Download A Quantum Mechanical Model for the Vibration and

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
ib ti
l modes.
d
• R
Rotation:
i
quantum mechanical
h i l model
d l iis rigid
i id rotor.
A molecule has 3 (2 for linear) principal axes of
rotation.
rotation
Degrees of Freedom
• vibration, rotation, and translation
• Ignoring coupling: basic motions are independent
of each other
•
Hˆ = Hˆ trans
+ Hˆ vibib + Hˆ rott
t
ψ = ψ transψ vibψ rot
E = Etrans + Evib + Erot
Classical Vibration
• Classical harmonic oscillator
Center of mass and reduced mass defined by
xCM
m1 x1 + m2 x2
m1m2
=
, μ=
m1 + m2
m1 + m2
QM Harmonic Oscillator
• Harmonic potential not bad approximation for low
energies
QM Harmonic Oscillator
• Schrödinger equation
2
= 2 d ψ n ( x ) kx 2
−
+
ψ n ( x ) = Enψ n ( x )
2
2μ
dx
2
Solutions:
ψ n ( x) =
⎛α ⎞
−α x 2 2
12
, n = 0,1, 2,3...
⎜ ⎟ H n (α x ) e
n
2 n! ⎝ π ⎠
1
14
where α = k μ / = 2 and H n (α 1 2 x ) is the Hermite polynomial
1⎞
⎛
En = hν ⎜ n + ⎟
2⎠
⎝
where ν =
1
2π
k
μ
QM Harmonic Oscillator
• Wave functions
f
i
are even functions
f
i
ffor n = 0, 2, 4…;
odd functions for n = 1, 3, 5…
QM Harmonic Oscillator
• Oscillations similar to particle in a box
• Equally spaced energy
levels ΔE = hν
• Zero point energy
E0 =
1
hν
2
• Finite valued wave function
in forbidden regions
Example
The infrared absorption spectrum of 1H35Cl has its
strongest band at 8.65×1013 Hz.
(1) calculate the force constant
(2) 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 CO2 and H2O
Degrees of Freedom
• vibration, rotation, and translation
• Ignoring coupling: basic motions are independent
of each other
•
Hˆ = Hˆ trans
+ Hˆ vibib + Hˆ rott
t
ψ = ψ transψ vibψ rot
E = Etrans + Evib + Erot
Rigid Rotor
• Rigid rotor model: A system of two atoms (m1 and
m2) and a fixed bond length r0 rotating around their
center
t off mass is
i reduced
d
d tto a single
i l reduced
d
d mass
rotating on the surface of a sphere of radius r0
m1m2
μ=
m1 + m2
•
V ( x, y, z ) = 0
2
2
2
2
⎛
⎞
1
=
∂
∂
∂
2
2
2
ˆ
H=
pˆ x + pˆ y + pˆ z ) = −
+ 2+ 2⎟
(
⎜
2
=
r
r
0
2μ
2μ ⎝ ∂x ∂y ∂z ⎠r = r
0
Rigid Rotor
• Spherical coordinates
r = x2 + y2 + z2
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
SPHERICAL COORDINATES
2
=
Hˆ = −
2μ r 2
⎡ 1 ∂ ⎛
∂ ⎞
1 ∂2
∂ ⎛ 2 ∂ ⎞⎤
sin
+
+
θ
⎟
⎜r
⎟
⎢ sin θ ∂θ ⎜
2
2
∂θ ⎠ sin θ ∂φ
∂r ⎝ ∂r ⎠ ⎥⎦
⎝
⎣
Rigid Rotor
• Schrödinger equation in spherical coordinates
=2
−
2 μ r02
2
⎡ 1 ∂ ⎛
∂Y (θ , φ ) ⎞
1 ∂ Y (θ , φ ) ⎤
⎢
⎥ = EY (θ , φ )
⎜ sin θ
⎟+ 2
2
∂θ ⎠ sin θ
∂φ
⎣ sin θ ∂θ ⎝
⎦
Rigid Rotor
2
d
Φ (φ ) )
d
θ
Θ
⎡
⎤
(
(
)
1
d
1
2
sin θ
⎢sin θ
⎥ + β sin (θ ) = −
dθ ⎣
dθ ⎦
dφ 2
Θ (θ )
Φ (φ )
d Θ (θ ) ⎤
1
d ⎡
2
2
sin θ
sin
sin
m
θ
+
β
θ
=
(
)
⎢
⎥
l
dθ ⎣
dθ ⎦
Θ (θ )
1
Φ (φ )
d 2 ( Φ (φ ) )
dφ 2
= −ml2
Rigid Rotor
• Solution − spherical harmonic function
2l + 1 ( l − ml ) ! ml
i lφ
Yl (θ , φ ) =
⋅
⋅ Pl ( cos θ ) eim
4π (l + ml )!
ml
Pl ml is a Legendre polynomial
ml = −l , − ( l − 1) , − ( l − 2 ) ,..., 00,..., ( l − 2 ) , ( l − 1) , l
Rigid Rotor
• Solution − spherical harmonic function
ˆ ml (θ , φ ) = EY ml (θ , φ )
HY
l
l
=2
E = l ( l + 1) , for l = 0,1, 2,3...
2I
I = μ r20 is the moment of inertia
Rigid Rotor
• l determines frequency (energy) of rotation
• ml determines orientation of rotational axis
• Energy is quantized and is given by
=2
El = l ( l + 1) , for l = 00,1,
1 2,3...
23
2I
• The degeneracy of each energy level is 2l + 1
ml = -l….+l
Rigid Rotor
• Angular momentum operators
lˆx
l = r × p ⇒ − i= x
lˆy
y
lˆz
z
∂
∂x
∂
∂y
∂
∂z
⎧ˆ
⎛ ∂
⎛
∂ ⎞
∂
∂ ⎞
=
=
φ
θ
φ
l
i
y
z
i
=
−
−
=
−
−
−
sin
cot
cos
⎪x
⎜ ∂z
⎟
⎜
⎟
y
θ
φ
∂
∂
∂
⎝
⎠
⎝
⎠
⎪
⎛
∂⎞
∂
∂ ⎞
⎪⎪ ˆ
⎛ ∂
⇒ ⎨l y = −i= ⎜ z − x ⎟ = −i= ⎜ cos φ
− cot θ sin φ ⎟
∂z ⎠
∂θ
∂φ ⎠
⎝ ∂x
⎝
⎪
⎪
⎛ ∂
⎛ ∂ ⎞
∂ ⎞
⎪lˆz = −i= ⎜ x − y ⎟ = −i= ⎜ ⎟
∂x ⎠
⎪⎩
⎝ ∂y
⎝ ∂φ ⎠
lˆ 2 = lˆx2 + lˆy2 + lˆz2
Rigid Rotor
S h i l harmonic
h
i ffunctions
i
i
f
i
• Spherical
are eigenfunctions
of lˆ 2 and lˆz
lˆ 2Yl ml (θ , φ ) = = 2l ( l + 1) Yl ml (θ , φ )
2
l
2 ˆ
ˆ
⎡
⎤
⇒ ⎣l , H ⎦ = 0 and E =
2I
lˆzYl ml (θ , φ ) = ml =Yl ml (θ , φ )
⇒ ⎡⎣lˆz , Hˆ ⎤⎦ = 0 and ⎡⎣lˆz , lˆ 2 ⎤⎦ = 0
Rigid Rotor
• S
Spherical
h i l harmonic
h
i ffunctions
i
− complex
l except ffor
ml = 0
• Chemistry conventions
Rigid Rotor
• Chemistry conventions
l = 0,1, 2,3, 4 → s, p, d, f, g orbital quantum number
ml is the angular momentum quantum number
s
px , p y , pz
d z 2 , d xz , d yz , d xy , d x2 − y 2
f x3 , f y3 , f z3 , f x z 2 − y 2 , f y z 2 − x2 , f z x2 − y 2 , f xyz
(
)
g z 4 , g z3 x , g z3 y , g z 2 xy , g z 2
(
(x
)
2
− y2
)
(
)
, g zx3 , g zy3 , g xy x2 − y 2 , g x4 + y 4
(
)
Rigid Rotor
• p functions
Rigid Rotor
• d functions
Example
At what
hat values
al es of θ does
maxima?
1
2
⎛ 5 ⎞
Y20 (θ , φ ) = ⎜ π ⎟ ( 2 cos2 θ − 1)
⎝ 16 ⎠
ha e
have
Example
The rotational
Th
i
l energy levels
l l off molecules
l
l are studied
di d b
by
microwave spectroscopy. If the frequency of the absorption
12 Hz
peak that corresponds
p
p
to transition of l = 0→1 is 3
3.65×10
5
for HCl molecule.
((1)) What
h is
i the
h bond
b d llength?
h
(2) If we deuterate H without affecting the bond length, what
will happen to the position of the absorption peak?