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 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 
. To minimize, we then need to compute the two
partial derivatives 
and 
, which will be real because f is always real.
Frequently, however, we are given the function f not in terms of 
and 
, but in terms of z and 
. Noting that 
and
, we may use the chain rule to take the derivative
of 
with respect to while treating z as a constant,
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!
