\[ \frac{\partial^2 \hat{\psi}_p }{ \partial \hat{Z}^2}\]
\[ \frac{\partial^2 \hat{\psi}_p }{ \partial \hat{Z}^2}= \hat{R}_0^{-1} P_\psi \Bigg\{2 c_3 -8 c_4 \bar{R}^2 +6 c_{10} \bar{Z}-24 c_{11} \bar{R}^2 \bar{Z}+c_6 (-24 \bar{R}^4 +96 \bar{R}^2 \bar{Z}^2) +c_5 (-18 \bar{R}^2 +24 \bar{Z}^2-24 \bar{R}^2 \ln{(\bar{R} )})+ c_{12} (160 \bar{Z}^3-480 \bar{R}^2 \bar{Z} \ln{(\bar{R} )}) +c_7 (150 \bar{R}^4 -1680 \bar{R}^2 \bar{Z}^2+240 \bar{Z}^4+360 \bar{R}^4 \ln{(\bar{R} )}-1440 \bar{R}^2 \bar{Z}^2 \ln{(\bar{R} )})\Bigg\} \]