Photoelectron Spectroscopy
• Lecture 4: Deconvolution of complex
ionization structure
– Band shapes with non-resolved vibrational
structure
– Other complications
– Deriving chemical meaningful information
This is the model we’ve defined…
Ionization Energy (eV)
18
H2 +
17
vertical
adiabatic
16
15
Lowest energy transition: Adiabatic
transition (ν0 ➔ ν0)
Most probable (tallest) transition: Vertical
transition
H2
0
0
1
r (Å)
2
Ground state vibrational population follows a
Boltzmann distribution:
e-E/kT
kT at room temperature is 0.035 eV (300 cm-1)
But things aren’t typically that simple!
2T
2g
Ionization of M(CO)6
S
S
M(CO)6
(OC)3Fe
Fe(CO)3
Cr
Mo
11
W
9.2
10
9
8
Ionization Energy (eV)
8.8
8.4
8.0
Ionization Energy (eV)
This lump contains seven ionizations!
Factors we have to consider for inorganic/organic
molecules of the type involved in your research:
• Complicated vibrational structure
– Multiple interdigitated modes
– Vibronic coupling between final states
• Additional final state effects
– Jahn-Teller splitting
– Spin-orbit splitting
• Congested spectra
– Inability to individually observe all ionizations of interest
• And we have to figure this all out in a way that gives
chemically meaningful information
Data Analysis of Spectroscopic Results
The Bible:
“Data Reduction and Error Analysis for the Physical
Sciences”, Philip R. Bevington and D. Keith
Robinson, 2nd Edition, McGraw-Hill, 1992
“We often wish to determine one characteristic y
of an experiment as a function of some other
quantity x. That is…we make a series of N
measurements of the pair (xi,yi), one for each of
several values of the index i, which runs from 1
to N. Our object is to find a function that
describes the relation between these two
measured variables.”
Fitting Data using WinFP
• Use a series of functions, each defined with some
number of degrees of freedom, to represent an arbitrary
function, the spectrum.
• Define an initial fit using your chemical intuition and
knowledge about the molecule.
• Have the computer perform a least-squares analysis to
arrive at a fit that then best matches the experimental
variables.
• The specific method we are using to search parameter
space, define conditions of convergence, and find a local
minima is the Marquardt Method. See Chapter 8 of
Bevington for details.
What function is appropriate?
• Poisson distribution
– Analytical form appropriate to measurements that describe a probability
distribution in terms of a variable x and a mean value of x.
– Appropriate for describing experiments in which the possible values of data
are strictly bounded on one side but not on the other.
– Non-continuous; only defined at 0 and positive integral values of the variable
x
• Guassian distribution
– more convenient to calculate that the Poisson distribution
– Continuous function defined at all values of x
– Limiting case for the Poisson distribution as the number of x variables
becomes large
• Lorentzian distribution
– appropriate for describing data corresponding to resonant behavior (NMR,
Mossbauer)
• Voigt Function: Combination of Lorentzian and Gaussian functions
– Used to describe Lorentzian data with Gaussian broadening.
Modeling a potential energy surface
with a symmetric Gaussian
Modeling a potential energy surface
with an asymmetric Gaussian
How do we fit data in a chemically
meaningful way?
• Think about the expected electronic structure
first!
• Consider how many valence ionizations are
likely to be clearly observed before the
overlapping sigma bond region (lower than
about 12 eV ionization energy).
• Luckily, these are usually the ionizations related
to the “interesting” orbitals of a molecule.
LCAO Model: The energies of the atomic orbitals are the starting point for the
energies of the molecular orbitals
Group
Period
1
2
3
4
5
6
7
1
H
s13.598
Li
s5.392
p3.54
Na
s5.139
p3.04
K
s4.341
p2.72
Rb
s4.177
p2.59
2
3
4
5
6
7
8
9
10
11
12
Be
s9.323
p6.60
Mg
s7.646
p4.93
Ca
Cu
Zn
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
s6.113
s7.726 s9.394
d3.03 d3.70 d4.37 d5.05 d5.72 d6.39 d7.07 d7.74
p4.23
p3.91 p5.36
Sr
Ag
Cd
Y
Zr
Nb
Mo
Tc
Ru
Rh
Pd
s6.62
s7.576 s8.994
d3.23 d3.96 d4.69 d5.42 d6.15 d6.88 d7.61 d8.34
p3.87
p3.80 p5.19
13
15
Lanthanides
Actinides
Rf
Db
Sg
Bh
Hs
Ce
5.47
Th
6.08
Pr
5.42
Pa
5.88
Nd
5.49
U
6.05
Pm
5.55
Np
6.19
Mt
Uun
Uuu
Uub
Sm
Eu
Gd
5.63 5.67 6.15
Pu
Am
Cm
6.06 5.974 6.02
Tb
5.86
Bk
6.23
Dy
5.93
Cf
6.3
C
N
s20.04 s27.15
p10.95 p13.60
Si
P
s14.35 s18.07
p7.94 p9.90
Ge
As
s15.04 s18.16
p7.60 p9.20
Sn
Sb
s13.61 s16.06
p7.06 p8.32
16
17
Ho
6.02
Es
6.42
O
s34.26
p16.26
S
s21.79
p11.85
Se
s21.28
p10.80
Te
s18.50
p9.59
18
He
24.587
Ne
s48.47
p21.564
Ar
s29.24
15.759
Kr
s27.51
p14.000
Xe
s23.40
p12.130
F
s41.37
p18.91
Cl
s25.52
p13.80
Br
s24.39
p12.40
I
s20.95
p10.86
At
Cs
Ba
Au
Hg
Tl
Pb
Bi
Po
Rn
La
Hf
Ta
W
Re
Os
Ir
Pt
s20.85
s3.894 s5.212
s9.226 s10.44 s12.6 s14.67 s16.73 s18.79
s22.91
d3.45 d4.13 d4.80 d5.48 d6.16 d6.84 d7.52 d8.20
p9.581
p2.44 p3.60
p4.27 p5.48 p6.108 p7.04 p7.96 p8.414
10.749
Fr
Ra
Ac
s4.073 5.278 i5.17
B
s12.93
p8.298
Al
s10.63
p5.986
Ga
s11.92
p5.999
In
s11.16
p5.786
14
Er
Tm
6.101 6.184
Fm
Md
6.5
6.58
Yb
6.254
No
6.65
Numbers with three decimal places are actual atomic ionization energies.
Numbers with two decimal places are interpolated.
Energies of unfilled p orbitals determined by excitation energy from the ground state.
Transition metal d orbital energies interpolated between ionization of d 1 configuration of group I element
and d10 configuration of group VIII element.
Lanthanides and actinides list ionization energies only.
Adapted by Dennis Lichtenberger from Craig Counterman
Lu
5.43
Lr
What Influences MO Ionization
Energies?
• Ionization energies of atomic orbitals
• Oxidation state, formal charge, charge
potential
• Bonding or anti-bonding interactions
• See MO Theory presentation by Dennis
Lichtenberger on the website for a detailed
discussion of how to estimate MO
ionization energies.
Some rough rules of thumb on the kinds of ionizations
clearly observed for larger molecules
• Transition metal d ionizations, 6-10 eV.
• Aryl HOMO ionizations
– Benzene 9.25 eV, doubly degenerate
• Main group p lone pair ionizations
– For halides, F ~14 eV, Cl ~11 eV, Br ~9 eV, I ~8 eV
(SO splitting large for Br, I)
• Metal-ligand dative bonds
Decide how many valence ionizations
should be clearly observed
• If band shapes = ionizations
– Begin fitting using that number of Gaussians
• If band shapes > ionizations
– Consider possible causes; vibrational structure, spin-orbit splitting, etc.
– Decide how this information should be chemically modeled.
• If band shapes < ionizations
– Fit using the minimum number of Guassians needed to define shape of
the spectrum
– Fit the spectra of a series of related molecules in a similar way so that
comparisons can be made.
• Least-squares analysis
• Look at results, possibly iterate through the steps again until
achieving a “good” fit that can be defended as giving
chemically meaningful information.
Conclusions
• Photoelectron band shapes can be modeled with
asymmetric Gaussians functions.
• Might not be able to analytically represent all data
content, but do want to represent data in a consistent,
chemically meaningful way.
• When analyzing data, think, then fit.