  It is proven that the exact excited-state wave function and energy may be
obtained by minimizing the energy expectation value of trial wave functions
that are constrained only to have the correct nodes of the state of interest.
This excited-state nodal minimum principle has the advantage that it requires
neither minimization with the constraint of wave-function orthogonality to all
lower eigenstates nor the antisymmetry of the trial wave functions. It is also
found that the minimization over the entire space can be partitioned into
several interconnected minimizations within the individual nodal regions, and
the exact excited-state energy may be obtained by a minimization in just one or
several of these nodal regions. For the proofs of the theorem, it is observed
that the many-electron eigenfunction (excited state as well as ground state),
restricted to a nodal region, is equivalent to a ground-state wave function of
one electron in a higher-dimensional space; and, alternatively, an explicit
excited-state energy variational expression is utilized by generalizing the
Jacobi method of multiplicative variation. In corollaries, error functions are
constructed for cases for which the nodes are not necessarily exact. The exact
nodes minimize the energy error functions with respect to nodal variations.
Simple numerical illustrations of the error functions are presented.
