It could be that Hermione was investigating a mysterious magical artifact in the Forbidden Forest on her own. She gets caught in a magical trap that ties her up. Snape, who is also in the forest for some reason (maybe collecting rare ingredients for potions), comes across her. He begrudgingly helps her out. Along the way, they start to have an unexpected conversation about magic and knowledge, and Snape realizes that Hermione is not just an ordinary student but someone with great potential and depth, leading to a newfound, albeit grudging, respect for her.
One possible plot could be that Hermione was set up by some dark wizards who wanted to use her as bait to lure in Harry Potter or other members of the Order of the Phoenix. Snape, being a double agent, stumbles upon her while on one of his secret missions. He might initially be conflicted between his loyalty to the Death Eaters (on the surface) and his true allegiance. He decides to help Hermione, using his knowledge of dark magic to free her from the bindings and then they have to work together to escape the place before the real villains return.
Perhaps it's a part of a complex scheme within Hogwarts itself. Some students, jealous of Hermione's intelligence and influence, tie her up in an abandoned classroom. Snape discovers her while patrolling the corridors. He is not one to show much sympathy openly, but he realizes that this could cause chaos in the school if not resolved. He unties her, gives her a stern lecture about being more careful, and warns her to keep it quiet so as not to undermine his authority in the school.
He might initially show annoyance. After all, Hermione is often seen as a know - it - all in his eyes. But his sense of justice and his underlying good nature (despite his cold exterior) would kick in. He'd untie her in a rather brusque manner, but then start to question her about how she got into this mess. Depending on her answer, he might offer some cryptic advice on how to avoid similar situations in the future.
Snape might be suspicious at first. He might think it's some sort of trick or part of a bigger plot. However, as he observes Hermione more closely and realizes the genuineness of her situation, his attitude could gradually change to one of reluctant assistance. He might even offer some words of wisdom, albeit in his typical sarcastic way, to help Hermione regain her strength.
A common theme could be redemption. Snape, who has a complex and often - dark past, might find in Hermione a chance to redeem himself in some way. Maybe he saves her from a danger in the dungeon, which starts his journey towards a more positive path. Another theme is the exploration of hidden knowledge. The dungeon is full of secrets, and both Snape and Hermione are curious minds. They could uncover something that changes their understanding of Hogwarts or the wizarding world.
One possible reason could be the stress from her studies. Hermione is always under a great deal of pressure to excel academically. Maybe she reached a breaking point due to the overwhelming workload. Another reason might be some kind of conflict with her friends. For example, if she had a big fight with Harry or Ron over something serious, it could leave her feeling broken.
One common plot is that Snape discovers Hermione's depression due to her overwork and the pressure of being a Muggle - born in a sometimes hostile wizarding world. He then takes it upon himself to mentor her in a way, teaching her how to better manage her stress and deal with the discrimination she faces. Another plot could be that Hermione is depressed because of a personal loss, like the death of a friend, and Snape, who has his own experiences with loss, helps her through it in an unexpected way.
One place to look could be Fanfiction.net. It has a vast collection of Harry Potter fanfictions. Just use the search bar and type in 'Snape finds depressed Hermione' and you might find some relevant stories. Another option is Archive of Our Own (AO3). It's known for its large and diverse range of fan works, and it has a good tagging system that can help you narrow down the search to find the specific type of story you're interested in.
Well, it might result in a confrontation. Snape has his own secrets and motives, and Hermione is not one to back down easily. If she's in the dungeon without permission, Snape could be quite harsh with her. However, some fanfictions take this in a different direction. Maybe Hermione discovers something about Snape's past while in the dungeon, and this knowledge makes Snape more vulnerable in front of her. It could be a turning point where their relationship goes from antagonistic to something more complex and nuanced.
No. Fanfiction that implies such a sexual or highly charged situation between a teacher (Snape) and a student (Hermione) goes against the general moral and ethical guidelines for young readers. The Harry Potter series, which these characters are from, is aimed at a wide range of ages, and this type of fanfiction strays far from the family - friendly nature of the original work.
Well, perhaps Hermione starts debating with Snape about magical theories during class. She might bring up new research or different interpretations that go against what Snape has been teaching for years. Like, when Snape is talking about the properties of a certain magical plant, Hermione could say 'But Professor, I've read in an old wizarding journal that the plant has additional properties that we're not considering.' This would surely annoy Snape as he's not used to being challenged in such a way.
She might be initially defensive. Hermione is a proud and independent character. So, when Snape finds out about her arm, she could be worried that he'll use it against her in some way, like taking away house points or using it as an excuse to give her extra work.