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Slide 1
Chapter 5
Molecular Vibrations andTime-Independent Perturbation Theory
Part A: The Harmonic Oscillator and Vibrations of Molecules
Part B: The Symmetry of Vibrations + Perturbation Theory + Statistical Thermodynamics
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Slide 2
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
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Slide 3
The Classical Harmonic Oscillator
Re
V(R
)
0
A BR
V(R) ½k(R-Re)2
Harmonic Oscillator Approximation
or V(x) ½kx2 x = R-Re
The Potential Energy of a Diatomic Molecule
k is the force constant
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Slide 4
m1 m2
x = R - Re
Hooke’s Law and Newton’s Equation
Force:21
2
dV df kx kx
dx dx
Newton’s Equation:2
2
d xf a
dt 1 2
1 2
m m
m m
where
Reduced Mass
Therefore:2
2
d xkx
dt
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Slide 5
Solution
2
2
d xkx
dt
2
2
d x kx
dt
Assume: ( ) s i n ( ) c o s ( )x t A t B t 2
2 22
sin ( ) cos( )d x
A t B td t
2 x
2 kx x
k
2
or1
2
k
or1
2
k
c
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Slide 6
Initial Conditions (like BC’s)
Let’s assume that the HOstarts out at rest stretchedout to x = x0.
( ) s i n ( ) c o s ( )x t A t B t
x(0) = x0
0
0t
dx
dt
cos( ) sin( )dx
A t B tdt
0
0t
dxA
dt
( ) c o s ( )x t B t
0( 0 )x x B and 0( ) c o s ( )x t x t
1
2 2
k
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Slide 7
Conservation of Energy
0 sin ( )dx
x td t
0( ) c o s ( )x t x tk
Potential Energy (V) Kinetic Energy (T) Total Energy (E)
21
2V kx
2 20
1cos ( )
2V kx t
21
2
dxT
dt
2 2 20
1sin ( )
2T x t
2 20
1sin ( )
2T kx t
E T V
20
1
2E kx
20
1
2V kx 0T 2
0
1
2E kx
0V 20
1
2E kx2
0
1
2T kx
t = 0, , 2, ... x = x0
t = /2, 3/2, ... x = 0
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Slide 8
Classical HO Properties
Energy: E = T + V = ½kx02 = Any Value i.e. no energy quantization
If x0 = 0, E = 0 i.e. no Zero Point Energy
Probability: E = ½kx02
x
P(x)
0 +x0-x0
Classical Turning Points
![Page 9: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/9.jpg)
Slide 9
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
![Page 10: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/10.jpg)
Slide 10
Math Preliminary: Taylor Series Solution of Differential Equations
Taylor (McLaurin) Series
( )
0
1( ) (0)
!n n
n
f x f xn
Example: ( ) xf x e
Any “well-behaved” function canbe expanded in a Taylor Series
2 3 41 1 11
2 6 24x x x x
0
1
!n
n
xn
![Page 11: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/11.jpg)
Slide 11
0 1
x ff f2 f3( )
( ) xf x e2
2 ( ) 12
xf x x
2 3
3 ( ) 12 6
x xf x x
x ff f3 f6( )
0 3
( ) xf x e2 3
3 ( ) 12 6
x xf x x
2 3 4 5 6
6 ( ) 12 6 24 120 720
x x x x xf x x
Example: ( ) xf x e 2 3 41 1 11
2 6 24x x x x
0
1
!n
n
xn
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Slide 12
A Differential Equation
( )( )
df xf x
dx
Solution: ( ) xf x A e
Example: ( ) xf x e
Taylor (McLaurin) Series
( )
0
1( ) (0)
!n n
n
f x f xn
Any “well-behaved” function can
be expanded in a Taylor Series
2 3 41 1 11
2 6 24x x x x
0
1
!n
n
xn
![Page 13: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/13.jpg)
Slide 13
Taylor Series Solution
( )( )
df xf x
dx
2 30 1 2 3
0
( ) nn
n
f x a a x a x a x a x
Our goal is to develop a “recursion formula” relatinga1 to a0, a2 to a1... (or, in general, an+1 to an)
SpecificRecursionFormulas:
21 2 3
( )2 3
d f xa a x a x
dx
2 21 2 3 0 1 2
( )2 3 ( )
d f xa a x a x a a x a x f x
d x
Coefficients of xn must be equal for all n
a1 = a0
2a2 = a1 a2 = a1/2 = a0/2
3a3 = a2 a3 = a2/3 = a0/6
1
1
mm
m
ma x
![Page 14: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/14.jpg)
Slide 14
a1 = a0
a2 = a0/2
a3 = a0/6
2 30 1 2 3( )f x a a x a x a x 2 3
0
1 11 ...
2 6a x x x
Power SeriesExpansion for ex
General Recursion Formula
( )( )
df xf x
dx
1
1 0
m nm n
m n
ma x a x
Coefficients of the two series may be equatedonly if the summation limits are the same.
Therefore, set m = n+1 n=m-1and m=1 corresponds to n = 0
10 0
( 1) n nn n
n n
n a x a x
1( 1 ) n nn a a or 1 ( 1)n
n
aa
n
![Page 15: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/15.jpg)
Slide 15
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
![Page 16: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/16.jpg)
Slide 16
The Quantum Mechanical Harmonic Oscillator
Schrödinger Equation
Particle vibrating against wall
m
x = R - Re
2 22
2
1
2 2
dkx E
m dx
Two particles vibrating against each other
2 22
2
1
2 2
dkx E
dx
m1 m2
x = R - Re
1 2
1 2
m m
m m
Boundary Condition: 0 as x
![Page 17: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/17.jpg)
Slide 17
Solution of the HO Schrödinger Equation
Rearrangement of the Equation2 2
22
1
2 2
dkx E
dx
2 22
2
10
2 2
dkx E
dx
22
2 2
2 10
2
dkx E
dx
22
2 2 2
20
d E kx
dx
Define
2
2 E
22
k
2 2
2
k
22 2
20
dx
dx
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Slide 18
2
2 E
22
k 2
22
22 2
20
dx
dx
Should we try a Taylor series solution?0
nn
n
a x
No!! It would be difficult to satisfy the Boundary Conditions.
Instead, it’s better to use:0
nn asymp
n
a x
( ) a s y m pH x
That is, we’ll assume that is the product of a Taylor Series, H(x),and an asymptotic solution, asympt, valid for large |x|
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Slide 19
The Asymptotic Solution
2
2 22
0d
xdx
For large |x|2
2 22
0asymptasympt
dx
dx
We can show that theasymptotic solution is
2 / 2asympt xdxe
dx
2 22
/ 2 2 2 / 22
asym pt x xde x e
dx
22
2 2 / 22
asympt xdx e
dx
2 22 2 / 2 2 2 / 2 0x xx e x e
2 / 2xa s y m p t e
Note that asympt satisfied the BC’s;
i.e. asympt 0 as x
Plug in
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Slide 20
A Recursion Relation for the General Solution
2
2 22
0d
xdx
2 / 2
0 0
n n xn asymp n
n n
a x a x e
Assume the general solution is of the form
2 / 2( ) xH x e
If one: (a) computes d2/dx2
(b) Plugs d2/dx2 and into the Schrödinger Equation
(c) Equates the coefficients of xn
Then the following recursion formula is obtained:
2
2 10, 1, 2, 3,
( 1)( 2)n n
na a n
n n
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Slide 21
2
2 10, 1, 2, 3,
( 1)( 2)n n
na a n
n n
Because an+2 (rather than an+1) is related to an, it is best to considerthe Taylor Series in the solution to be the sum of an even and odd series
2 2/ 2 2 4 3 5 / 20 2 4 1 3 5
0
n x xn
n
a x e a a x a x a x a x a x e
Even Series Odd Series
From the recursion formula: 0 2 4a a a Even Coefficients
1 3 5a a a Odd Coefficients
a0 and a1 are two arbitrary constants
So far: (a) The Boundary Conditions have not been satisfied
(b) There is no quantization of energy
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Slide 22
Satisfying the Boundary Conditions: Quantization of Energy
2
2 E
2
2E
So far, there are no restrictions on and, therefore, none on E
Let’s look at the solution so far:
Does this solution satisfy the BC’s: i.e. 0 as x ?
It can be shown that:
2 2/ 2 2 4 3 5 / 20 2 4 1 3 5
0
n x xn
n
a x e a a x a x a x a x a x e
Even Series Odd Series
2 / 2lim 0n xx x e unless n
Therefore, the required BC’s will be satisfied if, and only if, boththe even and odd Taylor series terminate at a finite power.
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Slide 23
2 2/ 2 2 4 3 5 / 20 2 4 1 3 5
0
n x xn
n
a x e a a x a x a x a x a x e
Even Series Odd Series
We cannot let either Taylor series reach x.
2
2 10, 1, 2, 3,
( 1)( 2)n n
na a n
n n
We can achieve this for the even or odd series by settinga0=0 (even series) or a1=0 (odd series).
We can’t set both a0=0 and a1=0, in which case (x) = 0 for all x.
So, how can we force the second series to terminate at a finitevalue of n ?
By requiring that an+2 = 0•an for some value of n.
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Slide 24
2
2 10, 1, 2, 3,
( 1)( 2)n n
na a n
n n
If an+2 = 0•an for some value of n, then the required BC willbe satisfied.
Therefore, it is necessary for some value of the integer, n, that:
2 10
( 1)( 2)
n
n n
2 1 0n
This criterion puts restrictions on the allowed values of and,therefore, the allowed values of the energy, E.
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Slide 25
Allowed Values of the Energy
22
k
2 2
2
2
2 E
2
2E
Earlier Definitions Restriction on
2 1 0n n = 0, 1, 2, 3,...
(2 1)n
n = 0, 1, 2, 3,...
Quantized Energy Levels
2
2E
2
(2 1)2
n
1
2E n
or 1
2E n h 2
2 1n
2 1
2
n
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Slide 26
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
![Page 27: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/27.jpg)
Slide 27
HO Energies and Wavefunctions
Harmonic Oscillator Energies
1 10, 1, 2 , 3,
2 2nE n n h n
Angular Frequency: k
Circular Frequency: 1
2
k
En
erg
y
1
2
3
2
5
2
7
2
Wavenumbers: 1
2
k
c
where 11
( ) ( / )cm
cm c cm s
![Page 28: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/28.jpg)
Slide 28E
ner
gy
1
2
3
2
5
2
7
2
Quantized Energies
Only certain energy levels are allowedand the separation between levels is:
E
The classical HO permits any value of E.
Zero Point Energy
The minimum allowed value for theenergy is:
0
1
2E
The classical HO can have E=0.
Note: All “bound” particles have a minimum ZPE. This is a consequence of the Uncertainty Principle.
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Slide 29
The Correspondence Principle
The results of Quantum Mechanics must not contradict those ofclassical mechanics when applied to macroscopic systems.
The fundamental vibrational frequency of the H2 molecule is 4200 cm-1. Calculate the energy level spacing and the ZPE, in J and in kJ/mol.
ħ = 1.05x10-34 J•sc = 3.00x108 m/s = 3.00x1010 cm/sNA = 6.02x1023 mol-1
2 2 c 1 0 12 3 .1 4 3 .0 0 1 0 / 4 2 0 0x c m s c m 1 4 17 . 9 2 1 0x s
Spacing: E 3 4 1 4 11 .0 5 1 0 7 .9 2 1 0x J s x s 2 08 . 3 1 0x J
2 3 1 2 06 .0 2 1 0 8 .3 1 0AN E x m o l x J
ZPE: 0
1
2E 34 14 11
1 .05 10 7 .92 102
x J s x s 2 04 . 1 5 1 0x J
45 . 0 0 1 0 / 5 0 /x J m o l k J m o l
2 3 1 2 0 40 6 . 0 2 1 0 4 . 1 5 1 0 2 . 5 1 0 / 2 5 /AN E x m o l x J x J m o l k J m o l
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Slide 30
2 4 1 4 12 3 .1 4 1 .0 0 1 0 6 .2 8 1 0x s x s
Spacing: E 3 4 4 1 3 01 .0 5 1 0 6 .2 8 1 0 6 .6 1 0x J s x s x J
ZPE: 0
1
2E 34 4 1 301
1 .0 5 1 0 6 .2 8 1 0 3 .3 1 02
x J s x s x J
ħ = 1.05x10-34 J•sc = 3.00x108 m/s = 3.00x1010 cm/sNA = 6.02x1023 mol-1
Macroscopic oscillators have much lower frequencies thanmolecular sized systems. Calculate the energy level spacing andthe ZPE, in J and in kJ/mol, for a macroscopic oscillator with afrequency of 10,000 cycles/second.
2 3 1 3 0 6 96 . 0 2 1 0 6 . 6 1 0 4 . 0 1 0 / 4 . 0 1 0 /AN E x m o l x J x J m o l x k J m o l
2 3 1 3 0 6 90 6 . 0 2 1 0 3 . 3 1 0 2 . 0 1 0 / 2 . 0 1 0 /AN E x m o l x J x J m o l x k J m o l
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Slide 31
Harmonic Oscillator Wavefunctions
Application of the recursion formula yields a different polynomial for each value of n.
These polynomials are called Hermitepolynomials, Hn
2 / 2( ) xn n nN H x e
2 / 2( ) zn n nN H z e
or
where z x
Some specific solutions
n = 02 2/ 2 / 2
0 0 0x zN e N e Even
n = 22 22 / 2 2 / 2
2 2 2( 4 2 ) ( 4 2 )x zN x e N z e Even
n = 12 2/ 2 / 2
1 1 1( 2 ) ( 2 )x zN x e N z e Odd
n = 32 23 / 2 3 / 2 3 / 2
3 3 3( 8 1 2 ) ( 8 1 2 )x zN x x e N z z e Odd
2 / 2
0
n xn
n
a x e
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Slide 32
0
1E
2
1
3E
2
2
5E
2
2
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Slide 33
E = ½x02
x
P(x)
0 +x0-x0
Classical Turning Points
Quantum Mechanical vs. Classical Probability
Classical
P(x) minimum at x=0
P(x) = 0 past x0
QM (n=0)
P(x) maximum at x=0
P(x) 0 past x0
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Slide 34
E = ½x02
x
P(x)
0 +x0-x0
Classical Turning Points
CorrespondencePrinciple
E = ½x02
x
P(x)
0 +x0-x0
Classical Turning Points
-6 -4 -2 0 2 4
vals.1
0.0
0.2
0.4
0.6
0.8
1.0
x 2 V zero
n=9
Note that as n increases, P(x)approaches the classical limit.
For macroscopic oscillatorsn > 106
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Slide 35
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
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Slide 36
Properties of the QM Harmonic Oscillator
2
0
1
2xe dx
2
0
1
2xxe dx
22
0
1
4xx e dx
2320
1
2xx e dx
2420
3
8xx e dx
Some Useful Integrals
Remember
If f(-x) = f(x)
0( ) 2 ( )f x d x f x d x
Even Integrand
If f(-x) = -f(x)
( ) 0f x dx
Odd Integrand
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Slide 37
Wavefunction Orthogonality
2 / 20 0
xN e - < x < 2 / 2
1 1 ( 2 ) xN x e - < x < 22 / 2
2 2 ( 4 2 ) xN x e - < x <
<0|1>
2 2/ 2 / 20 1 0 1 2
x xN e N x e d x
2
0 1 0 1 2xN N xe d x
Odd Integrand
0 1 0
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Slide 38
Wavefunction Orthogonality
2 / 20 0
xN e - < x < 2 / 2
1 1 ( 2 ) xN x e - < x < 22 / 2
2 2 ( 4 2 ) xN x e - < x <
<0|2>
2 2/ 2 2 / 20 2 0 2 4 2x xN e N x e d x
0 2 0 2
1 14 2 2 2
4 2N N
2
0
1
2xe dx
22
0
1
4xx e dx
2 220 2 0 2 4 2x xN N x e d x e d x
0 2 0
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Slide 39
Normalization
We performed many of the calculations below on 0 of theHO in Chap. 2. Therefore, the calculations below will beon 1. 2 / 2
1xA x e - < x <
22
/ 21 1 1 1xA xe dx
fromHW Solns.
1/ 4
2A
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Slide 40
Positional Averages
22
/ 21 1 1
xx x x A xe d x
fromHW Solns.
<x>
0x
22
2 2 2 / 21 1 1
xx x x A xe d x
fromHW Solns.
<x2>
2 3
2x
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Slide 41
Momentum Averages
2
2
/ 21/ 2
1 1 1ˆ
x
xd A xe
p p A xe dxi dx
fromHW Solns.
<p>
0p
fromHW Solns.
<p2>
2 23
2p
2
2
2 / 212 2 / 2 2
1 1 1 2
x
xd A xe
p p A xe dxdx
^
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Slide 42
Energy Averages
Kinetic Energy
221
2 2
pT p
21 3
2 2T
fromHW Solns.
3
4T
Potential Energy
2 21 1
2 2V kx k x
1 3
2 2V k
fromHW Solns.
3
4V
Total Energy
1 3
2 2E n
T V
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Slide 43
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
![Page 44: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/44.jpg)
Slide 44
The Vibrations of Diatomic Molecules
Re
V(R
)
0A B
R
The Potential Energy
Define: x = R - Req
Expand V(x) in a Taylor series about x = 0 (R = Req)
0 0
1( )
!
nn
nn x
d VV x x
n dx
2 3 4
0 1 2 3 42 3 40
0 0 0 0
1 1 1 1 1( )
0 ! 1! 2 ! 3! 4 !xx x x x
dV d V d V d VV x V x x x x x
dx dx dx dx
2 3 4
2 3 42 3 40
0 0 0 0
1 1 1( )
2 6 24xx x x x
dV d V d V d VV x V x x x x
dx dx dx dx
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Slide 45
Re
V(R
)
0
2 3 4
2 3 42 3 40
0 0 0 0
1 1 1( )
2 6 24xx x x x
dV d V d V d VV x V x x x x
dx dx dx dx
00
xV
By convention
0
0x
dV
dx
By definitionof Re as minimum
2
2
0x
d Vk
dx
3
3
0
1
6x
d V
dx
4
4
0
1
24x
d V
dx
2 3 41( )
2V x kx x x
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Slide 46
Re
V(R
)
0
2 3 41( )
2V x kx x x
21( )
2V x kx
The Harmonic Oscillator Approximation
2
2
0x
d Vk
dx
where
Ignore x3 and higher order terms in V(x)
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Slide 47
The Force Constant (k)
The Interpretation of k
Re
V(R
)
0
2
2
0x
d V d dV dk Slope
dx dx dx dx
i.e. k is the “curvature” of the plot, and represents the “rapidity” with which the slope turns from negative to positive.
Another way of saying this is the k represents the “steepness”of the potential function at x=0 (R=Re).
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Slide 48
ReV(R
)
0
Do
ReV(R
)
0
Do
Do is the Dissociation Energy of the molecule, andrepresents the chemical bond strength.
There is often a correlation between k and Do.
Small kSmall Do
Large kLarge Do
Correlation Between k and Bond Strength
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Slide 49
0 2 4 6 8 10 12
0
500
1000
1500
2000
2500
HIHBr
H2
O2
NO
CO
N2
Cl2
k [
N/m
]
Do [eV]
Note the approximately linear correlation betweenForce Constant (k) and Bond Strength (Do).
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Slide 50
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
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Slide 51
Vibrational Spectroscopy
Energy Levels and Transitions
1 1
2 2nE n n h
1
2 2
k
Selection Rule
n = 1 (+1 for absorption,-1 for emission)
IR Spectra: For a vibration to be IR active, the dipole moment must change during the course of the vibration.
Raman Spectra: For a vibration to be Raman active, the polarizability must change during the course of the vibration.
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Slide 52
Transition Frequency
n n+1
1 11
2 2E n h n h h
Wavenumbers: 1Ecm
hc c
Note:1
2
k
c
The Boltzmann Distribution
/nE k Tn nN g e
The strongest transition corresponds to: n=0 n=1
/nE k Te
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Slide 53
Calculation of the Force Constant
Units of k
2
2
0x
d Vk
dx
2
Jk
m
2
2
2
kg ms
m
2
kg
s
2kg m
sm
N
m
Calculation of k
1
2
k
c
22k c
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Slide 54
The IR spectrum of 79Br19F contains a single line at 380 cm-1
Calculate the Br-F force constant, in N/m. h=6.63x10-34 J•sc=3.00x108 m/sc=3.00x1010 cm/sk=1.38x10-23 J/K1 amu = 1.66x10-27 kg1 N = 1 kg•m/s2
Br F
Br F
m m
m m
79 19
79 19
amu amu
amu amu
1 5 . 3 2 a m u 2 7 2 61 . 6 6 1 0 / 2 . 5 4 1 0x k g a m u x k g
1
2
k
c
22k c 2
1 0 1 2 62 3 .1 4 3 1 0 / 3 8 0 2 .5 4 1 0x cm s cm x k g
21 3 0 /k k g s 21 3 0 / /k g m s m 1 3 0 /N m
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Slide 55
Calculate the intensity ratio, ,for the 79Br19F vibration at 25 oC.10
21
II
R
h=6.63x10-34 J•sc=3.00x108 m/sc=3.00x1010 cm/sk=1.38x10-23 J/K
= 380 cm-1~
1 2 1
0 1 0
I N
I N
1
0
/
/
E kT
E kT
e
e
1 0 /E E k Te /h c k Te
34 10 1
23
(6.63 10 ) 3.00 10 / 380
1.38 10 / 298
x J s x cm s cmhc
kT x J K K
1.841 2
0 1
0.16I
eI
= 1.84
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Slide 56
1 2
0 1
IR
I
The n=1 n=2 transition is called a hot band, because itsintensity increases at higher temperature
Dependence of hot band intensity on frequency and temperature
Molecule T R~
79Br19F 380 cm-1 25 oC 0.16
79Br19F 380 500 0.49
79Br19F 380 1000 0.65
H35Cl 2880 25 9x10-7
H35Cl 2880 500 0.005
H35Cl 2880 1000 0.04
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Slide 57
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
![Page 58: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/58.jpg)
Slide 58
Vibrational Anharmonicity
Re
V(R
)
0
2 3 41( )
2V x kx x x
Harmonic OscillatorApproximation: 21
( )2
V x kx
The effect of including vibrational anharmonicity in treating the vibrationsof diatomic molecules is to lower the energy levels and decrease thetransition frequencies between successive levels.
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Slide 59
Re
V(R
)
0
2 3 41( )
2V x kx x x
A treatment of the vibrations of diatomic molecules which includesvibrational anharmonicity [includes higher order terms in V(x)]leads to an improved expression for the energy:
21 1
2 2n
e
En n x
hc
is the harmonic frequency and xe is the anharmonicity constant.~
Measurement of the fundamental frequency (01) and firstovertone (02) [or the "hot band" (1 2)] permits determination of and xe.
~See HW Problem
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Slide 60
Energy Levels and Transition Frequencies in H2
E/h
c [c
m-1]
HarmonicOscillator
Actual
2,2000
6,6001
11,0002
15,4003
2,1700
6,3301
10,2502
13,9303
14 4 0 0~ c m
14 4 0 0~ c m
14 4 0 0~ c m
14 1 6 0~ c m
13 9 2 0~ c m
13 6 8 0~ c m
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Slide 61
Anharmonic Oscillator Wavefunctions
-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7
vals.1
0.0
0.2
0.4
0.6
0.8
1.0
xang V zero
-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7
vals.1
0.0
0.2
0.4
0.6
0.8
1.0
xang 2 V zero
2n = 0
-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7
vals.1
-1.0
-0.5
0.0
0.5
1.0
xang V zero
2
-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7
vals.1
0.0
0.2
0.4
0.6
0.8
1.0
xang 2 V zero
n = 2
n = 10
-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7
vals.1
-1.0
-0.5
0.0
0.5
1.0
xang V zero
2
-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7
vals.1
0.0
0.2
0.4
0.6
0.8
1.0
xang 2 V zero
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Slide 62
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
![Page 63: Slide 1 Chapter 5 Molecular Vibrations and Time-Independent Perturbation Theory Part A: The Harmonic Oscillator and Vibrations of Molecules Part B: The](https://reader035.vdocuments.us/reader035/viewer/2022062307/5516c89a550346fc4e8b4672/html5/thumbnails/63.jpg)
Slide 63
The Two Dimensional Harmonic Oscillator
The Schrödinger Equation
2 21 1( , )
2 2x yV x y k x k y
2 2 22 2
2 2
1 1( , ) ( , )
2 2 2x yk x k y x y E x yx y
2 2 2 22 2
2 2
1 1( , ) ( , )
2 2 2 2x yk x k y x y E x yx y
The Solution: Separation of Variables
The Hamiltonian is of the form: ( , ) ( ) ( )x yH x y H x H y
Therefore, assume that is of the form: ( ) ( )x y( x , y ) x y
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Slide 64
2 2 2 22 2
2 2
1 1( ) ( ) ( ) ( )
2 2 2 2x y x y x yk x k y x y E x yx y
222 22 2
2 2
1 1( ) ( )
2 2 2 2yx
y x x x y y x y
ddk x k y E x y
dx dy
222 22 2
2 2
1 1 1 1
2 2 2 2yx
x x y yx y
ddk x k y E
dx dy
The above equation is of the form, f(x) + g(y) = constant.Therefore, one can set each function equal to a constant.
E = Ex + Ey
=
Ex
=
EyEx
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Slide 65
222
2
1 1
2 2x
x x xx
dk x E
dx
222
2
1 1
2 2y
y y yy
dk x E
dx
222
2
1
2 2x
x x x x
dk x E
dx
22
22
1
2 2y
y y y y
dk x E
dx
The above equations are just one dimensional HO Schrödinger equations.Therefore, one has:
2 / 2( ) x xxn x n x n xN H x e
1
2x x xE n
xx
k
xx
2 / 2( ) y yyn y n y n yN H y e
1
2y y yE n
yy
k
yy
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Slide 66
Energies and Degeneracy
1 1
2 2x y x x y yE E E n n
yxx y
kk
Depending upon the relative values of kx and ky, one may havedegenerate energy levels. For example, let’s assume that ky = 4•kx .
2y x
12 1
2x x y xE n n
32
2x y xE n n
E [
ħ
]
3/20 0
g = 1
5/21 0
g = 1
7/22 0 0 1
g = 2
9/23 0 1 1
g = 2
11/24 0 0 2 2 1
g = 3
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Slide 67
• The Classical Harmonic Oscillator
• Math Preliminary: Taylor Series Solution of Differential Eqns.
• The Vibrations of Diatomic Molecules
• The Quantum Mechanical Harmonic Oscillator
• Vibrational Spectroscopy
• Harmonic Oscillator Wavefunctions and Energies
• Properties of the Quantum Mechanical Harmonic Oscillator
• Vibrational Anharmonicity
• The Two Dimensional Harmonic Oscillator
• Vibrations of Polyatomic Molecules
Part A: The Harmonic Oscillator and Vibrations of Molecules
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Slide 68
Vibrations of Polyatomic Molecules
I’ll just outline the results.
If one has N atoms, then there are 3N coordinates. For the i’th. atom,the coordinates are xi, yi, zi. Sometimes this is shortened to: xi. = x,y or z.
The Hamiltonian for the N atoms (with 3N coordinates), assumingthat the potential energy, V, varies quadratically with the change incoordinate is:
2 2
,2
1( )( )
2 2eq eq
i j i i j ji i ji i
H F x x x xm x
2
,
,eq eqi j
i ji j x x
VF
x
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Slide 69
After some rather messy algebra, one can transform the Cartesiancoordinates to a set of “mass weighted” Normal Coordinates.
, ( )eqk i i k i i
i
Q m L x x
Each normal coordinate corresponds to the set of vectors showingthe relative displacements of the various atoms during a given vibration.
There are 3N-6 Normal Coordinates.
For example, for the 3 vibrations of water, the Normal Coordinatescorrespond to the 3 sets of vectors you’ve seen in other courses.
H H
O
H H
O
H H
O
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Slide 70
In the normal coordinate system, the Hamiltonian can be written as:
2 23 62 2
21
1
2 2
N
k kk k
H QQ
2 2 2 2 2 22 2 2 2 2 21 1 2 2 3 32 2 2
1 2 3
1 1 1
2 2 2 2 2 2H Q Q Q
Q Q Q
Note that the Hamiltonian is of the form:3 6
1 1 2 2 3 31
( ) ( ) ( ) ( )N
i ii
H H Q H Q H Q H Q
Therefore, one can assume:3 6
1 1 2 2 3 31
( ) ( ) ( ) ( ) ...N
i ii
Q Q Q Q
and simplify the Schrödinger equation by separation of variables.
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Slide 71
One gets 3N-6 equations of the form:22
2 21 12
1
1
2 2i
i i iQ EQ
The solutions to these 3N-6 equations are the familiar HarmonicOscillator Wavefunctions and Energies.
As noted above, the total wavefunction is given by:3 6
1
( )N
i ii
Q
The total vibrational energy is:
3 6
1 2 3 1 1 2 2 3 31
1 1 1 1( )
2 2 2 2
N
i ii
E n n n n n n n
3 6
1 2 31
1( )
2
N
i ii
E n n n n h
Or, equivalently:
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Slide 72
3 11 2 3
1
( ) 1
2
N
i ii
E n n nn
hc
In spectroscopy, we commonly refer to the “energy in wavenumbers”,which is actually E/hc:
The water molecule has three normal modes, with fundamental
frequencies: 1 = 3833 cm-1, 2 = 1649 cm-1, 3 = 3943 cm-1.~ ~ ~
What is the energy, in cm-1, of the (112) state (i.e. n1=1, n2=1, n3=2)?
1 2 3
(11 2 ) 1 1 11 1 2
2 2 2
E
hc
1 1 13 3 53 83 3 16 49 3 94 3
2 2 2cm cm cm
11 8 , 0 8 1 c m
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Slide 73
The water molecule has three normal modes, with fundamental
frequencies: 1 = 3833 cm-1, 2 = 1649 cm-1, 3 = 3943 cm-1.~ ~ ~
What is the energy difference, in cm-1, between (112) and (100),i.e. E(112)/hc – E(100)/hc ?
1(11 2 )18, 081
Ecm
hc
1 2 3
(100 ) 1 1 11 0 0
2 2 2
E
hc
1 1 13 1 13 83 3 16 49 3 94 3
2 2 2cm cm cm 18 , 5 4 6 c m
1 1 1(11 2 ) (1 0 0 )1 8 , 0 8 1 8 , 5 4 6 9 , 5 3 5
E Ecm cm cm
h c h c
This corresponds to the frequency of the combination band in whichthe molecule’s vibrations are excited from n2=0n2=1 andn3=0n3=2.