What conditions are required for the linear combination of atomic orbitals?

The conditions required for linear combination of atomic orbitals include: same energy of combining orbitals, correct overlap between the atomic orbitals, and the identical symmetry of the molecular axis. 

1. It is crucial that the orbitals involved possess either identical or very close energy levels. For example, a 2s orbital can effectively combine with another 2s orbital, but attempting to combine it with a 1s orbital is impractical due to the substantial difference in energy levels. 
2. Successful formation of molecular orbitals also requires sufficient overlap between the atomic orbitals involved. The extent of overlap directly influences the nuclear density between the two atomic nuclei. Essentially, the more extensive the overlap, the greater the concentration of electron density between the nuclei. 
3. Lastly, the atomic orbitals being combined must exhibit matching symmetry around the molecular axis. For instance, the 3dz orbital of one atom can only combine with the 3dz orbital of another atom; orbitals with different axes of symmetry (such as the 3d x or 3d y orbitals) cannot effectively combine. The molecular axis of symmetry is typically defined along the z-axis.