Hylleraas- and Kinoshita-type wave functions with correct cusp conditions

The finite terms Hylleraas- and Kinoshita-type variational wave functions are considered for three-body systems. In the Coulombic case local properties of wave functions are restricted by the Kato's cusp conditions. It is showed that Kato's cusp conditions restrict the possible terms in variational calculations. Constraints for the linear expansion coefficients are also derived and a recursion-type solution is given. Two trial functions with correct cusp conditions are determined for the ground state of the He atom. Local and global properties of these wave function are studied through calculations of mean values, local energy, and quantities related to double photoionization.

Figure 1

The cusp ratio functions for the trial functions K10M14 and K20M26. The upper part shows the e-e and the lower part displays the e-n cusp ratio functions.

Figure 2

Errors in the local energy as a function of $\theta$ when $r_1=r_2=1$ a.u. Divergence at $\theta =0$ can be noticed for wave functions K10M14 and K20M26. For wave functions with correct cusp conditions there is no divergence at the two-particle coalescence point. The inset shows an enlarged region at around $\theta =0$.

Figure 3

Errors in the local energy away from the two-particle coalescence lines as a function of $\theta$, in the upper part $r_1=1$ a.u. and $r_2=2$ a.u. and in the lower part $r_1=0.5$ a.u. and $r_2=1$ a.u.