In Au (111) single crystalline as shown in attached file, is there a difference the way electrons are excited when they are excited from perpendicular and diagonal orientation as shown in the figure (attached file). Any help would be much appreciated.
Yes, the direction matters.
Take a look at this electronic band structure diagram for gold.
If there was only one spatial dimension (like a wire), the diagram would just have energy vs momentum. In higher dimensions, instead the momentum along certain directions is used to try to "flatten" the band structure onto a diagram we can easily visualize. Gold has a fcc (face centered cubic) crystal structure (which turns out to have a bcc lattice in momentum space), so the letters used (by convention) are $\Gamma$ is the center, and $X,W,K,L,U$ are all symmetry points on the crystal structure (well technically, the first Brillouin Zone in momentum space).
This has a nice description as well as images relating the "real space lattice" and the "momentum space lattice". britneyspears.ac/physics/crystals/wcrystals.htm
$K$ is the line you drew in the direction connecting nearest neighbors.
The other direction you drew I believe doesn't correspond to a nice point on the cube. It would be a direction somewhere between $L$ and $U$ (which unfortunately isn't included in the structure diagram I listed, but I haven't been able to find a diagram that includes those directions as well).