B. Hemisphere representation of hydrogen atoms: model H Ref. shows that Cartesian coordinates of the two hydrogen atoms (r H1 and r H2 ) of a water molecule r mol = [r O , r H1 , r H2 ] can be efficiently represented by a set of independent internal coordinates (x mol ), consisting of two bond lengths (d 1 , d 2 ), one bond angle (θ), and one (flip invariant) unit-quaternion (±q ∈ S 3 ⊂ R 4 ), as well as r O . For brevity, only the inverse of this transformation is discussed here. The configuration of a water molecule can be reconstructed from these coordinates using the following equation, where we adopt the approach detailed in Ref. to transform a unitquaternion into a rotation matrix R(q): 44,45