\[\frac{\partial^2 \hat{\psi}_p }{ \partial \hat{R} \partial\hat{Z}}\]
\[\frac{\partial^2 \hat{\psi}_p }{ \partial \hat{R} \partial\hat{Z}}=
\hat{R}_0^{-1} P_\psi \Bigg\{2 c_9 \bar{R} -16 c_4 \bar{R} \bar{Z}+c_{11}
(12 \bar{R}^3 -24 \bar{R} \bar{Z}^2)+c_6 (-96 \bar{R}^3 \bar{Z}+64 \bar{R} \bar{Z}^3)
+ c_{10} (-3 \bar{R} -6 \bar{R} \ln{(\bar{R} )})
+c_5 (-60 \bar{R} \bar{Z}-48 \bar{R} \bar{Z} \ln{(\bar{R} )})
+c_{12} (-120 \bar{R}^3 -240 \bar{R} \bar{Z}^2+
240 \bar{R}^3 \ln{(\bar{R} )}-480 \bar{R} \bar{Z}^2 \ln{(\bar{R} )})
+c_7(960 \bar{R}^3 \bar{Z}-1600 \bar{R} \bar{Z}^3+1440 \bar{R}^3 \bar{Z} \ln{(\bar{R}
)}-960 \bar{R} \bar{Z}^3 \ln{(\bar{R} )})\Bigg\} \]