Electronic transitions

Download Report

Transcript Electronic transitions

Lecture 14: Molecular structure
o Rotational transitions
o Vibrational transitions
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
o Electronic transitions
QuickTime™ and a
TIFF (Uncompressed) decompre
are needed to see this pictur
PY3P05
Bohn-Oppenheimer Approximation
o
Born-Oppenheimer Approximation is the assumption that the electronic motion and the
nuclear motion in molecules can be separated.
o
This leads to molecular wavefunctions that are given in terms of the electron positions (ri) and
the nuclear positions (Rj):
 molecule(rˆi , Rˆ j )   electrons( rˆi , Rˆ j ) nuclei (Rˆ j )
o
Involves the following assumptions:

o Electronic wavefunction depends on nuclear positions but not upon their velocities, i.e.,
the nuclear motion is so much slower than electron motion that they can be considered to
be
fixed.
o The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fastmoving electrons.
PY3P05
Molecular spectroscopy
o
Electronic transitions: UV-visible
o
Vibrational transitions: IR
o
Rotational transitions: Radio
E
Electronic
Vibrational
Rotational
PY3P05
Rotational motion
o
Must first consider molecular moment of inertia:
I   mi ri2
i
o
At right, there are three identical atoms bonded to
“B” atom and three different atoms attached to “C”.

o
Generally specified about three axes: Ia, Ib, Ic.
o
For linear molecules, the moment of inertia about the
internuclear axis is zero.
o
See Physical Chemistry by Atkins.
PY3P05
Rotational motion
o
Rotation of molecules are considered to be rigid rotors.
o
Rigid rotors can be classified into four types:
o Spherical rotors: have equal moments of intertia (e.g., CH4, SF6).
o Symmetric rotors: have two equal moments of inertial (e.g., NH3).
o Linear rotors: have one moment of inertia equal to zero (e.g., CO2, HCl).
o Asymmetric rotors: have three different moments of inertia (e.g., H2O).
PY3P05
Quantized rotational energy levels
o
The classical expression for the energy of a rotating body is:
E a  1/2Ia a2
where a is the angular velocity in radians/sec.

E  1/2Ia a2  1/2Ib b2  1/2Ic c2
o
For rotation about three axes:
o
In terms of angular momentum (J = I):

o
o
J a2
J b2
J c2
E


2Ia 2Ib 2Ic
We know from QM that AM is quantized:

J(J 1)
Therefore, E J 
2I
J  J(J  1)
2
, J = 0, 1, 2, …
, J = 0, 1, 2, …


PY3P05
Quantized rotational energy levels
o
Last equation gives a ladder of energy levels.
o
Normally expressed in terms of the rotational constant,
which is defined by:
hcB 
o
2I
 B 
4 cI
Therefore, in terms of a rotational term:

o
2
F(J)  BJ(J 1) cm-1
The separation between adjacent levels is therefore

F(J) - F(J-1) = 2BJ
o
As B decreases with increasing I =>large molecules
have closely spaced energy levels.
PY3P05
Rotational spectra selection rules
o
Transitions are only allowed according to selection rule for
angular momentum:
J = ±1
o
Figure at right shows rotational energy levels transitions and
the resulting spectrum for a linear rotor.
o
Note, the intensity of each line reflects the populations of the
initial level in each case.
PY3P05
Molecular vibrations
o
Consider simple case of a vibrating diatomic molecule,
where restoring force is proportional to displacement
(F = -kx). Potential energy is therefore
V = 1/2 kx2
o
Can write the corresponding Schrodinger equation as
QuickTime™ and a
Graphics decompressor
are needed to see this picture.
d 2
 [E  V ]  0
2 dx 2
2
d 2
 [E 1/2kx 2 ]  0
2
2 dx
2
where
o

m1m2
m1  m2

The SE results in allowed energies
k 
   
 
1/ 2
Ev  (v 1/2) 

v = 0, 1, 2, …
PY3P05

Molecular vibrations
o
The vibrational terms of a molecule can therefore
be given by
G(v)  (v  1/2)v˜
1 k 
v˜ 
 
2c  
1/ 2

o
Note, the force constant is a measure of the
curvature of the potential energy close to the

equilibrium extension of the bond.
o
A strongly confining well (one with steep sides, a
stiff bond) corresponds to high values of k.
PY3P05
Molecular vibrations
o
The lowest vibrational transitions of diatomic
molecules
approximate
the
quantum
harmonic oscillator and can be used to imply
the bond force constants for small
oscillations.
o
Transition occur for v = ±1
o
This potential does not apply to energies
close to dissociation energy.
o
In fact, parabolic potential does not allow
molecular dissociation.
o
Therefore
oscillator.
more
consider
anharmonic
PY3P05
Anharmonic oscillator
o
A molecular potential energy curve can be
approximated by a parabola near the bottom of the
well. The parabolic potential leads to harmonic
oscillations.
o
At high excitation energies the parabolic
approximation is poor (the true potential is less
confining), and does not apply near the
dissociation limit.
o
Must therefore use a asymmetric potential. E.g.,
The Morse potential:
V  hcDe 1 ea(R R e ) 
2
where De is the depth of the potential minimum
and

  2 1/ 2
a  

2hcDe 
PY3P05

Anharmonic oscillator
o
The Schrödinger equation can be solved for the Morse potential, giving permitted energy
levels:
G(v)  (v  1/2) v˜  (v˜  1/2) 2 x e v˜
where xe is the anharmonicity constant:
o
o

a2
xe 
2
The second term in the expression for G increases with v => levels converge at high quantum
numbers.

The number of vibrational levels for a Morse
oscillator is finite:
v = 0, 1, 2, …, vmax
PY3P05
Vibrational-rotational spectroscopy
o
Molecules vibrate and rotate at the same time =>
S(v,J) = G(v) + F(J)
S(v,J)  (v  1/2)v˜  BJ(J  1)
o
Selection rules obtained by combining rotational
selection rule ΔJ = ±1 with vibrational rule Δv = ±1.

o
o
When vibrational transitions of the form v + 1  v
occurs, ΔJ = ±1.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Transitions with ΔJ = -1 are called the P branch:
v˜P (J)  S(v 1,J 1)  S(v,J)  v˜  2BJ
o

o

Transitions with ΔJ = +1 are called the R branch:
v˜R (J)  S(v 1,J 1)  S(v,J)  v˜  2B(J 1)
Q branch are all transitions with ΔJ = 0
PY3P05
Vibrational-rotational spectroscopy
o
Molecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 –
4000cm-1 0.01 to 0.5 eV).
o
Vibrational transitions accompanied by rotational transitions. Transition must produce a
changing electric dipole moment (IR spectroscopy).
Q branch
P branch
R branch
PY3P05
Electronic transitions
o
Electronic transitions occur between molecular
orbitals.
o
Must adhere to angular momentum selection
rules.
o
Molecular orbitals are labeled, , , , …
(analogous to S, P, D, … for atoms)
o
o
For atoms, L = 0 => S, L = 1 => P
For molecules,  = 0 => ,  = 1 => 
o
Selection rules are thus
 = 0, 1, S = 0, =0,  = 0, 1
o
Where  =  +  is the total angular momentum
(orbit and spin).
PY3P05
The End!
o
All notes and tutorial set available from
http://www.physics.tcd.ie/people/peter.gallagher/lectures/py3004/
o
Questions? Contact:
o [email protected]
o Room 3.17A in SNIAM
PY3P05