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For each country, there is one, unique political center. So we use the when the context is a single country, because it's unique in that context. So London is a capital city, but it's the capital city of England.
While it could technically be used that way, the first sense is so common and idiomatic that you would likely be misunderstood if you tried to say that.
@JohnLawler I think the preposition "by" suggests multiplication rather than repetition. You wouldn't say "He went to the store by many times". As well, the context of the thing being described would make it clear whether repetition or multiplication makes sense.
