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

Despite the fact that the orbitals 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 of a single complex variable . One way to think of this problem as minimizing a real function of two independent real variables, namely the real and imaginary parts of . To minimize, we then need to compute the two partial derivatives and , which will be real because is always real.

Frequently, however, we are given the function not in terms of and , but in terms of and . Noting that and , we may use the chain rule to take the derivative of with respect to while treating as a constant,

 (15)

Thus, the real and imaginary components of give us both derivatives and simultaneously. In particular, to minimize over all possible values of , we need just one equation!

Tomas Arias 2004-01-26