Assume g_{ij}=0 for i \neq j and verify:

R_{hiih}=\sqrt{g_{ii}}\sqrt{g_{hh}}[\frac{\partial}{\partial x^h}(\frac 1{\sqrt{g_{hh}}} \frac{\partial \sqrt g_{ii}}{\partial x^h})+\frac{\partial}{\partial x^i}(\frac 1{\sqrt{g_{ii}}} \frac{\partial \sqrt g_{hh}}{\partial x^i})+\sum^n_{m=1,m \neq i,m \neq h} \frac{\partial \sqrt{g_{ii}}}{\partial x^m} \frac{\partial \sqrt{g_{hh}}}{\partial x^m}] where  h \neq i

Now I am so close, as I get:

R_{hiih}=\sqrt{g_{ii}}\sqrt{g_{hh}}[..........+\sum^n_{m=1,m \neq i,m \neq h} \frac{\partial \sqrt{g_{ii}}}{\partial x^m} \frac{\partial \sqrt{g_{hh}}}{\partial x^m}g^{mm}] where  h \neq i

I think I am right so far, so how do I justify the next step?