]. Change the choice of direction X 'axis so that it passes not by the bisector θ (angle between 1 r and 2 r ), but by the median of the corresponding triangle .…”

“…Number of exact and approximate analytical methods of many-electron atoms, are created: decomposition Fock wave function of the bound state of two-electron atoms [3], the method of hyperspherical coordinates [4], algebraic methods of group theory [5] and semiclassical variational methods. In the present work, based on the approach developed previously by Ostrovsky and Nikitin [6,7], the obtained system of differential equations was solved semi classically in the vicinity of Wannier-ridge resonances. …”

Semiclassic approach within the He atom sphere model

“…The dynamics of highly doubly excited quantum states with moderate total angular momentum are thus semiclassically connected to a quantization of the planar configuration L ជ ϭ 0 in the scalar variables. The general Schrodinger equation obtained after the variables' division was first used in [6]. One of its solutions for several cases was presented in our work [7].…”

“…The electron positions can be determined by three external coordinates and three Euler angles. In 6 a new choice of Euler angles was presented. The two‐electron wave function is expanded in a series of Wigner D ‐functions depending on the Euler angles (α, β, γ).…”