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

Luke Campbell Shaul Mukamel Daniel Healion Rajan 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.

  4  2 3 

c unocc

.



f f

 ˆ .

p

i

2  (

E f



E i

  )  µ(ω): absorption coefficient, intensity 

e



x

for depth

x.

 σ(ω): absorption cross section .



i

: initial state with energy

E i

.



f

: final state with energy

E f

; only transitions to unoccupied states are allowed.

  ˆ.

p

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

• Localized core → only local DOS contributes.

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.

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

H



val



lm

 †

c c

lm l m



val



jklm

V

jklm

† †

c c c c

j k m l



core



g



g

†

c c

g g



core val



g lm

U

†

c c c c

m g g

† Valence • Electron-electron interaction Core Interaction

V jklm

 

x x

1 2

j

x

1

k

x

2

r

1 1 

r

2

x

1

l

x

2

m

x

 • One-electron valence terms 

lm



l

 2 2 

nucl



a Z a



r

a m



core



g

[

V

lg

mg



V

lg

gm

]

g g

• Core hole potential → use Z+1 approximation, core hole approximated as point charge → equivalent to nuclear charge increased by 1.

U



l

1 

0

m

Fermi’s Golden Rule gives the absorption cross section: 

abs



4

 

c

2 

f



i

  

f

2  

E i



E f

 Dipole operator in ν direction    

gj

 

gj c c g

†

j

Dipole matrix element  

gj



g

 .

j



abs



4

 

c

2 

f

  lg 

gm



i c c l g

† 

f



f c c g m

†  

i



E i



E f

 

i

→ Initial wavefunction with energy

E i

.



f

→ Final wavefunction with energy

E f

.

c c l l

† ( ) → 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 In the Z+1 approximation:

  

i N G

0 ,





i N

→ Initial valence wavefunction.

  

f



N

 1

G

0 → Fully occupied core wavefunction.



f



N

 1 → Final valence wavefunction with core hole potential present.

G g



G



g

→ Core wavefunction with orbital g unoccupied.

• Effective valence Hamiltonians

H



G H G

0

i

0  

g g



f g

• Core filled (initial state) valence Hamiltonian:

G

g



H

i

 

g

 

g

 

val



lm

 †

c c

lm l m



val



jklm

V

jklm

† †

c c c c

j k m l

• Valence Hamiltonian with core hole in orbital g:

H

f g



g

 

g



g

 

val



lm



lm



U

†

c c

l m



val



jklm

V

jklm

† †

c c c c

j k m l



abs

• The absorption spectrum: 

4



3



c

 

f

  lg 

gm



i N c l



f



N

 1

(

 

E i



f



N

 1 †

c m



E f

)

2   2

i

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: • • singlet spin 5-15 singlet and/or triplet excited states with TDDFT or TDHF  Core excited state: • • • Z+1 approximation doublet spin 50 or more excited states with TDDFT/TDHF

[Ru(bpy)

3

]

2+

Experimental XANES

L3-Edge

• 1 eV valence shift of main peak (B → B ' ) after photoexcitation to

3

MLCT state.

• Appearance of new peak A ' after photoexcitation.

[Ru(bpy)

3

]

2+

SOS Simulated XANES

L3-Edge B3LYP/3-21G

• • Ground state XANES MLCT XANES (solid line) shows peak B.

(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).

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.

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).

l

• 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

(1) (a)

l

(2) (b) (a) (b) (c)

XANES 1.90 eV XANES spectra of water (O K-edge) 4a 1

h

 2b 2 HF/6-311++G** H O X-ray photon H

Ionization potential

Absorption H H O Water monomer

Peak splitting between the lowest transitions corresponding to 1a

1

→ 4a

1

and 1a

1

→ 2b

2 1.90 eV 1.92 eV 2.04 eV 1.83 eV

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)

Methyl Alcohol O K-Edge

SOS Transition Potential

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).

X-Ray Fluorescence

Hamiltonians in the Z+1 approximation:

H

  

g



g

 

val



lm

 †

c c

lm l m



val



jklm

V

jklm

† †

c c c c

j k m l i e



L

H

e g



g

 

g



g



val



lm



lm



U

†

c c

l m



val



jklm

V

jklm

† †

c c c c

j k m l



S f S

(  

L S

)  

 

      lg 

gm

 

N



c l E

   

N

  1 

E

     

N

 1  

L

†

c m

 

N

2   (

E

 

E

    

L



S

)

1b

2

Fluorescence Spectrum of Water Molecule Excitation at O K-edge 1b

1

Method/Basis 3a

1

SOS (HF)/D95V+* Ref: J.-H. Guo et al. Phys. Rev. Lett., Vol 89, 137402 (2002).

HF/Sadlej using Dalton program

Methyl Alcohol HF/Sadlej

Fluorescence Spectra of Methyl Alcohol

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.