And also,I would like to add, the expression "αυτήν τη υπόσχεση" is used more by people who use the scholar type of Greek (it's called "καθαρεύουσα",you can search it on google) than from people who speak the folk Greek.

Hello,Greekaz! Yes,you noted a detail! The normal is "αυτή την υπόσχεση".In ancient Greek, we added -ν in the end of the articles in the accusative case but in modern Greek,we put out -ν and we kept it for all masculines and only for fenimins in front of the letters κ,π,τ,ξ,ζ,ψ,μπ,τζ,τσ,γκ. But with pronouns,we contnued to put out -ν both in masculin and feminin in the accusative case. So,we say "αυτή την υπόσχεση" και "αυτό τον άνθρωπο".It's only for articles. But,there are people who conserve the ending -ν in αυτόν,αυτήν which was the ending of the accusative case of pronouns in ancient Greek because this defines better the noun that follows than without -ν.,the feminin and more the masculin noun. This is why you hear also "αυτήν την υπόσχεση". In modern Greek,we want to conserve somethings from the past,from ancient Greek language, which we thought that they were more useful than in modern Greek language. I think this is the right explanation for the subject.