Ionization Yields from TDDFT

It is apparent from Fig. 4.7 that a simple sequential mechanism is insufficient to describe the double ionization of helium. In this section we will show how one can try to go beyond this simple picture with the use of TDDFT [73]. 

To calculate the helium yields we invoke a geometrical picture of ionization. We divide the three-dimensional space, IR , into a (large) box, A, containing the helium atom, and its complement, B = IR A. Normalization of the (two-body) wave function of the helium atom, f (ri, r2,t), then implies 

To this point of the derivation we have utilized the many-body wave-function to define the ionization probabilities. Our goal is however to construct a density functional. For that purpose, we introduce the pair-correlation function 

We recall that by virtue of the Runge-Gross theorem is a functional of the time-dependent density. Separating g into an exchange part (which is simply 1/2 for a two electron system) and a correlation part, 

In Fig. 4.8 we depict the probability for double ionization of helium calculated from (4.123) by neglecting the correlation part of g. It is clear that all functionals tested yield a significant improvement over the simple sequential model. Due to the incorrect asymptotic behavior of the ALDA potential, the ALDA overestimates ionization The outermost electron of helium is not sufficiently bound and ionizes too easily. 

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