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

Molecules

Luke CampbellShaul MukamelDaniel HealionRajan Pandey

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.

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.

2 . 24 ˆ( ) ( ) . ( )3

unocc

f if

f i E Ec

p

µ(ω): absorption coefficient, intensity

σ(ω): absorption cross section.

: initial state with energy Ei.

: final state with energy Ef ; only transitions to unoccupied states are allowed.

: dipole operator (core size much smaller than x-ray wavelength).

( )xe for depth x.

i

f

ˆ. p

• Localized core → only local DOS contributes.

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.

Sum Over States Method (SOS):

Computationally expensive, requires explicit calculation of excited states.

Methodology

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.

Simulation of x-ray absorption near edge spectra (XANES) of molecules

• Start with the Deep Core Hamiltonian• Neglect valence-core exchange

† † † † † †,

val val core core val

lm l m jklm j k m l g g g lm g l m g glm jklm g g lm

H c c V c c c c c c U c c c c

1 2 1 2 1 21 2

1jklmV d d j k l m

x x x x x xr r

2†

lg lg

ˆ[ ]

ˆ2

nucl corea

lm mg gm g ga ga

Zpl m V V c c r r

,1

ˆlm gU l m 0r r

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.

( , )sx r

Fermi’s Golden Rule gives the absorption cross section:

2 24( , )abs i f i f

f

E Ec

†gj g j

gj

c c ˆ.gj g j p

2† †

lg,

4( , )abs gm i l g f f g m ig lm f

i f

c c c cc

E E

Dipole operator in ν direction Dipole matrix element

i

f†( )l lc c

→ Initial wavefunction with energy Ei.

→ Final wavefunction with energy Ef.

→ Electron annihilation (creation) operator for orbital l.

Core-valence separation

• Deep core Hamiltonian → separate eigenvalue problem for valence and core electrons → can represent as product space

0 ,Ni G 1N

f gG In the Z+1 approximation:

Ni

0G

1Nf

gG

→ Initial valence wavefunction.

→ Fully occupied core wavefunction.

→ Final valence wavefunction with core hole potential present.

→ Core wavefunction with orbital g unoccupied.

† † †val val

i g lm l m jklm j k m lg lm jklm

H c c V c c c c

† † †,

val valgf g lm lm g l m jklm j k m l

g g lm jklm

H U c c V c c c c

1 1 †

lg 2 2,

4( )3 ( )

N N N Ni l f f m i

abs gmg lm f i f

c c

c E E

0 0g

i g f gg

H G H G G H G • Effective valence Hamiltonians

• Core filled (initial state) valence Hamiltonian:

• Valence Hamiltonian with core hole in orbital g:

• The absorption spectrum:

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

First principles computation of ground and excited state XANESOf 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.

[Ru(bpy)3]2+ Experimental XANES

• 1 eV valence shift of main peak (B → B') after photoexcitation to 3MLCT state.• Appearance of new peak A' after photoexcitation.

L3-Edge

[Ru(bpy)3]2+ SOS Simulated XANES

• 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).

B3LYP/3-21GL3-Edge

Excited State Effects on X-ray Absorption

Charge transfer to or from the absorbing atom can alter the energies andintensities of transitions to the bound states.

• 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.

Examples:

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 l

l

(a)

(a)

(b)

(b) (c)

(1) (2)

O

HHX-ray photon

1.90 eV

O

H H

Absorption

Ene

rgy

Water monomer

Ionization potential

4a1

2b2

h

XANES

XANES spectra of water (O K-edge)

HF/6-311++G**

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)

Peak splitting between the lowest transitions corresponding to 1a1 → 4a1 and 1a1 → 2b2

1.83 eV2.04 eV

1.92 eV

1.90 eV

Methyl Alcohol

SOS

TransitionPotential

O K-Edge

XANES of Benzonitrile (N K-edge)

TDDFT (B3LYP)/D95**

TDHF/D95**

Method/Basis

Gives good agreement for the intensity ratio. However, peak splitting is not exact.

Gives good agreement in the peaksplitting. However, the intensityratio is different than experiment.

Ref: S. Carniato et al. Phys. Rev A 58, 022511 (2005).

X-Ray Fluorescence

† † †,

val val

i f g lm l m jklm j k m lg lm jklm

H c c V c c c c

† † †,

val valge g lm lm g l m jklm j k m l

g g lm jklm

H U c c V c c c c

21 1 †

lg,

( , )

( )

N N N Nl m

L S gmg lm L

L S

c cS

E E i

E E

Hamiltonians in the Z+1 approximation:

if

e

L S

Fluorescence Spectrum of Water Molecule

Method/Basis

SOS (HF)/D95V+*

HF/Sadlej using Dalton program

Ref: J.-H. Guo et al. Phys. Rev. Lett., Vol 89, 137402 (2002).

1b1

3a1

1b2

Excitation at O K-edge

Methyl Alcohol

HF/Sadlej

Fluorescence Spectra of Methyl Alcohol

Theoretical Challenges of Femtosecond X-Ray Simulations

Fast codes for excited state dynamics.

Codes for computing current profiles within molecules.

Study of X-ray fluorescence and four wave mixing when the molecule is initially in the optically excited state.

Time Resolved Geometry Changes

• photodissociation• changes in conformation• vibrations

Immediately after electronic excitation, the molecule will begin to relax to a new equilibrium structure. This can involve:

Simulate quantum molecular dynamics to find forces on atoms in excited state.

Use mixed quantum/classical molecular dynamics for solvent.