  We present a novel and simple algorithm in the variation after projection
(VAP) approach for the non-yrast nuclear states. It is for the first time that
the yrast state and non-yrast states can be varied on the same footing. The
orthogonality among the calculated states is automatically fulfilled by solving
the Hill-Wheeler equation. This avoids the complexity of the frequently used
Gram-Schmidt orthogonalization, as adopted by the excited VAMPIR method. Thanks
to the Cauchy's interlacing theorem in the matrix theory, the sum of the
calculated lowest projected energies with the same quantum numbers can be
safely minimized. Once such minimization is converged, all the calculated
energies and the corresponding states can be obtained, simultaneously. The
present VAP calculations are performed with time-odd Hartree-Fock Slater
determinants. It is shown that the calculated VAP energies (both yrast and
non-yrast) are very close to the corresponding ones from the full shell model
calculations. It looks the present algorithm is not limited to the VAP, but
should be universal, i.e., one can do the variation with different forms of the
many-body wavefunctions to calculate the excited states in different quantum
many-body systems.
