1. Photoelectron Spectroscopy
Lecture 4: Deconvolution of complex ionization structure
Band shapes with non-resolved vibrational structure
Other complications
Deriving chemical meaningful information

2. This is the model we’ve defined…18 17 16 15
Ionization Energy (eV)
vertical
adiabatic
Lowest energy transition: Adiabatic transition (ν0 ➔ ν0)
Most probable (tallest) transition: Vertical transition
r (Å) 1 2 H2
Ground state vibrational population follows a Boltzmann distribution:
e-E/kT
kT at room temperature is eV (300 cm-1)

3. But things aren’t typically that simple!
2T2g Ionization of M(CO)6
9.2
8.8
8.4
8.0
Ionization Energy (eV)
M(CO)6 Cr Mo W
This lump contains seven ionizations!

4. Complicated vibrational structure
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

5. 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.”

6. 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.

7. 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.

8. Modeling a potential energy surface with a symmetric Gaussian

9. Modeling a potential energy surface with an asymmetric Gaussian

10. 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.

11. 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 8 9 10 11 12 13 14 15 16 17 18
H s13.598 He
Li s5.392 p3.54
Be s9.323 p6.60
B s12.93 p8.298
C s20.04 p10.95
N s27.15 p13.60
O s34.26 p16.26
F s41.37 p18.91
Ne s48.47 p21.564
Na s5.139 p3.04
Mg s7.646 p4.93
Al s10.63 p5.986
Si s14.35 p7.94
P s18.07 p9.90
S s21.79 p11.85
Cl s25.52 p13.80
Ar s
K s4.341 p2.72
Ca s6.113 p4.23
Sc d3.03
Ti d3.70
V d4.37
Cr d5.05
Mn d5.72
Fe d6.39
Co d7.07
Ni d7.74
Cu s7.726 p3.91
Zn s9.394 p5.36
Ga s11.92 p5.999
Ge s15.04 p7.60
As s18.16 p9.20
Se s21.28 p10.80
Br s24.39 p12.40
Kr s27.51 p14.000
Rb s4.177 p2.59
Sr s6.62 p3.87
Y d3.23
Zr d3.96
Nb d4.69
Mo d5.42
Tc d6.15
Ru d6.88
Rh d7.61
Pd d8.34
Ag s7.576 p3.80
Cd s8.994 p5.19
In s11.16 p5.786
Sn s13.61 p7.06
Sb s16.06 p8.32
Te s18.50 p9.59
I s20.95 p10.86
Xe s23.40 p12.130
Cs s3.894 p2.44
Ba s p3.60
La d3.45
Hf d4.13
Ta d4.80
W d5.48
Re d6.16
Os d6.84
Ir d7.52
Pt d8.20
Au s9.226 p4.27
Hg s10.44 p5.48
Tl s12.6 p6.108
Pb s14.67 p7.04
Bi s16.73 p7.96
Po s18.79 p8.414
At s20.85 p9.581
Rn s
Fr s4.073
Ra 5.278
Ac i5.17 Rf Db Sg Bh Hs Mt
Uun
Uuu
Uub
Lanthanides
Ce 5.47
Pr 5.42
Nd 5.49
Pm 5.55
Sm 5.63
Eu 5.67
Gd 6.15
Tb 5.86
Dy 5.93
Ho 6.02
Er 6.101
Tm 6.184
Yb 6.254
Lu 5.43
Actinides
Th 6.08
Pa 5.88
U 6.05
Np 6.19
Pu 6.06
Am 5.974
Cm 6.02
Bk 6.23
Cf 6.3
Es 6.42
Fm 6.5
Md 6.58
No 6.65 Lr
  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 d1 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

12. 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.

13. 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

14. 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.

15. 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.