where is the 27-element hyperpolarizability tensor of the n th molecule in the system, i, j, and k range over the three-dimensional Cartesian coordinates (x, y, z) of the molecular frame, R n is an Euler rotation matrix projecting the SFG response from the n th molecular frame to the laboratory frame, and , , and are the Euler angles of the n th molecule projecting on to the laboratory coordinate (X, Y, Z). The molecular hyperpolarizability (b) arises from the Raman tensor ( ) and the transition dipole ( ) of the n th molecule according to