Running over a dog

I accidentally ran over a dog, and since then I have been filled with immense fear. I love dogs very much, and I feel that I have committed a terrible transgression. How can I seek atonement for this dreadful act?

Greetings,
When the Torah commands us not to deliberately kill a living creature without reasonable cause, this is because by doing that we would disregard the blessed Creator and His handiwork. However, when it is killed accidentally and without intent, there is no prohibition involved. The Creator, in His great wisdom, knows that the dog has fulfilled its purpose in the world, and there is no longer a need for it. Additionally, a dog does not possess a soul, and therefore, once it is trampled, there is no soul that laments its nonexistence. In contrast, when it comes to the killing of a human being, God forbid, the soul remains eternal.
Only in the case of unintentional killing of a human being, even though the Creator precisely times the event, nevertheless, the blessed Creator “engenders the guilt by the means of the guilty”, and therefore, atonement is necessary for this act that occurred unintentionally.

Noda BiYehuda (Kama, Yoreh De'ah, Ch. 83).