Here one works in a harmonic oscillator basis, with each major oscillator shell assigned a principal quantum number \( N=0,1,2,3,... \). The \( N\hbar\Omega \) or \( N_{max} \) truncation: Any configuration is given an noninteracting energy, which is the sum of the single-particle harmonic oscillator energies. (Thus this ignores spin-orbit splitting.)

Excited state are labeled relative to the lowest configuration by the number of harmonic oscillator quanta.

This truncation is useful because if one includes all configuration up to some \( N_{max} \), and has a translationally invariant interaction, then the intrinsic motion and the center-of-mass motion factor. In other words, we can know exactly the center-of-mass wavefunction.

In almost all cases, the many-body Hamiltonian is rotationally invariant. This means it commutes with the operators \( \hat{J}^2, \hat{J}_z \) and so eigenstates will have good \( J,M \). Furthermore, the eigenenergies do not depend upon the orientation \( M \).

Therefore we can choose to construct a many-body basis which has fixed \( M \); this is called an \( M \)-scheme basis.

Alternately, one can construct a many-body basis which has fixed \( J \), or a \( J \)-scheme basis.