\[ \frac{\partial^2 \hat{\psi}_p }{ \partial \hat{R}^2}\]
\[ \frac{\partial^2 \hat{\psi}_p }{ \partial \hat{R}^2}=
\hat{R}_0^{-1} P_\psi \Bigg\{ 2 c_2 +(3 \hat{\bar{R}}^2 )/2+2 c_9 \bar{Z}+c_4 (12 \bar{R}^2 -8 \bar{Z}^2)+c_{11}
(36 \bar{R}^2 \bar{Z}-8 \bar{Z}^3)
+c_6 (30 \bar{R}^4 -144 \bar{R}^2 \bar{Z}^2+16 \bar{Z}^4)+c_3 (-3 -2 \ln{(\bar{R}
)})+
A ((3 )/2-(3 \bar{R}^2 )/2+ \ln{(\bar{R} )})
+c_{10} (-9 \bar{Z}-6
\bar{Z} \ln{(\bar{R} )})+c_5 (21 \bar{R}^2 -54
\bar{Z}^2+36 \bar{R}^2 \ln{(\bar{R} )}-24 \bar{Z}^2 \ln{(\bar{R} )})
+c_{12} (-120 \bar{R}^2 \bar{Z}-240 \bar{Z}^3+720 \bar{R}^2 \bar{Z} \ln{(\bar{R} )}
-160 \bar{Z}^3 \ln{(\bar{R} )})
+ c_7 (-165 \bar{R}^4 +2160 \bar{R}^2 \bar{Z}^2-640 \bar{Z}^4-450 \bar{R}^4 \ln{(\bar{R} )}+2160 \bar{R}^2 \bar{Z}^2
\ln{(\bar{R} )}-240 \bar{Z}^4 \ln{(\bar{R} )})\Bigg\}\]