Molecules and electronic, vibrational and rotational structure
Max Born
Nobel 1954
Robert Oppenheimer
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Gerhard Herzberg
Nobel 1971
Hamiltonian for a molecule
 
 
2
2 2
2
H 
  i   2 M  A  V R, r
2m i
A
A
Born-Oppenheimer: the derivative
of electronic wave function w.r.t
nuclear coordinates is small:
i refers to electrons, A to nuclei;
 A el  0
Potential energy terms:
 
 
Z Ae 2
Z AZ Be2
e2
V R, r   
 

4

r
4

R

R
0 Ai A B
0 A
B i  j 4 0 rij
A,i
Nuclei can be considered stationary.
Then:
2A el  nuc   el 2A  nuc
Assume that the wave function of the system is
separable and can be written as:


 
Separation of variables is possible.


 
 
mol ri , RA   el ri ; R  nuc R
Assume that the electronic wave function
can be calculated for a particular R
Then:
 


 

 el ri ; R
 


 
 
i2 el ri ; R  nuc R   nuc R i2 el ri ; R
2A el  nuc   el 2A  nuc  2 A el  A  nuc   2A el
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Insert results in the Schrödinger
equation:
H mol el  nuc  Emol el  nuc
Separation of variables in the molecular Hamiltonian
  2
e2
Z Ae 2 
2
Hmol   nuc 
i  

 el 

4 0 rij A,i 4 0 rAi 
 2m i
i j



Z AZ Be2
2
 el  

 2A   nuc  Etotal mol
2M A
 A B 4 0 R A  RB

A
The wave function for the electronic part can be
written separately and “solved”; consider this as
a problem of molecular binding.

 
 
e2
Z Ae 2 
 2

2

 i   4 r   4 r  el ri ; R  Eel el ri ; R
2
m
0 ij A,i
0 Ai 

i
i j


 
 
Solve the electronic problem for each R and insert
result Eel in wave function.
This yields a wave equation for the nuclear motion:

 
2
Z AZ B e2


2
A  
 Eel ( R) nuc  Etotal  nuc
 
4 0 RA  RB


A B
 A 2M A

Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Schrodinger equation for the nuclear motion
The previous analysis yields:

 
2
Z AZ B e2


2
A  
 Eel ( R) nuc  Etotal  nuc
 
4 0 RA  RB


A B
 A 2M A

This is a Schrödinger equation with a
potential energy:
   4Z ARZ Be R
0 A
B

V R 
2
A B
Typical potential energy curves
in molecules


 Eel R
nuclear repulsion chemical binding
Now try to find solutions to the
Hamiltonian for the nuclear motion







2
2

 A  nuc R  V ( R)  nuc R  E nuc R
2
M
A
A
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Quantized motion in a diatomic molecule
Quantummechanical two-particle problem
Transfer to centre-of-mass system

 R 
M AM B
M A  MB
1   2  
R

2 R 
R 
R
 
 
1
2

 sin 
 2
2





 R sin   2
R sin 
1
Single-particle Schrödinger equation

Laplacian:






2 

  nuc R  V ( R)  nuc R  E nuc R
2 R
Consider the similarity and differences
between this equation and that of the
H-atom:
- interpretation of the wave function
- shape of the potential
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Angular part is the well-know equation
with solutions:
Angular momentum operators
Spherical harmonic wave functions !
Angular momentum in a molecule
Solution:
2
N N , M   2 N ( N  1) N , M
And angular wave function
N , M  YNM  ,  
N z N , M  M N , M
Hence the wave function of the molecule:
with
N  0,1,2,3...
M   N , N  1,..., N
 nuc R, ,    RYNM  ,  
Reduction of molecular Schrödinger equation
 2   2  

1 2
N  V ( R)  nuc ( R)  Evib,rot  nuc ( R)

R

2 R 
2

R
 2R
 2R

Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Eigenenergies of a “Rigid Rotor”
Rigid rotor, so it is assumed that R = Re = constant
Choose:
V R   V ( Re )  0
All derivates
 yield zero
R
Insert in:
 2   2  

1 2
N  V ( R)  nuc ( R)  Evib,rot  nuc ( R)

R

2 R 
2

R
 2R
 2R

 1  2
N   nuc ( R)  Evib,rot  nuc ( R)

2
 2Re

So quantized motion of rotation: Erot   2 N ( N  1)  BN ( N  1)
2Re2
With B the rotational constant
Deduce Re from spectroscopy
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
isotope effect
Vibrational motion
 2   2  

1 2
N  V ( R)  nuc ( R)  Evib,rot  nuc ( R)

R

2 R 
2

R
 2R
 2R

Non-rotation: N=0
Insert :
R  
 2 d 2

 V ( R)Q( R)  EvibQ( R)

2
2

dR


Q( R )
R
Make a Taylor series expansion around r=R-Re
dV
1 d 2V
2
V ( R)  V ( Re ) 
r
r
 ...
dR R
2 dR 2
e
Re
V ( Re )  0
by choice
dV
0
dR R
e
at the bottom of the well
Hence:
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
V ( R)  k R  Re 2
harmonic potential
Vibrational motion
 2 d 2 1 2 
 kr Q( r )  EvibQ( r )

2 2
2

dr


So the wave function of a vibrating molecule
resembles the 1-dimensional harmonic
oscillator, solutions:
Qv ( r ) 

2v / 2  1 / 4
with:  
1

exp  r 2  H v  r
2

v! 1 / 4
e

and
e 

k

Energy eigenvalues:
1
k
1

Evib  e  v    
v  
2
  2

isotope effect
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Finer details of the rovibrational motion
Centrifugal distortion:
Erot  BN ( N  1)  DN 2 ( N  1) 2
Anharmonic vibrational motion
2
1
1


Evib  e  v    e xe  v    ...
2
2


Dunham expansion:
k
1

EvN   Ykl  v   N l N  11
2

k ,l
Vibrational energies in the H2-molecule
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Energy levels in a molecule: general structure
v=2
B
v=1
J
v=0
v=2
A
v=1
J
v=0
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Rovibrational
structure
superimposed on
electronic
structure
Radiative transitions in molecules
The dipole moment in a molecule:


  e   N   eri   eZ A RA
i
A
In a molecule, there may be a:
- permanent or rotational
dipole moment
- vibrational dipole moment
1  d 2   2
 d 
 N  0     r 
 r
2


dR
2

 Re
 dR



In atoms only electronic transitions,
in molecules transitions within
electronic state
Note for transitions:
2
e 2
B


Einstein coefficient
2 ij
3 0
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs


 


 
 
mol ri , RA   el ri ; R  vib R
Dipole transition between two states
if   '" d
Two different types of transitions
'
"
e   N  el"  vib
if   el'  vib
d 

 

' " 
'
"
 el el dr  vib N vibdR

'
"  '
"
 el e el dr  vib vibdR 
Electronic transitions
Rovibrational transitions
The Franck-Condon principle for electronic transitions in molecules
1st term:
if  
Intensity of electronic transitions



'
"  '
"
 el e el dr  vib vibdR
I  if
2

2
'
"
 vib vibdR

 v' v"
2
Only contributions if (parity selection rule)
"
 el'   el
Franck-Condon approximation:
The electronic dipole moment independent
of internuclear separation:
" 
M e ( R)   el' e el
dr
Hence

'
"
if  M e ( R)  vib vibdR

Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Intensity proportional to the square
of the wave function overlap
Rotational transitions in molecules
2nd term in the transition dipole
if  N 'M '  NM
Projection of the dipole moment vector
on the quantization axis.
For rotation take the permanent dipole.
sin  cos  

  0  sin  sin    0Y1m
 cos  
if  0   YN ' M 'Y1mYNM d
 N ' 1 N  N ' 1 N 


 
0
0
0
M
'
m
M



Selection rules
N  1
M  0,1
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Purely rotational spectra
Level energies:
Fv  Bv N ( N  1)  Dv N 2 ( N  1) 2
Transition frequencies:
v  Fv N '  Fv ( N " ) 
Bv N ' ( N '1)  N " ( N "1)

 Dv N '2 ( N '1) 2  N "2 ( N "1) 2

Ground state with N” and excited N’
Absorption in rotational ladder: N’=N”+1
vabs  2Bv ( N "1)  4Dv ( N "1)3
spacing between lines ~ 2B
Rotational spectrum in a diatomic molecule
Ground state with N” and excited N’
Absorption in rotational ladder: N’=N”+1
vabs  2Bv ( N "1)  4Dv ( N "1)3
spacing between lines ~ 2B
In an absorption spectrum: R-lines
In an emission spectrum: P-lines
Homonuclear molecule



 N   eZ A RA  eZ A ( RA  RA )  0
A
For permanent dipole
No rotational spectrum
l
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Vibrational transitions in molecules
2nd term in the transition dipole


"
'
"
if   el'  el
dr  vib
 N vib
dR
Permanent dipole does not produce a
vibrational spectrum
if  v' 0 v"  0 v' v"  0
Within a certain electronic state:
 el'
"
  el
" 
 el'  el
dr  1

Line intensity:
if  v' vib v"
Permanent and induced dipole moments;
1  d 2   2
 d 
 N  0     r 
 r
2  dR 2 
 dR  Re


Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Wave functions for one electronic state
are orthogonal.
 d 
  r v"  v' r v"
 dR  Re
if  v' 
Dipole moment should vary with
Displacement  vibrational spectrum
Vibrational transitions in molecules
Permanent and induced dipole moments;
if  v' vib v"  v' ar  br v"
2
First order: the vibrating dipole
moment
In the harmonic oscillator approximation;
n r k   Qn r rQk r dr 

  n
n 1




k , n 1
k , n 1 
  2
2

Selection rules: v  v'v  1
if  v' vib v"  v' r v"
Homonuclear molecule



 N   eZ A RA  eZ A ( RA  RA )  0
A
For all derivatives
No vibrational dipole spectrum
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Higher order transitions (overtones) from:
- anharmonicity in potential
- induced dipole moments
Rovibrational spectra in a diatomic molecule
Level energies:
T  G(v)  Fv ( N )
with:
Fv  Bv N ( N  1)  Dv N 2 ( N  1) 2
2
1
1


G(v)  e  v    e xe  v  
2
2


Transitions
R-branch (N’=N”+1) – neglect D
 R   0  2Bv'  3Bv'  Bv" N  Bv'  Bv" N 2
For
 R   0  2Bv' N  1
Transitions from
v” (ground) to v’ (excited)
 v'v"   0  Fv' ( N ' )  Fv" N "
with
 0  G(v' )  Gv"
the band origin; the rotationless
transition (not always visible)
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Bv'  Bv"
P-branch (N’=N”-1) – neglect D
 P   0  Bv'  Bv" N  Bv'  Bv" N 2
For
Bv'  Bv"
 R   0  2Bv' N  1
Rovibrational spectra
 R   0  2Bv'  3Bv'  Bv" N  Bv'  Bv" N 2
 P   0  Bv'  Bv" N  Bv'  Bv" N 2
More precisely spacing between lines:
 R ( N  1)   R ( N )  3Bv'  Bv"  2Bv'
 P ( N  1)   P ( N )  Bv'  Bv"  2Bv'
3
2 N’
01
'
R-branch
R2
R1
P1
R0
P2
P3
0
Bv'  Bv"
Rotational constant in excited state
is smaller.
P-branch
3
"
If, as usual:
2 N"
1
0
Spacing in P branch is larger
Band head formation in R-branch
Spacing between R(0) and P(1) is 4B
“band gap”
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Example: rovibrational spectrum of HCl; fundamental vibration
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Example: rovibrational spectrum of HBr; fundamental vibration
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Rovibronic spectra
Vibrations  governed by the
Franck-Condon principle
v=2
B
v=1
TB
J
Rotations  governed by angular momentum
selection rules
Transition frequencies
v=0
v  T 'T "
T '  TB  G' (v' )  Fv ' ( N ' )
T "  TA  G" (v" )  Fv " ( N " )
R and P branches can be defined
in the same way
v=2
A
TA
v=1
J
v=0
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
 R   0  2Bv'  3Bv'  Bv" N  Bv'  Bv" N 2
 P   0  Bv'  Bv" N  Bv'  Bv" N 2
Band head structures and Fortrat diagram
 R   0  2Bv'  3Bv'  Bv" N  Bv'  Bv" N 2
 P   0  Bv'  Bv" N  Bv'  Bv" N 2
Define:
m  N 1
m  N
for the R-branch
for the P-branch
then for both branches:
   0  Bv'  Bv" m  Bv'  Bv" m2
if:
Bv'  Bv"
   0  m  m2
A parabola represents both
branches
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
- no line for m=0 ; band gap
- there is always a band head, in one branch
Population distributions; vibrations
Probability of finding a molecule
in a vibrational quantum state:
P (v ) 
P(v)/P(0)
e  E (v ) / kT
 e E (v) / kT
v

1
Z
e (v 1 / 2)
kT
e
Boltzmann distribution
Note: not always thermodynamic equilibrium
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Population distributions; rotational states in a diatomic molecule
Probability of finding a molecule
in a rotational quantum state:
P( J ) 
2 J  1e Erot / kT
 (2 J '1)e Erot / kT
J'

1
Z rot
(2 J  1)e
 BJ ( J 1)  DJ 2 ( J 1) 2
Find optimum via
dP( J )
0
dJ
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
HCl
T=300 K
K sensitivity coefficients to -variation for Lyman- transition
Definition of sensitivity coefficient:


 K


Calculation for Lyman- transition


E2  E1 3
 R c red
h
4
 me



with
red
me

M p / me
1  M p me 



me  M p  me  1  M p / me 1  
So (note energy scale drops out !):
  


 E2  E1  1     1  
 / 




 K

E2  E1
 / 1   
1    

K 
1
~ 5.4 104
1    
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Isotope effects in molecules + sensitivity for -variation
Electronic
Born-Oppenheimer: the derivative of electronic wave function
w.r.t nuclear coordinates is small:
 A el  0
Electronic wave functions and energies do not depend on nuclear masses
(compare the case of the atom)
Mass dependences
In the above mass dependences expressed as “reduced mass”;
Note that we assume:
  red
Proportionality with “baryonic mass” (neutrons and protons)
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Isotope effects in molecules + sensitivity for -variation
Vibrational energy:
Evib  
k
1
v  
  2
K-coefficient for purely vibrational transition (overtone included):
 En  Em 



En  Em
 1
1
n  1 / 2  m  1 / 2  1 n  1 / 2  m  1 / 2
  



1
1 n  1 / 2  m  1 / 2
  

1 

1  K
2 

So:
K  
1
2
For ALL vibrational transitions
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Isotope effects in molecules + sensitivity for -variation
Rotational energy:
Erot   2
N ( N  1)
2Re2


C


N
(
N

1
)

N
(
N

1
)

2
2
1
1
2Re2

K-coefficient for purely rotational transition :


So:
 K


K 
   d


   C 2   1
   d
C
K   1
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Isotope effects in molecules + sensitivity for -variation
For H2
Erotational
Evibrational
Etotal
Lyman bands
B1S+u - X1S+g
Eelectronic
Erotational
Werner bands
C1Pu - X1S+g
Evibrational
94 nm
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
110 nm
Ki different for H2 lines
red shifter
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
anchor blue shifter
Electronic spectra of H2
Composition of the universe:
80 % hydrogen H/H2
20 % helium
<0.1% other elements
H2
2p
2p
H (Lyman-) ~ 121 nm
H2, Lyman en Werner BANDS
~90 - 110 nm
Extreme Ultraviolet Wavelengths
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Empirical search for a change in 
• Spectroscopy
• Compare H2 spectra in different epochs:
Lab
QSO
today
12 Gyr ago
90-112 nm
~275-350 nm
li
 1  zi
0
li
Cosmological reshift
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Atmospheric transmission
only for z>2
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Quasars: Ultrazwakke objecten
Q2348-011
z = 2.42
Magnitude
18.4
1 arcsecond
ESO-VLT
2007
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
UVES:
Ultraviolet –
Visual
Echelle
Spectrograph
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Empirical search for a change in : quasars
H2
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
J2123 from HIRES-Keck at Hawaii
(normalized)
Resolution 110000 ; zabs=2.0593
H2
Keck telescope, Hawaii
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Keck:
/ = (5.6 ± 5.5stat ± 2.9syst) x 10-6
VLT:
/ = (8.5 ± 3.6stat ± 2.2syst) x 10-6
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
How much H2 at high redshift?
Done
Done
Done
Done
Analysis
Analysis
?
? (+ CO)
VLT 2013
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Done 2012
The rise of a giant
Cerro Amazones
39m
dish
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Outlook: More sensitive molecules
Quantum tunneling
K = -4.2
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Outlook: More sensitive molecules
Quantum tunneling
Calculations
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Extreme shifters
Extreme shifters; towards radio astronomy
48372.4558 MHz; K=-1
48376.892 MHz; K=-1
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
12178.597 MHz; K=-33
60531.1489 MHz; K=-7
Effelsberg Radio Telescope
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Effelsberg Radio Telescope
PKS-1830-211
at z=0.88582
(7 Gyrs look-back)
K=-33
K=-1
K=-7
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs