  We extend the orbital-dependent electron tunneling model implemented within
the three-dimensional (3D) Wentzel-Kramers-Brillouin (WKB) atom-superposition
approach for simulating scanning tunneling microscopy (STM) by including
arbitrary tip orientations. The orientation of the tip is characterized by a
local coordinate system centered on the tip apex atom obtained by a rotation
with respect to the sample coordinate system. The rotation is described by the
Euler angles. Applying our method, we highlight the role of the real-space
shape of the electron orbitals involved in the tunneling, and analyze the
convergence and the orbital contributions of the tunneling current above the
W(110) surface depending on the orientation of a model tungsten tip. We also
simulate STM images at constant-current condition, and find that their quality
depends very much on the tip orientation. Some orientations result in
protrusions on the images that do not occur above W atoms. The presence of such
apparent atom positions makes it difficult to identify the exact position of
surface atoms. It is suggested that this tip orientation effect should be
considered at the evaluation of experimental STM images on other surfaces as
well. The presented computationally efficient tunneling model could prove to be
useful for obtaining more information on the local tip geometry and orientation
by comparing STM experiments to a large number of simulations with
systematically varied tip orientations.
