  We compute the color-singlet and color-octet nonrelativistic QCD (NRQCD)
long-distance matrix elements for inclusive production of $P$-wave quarkonia in
the framework of potential NRQCD. In this way, the color-octet NRQCD
long-distance matrix element can be determined without relying on measured
cross section data, which has not been possible so far. We obtain inclusive
cross sections of $\chi_{cJ}$ and $\chi_{bJ}$ at the LHC, which are in good
agreement with data. In principle, the formalism developed in this Letter can
be applied to all inclusive production processes of heavy quarkonia.
