  We present a general class of operators resembling compact tetraquarks which
have a range of colour-flavour-spin structures, transform irreducibly under the
symmetries of the lattice and respect other relevant symmetries. These
constructions are demonstrated in lattice QCD calculations with light quarks
corresponding to $m_\pi =$ 391 MeV. Using the distillation framework,
correlation functions involving large bases of meson-meson and tetraquark
operators are computed in the isospin-1 hidden-charm and doubly-charmed
sectors, and finite-volume spectra are extracted with the variational method.
We find the spectra are insensitive to the addition of tetraquark operators to
the bases of meson-meson operators. For the first time, through using diverse
bases of meson-meson operators, the multiple energy levels associated with
meson-meson levels which would be degenerate in the non-interacting limit are
extracted reliably. The number of energy levels in each spectrum is found to be
equal to the number of expected non-interacting meson-meson levels in the
energy region considered and the majority of energies lie close to the
non-interacting levels. Therefore, there is no strong indication for any bound
state or narrow resonance in the channels we study.
