Double photoionization of helium: The hypersphericalR-matrix method with semiclassical outgoing waves

We present an ab initio scheme for computing the wave function of the pair of electrons projected into the double continuum of the helium atom by a one-photon dipolar excitation. The scheme uses the Floquet approach to convert the time-dependent Schrödinger equation into an infinite set of coupled time-independent equations. The latter reduces to a single stationary inhomogeneous Schrödinger equation in the weak-field limit, which is relevant here. This equation is solved using an $\mathcal{R}$ matrix approach to combine a quantum treatment of all variables within the hypersphere $R<~R_0$ with a semiclassical treatment of the R motion outside this hypersphere. We apply this approach to a model helium atom where $r_1{=r}_2.$ We thus demonstrate the feasibility of the method, and obtain insights into the dynamics of the double photoionization of helium at equal sharing of a low excess energy.