As I'm sure many people who MOC do, I looked to random image searches for a little bit of inspiration. Oddly enough, I ended up finding something close to what I was shooting for with this project from a post about spheres. This helped give me some insight in to how to get at least a portion of the effect I was looking for.
I started construction of the dome by figuring out the basic footprint I was shooting for, which ended up being 8x8 studs. From there, I started building a layer of staggered plates up the top, ending up with something resembling a pyramid. Then it was simply a matter of using a combination of 1x4 and 1x2 plates to create kind of a terrace effect from the bottom to the top. This left me with a roundish surface when I was done.
I decided to use a SNOT approach to help cascade off the sides, since the dome of the Capitol is more elongated than spherical. I used a pair of 1x1 bricks with a stud on the side on each side of the building, which gives the plate two firm connection points. You could use something else, but I had a ton of these on hand already from another stage of the building process.
Once the snot plates were attached, I used a 1x8 modified plate with door rail to help smooth out the edges.From there, I was able to just do a solid wall on each side with some windows and other design features to add some texture. I think the finished product ended up looking pretty good for my first try at a domed structure.
Check out previous installments of this series: Part 1 - Part 2 - Part 3