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* -7.88

0.5*EXHF + 0.5*EX

LSDA + 0.5*ΔEXBecke88 + EC

LYP BHandHLYP -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

Veillard et al. Theor. Chim. Acta 1972, 27, 281-287.

a1g-16.6

a2u-16.0

e2g-14.4

e1g-11.9

e1u-11.7

15.5

11.2

11.1

2A1g10.1

2E 2g8.3

2E1u

2E1g

2A2u

Ionization Energy (eV)7111315 9

He I

He II

Ferrocene

2E2g2A1g

2E1u

2E1g

DFT: Better Correlation, Fewer Problems with Relative Relaxation Energy

a1g

-7.17

e2g

-6.69

e1g

-6.00

e1u

-5.15

Ionization Energy (eV)7111315 9

He I

He II

Ferrocene

2E2g2A1g

2E1u

2E1g

a1g

e2g

e1g

e1u

-1.75

-8.92

-8.44

-7.75

-6.90

Condensed phase: ΔE SCF

Calculating Molecules in a Solvent

Gas phase: ΔE SCF

M

IE (M)

M++

COSMO: COnductor-like Screening MOdel

The solvent is modeled by a dielectric continuum.

The molecule creates a cavity of specific size and shape.

IE (Msolv)

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.