Job search question regarding following up after just an application

I recently saw a job posted at a medium to large sized company which looked to be a near perfect fit for me, so I applied via their website. I now see it posted on LinkedIn as well, including an actual contact person, and I’m tempted to follow up with them directly, but I want to make sure I’m not doing anything weird or that might hurt my chances. I applied at the beginning of the month, so it really hasn’t been that long, and I assume I could still be in consideration, however considering that I applied 100% online I’m thinking there’s always the chance the automated system didn’t select me or I missed a keyword or two on my resume or something. Of course they also may have reviewed my application and decided I’m not the fit I think I might be, I’m not trying to overlook that fact or sound cocky or over-confident, but I really do think the description fits me almost perfectly and if I were them I’d certainly want to talk to me about it.

Anyway, my question is would it be appropriate for me to send a message (via LinkedIn InMail) to the HR contact who posted it to LinkedIn? I definitely wouldn’t want to do anything to hurt my chances or seem like I’m bugging them or whatever, especially if my application may already be on the path to getting an interview. I guess if they’ve already taken a pass on my application I have nothing to lose with a follow up, so my main concern is that if a follow up from them may already be on the way, I don’t want to screw that up by being too persistent. Should I follow up via LinkedIn, or just wait and see if the online application gets me anywhere? Could sending a quick note on LinkedIn regarding the application I made end up reasonably hurting me?

At this point, they could still be in “harvesting” mode and have not begun “reviewing” mode. So, a reasonable followup on LinkedIn is not going to hurt your chances. As you noted, as long as you’re not bugging them repeatedly you should be OK.

If you get a job there, do you expect to sit like a lump and do what you are told only, or do you expect to show some initiative and come up with some new ideas and be proactive? If the latter, which I trust is the right answer, why do you think a prospective employer would object to you being proactive in the job search.

Obviously sending an email every day is a bad idea, but one is not going to hurt. And of course use it to mention that you applied the usual way but then make your case about how you can help them.
I always hated wading through dozens of nearly equivalent resumes that we got from our online post. If someone contacted me directly, and was a good match, I’d be happy to just do it.
Your odds of getting somewhere using personal contacts is far, far better than being the 51st resume in the stack. And if this company hates initiative, do you really want to work there?

A follow up will never hurt your chances, as long as you are reasonable about it. A single professional follow-up (“I just wanted to make sure that my application was submitted correctly and is being processed”) is fine; avoid repeated follow-ups, whining, and accusations (“Why haven’t you responded yet?”).