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.
2
4 2 unocc.
 ( )   ( ) 
f ˆ.p i  ( E f  Ei   )

3c f
 µ(ω): absorption coefficient, intensity
 e  x ( ) for depth x.
 σ(ω): absorption cross section.
 i : initial state with energy Ei.
 f : final state with energy Ef ; 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
val
val
core
lm
jklm
g
core val
H    lmcl†cm   V jklmc†j ck†cmcl    g cg† cg  U lm, g cl†cmcg cg†
Valence
g
lm
Core
Interaction
• Electron-electron interaction
V jklm    dx1dx2 j x1 k x2
1
x1 l x2 m
r1  r2
x  (r, s)
• One-electron valence terms
 lm
core
pˆ 2 nucl Z a
 l

m   [Vlg mg  Vlg gm ]cg† cg
ˆ  ra
2
a r
g
• Core hole potential → use Z+1 approximation, core
hole approximated as point charge → equivalent to
nuclear charge increased by 1.
U lm , g  l
1
m
rˆ  r0
Fermi’s Golden Rule gives the absorption cross section:
4 2
 abs ( , ) 
c


i   f
   Ei  E f 
2
f
Dipole operator in ν direction
Dipole matrix element
   c c
gj  g  .pˆ j


†
gj g j
gj
4 2
 abs ( , ) 
c
 
†



c
c
  lg gm i l g  f
 f cg cm†  i 
g ,lm f
   Ei  E f 
i
→ Initial wavefunction with energy Ei.
f
→ Final wavefunction with energy Ef.
cl (cl† ) → 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 G0 , 
 f   N 1 Gg
 iN
→ Initial valence wavefunction.
G0
→ Fully occupied core wavefunction.
fN 1
Gg
→ Final valence wavefunction with core hole potential
present.
→ Core wavefunction with orbital g unoccupied.
• Effective valence Hamiltonians
H  G0 H i G0   Gg H fg Gg
g
• Core filled (initial state) valence Hamiltonian:
val
val
lm
jklm
Hi    g     lmcl†cm   V jklm c†j ck†cmcl
g
• Valence Hamiltonian with core hole in orbital g:
H fg 
val
val
lm
jklm
†
† †






U
c
c

V
c
 g   lm lm,g  l m  jklm j ck cmcl
g  g
• The absorption spectrum:
4
 
 abs ( ) 


lg  gm
3c g ,lm f 
iN cl fN 1 fN 1 cm† iN 
(  Ei  E f )2   2
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 3MLCT state.
• Appearance of new peak A' after photoexcitation.
[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).
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
(b)
(a)
• 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).
(a)
l
l
(1)
(2)
(b)
(c)
XANES spectra of water (O K-edge)
1.90 eV
HF/6-311++G**
2b2
4a1
h
Energy
XANES
Ionization potential
H
H
X-ray photon
Absorption
H
H
O
O
Water monomer
Peak splitting between the lowest transitions corresponding to
1a1 → 4a1 and 1a1 → 2b2
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
e
Hamiltonians in the Z+1 approximation:
S
L
val
val
Hi, f    g    c c  V
g
H eg 
†
lm l m
lm
f
i
† †
jklm j k m l
ccc c
jklm
val
val
lm
jklm
†
† †






U
c
c

V
c
 g   lm lm, g  l m  jklm j ck cmcl
g  g
S (L , S )      lg  gm

 
  g ,lm 

 c 
N
  l
N 1


N 1

c 
†
m
E  E   L  i
N
2

 ( E  E   L  S )
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
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.