chapter5–!vibrational!motion!scienide2.uwaterloo.ca/~nooijen/chem356/chem+356+pdf/ch_5.pdfn...

Fall 2014 Chem 356: Introductory Quantum Mechanics

Chapter 5 – Vibrational Motion 65

Chapter 5 – Vibrational Motion ................................................................................................................. 65

Potential Energy Surfaces, Rotations and Vibrations ............................................................................ 65

Harmonic Oscillator ............................................................................................................................... 67

General Solution for H.O.: Operator Technique .................................................................................... 68

Vibrational Selection Rules .................................................................................................................... 72

Poly-‐atomic Molecules .......................................................................................................................... 73

Beyond the harmonic oscillator approximation .................................................................................... 74

Chapter 5 – Vibrational Motion

Potential Energy Surfaces, Rotations and Vibrations

Suppose we assume the nuclei of a molecule are fixed, then we can solve the Schrodinger equation for the electrons

1 2 1 2ˆ ( , ,.... ) ( , ,... )N NH r r r E r r rψ ψ=

This is a complicated problem, that will be discussed later (Chapter 7 and beyond)

We would get the (ground state) energy at a particular nuclear configuration { R }

Hence, assume we can solve this

We can fit a curve through the points → ({ })V R

This ({ })V R is called the potential energy surface (PES) and is a crucial concept in chemistry, eg.



Minima on the PES we associate with different isomers, for example with the reactionA B C D+ → + . Saddle points on the PES we associate with transition states

Obtaining the minima (+ curvature) we can get thermochemical information. From TS we get into rates of reactions (Chem254, Chem 350) Ideally, once we have obtained PES, we should solve for the Quantum energies involving the nuclei. This is very complicated, and today can only be done for small molecules (up to 5 atoms). What can be done is solving for molecular rotations + vibrations in the harmonic oscillator + Rigid Rotor approximation

The crux is to make a quadratic approximation to the PES around each minimum ( = molecular species)

V (!R) ≈V (Re )+ 1

2kαβ (R − Re )α (R − Re )β

αβ∑



A molecule with N atoms has 3N degrees of freedom -‐ 3 overall translations -‐ 3 overall rotations or 2 rotations for linear molecules

(3 6)N − Vibrations , [3 5N − for linear molecule]

kαβ : α ,β = 1....3N

All of the information V (!Re ) ,

!Re and

kαβ (!Re ) is obtained from an electronic structure program (like

Gaussian, Gamess, Turbomole…) Once we have these, one can use ‘exact’ calculations to solve for Harmonic oscillator + Rigid Rotor energies. The harmonic (quadratic) potential is an approximation, but often works well, certainly for stiff molecules/ degrees of freedom. What we will do next is: discuss harmonic oscillator for diatomic molecules

-‐ Ground state of harmonic oscillator -‐ All excited states using operator technique -‐ Generalize to polyatomic molecules -‐ More accurate solution for diatomic (Beyond H.O.) -‐ Selection rules, vibrational spectroscopy

Harmonic Oscillator In the following chapter we will go on to discuss rotations

For a general polyatomic molecule we can define H.O. Hamiltonian as

H = !2

2Mα

Pα2 +

α∑ 1

2kab(R − Re )a (R − Re )b

a,b=1...3N∑

For a diatomic this can be reduced to

H = − 1

2!2

µd 2

dx2 +12

kx2

Here µ =

M1M2

M1 + M2

is the [kg] reduced mass

( )ex R R= − is deviation [m] from equilibrium

k is the force constant [Nm-‐1]



In McQuarrie this is derived using classical equations of motion. I will post on the Website a general derivation to get the H.O. form for a general polyatomic, but will not discuss in class. Here I will simply use the form for H.

The ground state solution has the form 2 /2xe α− . Let us determine the constant α , and the energy. Later

on I will discuss the full solution.

2 2/2 /2x xd e xe

dxα αα− −= −

2 2 2

2/2 /2 2 2 /2

2x x xd e e x e

dxα α αα α− − −= − +

− !

2

2µd 2

dx2 +12

kx2⎡

⎣⎢

⎤

⎦⎥e−αx2 /2 = !2

2µα − !

2

2µα 2x2 + 1

2kx2⎡

⎣⎢

⎤

⎦⎥e−αx2 /2 = E0e

−αx2 /2

E0 =

!2

2µα

12

k − 12!2α 2

µ= 0

α = 1!

µk

E0 =

!2

2µµk!

= 12!

kµ= 1

2!ω

ω = k

µ ;

α = µ!ω

General Solution for H.O.: Operator Technique (see appendix 5 in McQuarrie)

H = −!2

2md 2

dx2 +12

kx2

Solution 2 /2xe α− α = µω / ! = µk / !

Define

q x α=

2 /2xe α− →

2 /2qe−

q = 1

!µk

⎛⎝⎜

⎞⎠⎟

12

x x = 1

!µk

⎛⎝⎜

⎞⎠⎟

−12

q



∂∂x

= ∂q∂x

∂∂q

= 1!

µk⎛⎝⎜

⎞⎠⎟

12 ∂∂q

H = −!2

2µµk!

∂2

∂q2 +12

k !

µkq2

= ! k

µ− 1

2∂2

∂q2 +12

q2⎛⎝⎜

⎞⎠⎟

= !ω − 1

2d 2

dq2 +12

q2⎛⎝⎜

⎞⎠⎟

‘q ’ are called dimensionless coordinates (Check that q = 1!

µk x is indeed dimensionless!)

Commutation Relation:

, 1dqdq

⎡ ⎤= −⎢ ⎥

⎣ ⎦ ( ) ( )d dq q f q f q

dq dq⎛ ⎞

− = −⎜ ⎟⎝ ⎠

Define new operators:

1ˆ2

db qdq

+ ⎛ ⎞= −⎜ ⎟

⎝ ⎠

1ˆ2

db qdq

⎛ ⎞= +⎜ ⎟

⎝ ⎠

Then !ωb+b = !ω

2q − d

dq⎛⎝⎜

⎞⎠⎟

q + ddq

⎛⎝⎜

⎞⎠⎟

= !ω

2q2 − d 2

dq2 + q, ddq

⎡

⎣⎢⎤

⎦⎥⎛⎝⎜

⎞⎠⎟

= H − 1

2!ω

→ 1ˆˆ2

H b bω +⎛ ⎞= +⎜ ⎟⎝ ⎠h

Commutation Relations, between b operators:

ˆˆ ˆ ˆ, , 0bb b b+ +⎡ ⎤ ⎡ ⎤= =⎣ ⎦ ⎣ ⎦

1ˆ ˆ, ,2

d db b q qdq dq

+ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎡ ⎤ = + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎝ ⎠ ⎝ ⎠⎣ ⎦

1 [ , ] , , ,2

d d d dq q q qdq dq dq dq

⎛ ⎞⎡ ⎤ ⎡ ⎤ ⎡ ⎤= + − −⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠



( )1 1 ( 1) 12

= + − − =

Hence ˆ ˆ, 1b b+⎡ ⎤ =⎣ ⎦ ˆ ˆ, 1b b+⎡ ⎤ = −⎣ ⎦

Using this form of the Hamiltonian and the commutation relations we can derive the eigenvalues and eigenfunctions of H.O.!!

a) If ( )n qψ is eigenfunction with eigenvalue nE then

i) ˆ ( )nb qψ+ is eigenfunction with En+1 = En + !ω

ii) ˆ ( )nb qψ is eigenfunction with En−1 = En − !ω

Proof:

i) H b+ψ n(q)( ) = −!ω b+b+ 1

2⎛⎝⎜

⎞⎠⎟

b+ψ n(q)

= !ω b+ b, b+⎡⎣ ⎤⎦ + b+ b+b+ 1

2⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ψ n(q)

= b+ !ω + H( )ψ n(q)

= (En + !ω )b+ψ n(q)

ii) H bψ n(q)( ) = !ω b+b+ 1

2⎛⎝⎜

⎞⎠⎟

bψ n(q)

= !ω b+ , b⎡⎣ ⎤⎦ + bb+ + 1

2⎛⎝⎜

⎞⎠⎟

bψ n(q)

= !ω −1⋅ b+ b b+b+ 1

2⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ψ n(q)

= b −!ω + H( )ψ n(q) = En − !ω( ) bψ n(q)

b+ and b are called the raising and lowering operators, or ladder operators



What about the ground state?

bψ n(q) = E0 − !ω( ) bψ n(q)

Still true, but E0 − !ω < 0E !!

Only way out: 0ˆ ( ) 0b qψ =

01 ( ) 02

dq qdq

ψ⎛ ⎞+ =⎜ ⎟

⎝ ⎠

Differential equation with solution 21

20 ( )

qq eψ

−=

2 21 1

2 2( ) 0q qdq e q q e

dq− −⎛ ⎞

+ = − =⎜ ⎟⎝ ⎠

Putting it all together

2

1142

01( )

qq eψ

π−⎛ ⎞= ⎜ ⎟⎝ ⎠

normalized

( ) 01 ˆ( ) ( )!

n

n q b qn

ψ ψ+= normalized

1ˆ2

db qdq

+ ⎛ ⎞= −⎜ ⎟

⎝ ⎠

12nE n ω⎛ ⎞= +⎜ ⎟⎝ ⎠h 0,1,2,3....n =

First couple of unnormalized functions in terms of q :



21

20 ( )

qq eψ

−=

2

21

/221 ( ) 2

q qdq q e qedq

ψ− −⎛ ⎞

→ − =⎜ ⎟⎝ ⎠

( )2 2/2 2 /22 ( ) 2 1q qdq q qe q e

dqψ − −⎛ ⎞

→ − = −⎜ ⎟⎝ ⎠

( ) ( )2 22 /2 3 /23 ( ) 2 1 4 6q qdq q q qe q q e

dqψ − −⎛ ⎞

→ − − = −⎜ ⎟⎝ ⎠

Etc. We can obtain all eigenfunctions by differentiation

( )n xψ → substitute x q α= + normalize

The wave functions take the form 21

2( )q

nH q e−

( )nH q are Hermite polynomials, they are either odd or even functions. (Each

polynomial contains only odd or only even terms)

Vibrational Selection Rules Later we will discuss more generally the radiation process. For now the transition dipole moment is used to define the strength of a spectroscopic translation

*( ) ( ) ( )n mx x x dxψ µ ψ∫ m n→

0

0( ) ( ) .......x

dx x xdxµµ µ ⎛ ⎞= + +⎜ ⎟⎝ ⎠

0µ is the dipole at equilibrium distance, ddxµ is the change in dipole with changing x

(internuclear-‐distance)

Eg. N2 : 0ddxµ =

HF: 0ddxµ ≠ large

CO: 0ddxµ ≠ small

*( ) ( ) ( )n mx x x dxψ µ ψ∫

0 * ( ) ( ) 0n mx x dxµ ψ ψ= →∫ (if n m≠ )



*( ) ( )n md x x x dxdxµ ψ ψ+ ∫

2

* ( ) ( )n md q q q dqdxµ α ψ ψ→ ∫

( )ˆ ˆ*( ) ( )2 n m

d q b b q dqdxµ α ψ ψ+→ +∫

1ˆ( ) ~m mbψ ψ − 1

ˆ ( ) ~m mb ψ ψ++

For Harmonic Oscillator we can only get allowed transitions if 1nΔ = + or 1nΔ = −

ΔE = ±!ω

Poly-‐atomic Molecules

H = −!2∇2

2µαα∑ + 1

2kαβ (R − Re )α (R − Re )β

α ,β∑

By some manipulation (see notes on webpage) this can be written as a sum of H.O. Hamiltonians…….tedious derivations

H = !

i∑ ω i − 1

2d 2

dqi2 +

12

qi2⎡

⎣⎢

⎤

⎦⎥

= !

i∑ ω i bi

+bi +12

⎡

⎣⎢⎤

⎦⎥

The coordinates q : are linear combinations of atomic displacement vectors

3 N coordinate → 3 translation, 3 rotation, (3 N -‐6) vibration

For linear molecule, rotation around axis is not a displacement → 2 rotations, and (3 N -‐5) vibrations Eg. Water

Normal modes : display symmetry of molecule



The eigenfunctions are simply products of 1 dimensional H.O. functions 1( )k qψ 2( )l qψ 3( )m qψ

Ek ,l ,m = k + 1

2⎛⎝⎜

⎞⎠⎟!ω1 + l + 1

2⎛⎝⎜

⎞⎠⎟!ω 2 + m+ 1

2⎛⎝⎜

⎞⎠⎟!ω3

Very simple solution. One needs to diagonalize the “mass weighted hessian”

1Mα

kαβ1Mβ

3N × 3N Matrix

Reasonable approximations to all vibrational levels Statistical Mechanics (recall Chem 350)

Beyond the harmonic oscillator approximation

The true potential is not quadratic. For large molecules this is not so easy to correct (people would use low order perturbation theory, based on a quartic force field) For smalls molecules, in particular diatomics, one can solve for the full vibrational problem

The exact energies are not equidistant. A better approximation is

E(n) = !ω e n+ 1

2⎛⎝⎜

⎞⎠⎟− !ω exe n+ 1

2⎛⎝⎜

⎞⎠⎟

2

The energy levels are usually called

g(v), v = n,vibrational quantum number There are a finite number of bound levels

Another effect is that transition moments can be non-‐zero even when 1nΔ ≠ ± . This leads to the observation of overtones. An often used form for the potential for a diatomic is the Morse potential



( )2( ) 1 xeV x D e β−= −

0

0x

Vx =

∂ =∂

0x = equilibrium geometry

2

22

12 e

V D kx

β∂ =∂

e

kD

β→ =

eD can be used from experiment ( 0D measurable)

This is still an approximate potential Today: Potential energy curves can be calculated using electronic structure programs.

chapter5–!vibrational!motion!scienide2.uwaterloo.ca/~nooijen/chem356/chem+356+pdf/ch_5.pdfn...

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