Highly accurate wavefunctions for two-electron systems using two parameteres

It is shown for two electron atoms that ground-state wavefunctions of the form \begin{equation} \Psi(\vec{r_{1}}, \vec{r_{2}})=\phi(\vec{r_{1}})\phi(\vec{r_{2}})(\cosh ar_{1}+\cosh ar_{2})(1+0.5 r_{12}e^{-b r_{12}}) \end{equation} where $\overrightarrow{r_1}$ and $\overrightarrow{r_2}$ are the coordinates of two electrons and $r_{12}=\mathrm{\mid }\overrightarrow{r_1}-\overrightarrow{r_2}\mathrm{\mid }$, can be made highly accurate by optimizing $a$, $b$ and $\varphi$. This is done by solving a variationally derived equation for $\varphi$ for a given $a$ and $b$ and finding $a$ and $b$ so that the expectation value of the Hamiltonian is minimum. For the set $\{a,b,\varphi \}$ the values for various quantities obtained from the above wavefunction are compared with those given by $204$-parameter wavefunction of Koga et al.[11] and are found to be matching quite accurately(within ppm) with them.