12+harmonic+oscillator

7/21/2019 12+Harmonic+Oscillator

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Harmonic Oscillator

The diatomic molecule can be modeled as harmonic oscillator oscillates around

center of mass

The potential energy of the harmonic oscillator is

The Schrdinger equation is

( )

This type of equation has known solutions

The wave function will be

Where called the Hermit polynomial

And the energy will be

( ) Where

Compare it with

Plank assumption

Compression

Stretching


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At n=0

The energy level diagram will be

Comments

1-

As n increase E increase

2-

Energy levels are equally spaced by

3-

Minimum possible energy =called the Zero-Point Energy (ZPE)4- Wave functions extend beyond separations allowed by classical mechanicsThe Probability Density

I- Classical Harmonic Oscillator

For the classical harmonic oscillator, the speed of motion will be low at the

compression and stretching positions (reach to zero) due to changing direction,

so the probability of finding the oscillator in these positions are high while at the

equilibrium region the speed of motion reaches the maximum value so the

probability of finding the oscillator in this position is very low.

ZPE =

35

Change direction

Minimum Speed

High Pr

Maximum Speed

Low Pr

Pr

EqCom Str


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Where

is the Anharmonicity constant

Comments

1-

As n increase the spacing decrease

2-

The spacing reach zero at high r

values3-

S.H.O. Fits well at low excitation

BUTFits poorly at high excitation

Prediction of Dissociation energy (D0)

The following figure shows the relation between

the dissociation energy, D0, and the minimum energy,

De, of a molecular potential energy curve.

So Simple Harmonic Oscillator Approximation (Polyatomic Molecules)

Using this approximation the wave number of the normal modes can be calculated

Normal modes = 3*N-6

The wave numbers

Compare it with

Plank assumptionFundamental

vibrations

Overtones

The Force constant


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Example the water molecule

Mode Experimental

cm-1

Calculated

cm-1

Difference

cm-1

3652 3806 154 3756 3928 1723

1595 1600 5

The lower excitation energy has low difference

Examples:

A1H

35Cl molecule has a force constant of 516 Nm

1. Calculate the vibrational

stretching frequency

Hydrogen bromide (1H81Br) absorbs infrared radiation of wavenumber 2649.7

cm-1

. Calculate the force constant of the1H

81Br bond. (The molar masses of

1H

and81

Br are 1.008 g mol-1

, and 80.916 g mol-1

, respectively.)


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The infrared absorption spectrum of1H

35Cl has its strongest band at

8.651013

Hz.

(1) Calculate the force constant

(2) Calculate the zero-point vibrational energy

(3) What happens to the strongest absorption band in IR spectrum if we deuterate

the molecule?

(4) What happens to the force constant if we deuterate the molecule?

12+harmonic+oscillator

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