molecules and electronic, vibrational and rotational structurewimu/educ/biaqp-mol-phys-2015.pdf ·...
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Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Molecules and electronic, vibrational and rotational structure
Max Born Nobel 1954
Robert Oppenheimer Gerhard Herzberg Nobel 1971
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Hamiltonian for a molecule
rRVMm
H A
A Ai
i
,22
22
22
ji ijBA BA
BA
iA Ai
A
r
e
RR
eZZ
r
eZrRV
0
2
0
2
, 0
2
444,
i refers to electrons, A to nuclei; Potential energy terms:
Assume that the wave function of the system is separable and can be written as:
RRrRr iAi
nucelmol ;,
Assume that the electronic wave function can be calculated for a particular R
Rri
;el
RrRRRr iiii
;; el
2nucnucel
2 Then:
el2
nucelnuc2
elnucel2 2 AAAAA
Born-Oppenheimer: the derivative of electronic wave function w.r.t nuclear coordinates is small: Nuclei can be considered stationary. Then: Separation of variables is possible. Insert results in the Schrödinger equation:
0el A
nuc2
elnucel2 AA
nucelmolnucelmol EH
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
moltotalnuc2
2
0
2
el
el
, 0
2
0
22
2
nucmol
24
442
EMRR
eZZ
r
eZ
r
e
mH
BA A
AABA
BA
iA Ai
A
ji iji
i
Separation of variables in the molecular Hamiltonian
The wave function for the electronic part can be written separately and “solved”; consider this as a problem of molecular binding.
RrERrr
eZ
r
e
mii
iA Ai
A
ji iji
i
;;
442elelel
, 0
2
0
22
2
Solve the electronic problem for each R and insert result Eel in wave function. This yields a wave equation for the nuclear motion:
nuctotalnuc0
22
2
)(42
ERERR
eZZ
MBA
elBA
BA
A
AA
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Schrodinger equation for the nuclear motion
The previous analysis yields:
nuctotalnuc0
22
2
)(42
ERERR
eZZ
MBA
elBA
BA
A
AA
This is a Schrödinger equation with a potential energy:
RERR
eZZRV
BA BA
BA
el0
2
4
nuclear repulsion chemical binding
Typical potential energy curves in molecules
Now try to find solutions to the Hamiltonian for the nuclear motion
RERRVRM
A
AA
nucnucnuc
22
)(2
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
BA
BA
MM
MM
Single-particle Schrödinger equation
RERRVRR
nucnucnuc
2
)(2
Consider the similarity and differences between this equation and that of the H-atom: - interpretation of the wave function - shape of the potential
Laplacian:
2
2
22
2
2
sin
1sin
sin
1
1
RR
RR
RRR
Angular part is the well-know equation with solutions: Angular momentum operators Spherical harmonic wave functions !
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Angular momentum in a molecule
MNMMNN
MNNNMNN
z ,,
,)1(, 22
Solution:
with
NNNM
N
,...,1,
...3,2,1,0
And angular wave function
,, NMYMN
Hence the wave function of the molecule:
,,,nuc NMYRR
Reduction of molecular Schrödinger equation
)()()(2
1
2nucrotvib,nuc
2
2
2
2
2
RERRVNRR
RRR
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: 0)( eRVRV
All derivates yield zero R
)()()(2
1
2nucrotvib,nuc
2
2
2
2
2
RERRVNRR
RRR
Insert in:
)()(2
1nucrotvib,nuc
2
2RERN
Re
So quantized motion of rotation: )1(2
)1(
2
2rot
NBN
R
NNE
e
With B the rotational constant
Deduce Re from spectroscopy isotope effect
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Vibrational motion
)()()(2
1
2nucrotvib,nuc
2
2
2
2
2
RERRVNRR
RRR
Non-rotation: N=0
Insert : R
RQR
)(
)()()(2
vib2
22
RQERQRVdR
d
Make a Taylor series expansion around r=R-Re
...2
1)()( 2
2
2
rr
ee RR
edR
Vd
dR
dVRVRV
0)( eRV by choice
0
eRdR
dV at the bottom of the well
Hence: 2)( eRRkRV harmonic potential
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Vibrational motion
)()(2
1
2vib
2
2
22
rrrr
QEQkd
d
So the wave function of a vibrating molecule resembles the 1-dimensional harmonic oscillator, solutions:
rr
r v
v
v Hv
Q
2
4/1
4/12/
2
1exp
!
2)(
with:
e and
ke
Energy eigenvalues:
2
1
2
1vib v
kvE e
isotope effect
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Finer details of the rovibrational motion
Centrifugal distortion:
22rot )1()1( NDNNBNE
...2
1
2
12
vib
vxvE eee
Anharmonic vibrational motion
Dunham expansion:
1
,
12
1
NNvYE l
k
lk
klvN
Vibrational energies in the H2-molecule
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Energy levels in a molecule: general structure
Rovibrational structure superimposed on electronic structure
J v=0
v=1
v=2
J v=0
v=1
v=2
A
B
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Radiative transitions in molecules
The dipole moment in a molecule:
A
A
A
i
iNe ReZre
In a molecule, there may be a: - permanent or rotational dipole moment - vibrational dipole moment
2
2
2
02
1rr
dR
d
dR
d
eRN
In atoms only electronic transitions, in molecules transitions within electronic state
RRrRr iAi
vibelmol ;,
Dipole transition between two states
dif "'
Rdrd
Rdrd
d
N
e
Neif
"vib
'vib
"el
'el
"vib
'vib
"el
'el
"vib
"el
'vib
'el
Two different types of transitions
Electronic transitions
Rovibrational transitions
2
20
2
3ij
eB
Note for transitions: Einstein coefficient
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
The Franck-Condon principle for electronic transitions in molecules
Rdrdeif
"vib
'vib
"el
'el
1st term:
Only contributions if (parity selection rule)
"el
'el
Franck-Condon approximation: The electronic dipole moment independent of internuclear separation:
rdRM ee"
el'el)(
RdRMeif
"vib
'vib)(
Hence
Intensity of electronic transitions
22"vib
'vib
2"' vvRdI if
Intensity proportional to the square of the wave function overlap
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Rotational transitions in molecules
NMMNif ''
2nd term in the transition dipole
Projection of the dipole moment vector on the quantization axis. For rotation take the permanent dipole.
mY100
cos
sinsin
cossin
MmM
NNNN
dYYY NMmMNif
'
1'
000
1'
1''0
Selection rules
1,0
1
M
N
Purely rotational spectra Level energies:
22 )1()1( NNDNNBF vvv
Transition frequencies:
2222 )1"(")1'('
)1"(")1'('
)"('
NNNND
NNNNB
NFNFv
v
v
vv
Ground state with N” and excited N’ Absorption in rotational ladder: N’=N”+1
3abs )1"(4)1"(2 NDNBv vv
spacing between lines ~ 2B
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Rotational spectrum in a diatomic molecule
Ground state with N” and excited N’ Absorption in rotational ladder: N’=N”+1
3abs )1"(4)1"(2 NDNBv vv
spacing between lines ~ 2B
In an absorption spectrum: R-lines In an emission spectrum: P-lines
l
Homonuclear molecule
0)( AAAA
A
AN RReZReZ
For permanent dipole No rotational spectrum
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Vibrational transitions in molecules
Rdrd Nif
"vib
'vib
"el
'el
2nd term in the transition dipole
Within a certain electronic state:
"el
'el 1"
el'el rd
"' vib vvif
Permanent and induced dipole moments;
2
2
2
02
1rr
dR
d
dR
d
eRN
Line intensity:
Permanent dipole does not produce a vibrational spectrum
0"'"' 00 vvvvif
Wave functions for one electronic state are orthogonal.
"'"' vvvdR
dv
eRif rr
Dipole moment should vary with Displacement vibrational spectrum
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Vibrational transitions in molecules
"'"' 2vib vbavvvif rr
Permanent and induced dipole moments; In the harmonic oscillator approximation;
1,1,2
1
2nknk
kn
nn
dQQkn
rrrrr
Selection rules: 1' vvv
Higher order transitions (overtones) from: - anharmonicity in potential - induced dipole moments
First order: the vibrating dipole moment
"'"' vib vvvvif r
Homonuclear molecule
0)( AAAA
A
AN RReZReZ
For all derivatives No vibrational dipole spectrum
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Rovibrational spectra in a diatomic molecule
)()( NFvGT v
Level energies:
22 )1()1( NNDNNBF vvv
with:
2
2
1
2
1)(
vxvvG eee
Transitions from v” (ground) to v’ (excited)
")'("' "'0 NFNFvv vv
with ")'(0 vGvG
the band origin; the rotationless transition (not always visible)
Transitions R-branch (N’=N”+1) – neglect D
2"'"''0 32 NBBNBBB vvvvvR
For "' vv BB
12 '0 NBvR
P-branch (N’=N”-1) – neglect D
2"'"'0 NBBNBB vvvvP
For "' vv BB
12 '0 NBvR
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Rovibrational spectra
2"'"''0 32 NBBNBBB vvvvvR
2"'"'0 NBBNBB vvvvP
01
2
3
01
2
3
0
R0
R1
R2
P1
P2
P3
"
'
R-branch P-branch
N"
N’
More precisely spacing between lines:
'"' 23)()1( vvvRR BBBNN
'"' 2)()1( vvvPP BBBNN
If, as usual: "' vv BB
Rotational constant in excited state is smaller.
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
J v=0
v=1
v=2
J v=0
v=1
v=2
A
B
Vibrations governed by the Franck-Condon principle Rotations governed by angular momentum selection rules Transition frequencies
TB
TA
)'(')'('' NFvGTT vB
)"(")"("" NFvGTT vA
"' TTv
R and P branches can be defined in the same way
2"'"''0 32 NBBNBBB vvvvvR
2"'"'0 NBBNBB vvvvP
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Band head structures and Fortrat diagram
2"'"''0 32 NBBNBBB vvvvvR
2"'"'0 NBBNBB vvvvP
Define:
1 Nm for the R-branch
Nm for the P-branch
2"'"'0 mBBmBB vvvv
then for both branches:
if: "' vv BB
20 mm
A parabola represents both branches
- no line for m=0 ; band gap - there is always a band head, in one branch
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Population distributions; vibrations
kT
v
v
kTvE
kTvE
e
eZ
e
evP
)2/1(
/)(
/)(
1
)(
Probability of finding a molecule in a vibrational quantum state:
Note: not always thermodynamic equilibrium
P(v)/P(0)
Boltzmann distribution
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Population distributions; rotational states in a diatomic molecule
22 )1()1(
'
/
/
)12(1
)1'2(
12)(
JDJJBJ
rot
J
kTE
kTE
eJZ
eJ
eJJP
rot
rot
Probability of finding a molecule in a rotational quantum state:
Find optimum via
0)(
dJ
JdP
HCl T=300 K
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
K sensitivity coefficients to -variation for Lyman- transition
KDefinition of sensitivity coefficient:
Calculation for Lyman- transition
e
red
mcR
h
EE
4
312
1/1
/1
ep
ep
ep
ep
ee
red
mM
mM
mM
mM
mmwith
So (note energy scale drops out !):
K
EE
EE
1
/
1/
11
12
12
4104.5~1
1
K
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Isotope effects in molecules + sensitivity for -variation
Born-Oppenheimer: the derivative of electronic wave function w.r.t nuclear coordinates is small:
0el A
Electronic wave functions and energies do not depend on nuclear masses (compare the case of the atom)
Electronic
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
2
1vib v
kE
Vibrational energy:
K-coefficient for purely vibrational transition (overtone included):
K
mn
mnmn
EE
EE
mn
mn
12
11
12/12/11
2/12/11
2/12/11
So: 2
1K For ALL vibrational transitions
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Isotope effects in molecules + sensitivity for -variation
Rotational energy:
K-coefficient for purely rotational transition :
K
1K
2
2
rot2
)1(
eR
NNE
So:
12
C
Cd
dK
C
NNNNRe
)1()1(2
11222
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Isotope effects in molecules + sensitivity for -variation
Eelectronic
Evibrational
Erotational
Etotal Lyman bands B1S+
u - X1S+g
Werner bands C1Pu - X1S+
g
94 nm 110 nm
For H2
Evibrational
Erotational
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
anchor blue shifter red shifter
Ki different for H2 lines
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Electronic spectra of H2
Composition of the universe: 80 % hydrogen H/H2
20 % helium <0.1% other elements
H2
H (Lyman-) ~ 121 nm
2p
2p
H2, Lyman en Werner BANDS ~90 - 110 nm Extreme Ultraviolet Wavelengths
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
QSO 12 Gyr ago
Lab today
90-112 nm ~275-350 nm
Cosmological reshift i
i
i z10l
l
Empirical search for a change in
• Spectroscopy
• Compare H2 spectra in different epochs:
Atmospheric transmission only for z>2
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Quasars: Ultrazwakke objecten Q2348-011 z = 2.42 Magnitude 18.4 ESO-VLT 2007
1 arcsecond
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
H2
J2123 from HIRES-Keck at Hawaii (normalized)
Resolution 110000 ; zabs=2.0593
Keck telescope, Hawaii
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
/ = (5.6 ± 5.5stat ± 2.9syst) x 10-6
/ = (8.5 ± 3.6stat ± 2.2syst) x 10-6
Keck:
VLT:
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
How much H2 at high redshift?
VLT 2013
Done 2012
? (+ CO)
? Analysis
Analysis
Done Done Done Done
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Cerro Amazones
The rise of a giant
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
Calculations Extreme shifters
Outlook: More sensitive molecules
Quantum tunneling
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
48372.4558 MHz; K=-1 48376.892 MHz; K=-1
12178.597 MHz; K=-33 60531.1489 MHz; K=-7
Extreme shifters; towards radio astronomy
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Effelsberg Radio Telescope
Lecture Notes Fundamental Constants 2015: Molecular Physics; W. Ubachs
Effelsberg Radio Telescope
K=-33
K=-1
K=-7
PKS-1830-211
at z=0.88582 (7 Gyrs look-back)