A Quantum Mechanical Model for the Vibration and Rotation of Molecules Harmonic Oscillator Rigid Rotor
Degrees of Freedom Translation: quantum mechanical model is particle in box or free particle. A molecule has 3 translational degrees of freedom. Vibration: quantum mechanical model is harmonic oscillator. A molecule has 3n 6 (3n 5 for linear) vibrational modes. Rotation: quantum mechanical model is rigid id rotor. A molecule has 3 ( for linear) principal axes of rotation.
Degrees of Freedom vibration, rotation, and translation Ignoring coupling: basic motions are independent of each other Hˆ = Hˆ + Hˆ + Hˆ trans vib rot ψ = ψ ψ ψ trans vib rot E = E + E + E trans vib rot
Classical Vibration Classical harmonic oscillator Center of mass and reduced mass defined by x CM mx 1 1 + mx mm 1 =, μ = m + m m + m 1 1
QM Harmonic Oscillator Harmonic potential not bad approximation for low energies
QM Harmonic Oscillator Schrödinger equation ( ) d ψ n x kx + ψ = μ dx Solutions: ψ ( x ) E ψ ( x ) n n n 14 1 α 1 α x n( x) = Hn( α x) e, n = 0,1,,3... n n! π ( ) 1 where α = kμ/ and Hn α x is the Hermite polynomial 1 E n = hν n+ 1 k where ν = π μ
QM Harmonic Oscillator f i f i f Wave functions are even functions for n = 0,, 4 ; odd functions for n = 1, 3, 5
QM Harmonic Oscillator Oscillations similar to particle in a box Equally spaced energy levels Δ E = hν Zero point energy E 0 = 1 hν Finite valued wave function in forbidden regions
Example The infrared absorption spectrum of 1 H 35 Cl has its strongest band at 8.65 10 13 Hz. (1) calculate the force constant () calculate the zero-point vibrational energy (3) what happens to the force constant if we deuterate the molecule? (4) What happens to the strongest absorption band in IR spectrum if we deuterate the molecule?
Vibrational Modes Vibrational modes of CO and H O
Degrees of Freedom vibration, rotation, and translation Ignoring coupling: basic motions are independent of each other Hˆ = Hˆ + Hˆ + Hˆ trans vib rot ψ = ψ ψ ψ trans vib rot E = E + E + E trans vib rot
Rigid Rotor Rigid rotor model: A system of two atoms (m 1 and m ) and a fixed bond length r 0 rotating around their center of mass is reduced d to a single reduced d mass rotating on the surface of a sphere of radius r 0 μ = mm 1 m + m 1 ( H p ) x py pz r= r0 V( x, y, z ) = 0 ˆ 1 = ˆ + ˆ + ˆ = + + μ μ x y z y r = r 0
Rigid Rotor Spherical coordinates r = x + y + z x = rsinθcos φ y = rsinθ sinφ z = rcosθ SPHERICAL COORDINATES Hˆ 1 1 = sinθ r μr + + sinθ θ θ sin θ φ r r
Rigid Rotor Schrödinger equation in spherical coordinates 1 Y ( θφ, ) 1 Y ( θφ, ) sin θ + EY, = μr0 sinθ θ θ sin θ φ ( θ φ )
Rigid Rotor ( θ ) ( Φ ( φ )) 1 d dθ 1 d sinθ sinθ + βsin ( θ) = Θ( θ) dθ dθ Φ( φ) dφ ( Φ ( φ )) ( θ ) 1 d dθ sinθ sinθ + βsin ( θ) = ml Θ( θ) dθ dθ 1 Φ ( φ) d dφ = m l
Rigid Rotor Solution spherical harmonic function l + 1 ( l m l )!)! i Yl ( θ, φ) = Pl ( cosθ) e 4 π ( l+ m )! m m im φ l l l l m l P l is a Legendre polynomial ( 1 ) ( ) 0 ( ) ( 1 ) m = l, l 1, l,...,0,..., l, l 1, l l
Rigid Rotor Solution spherical harmonic function ml ( θφ ) = EY ( θφ ) ˆ ml HY,, l l E = l( l+ 1, ) for l = 0,1,,3... I I = μ r 0 is the moment of inertia
Rigid Rotor l determines frequency (energy) of rotation m l determines orientation of rotational axis Energy is quantized and is given by El = l ( l + 1 ), for l = 01 0,1,,3... I The degeneracy of each energy level is l + 1 m l = -l.+l
Rigid Rotor Angular momentum operators lˆ lˆ lˆ x y z l=r p i x y z x y z lˆ x = i y z = i sinφ cotθcosφ z θ y φ lˆ y = i z x = i cosφ cotθsinφ x z θ φ lˆz = i x y = i y x φ l ˆ = l ˆ + l ˆ + l ˆ x y z
Rigid Rotor Spherical harmonic functions are eigenfunctions i of ˆ l and lˆ z ˆ ml l, 1 l, ml ( θφ) = ( + ) ( θφ) ly l l Y ˆ ˆ l, H = 0 and E = l I ˆ ml ly,, ml ( θφ) = m Y ( θφ) z l l l lˆ, 0 and ˆ, ˆ z H l l = = 0 ˆ 0 and z 0
Rigid Rotor Spherical harmonic functions complex except for m l = 0 Chemistry conventions
Rigid Rotor Chemistry conventions l = 0,1,,3, 4 s, p, d, f, g orbital quantum number m l is the angular momentum quantum number s p, p, p x y z d, d, d, d, d z xz yz xy x y f 3, f 3, f 3, f, f, f, f x y z xz ( y ) yz ( x ) zx ( y ) g, g, g, g, g,,,, z ( x y ) g g g xy( x y ) g + 4 3 3 3 3 4 4 z z x z y z xy zx zy x y xyz
Rigid Rotor p functions
Rigid Rotor d functions
Example 0 At what values of θ does Y ( θφ, ) = π ( cos θ 1 ) have 16 maxima? 5 1
Example The rotational energy levels l of molecules l are studied d by microwave spectroscopy. If the frequency of the absorption peak that corresponds to transition of l = 0 1is 3.65 10 1 Hz for HCl molecule. () (1) What is the bond length? () If we deuterate H without affecting the bond length, what will happen to the position of the absorption peak?