The Quantum Mechanical Harmonic Oscillator

Motivation:

The quantum mechanical treatment of the

harmonic oscillator serves as a good model for

vibrations in diatomic molecules.

The harmonic oscillator model accounts for the

infrared spectrum a diatomic molecule.

Using “normal mode analysis”, it also serves as the basis for

vibrational analysis in larger molecules which are commonly used in

chemistry today to interpret infrared spectra.

Potential Energy Surface of a Typical Diatomic Molecule

r

m2m1

R0 (equilibrium bond length)

pote

ntial energ

y V

(r)

internuclear distance, r

bond breakage

The potential energy surface relates the geometry of a molecule (given

by its nuclear coordinates) to the molecule’s potential energy.

The Harmonic Oscillator

2

0 )(2

1)( RrkrV

R0 is the equilibrium bondlength

k is the force constant.

R0 and k are different forvarious diatomic molecule, i.e.

HCl, N2, O2, …

With the harmonic oscillator, we approximate the true internuclear

potential with a harmonic (parabolic) potential given by:

The harmonic oscillator is usually a good approximation for r values close

to R0 (for example, “low” temperatures such as room temperature).

m2

internuclear distance, r

energ

y, E

R0

r

m1

The Classical Picture of the Harmonic Oscillator

If we took our classical harmonic oscillator and stretched it to a length A at

time zero, and let it go with zero velocity at time t = 0, then the harmonicoscillator would vibrate according to:

tAtr cos)(

The total energy of the system can assume any value depending on how

far we initially displace the spring:

A

+A

tr(t) classical turning pointsat r = A and r = A

The displacements of the masses never go beyond amplitude |A|.

2

2

1kAE continuous energy

The system can have zero energy if the oscillator is not moving (A = 0).

Reduction of the two-body problem to two one-body problems

Consider the motion of the diatomic molecule

in one dimension. The classical Hamiltonian

function is given by:

rm2

m1

x1 x2Xcm

But wait!!! We can reduce this two-body problem to a one-body problem

for the translational motion of the center of mass, Xcm, of the diatomic and

the vibrational motion in terms of r, the distance between the nuclei.

21

2211CM

mm

xmxmX

12 xxr

2

12

2

22

2

11 )(2

1

2

1

2

1oRxxkxmxmH

center of mass coordinate

internuclear distance coordinate

2 1r x x

21

2211CM

mm

xmxmX

12 xxr

Using these two equations, we can also express x1 and x2 in terms of Xcmand r. (i.e., two equations and two unknowns)

rmm

mXx

21

2CM1

r

mm

mXx

21

1CM2

Use these relationships to derive the kinetic energy in terms of the center

of mass coordinate and the bond distance coordinate (next tutorial!).

The ultimate goal is to separate the translational motion from the

vibrational motion.

2 21 21 21 2

1 1

2 2CM

m mT m m X r

m m

After some straightforward algebra, we end up with:

The first term can be interpreted as the kinetic energy of translational

motion of the whole molecule (m1 + m2) through space.

The second term is the internal kinetic energy of the relative motion of

the two nuclei, or the vibrational motion. To simplify things, we define

the reduced mass:

21

21

mm

mm

2

Vib 2

1rT

The vibrational kinetic energy therefore reduces from a two body

problem with masses m1 and m2 to a one-body problem with an

‘effective’ mass equal to .

rm2

m1We now consider the vibrational motion of the

molecule. The classical Hamiltonian function

for vibrational motion only is:

22 )(2

1

2

1oRrkrH

where r is the internuclear distance and is the reduced mass.

It is reasonable to assume that the vibrational motion is independent of

the motion of the center of mass as long as the molecule is isolated and

does not interact with anything. The vibration of the molecule will then be

unaffected by how fast or slow the center of mass is moving.

Separation of Translational and Vibrational Motion

We have already examined the quantum mechanical treatment of the

translational motion in one dimension. This was the ‘free particle’.

2

0

2 )(2

1

2

1RrkrH

Let’s make a further simplification by defining a new coordinate, x:

120 xxRrx

In this way our classical Hamiltonian becomes:

22

2

1

2

1kxxH

recall that

rt

trRtr

tx

d

)(d))((

d

d0

so 22 rx

Note: this new coordinate x (the displacement fromthe equilibrium internuclear separation) is not the

same as the x1 and x2 coordinates used previously.

r

m2m1

2

0 )(2

1)( RrkrV

R0

internuclear distance, r

pote

ntia

l energ

y, V

When x is negative, the bond is compressed, and when x is positive, thebond is stretched.

0Rrx

2

2

1)( kxxV

x = 0pote

ntial energ

y, V

+x-x

displacement:

We have now reduced our vibrational motion of two bodies into the

effective motion of a single particle. Before we look at solving for the

quantum mechanical behavior, let’s relate what we’ve just done to the

particle-in-a-box problem.

L

V = V = V = 0

x = 0 x = L

V(x) = 0 0< x < L

V(x) = 0 x and x L

0 < x < L

2

2

1)( kxxV

x = 0 pote

ntial energ

y, V

+x-x

- < x < +

The Quantum Mechanical Harmonic Oscillator

Now solve for the wave function of the Harmonic Oscillator. We first need

the Hamiltonian operator. We know the classical Hamiltonian is

22

2

1

2

1kxxH

2

2

22

2

1

d

d

2ˆ kx

xH

So our quantum mechanical Hamiltonian operator is given by

Notice the reduced mass is retained.

21

21

mm

mm

)()(2

1)(

d

d

2

2

2

22

xExkxxx

The Schrödinger equation for the harmonic oscillator is therefore

0)(2

12)(

d

d 222

2

xkxEx

x

We again have a ordinary differential equation. Rearranging into a more

standard form, we get

E – V is not constant in this

case, but a function of x

When E – V is variable, there is no general way of solving this type ofdifferential equation. Each case must be studied individually, depending

of the function.

It turns out that we can use a power series method. This is a bit tedious

and involves math we have not yet covered.

For the moment, the solutions will only be presented and interpreted.

x

The Quantum Mechanical Harmonic Oscillator

)()(2

1)(

d

d

2

2

2

22

xExkxxx

The Harmonic Oscillator Energies

21

2/1

n

kEn

n = 0, 1, 2 ,3 ….

The energy levels are quantized: E0, E1, E2, E3, …

The states are expressed in terms of the quantum number n.

is the reduced mass defined by:

k is the force constant (characteristic of each diatomic bond).

21

21

mm

mm

This is usually rewritten as

21 nEn n = 0, 1, 2 ,3 ….

2/1

k

where is classically interpreted as the vibration frequency inradians per second (the angular vibration frequency).

Many textbooks also use:

21 hEn = 0, 1, 2 ,3 ….

22

12/1

k

where (“Nu”) is the frequency in cycles per second (Hz).

where (“Upsilon”) is the harmonic oscillator quantum number.

20

E

n = 0, 1, 2 ,3 ….

The Harmonic Oscillator Energies

• Energy levels are all equally spaced.

spacing

• Notice that the ground state in this case has

an index of n = 0, not n = 1 as with the particlein a box.

• Again there is a zero-point energy with the

ground state having a positive non-zero

energy:

vibrational zero-point

energy

n = 0

n = 1

n = 2

n = 3

n = 4

n = 5

E

21 nEn

The Harmonic Oscillator Wave Functions

Solving the harmonic oscillator time-independent Schrödinger equation

gives time-independent wave functions of the following form:

2/ )()(2xeHNx nnn

(normalization

constant)

k

4/1

!2

1

nN

nn

x2/1 )(nH are Hermite polynomials, functions of “xi” =

n = 0, 1, 2 ,3 ….

1 )(0 H 2 )(1H 12)(3

3 8 H

)(nH is an nth order polynomial in and therefore an nth orderpolynomial in x.

12 xxx

)2/exp( 241

0 αxπ

αψ

/

)2/exp(4 2

413

1 αx xπ

αψ

/

)2/exp( 124

22

41

2 αxx π

αψ

/

)2/exp( 329

23

413

3 αxxx π

αψ

/

The First Four Harmonic-Oscillator Wave Functions

k

2

1 4

02

/αx

αψ e

π

2124

2

2

41

2

αx

ex π

αψ

/

24

2

413

1

αx

xeπ

αψ

/

2329

2

3

413

3

αx

exx π

αψ

/

x

x = 0

Harmonic Oscillator Wave Functions

As n increases, the wave function ‘widens’ (i.e., it has moreenergy and can “stretch out more” than the classical oscillator).

The harmonic oscillator wave functions are often superimposed on

the potential.

5

4

3

2

1

E

00( )ψ x

1( )ψ x

2( )ψ x

3( )ψ x

4( )ψ x

Verify that 0(x) is normalized.

2

2

41

0

αx

eπ

αψ

/

Verify that 0(x) and 1(x) are orthogonal.

24

2

413

1

αx

xeπ

αψ

/

Integrals involving Even and Odd Functions

An even function is a function that satisfies: )()( xfxf

Even functions are ‘symmetric’ about the origin (e.g., cos(x) or x2).

An odd function is a function that satisfies: )()( xfxf

Odd functions are ‘antisymmetric’ about the origin (e.g., sin(x) or x3).

AA

Adxxfdxxf

0)(2)(

0)(2)( dxxfdxxf

If a function f(x) is even, then:

0)( A

Adxxf

0)(

dxxf

Notice that the Harmonic Oscillator wave functions alternate

between even and odd functions.

The harmonic oscillator wave functions with n odd are odd functions.

The harmonic oscillator wave functions with n even are even functions.

even

odd

even

odd

even

x = 0

5

4

3

2

1

E

0

( )nψ x

An odd function multiplied by an odd function is an even function.

An odd function multiplied by an even function is an odd function.

odd times odd = even

odd times even = odd

An even function multiplied by an even function is an even function.

even times even = even

We can use these properties to help us evaluate certain properties.

x

xp

Harmonic Oscillator and Barrier Penetration

The harmonic oscillator wave functions show that there is penetration

into the classically forbidden regions.

The harmonic function (dotted line) shows the classical turning point for

each energy level.

Because the system is in a bound state, the particle cannot escape, and

therefore this is barrier penetration, not tunneling (“getting through”).

wave functions

5

4

3

2

1

0

probability density

E

5

4

3

2

1

0

E

x

( )ψ x2

( )ψ x

Determine where the classical turning point is for the quantum

mechanical Harmonic Oscillator in the ground state.

Also notice that as n increases, theprobability increases at the classical

turning point.

Plotted below is the probability density for the n = 10 state.

The dotted line shows the probability

density derived from the classical

harmonic oscillator of the same energy.

It shows that the probability of finding

the system is greatest at the turning

point, where the particles slow down

and change direction.

x = 0

n = 0

= 6n

Harmonic Oscillator Model Accounts for the Infrared

Spectrum of Diatomic Molecules

What are the spectroscopic predictions of the Harmonic Oscillator model?

To rigorously examine spectroscopic transitions, we would have to learn

more quantum mechanics, namely time-dependent theory, which is

covered in more detail in advanced courses.

So some of the statements will not be derived rigorously. Be aware that

they were derived using the Harmonic Oscillator wave functions (an

approximation of real molecules).

A diatomic molecule can make transitions from

one vibrational energy state to another by

absorbing or emitting photons or light.

n = 5

energ

y

n = 0

n = 1

n = 2

n = 3

n = 4

n = 0, 1, 2 ,3 …. 21 nEn

But not all transitions are allowed.

From quantum mechanical time-dependent perturbation theory, a

transition probability (or intensity) can be determined as:

2* ˆf iI x d

It can be shown (but not here!) that the harmonic oscillator model only

allows transitions between adjacent energy states with:

1n

The above is condition is an example of a spectroscopic selection rule.

This is known as the fundamental vibrational frequency. In units of cm-1

(“wavenumbers”) it is given by: 2/1

obs2

1~

k

c

(c is the speed of light in cm/s)

1n

Given that only the above transitions are allowed,

how many peaks should we see?

n = 5

energ

y

n = 0

n = 1

n = 2

n = 3

n = 4

21 nEn

Example:

The infrared spectrum of HCl has a very intense line at 2886 cm-1.

What is the force constant of the H-Cl bond?

From the IR spectra, we can determine some information about the

strength of the H-Cl bond (because we know the masses of the nuclei).

Furthermore, for a vibration to be infrared ‘active’, the dipole moment of

the molecule must change as the molecule vibrates.

For diatomic molecules, this means that the molecule must have a

permanent dipole moment to be infrared active. In other words, for

diatomics, only hetero-diatomic molecules are infrared active.

e.g. HCl absorbs in the IR but not N2.

Why?

molecule

fundamental

frequency (cm-1)

force constant

(N/m)

H2 4159 520

H35Cl 2886 482

H79Br 2559 385

H127I 2230 293

12C16O 2143 1870

14N16O 1876 1550

Why are we so concerned about the isotopes?

Will the isotope change the force constant?

R0

internuclear distance, r

energ

y, E

The anharmonicity has the following

consequences:

The Harmonic Oscillator Model is an Approximation

What have approximated the true diatomic potential with a harmonic

potential. However, the true potential is anharmonic.

The energy levels are not all equally

spaced. The spacings get smaller

and smaller until the bond breaks.

The selection rule n = ±1 is notrigorously obeyed.

IR intensities of non-fundamental

transitions are typically small.

(Shows that the harmonic oscillator

approximation is a good one.)

The Vibrational Zero-Point Energy Is Not Negligible

2/1

022

kE

vibrational zero-pointenergy

As with the particle in a box, the harmonic oscillator has a zero-point

energy correction. Meaning the lowest energy state of the particle has a

positive, non-zero energy.

The vibrational zero-point energy has significant implications for

dissociation energies and the kinetic isotope effect.

Bond Dissociation Energies

en

erg

y, E

r

E0 = zero point energy

D0 = experimentaldissociation energy

e.g., for H2

D0 = 103.2 kcal/mol

ZPE = 6.5 kcal/mol

What happens with D2?

Vibrations in Polyatomic Molecules can be

Approximated by Harmonic Oscillator Solutions

A generalization of our simple quantum mechanical treatment of the

harmonic oscillator is frequently used to approximate vibrations in

polyatomic molecules.

Simulation of molecular vibrations are used today by quantum chemists

and experimental chemists to:

• interpret and predict IR and Raman spectra.

• calculate zero-point energies for accurate determination

of thermodynamic data.

The same approximation is made, in that we are at low temperatures and

close to the equilibrium geometry. (The approximation also breaks down

with very weak bonds, such as hydrogen bonds).

In our one-dimensional Harmonic Oscillator problem we

reduced the two body problem into an effective one

body problem.

We made a coordinate transformation from Cartesians

to CM (center of mass) and relative coordinates.

2

2

22

vib2

1

d

d

2ˆ kx

xH

2

0122

2

2

2

2

2

1

2

1

2

)(2

1

d

d

2d

d

2ˆ Rxxk

xmxmH

2

2

21

2

transd

d

)(2ˆ

CMXmmH

If we assume the translational (and rotational) motion are independent of

the vibrational motion we can solve for the vibrational Schrödinger

Equation, which gives:

n = 0, 1, 2 ,3 …. 21 nEn )(xn

r m2m1

x1 x2Xcm

A 3-dimensional and polyatomic (more than 2 atoms) extension of this

procedure can be made. A coordinate transformation to what are

known as normal coordinates gives:

VIBN

i

ii

ii

QkQ

H1

2

2

22

vib2

1

d

d

2ˆ

where the Qi’s are the normal coordinates. (They can be determinedfrom straightforward matrix algebra.)

The sum is over all molecular vibrations (vibrational degrees of freedom).

Each vibration has its own effective mass and force constant.

The vibrational energy of the whole molecule becomes:

vib

1

21

vib

N

i

ii nhE

There is a quantum number ni for each vibration.

Normal Modes of Water

Predicted Infrared spectra of Pyridine

The vibrations along the normal coordinates are called normal modes.

wave number (cm-1)

IRin

ten

sity

rms deviation from experimental is about 30 cm-1

Below is a spectra for pyridine determined from a quantum mechanical

calculation using the generalized ‘harmonic oscillator’ approximation.

N

O

HH

O

HH

O

HH

3657 cm-1 1595 cm-1 3756 cm-1

HF/6-31G(d)

simulated spectra from quantum chemistry