I strongly suggest the book by David J Tannor entitled "Introduction to Quantum Mechanics: A Time-Dependent Perspective" (University Science Books, Saulito, 2007).
For my understanding of quantum mechanics and chemistry, these books helped a lot:
1) P. W. Atkins - Molecular Quantum Mechanics
2) D.A. McQuarrie, J.D. Simon - Physical Chemistry: A Molecular Approach
I think, the question is also if you want to know about "quantum mechanics and chemistry" or "applications of quantum mechanics on chemistry" which possibly would rather mean theoretical or computational chemistry. My answer tackles more the first point, some of the above (e.g. the book by Cramer) more the second.
I really like Merzbacher (see other questions in this topic) best to understand the basics of QM but think he is a little to complex for the questioner, asking for a more chemical basis. Leads to the next question
We had this discussion before about a questioner; this time I like to ask you: which book do you like best? This is of interest for advising others here.
By the way, Merzbacher is also freely down-loadable from the web
Harry, I guess you're right. I had Maths, Physics and Chemistry majors but obtained my PhD in Physics. Our prescribed Quantum textbook in Chemistry was "Wave Mechanics and Valency" by JW Linnett. I found it clear and concise, but somewhat dull, given where my real interests lay.
Hi, I am a student in physics. I guess that if you are familiar with quantum mechanics, you could really go on and choose any book on some specific topic in chemistry. If you are not familiar with quantum mechanics, I strongly recommend you to read Shankar 's book called Principle of Quantum Mechanics, which explained from every detail of assumption to linear algebra calculation. It might be tedious, if you are familiar with it. If you are not familiar with it, it will be a great book for someone who has no knowledge of Quantum mechanics at all.