Molecular Vibrations and Hooke’s Law (14.13) Bonds between atoms form molecules and these bonds also cause the atoms to vibrate within the molecule These bonds behave as springs as long as the atoms are not too far apart from one another (or too close) Assuming Hooke’s Law for springs holds, a simple potential energy function can be used to describe molecular vibrations A force constant (k) dictate how stiff the spring is (or how strong the bond is) The frequency of oscillation is dependent on the strength of the bond and the mass of the particles involved in the bond Stronger bonds vibrate more rapidly, but heavier atoms slow the vibrations down Reduced mass (μ) is used to simplify the result

Harmonic Oscillators (HO) (14.13) Solving the SE for the Hooke’s Law potential gives some expected and unexpected results The potential resembles a well, so the wavefunction has many qualities of previously studied systems: oscillatory, node distribution, penetrates potential wall The energy is only linearly proportional to the quantum number (not squared as in PIB) Lowest quantum number is zero, not one The wavefunction is a product of functions that give oscillatory behavior Gaussian function (exponential) gives the wavefunction its “lumps” Hermite polynomials give the wavefunction its oscillatory behavior and is a function of the quantum number (which determines the number of nodes) Probability distribution function shows similarities and differences from previous models Most probable position in ground state is in middle of the well (not classical behavior) As quantum number increases, most probable positions shift to edges of the well (classical turning points)

HO Energies and Vibrational Spectroscopy (14.13, 18.3) Harmonic oscillator energy levels are evenly spaced due to linear dependence on quantum number (n) Harmonic oscillators always have a nonzero energy (zero point energy or ZPE) Infrared light is used to probe transitions between vibrational states in a molecule In the HO approximation, only transitions between adjacent vibrational levels is possible (allowed transitions) Energy absorbed is a measure of frequency of vibration (when n=0, this is called the fundamental transition) In reality, potential energy is not strictly parabolic (has anharmonicities) Transitions between nonadjacent energy levels are possible (called overtones) Energy levels are not evenly spaced, they start to bunch up near top of well

Molecular Potential Energy Functions

Harmonic Oscillator Wavefunctions

Harmonic Oscillator Probability Distribution Functions

Anharmonic Potential Energy Curve