It seems to me that all remaining analytic derivatives except first three are right. All first three are wrong similar way.
Because g(4,0) = 3/8 * x^4 + 3/4 * x^2y^2 - 3 * x^2z^2 + 3/8 * y^4 - 3 * y^2z^2 + z^4 is symmetric upon x,y permutation, so dg(4,0)/dx + dg(4,0)/dy also should be but doesn't.
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t11=threealpha*x4py4 t12=zz*(24.d0+eightalphaz2) t13=6.d0*yy*(1.d0+fouralphaz2) t14=6.d0*xx*(alphay2+fouralphaz2-1.d0) t15=-0.25d0*(t11+t12-t13+t14) pgra(1)=x*t15 pgra(2)=y*t15
t11 - symmetric
t12 - symmetric
-t13+t14= - 6.d0*yy*(1.d0+fouralphaz2) + 6.d0*xx*(alphay2+fouralphaz2-1.d0) = 6.0*(xx*(alphay2+fouralphaz2-1.d0)-yy*(1.d0+fouralphaz2)) - not symmetric
same for dg(4,0)/dz.