Derivative of coefficients in linear variational method
Solution 1:
you can see this derivation as a "standard" partial derivative of G with respect to the variable $c_i^*$ An important fact to note is that when you do this derivation you should consider $c_i^* $ and $c_i$ as independent parameters, leading to tour remark (absence of imaginary complex conjugate coefficients)
The key formula to use is then
$\frac{\partial c_j}{\partial c_i^*}=0$ and $\frac{\partial c_j^*}{\partial c_i^*}=\delta_{ij}$