1. MS310 Quantum Physical Chemistry
Ch 7. A Quantum Mechanical Model for the Vibration and Rotation of Molecules
Schrödinger eq. for the Q.M. harmonic oscillator
Described by energy spectrum and energy eigen-
functions of molecules having translational,
vibrational, and rotational degrees of freedom
Schrödinger eq. for rotation in 2-D and 3-D
Angular monentum to consider orbitals…
MS310 Quantum Physical Chemistry

2. MS310 Quantum Physical Chemistry
7.1 Solving the Schrödinger equation for the Q.M Harmonic Oscillator
Example of vibration in Q.M : chemical bond
Bonding electron in the simple potential, and equilibrium distance is determined by bond length.
Real potential : anharmonic oscillator(not ideal)
At 300K, 1 or 2 state of vibration occupied → can approximate V(x) as a harmonic oscillator
MS310 Quantum Physical Chemistry

3. MS310 Quantum Physical Chemistry
Schrödinger equation is given by
Find the form of solution
E term can be ignored because of x2 >> E when y2 → ∞
Multiply the 2dψ/dx both side and use the product rule
MS310 Quantum Physical Chemistry

4. MS310 Quantum Physical Chemistry
Text p.104
MS310 Quantum Physical Chemistry

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Assume right term is much smaller than left term
Solution : ‘Gaussian’ form
Therefore, we can assume the solution
MS310 Quantum Physical Chemistry

7. MS310 Quantum Physical Chemistry
Schrödinger equation is rewritten by
Hermite equation is already solved : hermite polynomials
Solution is given by
Even state(n:even) : ψ(-x)=ψ(x) : even function
Odd state(n:odd) : ψ(-x)=-ψ(x) : odd function
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8. MS310 Quantum Physical Chemistry
Eigenvalue is given by
There are 2 different phenomenon to classical H.O
1) energy of ground state is not zero : ZPE
2) particle can be found in the classical forbidden region
Probability in the interval ∆x :
MS310 Quantum Physical Chemistry

9. Probability density of 12th state of H.O
Time-dependent solution
: standing wave
Probability density of 12th state of H.O
MS310 Quantum Physical Chemistry

10. MS310 Quantum Physical Chemistry
7.2 Solving the Schrödinger equation for rotation in 2-dimensions
Neglect the coupling, hamiltonian operator is sum of individual operators for the degrees of freedom for the molecule
And, total energy also can divide to each energy
Finally, total wavefunction is product of eigenfunctions of each operator
MS310 Quantum Physical Chemistry

11. MS310 Quantum Physical Chemistry
Set V=0 : no vibration(make easier problem)
Rotation : internal motion → motion of reduced mass
Laplacian in 2-dimension
Fixed r : radial term canceled
Solution is given by angular term
It means clockwise and counterclockwise rotation
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12. MS310 Quantum Physical Chemistry
Ex) 7.4
Normalize the rotational wavefunctions in 2-dimension
Sol)
MS310 Quantum Physical Chemistry

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Boundary condition : ‘quantization’ of angular momentum
Angular momentum must be periodic function because of φ+2π= φ always satisfies.
→ indistinguishable values φ and φ+2nπ
Use Euler’s relation
condition of ml : 0, ±1, ±2, ±3, …
: quantization of angular momentum
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Energy of the rotation
State of +ml and –ml : same energy and orthogonal each other
→ 2-fold degenerate with ml ≠ 0 level
l : angular momentum vector, : angular momentum operator
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Φ+(φ) , Φ-(φ) : eigenfunction of both of hamiltonian and momentum operator
Eigenvalue of momentum operator : +ℏml and -ℏml
Then, we can obtain the similar form as the C.M
Probability of angular motion
: same for all region
MS310 Quantum Physical Chemistry

16. MS310 Quantum Physical Chemistry
7.3 Solving the Schrödinger equation for rotation in 3-dimension
3-dimensional rigid rotor : similar than 2-dimensional problem
Laplacian in spherical coordinate is given by
Rigid rotor : ‘fixed r’ → r term canceled
Like the 2-dimension problem, we can write the Schrödinger equation the
MS310 Quantum Physical Chemistry

17. MS310 Quantum Physical Chemistry
Define the β=2μr02E/ℏ2
Equation is changed by
Use the separation of variable : Y(θ,φ) = Θ(θ)Φ(φ)
Solve it by the left part and right part is ‘constant’
MS310 Quantum Physical Chemistry

18. MS310 Quantum Physical Chemistry
Right part : similar to 2-dimensional problem : set c = ml2
Equation can be change to two ODEs.
Second equation : same as the 2-dimensional problem
φ part of Y(θ,φ) : depends on ml
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Solve the first equation : Legendre’s equation
Set z = cos θ and use it, equation change to
Case of ml = 0 : Legendre’s equation
Use the power series, write the solution P(z) instead of Θ(z)
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Recurrence relation is given by
If β ≠ integer, this series will not terminate.
However, it cannot be solution because it diverge at z=1.
Why?
By the ratio test, series diverges at z=1 and it cannot the solution of wavefunction!
If β = integer, well-behaved wavefunction exists and eigenvalue of equation is given by β = l(l+1) (set n=l)
P(z) is called the Legendre polynomials.
MS310 Quantum Physical Chemistry

21. MS310 Quantum Physical Chemistry
Case of ml ≠ 0 : Associated Legendre polynomial
Write the solution as the Plml(z)
set Plml(z) = (1 - z2)m/2F(z) and equation is given by
MS310 Quantum Physical Chemistry

22. MS310 Quantum Physical Chemistry
Finally, equation is changed to
Solution is given by
ml must be | ml | ≤ l : if more than l times of differentiation, wavefunction becomes zero and it is not allowed state.
Therefore, quantum number is given by
β = l(l+1), l = 0,1,2,3…
ml = -l, -(l-1), … , -1, 0, 1, …, l-1, l
Wavefunction Ylml(θ,φ) is given by
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Energy of angular momentum
And this notation satisfies, too.
Case of 2-dimensional rotation : 2-fold degenerecy
In this problem(3-dimentional rotation) : 2l+1 degenerecy
There are 2l+1 ml values per one l value, and these states have same energy!
MS310 Quantum Physical Chemistry

24. MS310 Quantum Physical Chemistry
7.4 The quantization of angular momentum
Energy of angular motion is given by
Difference between |l2| and E : divide by 2I
Therefore, hamiltonian and operator also satisfy same relationship.
Total energy quantized → |l|2 quantized
We can write for operator
Therefore, value of |l| is given by
commute hamiltonian, not
Then, has 3 component : lx,ly,lz, obtained by the l = r x p
MS310 Quantum Physical Chemistry

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Can calculate this formula
Angular momentum operator in spherical coordinate is
Commutator relation is given by
lx, ly, lz are not commute.
MS310 Quantum Physical Chemistry

26. MS310 Quantum Physical Chemistry
How can obtain the component of angular momentum?
→ see lz : simplest form(only depends on φ)
Ylml(θ,φ) is eigenfunction of lz
→ Ylml(θ,φ) is eigenfunction of both and lz
Therefore, we can choose and lz can solve the problem easily. Also, we can know the length of angular momentum l and value of z-component lz, but we cannot know the value of x and y component.
Why z component is special?
→ no special!
We can choose another direction and it also commute to . It means z component is simple only in the spherical coordinate and ‘only 1’ component of angular momentum is commute with .
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7.5 The spherical harmonic functions
We see the spherical harmonic functions
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Spherical harmonic functions : ‘complex’
Make the function ‘real’ by the linear combination
Real wavefunctions(we called it ‘orbital’) are orthonormal, too.
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Shape of p and d orbitals. We can see the each orbital is perpendicular.
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Superposition of p and d orbital
Magnitude of pz and py orbital
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7.6 The classical harmonic oscillator
Example of oscillator : two masses connect by spring
Introduce the center-of-mass coordinate and relative position
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Differentiate by time, we can obtain the center-of-mass velocity and relative velocity
Total energy of 2-mass system
Change the total energy by the vCM and v.
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vCM : velocity of ‘whole system’ : independent to motion of internal system
v : relative velocity : dependent to motion of internal system
Restoring force act to ‘internal system’ → restoring force act to reduced mass
MS310 Quantum Physical Chemistry

34. MS310 Quantum Physical Chemistry
Therefore, we can divide this motion by two motions.
(whole motion : motion of center of mass + motion of reduced mass)
Our focus is motion by restoring force.
If oscillator is harmonic oscillator, force is given by F = -kx
Solution is given by
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35. MS310 Quantum Physical Chemistry
If initial condition is x(0)=0,v(0)=v0
Potential & kinetic energy of harmonic oscillator
Classical harmonic oscillator : ‘continuous energy spectrum’
MS310 Quantum Physical Chemistry

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7.7 Angular motion and the classical rigid rotor
Rotation of 2-particle : centered at center of mass
Consider the constant r = r1+r2
Kinetic energy of system is given by
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Centripetal acceleration
Angular velocity and angular acceleration is given by
MS310 Quantum Physical Chemistry

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Direction of angular velocity and angular acceleration : right-hand rule
Case of constant acceleration
I = μr2 : moment of inertia
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Angular momentum l is defined by
x : cross product
Magnitude or Angular momentum is
φ : angle between p and r
Kinetic energy is given by
Classical rigid rotor : continuous energy
MS310 Quantum Physical Chemistry

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7.8 Spatial quantization
See the angular momentum.
First, we see the semiclassical description
angular momentum cannot lie on the z-axis. Why?
| ml | ≤ l is condition of ml and magnitude of l is given by
Therefore, if the case of ml = l (extreme case)
→ z-component cannot be same as the magnitude of angular momentum.
Angular momentum lie on the z-axis :
x, y component = 0
→ know 3 component simultaneously
But it cannot be possible because commutator is not zero!
MS310 Quantum Physical Chemistry

42. MS310 Quantum Physical Chemistry
If we know the total angular momentum and z-component, then we cannot know the x and y component and only we know the
→ cone has an open end
Finally, we can see the l=2 case(d orbital, too), vector model of angular momentum
Vector of angular momentum only have certain orientation in space.
→ spatial quantization
c.f) classical case : possible l values make the surface of sphere, not a cone
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43. MS310 Quantum Physical Chemistry
Summary
Quantum mechanics is used to study the vibration and rotation of a diatomic molecules.
Vibrational degree of freedom modeled by the harmonic oscillator was considered.
The harmonic oscillator has a discrete energy spectrum like the particle in the box in Q.M
The Q.M model for rotational motion providing a basis for understanding the orbital motion of electrons around the nucleus of an atom as well as the rotation of a molecule about its principal axes was formulated.
MS310 Quantum Physical Chemistry