neglect

Larry neglected his children because he was an alcoholic.
The soldier neglected his duties and was discharged from the military.
Her husband was neglecting her (i.e. not spending time with her, not paying attention to her, ignoring her) and for this reason she decided to leave.

Oh, man. You couldn't have chosen a more difficult word to translate! :) Discouraging aside, let's see what I can do to help.

Ларри не уделял своим детям никакого внимания, потому что он был алкоголиком. A bit better: Ларри был алкоголиком и не уделял своим детям никакого внимания.

Солдат не выполнил свои обязанности и был уволен из вооруженных сил. One thing to note, in Russia soldiers are not "hired" they join out of the obligation for a mandatory period of time, it's one year now, hence срочник is the word to refer to such soldiers. Anyone who decides to sign up a contract for an official job in the military is higher in rank than a simple soldier. So I guess, the soldier in your variant will be court-martialed or disciplined at least. Now, the tricky part, Russian солдат is not always the same is English soldier. Солдат in Russia is always about rank, soldier in English can just mean that someone is a part of military. So if you want to convey this meaning, use военный

Военный не выполнил свои обязанности и был уволен из вооруженных сил.

A bit of a side note: if someone neglects their duties on a regular basis or for some time, there is a good set phrase to use: уклонятся от выполнения обязанностей (or обязательств). I suppose you can use perfective уклонится in your case, but that's not that common.

Её муж не уделял ей должного внимания, и поэтому она решила от него уйти. Note that I used никакого внимания in the first case, but went for должного внимания in the unhappy wife sentence. There is not much difference there, you can use either of the two for both sentences. It's mostly just style, as an alcoholic is really supposed to completly ignore his children. A bad husband is just like he only cares about himself :)

There is a phrase "проявлять преступную халатность",
I think it's very close to "neglect one's duties"

Наиболее близкое по значению слово - "пренебрегать". Оно подходит ко всем примерам

One can say also игнорировать (кого/что).

There is also an old-fashioned word манкировать (кем/чем) (from french manquer). Манкировать обязанностями.