A geometric setting for internal motions of the quantum three-body system

Quantum mechanics for internal motions of the three-body system is set up on the basis of the complex vector bundle theory. The three-body system is called a triatomic molecule in the Born–Oppenheimer approximation. The internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator. This bundle is equipped with a linear connection, which is a natural consequence of a geometric interpretation of the so-called Eckart condition. The coupling of the internal motion with the rotation is understood naturally in terms of this connection. The internal Hamiltonian operator is obtained which includes the internal motion–rotation coupling and a centrifugal potential. The complex vector bundle for the triatomic molecule proves to be a trivial bundle, though the geometric setting for the internal motion is independent of whether the bundle is trivial or not.