  On the occasion of its 25th anniversary, we revise Chen's derivative rule for
electron tunneling [C.J. Chen, Phys. Rev. B 42, 8841 (1990)] for the purpose of
computationally efficient simulations of scanning tunneling microscopy (STM)
based on first principles electronic structure data. The revised model allows
the weighting of tunneling matrix elements of different tip orbital characters
by an arbitrary energy independent choice or based on energy dependent
weighting coefficients obtained by an expansion of the tip single electron
wavefunctions/density of states projected onto the tip apex atom. Tip-orbital
interference in the STM junction is included in the model by construction and
can be analyzed quantitatively. As a further advantage, arbitrary tip
geometrical orientations are included in the revised model by rotating the
coordinate system of the tip apex using Euler angles and redefining the
weighting coefficients of the tunneling matrix elements. We demonstrate the
reliability of the model by applying it to two functionalized surfaces of
recent interest where quantum interference effects play an important role in
the STM imaging process: N-doped graphene and a magnetic Mn2H complex on the
Ag(111) surface. We find that the proposed tunneling model is 25 times faster
than the Bardeen method concerning computational time, while maintaining good
agreement. Our results show that the electronic structure of the tip has a
considerable effect on STM images, and the Tersoff-Hamann model does not always
provide sufficient results in view of quantum interference effects. For both
studied surfaces we highlight the importance of interference between s and pz
tip orbitals that can cause a significant contrast change in the STM images.
