Photoelectron Spectroscopy
• Lecture 11: Comparison of electronic
structure calculations to experiment
Why Calculations?
• Can be used to predict chemical/physical behavior, including
ionization energies.
• Can help in assignment of experimental results, particular from
spectroscopy.
• Can often get computational information that would be hard to
gather experimentally.
• The era of computing chemists, when hundreds if not thousands of
chemists will go to the computing machine instead of the
laboratory…is already at hand. There is only one obstacle, namely,
that someone must pay for the computing time. Robert S. Mulliken
(1966)
A Few Caveats
• Every attempt to employ mathematical methods in the study of
chemical questions must be considered profoundly irrational and
contrary to the spirit of chemistry. If mathematical analysis should
ever hold a prominent place in chemistry - an aberration which is
happily almost impossible - it would occasion a rapid and
widespread degeneration of that science. A Compte (1830)
• It is more important to have beauty in one's equations than to have
them fit experiment. Paul Dirac (1963)
• Experiments are the only means of knowledge at our disposal. The
rest is poetry, imagination. Max Planck
Important points to consider for calculations:
• What general type of calculation is being performed?
– This will explain the general approximations being made.
• What specific method is being used?
– Even more information on the approximations
• What basis set is being used?
– How are we modeling orbitals, polarization, etc.
Type 1: Molecular Mechanics
• Molecular equilibrium geometries are described by force
fields based on classical mechanics: atoms are treated
as balls, bonds are springs.
– Electrons are completely ignored.
• Computationally undemanding.
– Time  N2 (N is number of atoms)
– Useful for geometry information on big molecules like proteins
• No electronic structure information
– Must be paramaterized for specific atom types or functional
groups
Type 2: ab Initio
• Quantum mechanical description of electrons. Methods are based
on finding solutions to the Schrodinger equation. Combined
approximations to calculate the many-body problem include:
– Born-Oppenheim approximation (separate nuclear motion from electron
motion)
– Hartree Theory (reduce many-electron problem to a series of single
electrons moving in a potential field)
– Fock Theory (“exchange symmetry” between electrons of different spin
in an orbital)
• Can be applied to any kind of system.
• Computationally demanding.
– Time N4 (N is number of electrons, degrees of freedom are 3N spatial
and N spin)
Type 3: Semi-empirical
• Quantum mechanical description of electrons based on same
principles as ab Initio, but with many (more) approximations built into
the equations to make calculations go faster.
– Also commonly contain some parameterization (design of computational
equations or input parameters) based on experimental (empirical) data.
• Calculations are faster than ab Initio, larger systems can be
handled.
– Results often agree well with experimental values because of
parameterization.
• Only reliable for systems for which a method is parameterized.
Type 4: Density Functional Theory
• Based on quantum mechanical calculation of the
electron density of a system.
– Typically solved in terms of one-electron orbitals (Kohn-Sham
orbitals) that bear some resemblance to one electron ab Initio
orbitals.
• Can be applied to any kind of system.
• Not as computationally demanding as ab Initio.
– Time  N3 (N is number of electrons, degrees of freedom are 3N
spatial)
• We don’t know how to actually calculate the full electron
density. Currently many approximations are used.
Type 5: Combine All The Previous
• Might have a large system that you wish to do a
calculation on, but need to model a portion at a high
level of theory.
• For example, for a heme protein:
– Calculate the geometry of the protein backbone using molecule
mechanics.
– Calculate the periphery of the heme center at a semi-empirical
level.
– Calculate the heme pocket, iron ligation site using density
functional theory.
• Referred to as QM/MM methods.
– Implemented in Gaussian as ONIOM.
Two Important Concepts:
Correlation and Exchange
• The position of each electron is correlated to
the position of all other electrons.
– if one electron moves, its electrostatic field will
influence the positions of any other electrons
– Ignored in Hartree-Fock Theory
• Exchange: use of the spin quantum number to
define spin symmetry
– handled explicitly in ab Initio calculations
– Only approximated in Density Functional Theory
Ab Initio and DFT Treat Exchange and Correlation
Differently: Example, Koopmans’ Theorem for Water
Experimental
12.62
Semi-empirical
Extended Huckel
-14.80
Semi-empirical
INDO
-16.06
Hartree-Fock Theory
HF/3-21g
-12.96
“Better” basis set
HF/6-31g*
-13.48
Correct for electron correlation
MP2/6-31g*
-13.48
DFT, Exchange: Local Density Approximation
XAlpha/6-31g*
-5.29
Exchange: Generalized Gradient Approximation
Becke88/6-31g*
-5.16
Exchange + Correlation
BLYP/6-31g*
-6.05
Hybrid: HF Exchange+DFT Exchange/Correlation B3LYP/6-31g*
0.5*EXHF + 0.5*EXLSDA + 0.5*ΔEXBecke88 + ECLYP
BHandHLYP
-7.88
-10.21
Calculating ionization energies as the difference in
self-consistent field energy of states: SCF
• Do a calculation for the ground state of the molecule.
• Do a second calculation at the equilibrium geometry
found in the first calculation, but with one less electron.
• Difference in total self-consistent field energy of the two
calculations is equal to the ionization energy.
Example: B3LYP SCF on Water
• Total Energy of H2O (1A1): -76.4080151 Hartree
• Total Energy of H2O+ (2B1): -75.9542981 Hartree
• Difference = 0.453717 Hartree
• * 27.2116 eV/Hartree = 12.35 eV
• But now we have to do a completely separate calculation
for each ionization energy we want to calculate!
Another Important Concept:
Relaxation/Reorganization Energy
• Koopmans’ Theorem is a “frozen orbital” approximation: assumes
that removing an electron from a system will not effect the remaining
electrons.
• In actuality, when as electron is removed the remaining electrons
reorganize, effectively instantaneously, as the charge potential
changes in the cation.
– The system relaxes to a lower energy state.
• As the number of electrons in the valence orbitals becomes larger
(like for d orbitals), correlation becomes more important, and
relaxation energies increase.
• Because of this, the orbital ordering calculated for transition metal
containing molecules is often incorrect.
Example: HF Calculation on Ferrocene
2E
2g
2A
1g
8.3
Ferrocene
He I
10.1
2
E1u
-11.7
-11.9
e1u
2E
1u
e1g
2E
1g
11.1
2
A1g E2g
2
E1g
2
11.2
He II
-14.4
-16.0
-16.6
e2g
a2u
2A
2u
15.5
15
a1g
13
11
9
7
Ionization Energy (eV)
Veillard et al. Theor. Chim. Acta 1972, 27, 281-287.
DFT: Better Correlation, Fewer Problems
with Relative Relaxation Energy
Ferrocene
-5.15
-6.00
He I
e2g
2
E1u
a1g
E1g
-1.75
-6.69
-7.17
2
A1g E2g
2
e1u
e1g
2
He II
e2g
a1g
-6.90
-7.75
15
e1u
e1g
13
11
9
Ionization Energy (eV)
-8.44
-8.92
7
Calculating Molecules in a Solvent
Condensed
Gas
phase: ΔE
phase:
SCFΔE SCF
COSMO: COnductor-like Screening MOdel
+
M+
The molecule creates a cavity
of specific size and shape.
IE IE
(Msolv
(M))
The solvent is modeled by
a dielectric continuum.
M
Solvation calculations can be
repeated for different solvents
Conclusions
• Molecular orbital calculations are a
powerful tool, but one should realize the
limitations of the tool before using.
Overall PES Conclusions
• Koopmans’ Theorem: the whole reason
calculations/orbitals/spectroscopy are related.
• Information content in photoelectron spectra include ionization
energies, relative geometries of ground and excited states, bonding
character, atomic character of molecular orbitals, and more.
• Don’t overanalyze experimental data—but don’t under analyze it,
either.
• When comparing measurements made in different ways, phase and
time scale matters.
• Calculations are a great tool—just be sure you understand the
theory you are using.