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Probing Excited States by Photoelectron
Imaging: Dyson Orbitals within Equation-of-
Motion Coupled-Cluster Formalism
Anna I. Krylov
University of Southern California, Los Angeles
Ohio Spectroscopy Meeting, 2007
Acknowledgements:
Dr. Melania Oana
Inspiration from experiments of:Albert Stolow, Hanna Reisler,Klaus Muller-Dethlefs
Hanna Reisler, USCAlbert Stolow,Steacie Institute for Molecular Sciences
Photoelectron spectroscopy: Probing energy levels, structure,Photoelectron spectroscopy: Probing energy levels, structure,and electronic wave functionsand electronic wave functions
Kin
etic
ene
rgy
of e
lect
ron
Ioni
zing
h
ionized state
initial state
Kinetic energy of the electrons: 1. Information about electronic states of the target; 2. Vibrational levels and structural changes. Angular distribution of photoelectrons (PAD): direct probe of electronic wave functions.
Excitation
laser
Molecular beam
ion detector
e- detector
Ionization laserPAD and electronic wave functions
Ion image from (NO)2 by Hanna Reiser
PAD from (NO)2 in the lab frame and the molecular frame by Albert Stolow
Challenges: - Character of the wave function from PAD; - PAD from ab initio electronic structure calculations.
Outline:
1. From wave-function to PAD: Dyson orbitals.
2. Dyson orbitals for the ionization from the ground and excited states of formaldehyde: numerical examples.
3. From Dyson orbitals to PADs: Selection rules and averaging.
4. Dyson orbitals for NO dimer ionization.
5. Conclusions.
Photoelectron wave functions and PADs
The probability of finding an electron in dV at {r, θ, φ}:
Angular part:Ylm – spherical harmonics
Ψel can be expanded in the basis of spherical waves:
dV
Radial part:Rkl ~ Bessel functions, Jl+1/2:
, with
|Cklm|2 - probability to find an ejected electron in the {klm} state.They are given by the ionization dipole moment matrix elements between the initial ( ΨN) and the final (ΨN-1 x Ψel) states:
Using permutational symmetry of the wave functions and integrating over N-1 coordinates:
where ri - spatial coordinates of the ith electron
PAD and Dyson Orbitals
where d(r) is Dyson orbital:
Dyson Orbitals: Summary
1. The “difference” or “overlap” between the N and N-1 e– wave functions of the neutral and the cation.2. Can be used to calculate probability to find the ionized electron in a particular state.3. Norm of the Dyson orbital ~ probability of the ionization event.4. Can be interpreted as an initial state of the ionized electron, e.g., for a one-electron system Dyson orbital is just the wave function.5. For Hartree-Fock wave functions and within the Koopmans approximation:
Φd = φk
i
j
k
l
M M+
Dyson Orbitals in EOM-CC Formalism
EOM-IP/EE-CCSD: ΨM+/M* = (R1 + R2) 0
Coefficients of Dyson orbitals in MO basis - analogous to transition density matrix element:
R1Ψref R2Ψref
M* M+ M
R1Ψref R2Ψref
RIPREE
Ψref
)exp()exp( THTH 00 ERRH
Formaldehyde Example
Dyson orbital for correlated (EOM-IP-CCSD/6-311G**(2+,2+)) wave functions - ground state ionization 1A1 1B1:
Φd = 98.7% φ2b1
CH2O CH2O+
2b1
3b2
1b2
5a1
π*
nπ
π
nσ
Formaldehyde Example
Dyson orbital for excited state ionization 1A2 1B1: (EOM-EE/IP-CCSD/6-311G**(2+,2+))
Φd = 4.2% φ2b2
+ 71.4% φ3b2
- 22.4% φ5b2
CH2O (1A2, n->*) CH2O+
2b1
3b2
- λ
5b2
1b2
5a1
Formaldehyde Example
Dyson orbital for excited state ionization 1B2 1B1:
Φd = 99.1% φ3a2
CH2O* CH2O+
3b2
- λ
5b2
1b2
2b1
5a1
PAD is the result of averaging over all possible molecular orientations:
Dyson orbitals, PADs, and molecular orientation
- Isotropic distribution spherically averaged PADs electronic structure information is lost;- Excitation laser: selects molecules cos or sin2 distributions (parallel vs perpendicular transitions) ;- PAD in Molecular Frame: more structured PAD, e.g., only azimuthal averaging in (NO)2 photodissociation experiments.
Laser beam
Molecular beam
ion detector
e- detector
dddelPAD 2
Electron Angular Momentum States: Selection Rules
0 x 0 x0 x
Dyson orbital r (x, y, z) Φd(r)·r RklYlm
s (l = 0)
l = 1
px (l = 1)
~ x x
y
z
x2
xy
xz
l = 0, 2
Allowed electron angular momentum states: Δl = ±1Molecular Dyson orbital: more angular momentum states
Higher angular momentum and diffuse orbitals
Rkl (E = 1eV)
-2.0E-01
0.0E+00
2.0E-01
4.0E-01
6.0E-01
0 5 10 15 20 25 30
r (A)
Rk0
Rk1
Rk2
Rk3
Rk4
Rk5
Diffuse orbitals – higher angular momentumHigher kinetic energy – higher angular momentum
O. Gessner, A.M.D. Lee, J.P. Shaffer, H. Reisler, S.V. Levchenko, A.I. Krylov, J.G. Underwood, H. Shi, A.L.L. East, D.M. Wardlaw, E.t.-H. Chrysostom, C.C. Hayden and A. Stolow, Femptosecond Multi-dimensional Imaging of a Molecular Dissociation, Science, 311, 219-222 (2006).
(NO)2 dissociation - 2 time scales are observed: (NO)2
* disappears 1=140+/-30 fs. NO appears 2=590+/-20 fs.
Nature of the intermediate state was controversial. Our calculations (Sergey Levchenko): two B2 states are involved. PADs: additional information about the electronic state.
Conclusions1. Dyson orbitals for the ground and excited state ionization are implemented within EOM-IP/EA/EE/SF-CCSD.2. Dyson orbitals for one-electron ionizations – obey Koopmans like rules.3. Quantitative analysis of Dyson orbitals: l,m angular momentum states accessible to the photoionized electron and the corresponding probabilities |Cklm|2.4. PAD modeling: - within RPA: |Cklm|2 ↔ PAD; - beyond RPA: need the interference contributions - cross terms C*
klmCkl’m.
5. Qualitative trends in molecular PADs: diffuse states – higher angular momentum; higher kinetic energy – higher angular momentum. 6. NO dimer: observed PADs are inconsistent with the A1 state. More detailed comparison needs to take into account kinetic energy distribution and the phases.
THANKS:My group;Ab initio packages:Our codes: available in Q-CHEM & SPARTANAdditional calculations: ACES II, GAMESS
Funding:1. Center for Computational Studies of Open-Shell and Electronically Excited Species (NSF): http://iopenshell.usc.edu bridging the gap between ab initio theory and experiment.
2. Department of Energy.3. National Science Foundation.4. WISE Research Fund (USC).5. NIH-SBIR (w/Q-Chem).
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