  The envelope theory, also known as the auxiliary field method, is a simple
technique to compute approximate solutions of Hamiltonians for $N$ identical
particles in $D$ dimensions. The quality of the approximate eigenvalues can be
improved by adding a free parameter in the characteristic global quantum number
of the solutions. A method is proposed to determine the value of this parameter
by comparing the eigenvalues computed with the envelope theory to the
corresponding ones computed with a $N$-body generalization of the dominantly
orbital state method. The accuracy of the procedure is tested with several
systems.
