Derivative of a real function of a complex variable and its conjugate

Despite the fact that the orbitals $\psi_i(\vec x)$ may be complex, the energy function (12) always turns out to be real. We can use this to take a very powerful short cut which is often used but infrequently explained.

For simplicity of notation, let us consider minimizing a real function $f(\ldots)$ of a single complex variable $z$. One way to think of this problem as minimizing a real function of two independent real variables, namely the real and imaginary parts of $z \equiv z_r+z_i \, i$. To minimize, we then need to compute the two partial derivatives $\left. \partial f(z_r,z_i)/\partial z_r \right\vert _{z_i}$ and $\left. \partial f(z_r,z_i)/\partial z_r \right\vert _{z_r}$, which will be real because $f$ is always real.

Frequently, however, we are given the function $f$ not in terms of $z_r$ and $z_i$, but in terms of $z$ and $z^*$. Noting that $z_r=(z+z^*)/2$ and $z_i=(z-z^*)/2i$, we may use the chain rule to take the derivative of $f(z,z^*)$ with respect to $z^*$ while treating $z$ as a constant,

$$\left. \frac{\partial f}{\partial z^*}\right\vert _{z} = \left. \frac{\partial z_r}{\partial z^*}\right\vert _{z} \left. \... ...z^*} \right\vert _{z} \left. \frac{\partial f}{\partial z_i} \right\vert _{z_r}$$ (15)

$$= \frac{1}{2} \left( \left. \frac{\partial f}{\partial z_r} \right\vert _{z_i} + i \left.\frac{\partial f}{\partial z_i} \right\vert _{z_r}\right).$$

Thus, the real and imaginary components of $\left. \partial f(z,z*)/\partial z^* \right\vert _{z}$ give us both derivatives $\partial f(z_r,z_i)/\partial z_r$ and $\partial f(z_r,z_i)/\partial z_r$ simultaneously. In particular, to minimize over all possible values of $z=z_r+i z_i$, we need just one equation!

$$0 = \left. \frac{\partial f}{\partial z^*}\right\vert _{z}.$$