1. 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

2. Acknowledgements: Dr. Melania Oana Inspiration from experiments of: Albert Stolow, Hanna Reisler, Klaus Muller-Dethlefs Hanna Reisler, USC Albert Stolow, Steacie Institute for Molecular Sciences

3. Photoelectron spectroscopy: Probing energy levels, structure, and electronic wave functions Kinetic energy of electron Ionizing 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.

4. Excitation laser Molecular beam ion detector e - detector Ionization laser PAD 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.

5. 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.

6. Photoelectron wave functions and PADs The probability of finding an electron in dV at {r, θ, φ}: Angular part: Y lm – spherical harmonics Ψ el can be expanded in the basis of spherical waves: dV Radial part: R kl ~ Bessel functions, Jl+1/2:, with

7. |C klm | 2 - probability to find an ejected electron in the {klm} state. They are given by the ionization dipole moment matrix elements between the initial ( Ψ N ) and the final (Ψ N-1 x Ψ el ) states: Using permutational symmetry of the wave functions and integrating over N-1 coordinates: where r i - spatial coordinates of the i th electron PAD and Dyson Orbitals where  d (r) is Dyson orbital:

8. 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: Φ d = φ k  i j k l M M+M+

9. Dyson Orbitals in EOM-CC Formalism EOM-IP/EE-CCSD: Ψ M+/M* = (R 1 + R 2 )  0 Coefficients of Dyson orbitals in MO basis - analogous to transition density matrix element: R 1 Ψ ref R 2 Ψ ref M*M* M+M+ M R 1 Ψ ref R 2 Ψ ref R IP R EE Ψ ref

10. Formaldehyde Example Dyson orbital for correlated (EOM-IP-CCSD/6-311G**(2+,2+)) wave functions - ground state ionization 1A 1  1B 1 : Φ d = 98.7% φ 2b1  CH 2 O CH 2 O + 2b 1 3b 2 1b 2 5a 1 π*π* nπnπ π nσnσ

11. Formaldehyde Example Dyson orbital for excited state ionization 1A 2  1B 1 : (EOM- EE/IP-CCSD/6-311G**(2+,2+)) Φ d = 4.2% φ 2b2 + 71.4% φ 3b2 - 22.4% φ 5b2  CH 2 O ( 1 A 2, n->  *) CH 2 O + 2b 1 3b 2 - λ 5b 2 1b 2 5a 1

12. Formaldehyde Example Dyson orbital for excited state ionization 1B 2  1B 1 : Φ d = 99.1% φ 3a2  CH 2 O* CH 2 O + 3b 2 - λ 5b 2 1b 2 2b 1 5a 1

13. 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 sin 2 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

14. Electron Angular Momentum States: Selection Rules 0 x 0 x 0 x Dyson orbitalr (x, y, z)Φ d (r)·rR kl Y lm s (l = 0) l = 1 p x (l = 1) ~ xxyzxyz x 2 xy xz l = 0, 2 Allowed electron angular momentum states: Δl = ±1 Molecular Dyson orbital: more angular momentum states

15. Higher angular momentum and diffuse orbitals Diffuse orbitals – higher angular momentum Higher kinetic energy – higher angular momentum

16. 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 B 2 states are involved. PADs: additional information about the electronic state.

17. Example: PADs for photodissociation from valence and Rydberg states of (NO) 2 2B 2 1B 2

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

19. Conclusions 1. 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 |C klm | 2. 4. PAD modeling: - within RPA: |C klm | 2 ↔ PAD; - beyond RPA: need the interference contributions - cross terms C * klm C kl’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 A 1 state. More detailed comparison needs to take into account kinetic energy distribution and the phases.

20. THANKS: My group; Ab initio packages: Our codes: available in Q-CHEM & SPARTAN Additional 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).