Philosophers have claimed that: (a) Born-Oppenheimer approximation methods for solving molecular Schrödinger equations violate the Heisenberg uncertainty relations; therefore, (b) ‘quantum chemistry’ is not fully quantum; and (c) therefore chemistry does not reduce to physics. This paper analyses the reasoning behind Born-Oppenheimer methods and shows that they are internally consistent and fully quantum mechanical, contrary to (a)-(c). Our analysis addresses important issues of mathematical rigour, physical idealization, reduction, and classicality in the quantum theory of molecules, and we propose an agenda for the philosophy of quantum chemistry more grounded in scientific practice.