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,
Tomas Arias 2004-01-26