As a non-native English learner, it came across my mind that I don't know a way to put the verb that's after can in negative form.
For instance, I can say:
"I am capable of not doing this thing"
But I can't do the same if I wanted to use can instead of capable of so what am I missing?
"I can not do this thing" is ambiguous between the two senses (1) "am not able to do this thing" and (2) "am able to not do this thing". If you want sense (1), that's straightforward -- you can simply contract, since "I can't do this thing" has only sense (1). It's a problem to get the other sense, unambiguously. You might try a long pause between "can" and "not", or special stress, but I find that these attempts wind up still ambiguous.
So I don't think there is a way to do what you want, short of a complete rephrasing: "I can leave this undone," for instance.
It is possible, as some people have said, to use "can" and have the meaning be clear.
An example where context is enough:
I can go to university, or I can not go to university. Those are my options.
Here "not go to university" is clearly a unit since it's contrasted with "go to university". But that structure could be extrapolated to a case with two different verbs, albeit with a little more ambiguity:
I can slog through university, or I can not have the career I want. Those are my options.
An example where emphasis/intonation is (almost) enough:
I'm not worried about the bungee thing. If I get scared, I can always not jump.
Also, in spoken conversation, one almost always uses "can't" for "to not be able to", which means that when you hear "can" and a separate, emphasized "not", it probably means "to be able not to".
It's also worth noting that the spelling is technically enough, because "cannot" is the normal spelling for "to not be able to", whereas "can not" is the normal spelling for "to be able not to".
But this is such a common source of errors, even among native speakers, that I wouldn't rely on it.
Finally, if you really want to clear up ambiguity, a very helpful verb is "choose".
I can go to university, or I can choose not to go.
If I get scared, I can choose not to jump.
I can choose not to do this thing.
If it's not directly up to choice, you could say "avoid" plus the gerund:
I can avoid doing this thing.
By the way, in colloquial speech of the current generation (25 and under), here's a common pattern of sarcastic conversation. It relies on context and emphasis for "can not" to be understood.
— "Just to let you know, I'm going to sing the Batman theme a capella for the next hour."
— "Can you please not?"
i.e., "Please choose not to."Someone is smoking in the hall of an apartment building. Another resident walks by and is put off by the smell. She says without preamble:
— "Oh my God, could you not?"
If the intent of the original sentence is to convey the inability to do a thing, then it would be easier to say, "I am incapable of doing this thing," or "I cannot do this thing," or "I can't do this thing." As the sentence is written ("I am capable of not doing this thing"), you are conveying that you could resist temptation and avoid doing the thing, which is awkward at best.
One way of achieving your objective is by using a verb, or verb+adjective, that denotes "not" doing something.
So, if you want to say: I can "not speak", instead say "I can remain quiet"
Or, if you wanted to say: I am capable of "not getting angry", instead say "I can restrain myself"
I am writing this under the assumption that you insist on using the introductory phrase "I can .... " or "I am capable of"
If this is not necessary then there are much better options.
At least one of my linguistics professors taught that "cannot" and "can not" are not the same, with "cannot" meaning "not able to" and "can not" meaning "able not to" -- but this distinction is probably generally lost if it even ever existed.
Although a non-native speaker asked the question, I think enough confusion exists among native speakers about matters of usage like this to post answers here.
[CAN [NOT DO]]instead of[NOT [CAN DO]], you have to use very peculiar stress and intonation, which can't be captured in the orthography. So it's effectively not possible in print. The workaround is to use a different modal, usually.