molecules and electronic, vibrational and rotational structurewimu/educ/biaqp-mol-phys-2015.pdf ·...

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


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


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


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


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 !


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


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


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


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


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


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


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


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


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


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


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


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


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


Rovibrational spectra


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”


Example: rovibrational spectrum of HCl; fundamental vibration


Example: rovibrational spectrum of HBr; fundamental vibration


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 NBBNBB vvvvP


Band head structures and Fortrat diagram


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


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


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


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


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)



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



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



Eelectronic

Evibrational

Erotational

Etotal Lyman bands B1S+

u - X1S+g

Werner bands C1Pu - X1S+

g

94 nm 110 nm

For H2

Evibrational

Erotational


anchor blue shifter red shifter

Ki different for H2 lines


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


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


Quasars: Ultrazwakke objecten Q2348-011 z = 2.42 Magnitude 18.4 ESO-VLT 2007

1 arcsecond


UVES: Ultraviolet – Visual Echelle Spectrograph


Empirical search for a change in : quasars

H2


H2

J2123 from HIRES-Keck at Hawaii (normalized)

Resolution 110000 ; zabs=2.0593

Keck telescope, Hawaii


/ = (5.6 ± 5.5stat ± 2.9syst) x 10-6

/ = (8.5 ± 3.6stat ± 2.2syst) x 10-6

Keck:

VLT:


How much H2 at high redshift?

VLT 2013

Done 2012

? (+ CO)

? Analysis

Analysis

Done Done Done Done


Cerro Amazones

The rise of a giant

39m dish


Outlook: More sensitive molecules

Quantum tunneling

K = -4.2


Calculations Extreme shifters

Outlook: More sensitive molecules

Quantum tunneling


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


Effelsberg Radio Telescope


Effelsberg Radio Telescope

K=-33

K=-1

K=-7

PKS-1830-211

at z=0.88582 (7 Gyrs look-back)

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