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Vibrational Spectroscopy of Diatomic Molecules

Dr.Kaushalendra Kumar, Associate Professor, Department of Chemistry, V.B.U.Hazaribag

IR spectroscopy which has become so useful in identification, estimation, and structure

determination of compounds draws its strength from being able to identify the various

vibrational modes of a molecule. A complete description ofvibrational normal modes, their

properties and their relationship with the molecular structure is important

Vibrations

The motion of two particles in space can be separated into translational, vibrational, and

rotational motions.

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Degree of freedom is the number of variables required to describe the motion of a particle

completely. For an atom moving in 3-dimensional space, three coordinates are adequate so its

degree of freedom is three. Its motion is purely translational. If we have a molecule made of N

atoms (or ions), the degree of freedom becomes 3N, because each atom has 3 degrees of

freedom. Furthermore, since these atoms are bonded together, all motions are not translational;

some become rotational, some others vibration.

The degrees of vibrational modes for

linear molecules

can be calculated using the formula:

3N−5

The degrees of freedom for

nonlinear molecules

can be calculated using the formula:

3N−6

n

is equal to the number of atoms within the molecule of interest. The following procedure

should be followed when trying to calculate the number of vibrational modes:

Atoms (very

symmetric)

Linear molecules (less

symmetric)

Non-linear molecules (most

unsymmetric)

Translation

(x, y,

and z)

3

3

3

Rotation

(x, y, and

z)

0

2

3

Vibrations

0

3

N

− 5

3

N

– 6

Total

(including

Vibration)

3

3

N

3

N

Properties of a Molecular Bond

What do we know about bonds from general chemistry?

1.

Breaking a bond always requires energy and hence making bonds always release energy.

2.

Bond length

3.

Bond energy (or enthalpy or strength)

The potential energy of a system of two atoms depends on the distance between them. At large

distances the energy is zero, meaning “no interaction”. At distances of several atomic diameters

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attractive forces dominate, whereas at very close approaches the force is repulsive, causing the

energy to rise. The attractive and repulsive effects are balanced at the minimum point in the

curve.

The internuclear distance at which the potential energy minimum occurs defines the

bond

length

. This is more correctly known as the

equilibrium

bond length, because the two atoms will

always vibrate about this distance.

Bond lengths depend mainly on the sizes of the atoms, and secondarily on the bond strengths, the

stronger bonds tending to be shorter. Bonds involving hydrogen can be quite short; The shortest

bond of all, H–H, is only 74 pm. Multiply-bonded atoms are closer together than singly-bonded

ones; this is a major criterion for experimentally determining the

multiplicity

of a bond. This

trend is clearly evident in the above plot which depicts the sequence of carbon-carbon single,

double, and triple bonds.

In general, the stronger the bond, the smaller will be the bond length.

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The internal motions of vibration and rotation for a two-particle system can be described by a

single reduced particle

with a reduced mass

μ

located at

r

. In the below figure, the vector

r

corresponds to the internuclear axis. The magnitude or length of

r

is the bond length, and the

orientation of

r

in space gives the orientation of the internuclear axis in space. Changes in the

orientation correspond to rotation of the molecule, and changes in the length correspond to

vibration. The change in the bond length from the equilibrium bond length is the

vibrational

coordinate

for a diatomic molecule.

The diagram shows the coordinate system for a reduced particle.

R

1 and

R

2 are vectors to

m

1

and

m

2

.

R

i

s the resultant and points to the center of mass. (b) Shows the center of mass as the

origin of the coordinate system, and (c) expressed as a reduced particle.

The Classical Harmonic Oscillator

Simple harmonic oscillators about a potential energy minimum can be thought of as a ball rolling

frictionlessly in a dish (left) or a pendulum swinging frictionlessly back and forth. The restoring

forces are precisely the same in either horizontal direction.

Recall that the Hamiltonian operator

H^

is the summation of the kinetic and potential energy in a

system. There are several ways to approximate the potential function

V

, but the two main means

of approximation are done by using a Taylor series expansion, and the Morse Potential. The

vibration of a diatomic is akin to an oscillating mass on a spring.

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.

The classical forces in chemical bonds can be described to a good approximation as spring-like

or Hooke's law type forces. This is true provided the energy is not too high. Of course, at very

high energy, the bond reaches its dissociation limit, and the forces deviate considerably from

Hooke's law. It is for this reason that it is useful to consider the quantum mechanics of a

harmonic oscillator.

We will start in one dimension. Note that this is a gross simplification of a real chemical bond,

which exists in three dimensions, but some important insights can be gained from the

one-dimensional case. The Hooke's law force is

where k is the force constant, a measure of the strength of the restoring force. Larger the value of k, larger the restoring force even for a small displacement. k is a property of the harmonic oscillator system. For a diatomic molecule, a large value of k means a strong bond. A smaller value of k means, for example, a weaker bond

The potential energy of the oscillator is given by V(x). When there is no displacement, V = 0.

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The restoring force at any point is the negative derivative of the potential energy at that point with respect to the displacement coordinate. i.e. V(0) = 0.

The kinetic energy of the harmonic oscillator due to its motion is

The total energy, or the Hamiltonian H of the harmonic oscillator is the sum of its potential and kinetic energies. Classically,

In order to understand the relation between the harmonic oscillator Hamiltonian and molecular vibrations, consider the diatomic molecule in the picture

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The kinetic energy of the diatomic molecule when it oscillates with a small amplitude is

Where x1 and x2 are vector displacements of atoms 1 and 2 (grey and red) from their equilibrium

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If we neglect the overall translational motion of the molecule (i.e. the centre-of-mass motion of the molecule), then the kinetic energy is given by

Where is the reduced mass.

Therefore the Hamiltonian for a vibrating diatomic molecule without the centre-of-mass translational motion is

The Schrodinger equations for the harmonic oscillator and the eigen values and eigen functions. The Schrodinger equations is obtained from the classical Hamiltonian by replacing both position (x) and momentum (p) by the appropriate operators. Since this is a one dimensionsional system,

and

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The equation is not solved here. Standard methods for solving this equation require power series expansion of the solution, slightly more advanced for this course. Hence only the solutions are given. The solutions are constructed usingHermite polynomials. In the process of solving the differential equation above, one can transform the Schrodinger equation in to the Hermite's differential equation whose solutions are known as Hermite polynomials. The polynomials are infinite in number and form the class of orthogonal polynomials. They are denoted by the symbol where V = 0,1,2,3,..., is the order of the polynomial and x is the variable.

The harmonic oscillator energies are quantized and are equidistant. What is meant by this statement is the following: the difference between any two adjacent levels is the same for all energy levels. The energy levels are equally spaced.

The quantities and are known as angular frequency and frequency respectively.

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The lowest quantum number is V=0, unlike the particle-in-a box. The plot of the wave functions and the squares of the wave functions are given below and indicate the number of nodes for each quantum number V. The number of noded of a wavefunction for quantum number V is V itself. The functions also display the odd-even characteristics, namely wavefunctions corresponding to odd quantum numbers are all odd function and the wave functions corresponding to even quantum numbers are all even functions.

Significance of the solutions.

1. The quantum mechanical harmonic oscillator has some energy even when the quantum number is zero. (i.e. it cannot have zero energy of vibration). Classically an oscillator having zero energy does not oscillate!!

2. Zero point energy (referring to the above) is a purely quantum mechanical concept and is fundamentally important for molecules and solids.

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The probability densities for the four lowest energy states of the harmonic oscillator.

4. The boundary for an oscillator is endless. What does the potential look like? It is a parabola and the classical oscillator can ONLY oscillate between the two ends of the parabola. The total energy of the oscillator is all potential energy at the turning points and all kinetic energy midway between the two points. However, in quantum mechanics the the wave function outside of this classical parabola is nonzero for all values of x, though very small. Thus, the square of the wave function which represents probability density is nonzero everywhere, except for a finite number of points. It goes to zero as x tends towards infinity. This leads to a very interesting phenomenon called quantum mechanical tunneling. This means that the quantum mechanical oscillator can be found to oscillate into regions which are

forbidden by classical mechanics. Experimentally tunneling phenomena have been amply demonstrated. This can be seen from the plots of the squares of wavefunctions which extend outside of the classically forbidden region, namely, outside the parabola for the potential energy. The scanning tunneling

microscope used by metallurgists and nanotechnologists worldwide is based on tunneling phenomenon. Harmonic oscillator provides a beautiful example for tunneling.

5. For small values of energy you notice that the oscillator is more likely to be found in the middle of the potential well. (The wave functions associated with lower energies have peaks in the centre of the parabolic curve. This is against classical intuition as it is in these regions that the oscillator has the maximum kinetic energy, classically speaking, and is least likely to be spotted!!

6. The diatomic molecular vibrational energy is quantized and the simplest model above explains the basic features of the vibrational spectra of most stable molecules. Replace the mass of the oscillator by the reduced mass of the diatomic molecule and the connection between the two systems is established immediately as you have seen in the previous page.

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Anharmonicity

Harmonic approximation, and all of the above valid for small amplitude

vibrations. For large amplitudes , or excitation by radiation to very high energy

levels - the potential energy of vibrational motion is

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It takes into account the possibility (eventuality) of the diatomic molecule dissociating eventually if its vibrational amplitude becomes quite large compared to the bond length.

The quantum mechanical results for the Morse oscillator model can be obtained by solving the Schrodinger equation for a Morse oscillator :

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The gap between a given vibrational level v and the next vibrational level (V+1) decreases as v increases. Therefore,

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Anharmonicity also implies vibrational transitions are no longer for only, transitions between any level and any other level is indeed nonzero. Thus the Morse oscillator model takes into account the dissociative possibility also. Because the energy differences are different even for nearby levels, the spectrum now contains more than a single line and has some characteristic features of the given molecule

De - dissociation energy from equilibrium geometry. Zero point energy of the molecule :

References

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