In general, I like to specialize my capital to be the science city--since as far as I can tell only two factors affect a city's long-term potential science output:
1) Amount of food (every citizen in a city makes beakers, and libraries have the biggest impact on large cities)
2) Proximity to a mountain--if you're beside a mountain you can build an observatory (+50% sci) later in the game.
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3) Jungles in nearby hexes--your universities will cause jungles to increase their yield by two beakers.
In my opinion the potential size of the city is the strongest of these factors, and the simplest and least-context dependent to prioritize.
So, since the maritime city states buff your capital the strongest, and your capital is therefore your easiest city to grow to a large population, I believe your capital is generally best suited to be your science city. What does this mean in terms of improvements?
1) Farms rather than trading posts. After civil service and later fertilizer you will have a tremendous amount of food always coming into the capital to keep its population high.
2) Settling your scientists (as academy tile improvements) is of debatable value--generally speaking a great scientist will bulb a tech in one turn even late into the game, so I generally use my scientists to bulb techs. However, you could settle him into an academy if you're still pretty early in the game as the long term beakers may be worth it. This will be map- and situation-dependent.
3) Buildings in the capital: anything that produces food, anything that produces beakers (library, university, observatory, research lab, national college if possible), and later in the game you want your hospital and medical lab in the capital as well to keep the population growth going.
Hope this is at least a place to get started.