An alternative way to calculate product operators from the
density matrix is to use spherical harmonics for the operator basis
instead of the cartesian basis matrices ^{n} basis matrices are needed for a system of *n* spins. For one
spin the basis matrices are the identity matrix

the Zeeman order population with the coherence order 0

the positive shift operator with the coherence order +1

and the negative shift operator with the coherence order -1

Products of operators like ^{+}I2^{+}^{+}I2^{-}^{+}I2z

The spherical tensor basis does not have a geometric interpretation like the vector model. They are based
on the spherical harmonics *Y _{l,m}* that are known from the quantum mechanical rotator and
the angular components of the hydrogen atom orbitals. Their usefulness comes from the fact that the
coherence order can be easily calculated as the sum of all coherence orders
of the basis operators in the product operator. Of the examples above the operator

All base matrices are real, the orientation of the magnetic moments must therefore be given by a complex
factor *A*⋅e^{iφ} with the magnitude *A* and the imaginary
phase *φ*.