Therefore, the symmetric stretch contribution is (7) and the asymmetric stretch contribution is (8) Relationships between these symmetric and asymmetric tensor contributions can be further simplified by considering the intramolecular coupling as a relatively minor perturbation to a system of two uncoupled local-mode O-H stretching motions. We can then express the molecular hyperpolarizability b elements in equation 6 in terms of the surviving hyperpolarizability elements in the two O-H frames illustrated in Figure . Prior to "turning on" coupling, each of the individual O-H motions is linear (local C¥-symmetry). Hence, the only surviving elements are and , where i = 1 or 2 (Figure ). In this perturbation theoretical framework, the signs and magnitudes of the symmetric and asymmetric motions can be generated by projecting the local-mode O-H motions from the two separate O-H bond frames (a1, b1, c1) and (a2, b2, c2) ) to the water molecular frame (x, y, z) (Figure ). The first O-H frame is related to the molecular frame by the transformation matrix of (9) where = 52.25°, which is the angle between the c1 axis and z axis (Figure ), and half the water molecule's H-O-H angle. The second O-H frame is related to the molecular frame with an opposite direction of rotation by the transformation matrix of (2)