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.

Example: PADs for photodissociation from valence and Rydberg states of (NO)2

2B2

1B2

Example: PADs for photodissociation from valence and Rydberg states of (NO)2

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).