1. Application of equation-of-motion coupled- cluster theory to photodetachment cross section calculations Takatoshi Ichino and John F. Stanton The University of Texas at Austin June 18, 2012

2. Why do we care about photodetachment cross section? ex. photoelectron spectrum of NO 3 ‾ Weaver et al., J. Chem. Phys. 94, 1740 (1991) Interpretation of photoelectron spectra can be facilitated with knowledge of cross sections.

3. Two approaches to photodetachment cross section calculations 1.Plane wave approximation of scattering electrons Transition of the electron from the Dyson orbital obtained with equation- of-motion coupled-cluster theory (EOMIP-CC) cf. Reed et al., J. Chem. Phys. 64, 1368 (1976). Öhrn and Born, Adv. Quantum Chem. 13, 1 (1981). Oana and Krylov, J. Chem. Phys. 131, 124114 (2009). 2.Direct ab initio calculations of electronic transition dipole moments for pseudostates Equation-of-motion coupled-cluster calculations of oscillator strengths (EOMEE-CC) to obtain the associated moments cf. Langhoff and Corcoran, J. Chem. Phys. 61, 146 (1976). Reinhardt, Comp. Phys. Comm. 17, 1 (1979). Müller-Plathe and Diercksen in “Electronic Structure of Atoms, Molecules and Solids” (1990).

4. First approach: Plane wave approximation differential cross section: photodetachment from an orbital : SCF orbital → Koopmans’ theorem : Dyson orbital (EOMIP-CC) : plane wave How to choose the orbital: : CC wavefunction : EOMIP-CC wavefunction Feynman-Dyson amplitudes 1) 2)

5. First approach: Plane wave approximation differential cross section: photodetachment from an orbital : plane wave Adjustments of the plane wavefunction: Partial orthogonalization: orthogonalized against Full orthogonalization: 1) 2) (SCF, Dyson) orthogonalized against all natural orbitals

6. First approach: Plane wave approximation Choice of the operator in transition moment calculations: EOMIP-CCSD calculations of Feynman-Dyson amplitudes Steps in cross section calculations: Dipole length: Analytic calculations of integrals (overlap, dipole, derivative) Analytic evaluation of angular integrals of transition moments 1) 2)Momentum: 1) 2) 3)

7. First approach: Plane wave approximation Result 1: Photodetachment from H anion SCF Dyson (dipole length) Dyson (momentum) Branscomb and Smith, Phys. Rev. 98, 1028 (1955) Smith and Burch, Phys. Rev. 116, 1125 (1959) Experiment (dots): EOMIP-CCSD aug-pVQZ + diffuse functions Transition dipole length calculations with the Dyson orbital give an excellent match with the experimental results.

8. First approach: Plane wave approximation Result 2: Photodetachment from Li anion SCF Dyson/full orthogonalization (momentum) EOMIP-CCSD/aug-pVTZ Transition dipole length calculations with the Dyson orbital and the fully orthogonalized plane wave give an excellent match with the experimental results. Dyson/partial orthogonalization Dyson/full orthogonalization (dipole length) Experiment (dots) : Kaiser et al., Z. Phys. 270, 259 (1974)

9. First approach: Plane wave approximation Result 3: Photodetachment from O radical anion SCF Dyson/full orthogonalization (momentum) Transition dipole length calculations with the Dyson orbital and the fully orthogonalized plane wave give an excellent match with the experimental results. Dyson/partial orthogonalization Dyson/full orthogonalization (dipole length) EOMIP-CCSD aug-pVTZ + diffuse functions Branscomb et al., Phys. Rev. 111, 504 (1958); J. Chem. Phys. 43, 2906 (1965) Experiment (dots):

10. First approach: Plane wave approximation Result 4: Photodetachment from F anion SCF Dyson/full orthogonalization (momentum) Higher correlation effects? Wrong experimental results? Dyson/partial orthogonalization Dyson/full orthogonalization (dipole length) EOMIP-CCSD aug-pVTZ + diffuse functions Experiment (dots): Mandl, Phys. Rev. A 3, 251 (1971)

11. Second approach: Electronic excitation to pseudostates The moments of the oscillator strength for the continuum states: EOMEE-CCSD calculations of oscillator strengths for pseudostates Steps in cross section calculations (Stieltjes imaging): Quadrature calculations utilizing the relation between moments and orthogonal polynomials Differentiation to obtain the oscillator strength density 1) 2) 3) approximated by the oscillator strengths for pseudostates Photodetachment cross sections calculated from the oscillator strength density:

12. Second approach: Electronic excitation to pseudostates Result 1: Photodetachment from H anion Dyson (dipole length) ▲: Stieltjes imaging Branscomb and Smith, Phys. Rev. 98, 1028 (1955) Smith and Burch, Phys. Rev. 116, 1125 (1959) Experiment (dots): EOMEE-CCSD aug-pVQZ + diffuse functions Stieltjes imaging gives an excellent match with the experimental results.

13. Second approach: Electronic excitation to pseudostates Result 2: Photodetachment from Li anion Dyson/full orthogonalization (dipole length) Experiment (dots) : Kaiser et al., Z. Phys. 270, 259 (1974) EOMEE-CCSD aug-pVQZ + diffuse functions ▲: Stieltjes imaging Stieltjes imaging gives an excellent match with the experimental results.

14. Second approach: Electronic excitation to pseudostates Result 3: Photodetachment from O radical anion Dyson/full orthogonalization (dipole length) Branscomb et al., Phys. Rev. 111, 504 (1958); J. Chem. Phys. 43, 2906 (1965) Experiment (dots): EOMEE-CCSD aug-pVQZ + diffuse functions ▲: Stieltjes imaging Stieltjes imaging gives a good match with the experimental results.

15. Second approach: Electronic excitation to pseudostates Result 4: Photodetachment from F anion Dyson/full orthogonalization (dipole length) Experiment (dots): Mandl, Phys. Rev. A 3, 251 (1971) EOMEE-CCSD aug-pVQZ + diffuse functions ▲: Stieltjes imaging Stieltjes imaging gives cross sections consistent with those from Dyson orbital calculations.

16. The Dyson orbitals obtained from equation-of-motion coupled-cluster calculations (EOMIP-CCSD) can successfully represent the initial states of photodetachment processes for atomic anions in the cross section calculations. The scattering electrons are described as plane waves orthogonalized against all natural orbitals of the anions. The transition moment calculations should be performed with the dipole length operator. The oscillator strengths for photoexcitation to discrete pseudostates of atomic anions obtained from equation-of-motion coupled-cluster calculations (EOMEE-CCSD) can successfully be utilized for the moments for the photodetachment processes. The associated quadrature calculations provide “smoothed” oscillator strength density, based on which photodetachment cross sections can be evaluated. It may be worth reexamining the photodetachment cross section of the F anion experimentally. Conclusions Thanks to: Professor John F. Stanton NSF, DOE, The Welch Foundation

17. Feynman-Dyson amplitude in EOMIP Stanton and Gauss, J. Chem. Phys. 101, 8938 (1994) Oana and Krylov, J. Chem. Phys. 127, 234106 (2007); 131, 124114 (2009) (ex) EOMIP-CCSD

18. Brauman’s treatment Koopmans’ theorem Partially orthogonalized plane wave Slater functions Reed et al., J. Chem. Phys. 64, 1368 (1976) continuum orbital reference orbital from which an electron is detached

19. What the program does … integral evaluation for the transition moment angular integration to collect all electrons Öhrn and Born, Adv. Quantum Chem. 13, 1 (1981)