  Line-graph (LG) lattices are known for having flat bands (FBs) from the
destructive interference of Bloch wavefunctions encoded in pure lattice
symmetry. Here, we develop a generic atomic/molecular orbital design principle
for FBs in non-LG lattices. Based on linear-combination-of-atomic-orbital
(LCAO) theory, we demonstrate that the underlying wavefunction symmetry of FBs
in a LG lattice can be transformed into the atomic/molecular orbital symmetry
in a non-LG lattice. We illustrate such orbital-designed topological FBs in
three 2D non-LG, square, trigonal, and hexagonal lattices, where the designed
orbitals faithfully reproduce the corresponding lattice symmetries of
checkerboard, Kagome, and diatomic-Kagome lattices, respectively.
Interestingly, systematic design of FBs with a high Chern number is also
achieved based on the same principle. Fundamentally our theory enriches the FB
physics; practically it significantly expands the scope of FB materials, since
most materials have multiple atomic/molecular orbitals at each lattice site,
rather than a single s orbital mandated in graph theory and generic lattice
models.
