Scaling laws for strongly anharmonic vibrational matrix elements

The Heisenberg correspondence between classical Fourier components and quan tal matrix elements is used to demonstrate remarkably accurate scaling laws , by which the matrix elements scale simply with the reduced energy ((E) ov er bar /B), where B is a characteristic energy parameter and (E) over bar i s the average energy of the relevant states. Applications to the Morse osci llator, a cylindrically symmetric double well (champagne bottle) potential and to the spherical pendulum are described. The relevance of the scaling r ules to modeling highly excited molecular vibrational states close to saddl e points on the potential surface is discussed. Significant differences bet ween Morse oscillator and vibron oscillator matrix elements are reported.