Simulation of X-ray Absorption Near Edge Spectroscopy (XANES) of Molecules

1. Simulation of X-ray Absorption Near Edge Spectroscopy (XANES) of Molecules Luke Campbell Shaul Mukamel Daniel Healion Rajan Pandey

2. Motivation • X-ray Absorption Near Edge Spectroscopy (XANES) is an • attractive tool for measuring local changes in electronic • structure due to geometry and charge distribution of • transient species. • Recent advances in ultrashort (femtosecond to attosecond) • x-ray pulses enable real time probing of optically induced • electron motions and chemical processes. • Time resolved XANES measures changes in geometry • and charge distribution during and after the excitation. • Theory can provide a guide for the design and interpretation • of these measurements.

3. Basic Physics of X-ray Absorption • X-ray absorption probes the unoccupied dipole allowed one electron • density of states of a molecule in the vicinity of the absorbing atom. • µ(ω): absorption coefficient, intensity • σ(ω): absorption cross section. • : initial state with energy Ei. • : final state with energyEf ; only transitions to unoccupied states are allowed. • : dipole operator (core size much smaller than x-ray wavelength). for depth x. • Localized core → only local DOS contributes.

4. Methodology Sum Over States Method (SOS): • Many-electron ground states (with and without core holes) are calculated using standard quantum chemistry codes. within density functional theory or Hartree-Fock approximation, (Z+1 approximation, where Z is the nuclear charge). • Electronically excited states are calculated using time dependent density functional theory (TDDFT) or time dependent Hartree-Fock (TDHF) theory. • Computationally expensive, requires explicit calculation of excited states.

5. Transition Potential Method: • Uses a reference system with partially filled orbitals (incorporated in the StoBe Demon code). • Represents systems with different numbers of core holes by different occupation numbers of a single set of reference orbitals. • Computationally less expensive than SOS. • Works well for core level spectroscopies of small molecules.

6. Simulation of x-ray absorption near edge spectra (XANES) of molecules • Start with the Deep Core Hamiltonian • Neglect valence-core exchange Valence Core Interaction • Electron-electron interaction • One-electron valence terms • Core hole potential → use Z+1 approximation, core • hole approximated as point charge → equivalent to • nuclear charge increased by 1.

7. Fermi’s Golden Rule gives the absorption cross section: Dipole operator in νdirection Dipole matrix element → Initial wavefunction with energy Ei. → Final wavefunction with energy Ef. → Electron annihilation (creation) operator for orbital l.

8. In the Z+1 approximation: → Initial valence wavefunction. → Fully occupied core wavefunction. → Final valence wavefunction with core hole potential present. → Core wavefunction with orbital g unoccupied. Core-valence separation • Deep core Hamiltonian → separate eigenvalue problem for valence • and core electrons → can represent as product space

9. Effective valence Hamiltonians • Core filled (initial state) valence Hamiltonian: • Valence Hamiltonian with core hole in orbital g: • The absorption spectrum:

10. First principles computation of ground and excited state XANES Of chemical species • Use quantum chemistry code (Gaussian 03) to find electronic structure of ground and excited states. • Find energies and intensities of transitions from a given initial ground or excited state to possible final excited states. • Basis set: Selection based on kind of chemical species in a molecule • Level of theory: Becke 3-parameter density functional with Lee-Yang-Parr correlation, Hartree-Fock approximation. • Code: GAUSSIAN-03 • Geometry: from x-ray crystallography data (complex molecules). • Ground state: • Core excited state: • singlet spin • 5-15 singlet and/or triplet excited states with TDDFT • or TDHF • Z+1 approximation • doublet spin • 50 or more excited states with TDDFT/TDHF

11. [Ru(bpy)3]2+ Experimental XANES L3-Edge • 1 eV valence shift of main peak (B → B') after photoexcitation to 3MLCT state. • Appearance of new peak A' after photoexcitation.

12. [Ru(bpy)3]2+ SOS Simulated XANES L3-Edge B3LYP/3-21G • Ground state XANES (solid line) shows peak B. • MLCT XANES (dottes) shows peak B' blue shifted by 1 eV and appearance • of peak A'. • Luke Campbell and Shaul Mukamel, J. Chem. Phys. 121, 12323 (2004).

13. Excited State Effects on X-ray Absorption Charge transfer to or from the absorbing atom can alter the energies and intensities of transitions to the bound states. Examples: • Removing an electron makes the atom more • positively charged, so more energy is needed • to excite the core electron to orbitals farther • from atom. • Absorption peaks shift position • When electrons are taken out of previously • filled orbitals, new core → valence transitions • are possible. • When electrons are put into previously empty • orbitals,peaks can disappear.

14. Single and Double Excitations Neglecting changes in orbitals due to core excitation: • From any initial optically excited state, the • final XANES state (a) can be reproduced • with two excitations from the lowest core • excited state (b). • From some initial states, such as the ground • state or HOMO to LUMO excitations, the • final XANES state can be represented by one • excitation from the lowest core excited state • (b). Transition (1) gives ground state XANES • (a), transition (2) gives HOMO to LUMO • excitation XANES (c). l (b) (a) l l (1) (2) (a) (b) (c)

15. XANES spectra of water (O K-edge) 1.90 eV HF/6-311++G** 2b2 4a1 XANES Energy Absorption Ionization potential H H H H X-ray photon O O Water monomer

16. 1.90 eV 1.92 eV 2.04 eV 1.83 eV Peak splitting between the lowest transitions corresponding to 1a1→ 4a1and 1a1→ 2b2 Sum Over States SOS (solid line) gives a good agreement with the experiment. Plots and numbers reproduced (except solid curve - SOS) from Ref: M. Cavalleri et al. J. Chem. Phys. Vol. 121, 10074 (2004)

17. Methyl Alcohol O K-Edge SOS Transition Potential

18. XANES of Benzonitrile (N K-edge) Method/Basis TDDFT (B3LYP)/D95** Gives good agreement for the intensity ratio. However, peak splitting is not exact. TDHF/D95** Gives good agreement in the peak splitting. However, the intensity ratio is different than experiment. Ref: S. Carniato et al. Phys. Rev A 58, 022511 (2005).

19. X-Ray Fluorescence Hamiltonians in the Z+1 approximation:

20. Fluorescence Spectrum of Water Molecule Excitation at O K-edge 1b1 Method/Basis 3a1 SOS (HF)/D95V+* 1b2 Ref: J.-H. Guo et al. Phys. Rev. Lett., Vol 89, 137402 (2002). HF/Sadlej using Dalton program

21. Methyl Alcohol HF/Sadlej

22. Fluorescence Spectra of Methyl Alcohol

23. Theoretical Challenges of Femtosecond X-Ray Simulations Time Resolved Geometry Changes • Immediately after electronic excitation, the molecule will begin to relax to a new equilibrium structure. This can involve: • photodissociation • changes in conformation • vibrations • Fast codes for excited state dynamics. • Codes for computing current profiles within molecules. • Simulate quantum molecular dynamics to find forces on atoms in excited state. • Use mixed quantum/classical molecular dynamics for solvent. • Study of X-ray fluorescence and four wave mixing when the molecule is initially in the optically excited state.