Application of equation-of-motion coupledcluster theory to photodetachment cross
section calculations
Takatoshi Ichino and John F. Stanton
The University of Texas at Austin
June 18, 2012
Why do we care about photodetachment cross section?
ex. photoelectron spectrum of NO3‾
Interpretation of photoelectron spectra
can be facilitated with knowledge of
cross sections.
Weaver et al., J. Chem. Phys. 94, 1740 (1991)
Two approaches to photodetachment cross section calculations
1.
Plane wave approximation of scattering electrons
• Transition of the electron from the Dyson orbital obtained with equationof-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).
First approach: Plane wave approximation
differential cross section:
: plane wave
photodetachment from an orbital
How to choose the orbital:
1)
: SCF orbital → Koopmans’ theorem
2)
: Dyson orbital (EOMIP-CC)
Feynman-Dyson amplitudes
: CC wavefunction
: EOMIP-CC wavefunction
First approach: Plane wave approximation
differential cross section:
: plane wave
photodetachment from an orbital
(SCF, Dyson)
Adjustments of the plane wavefunction:
1) Partial orthogonalization:
2) Full orthogonalization:
orthogonalized against
orthogonalized against all natural orbitals
First approach: Plane wave approximation
Choice of the operator in transition moment calculations:
1) Dipole length:
2) Momentum:
Steps in cross section calculations:
1) EOMIP-CCSD calculations of Feynman-Dyson amplitudes
2) Analytic calculations of integrals (overlap, dipole, derivative)
3) Analytic evaluation of angular integrals of transition moments
First approach: Plane wave approximation
Result 1: Photodetachment from H anion
SCF
Dyson (dipole length)
Dyson (momentum)
EOMIP-CCSD
aug-pVQZ + diffuse functions
Transition dipole length calculations with the Dyson orbital give an excellent match
with the experimental results.
Experiment (dots): Branscomb and Smith, Phys. Rev. 98, 1028 (1955)
Smith and Burch, Phys. Rev. 116, 1125 (1959)
First approach: Plane wave approximation
Result 2: Photodetachment from Li anion
SCF
EOMIP-CCSD/aug-pVTZ
Dyson/partial orthogonalization
Dyson/full orthogonalization
(dipole length)
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.
Experiment (dots) : Kaiser et al., Z. Phys. 270, 259 (1974)
First approach: Plane wave approximation
Result 3: Photodetachment from O radical anion
Dyson/partial orthogonalization
Dyson/full orthogonalization
(momentum)
SCF
Dyson/full orthogonalization
(dipole length)
EOMIP-CCSD
aug-pVTZ + diffuse functions
Transition dipole length calculations with the Dyson orbital and the fully
orthogonalized plane wave give an excellent match with the experimental results.
Experiment (dots):
Branscomb et al., Phys. Rev. 111, 504 (1958); J. Chem. Phys. 43, 2906 (1965)
First approach: Plane wave approximation
Result 4: Photodetachment from F anion
Dyson/partial orthogonalization
SCF
Dyson/full orthogonalization
(momentum)
Dyson/full orthogonalization
(dipole length)
EOMIP-CCSD
aug-pVTZ + diffuse functions
Higher correlation effects?
Wrong experimental results?
Experiment (dots): Mandl, Phys. Rev. A 3, 251 (1971)
Second approach: Electronic excitation to pseudostates
The moments of the oscillator strength for the continuum states:
approximated by the oscillator
strengths for pseudostates
Photodetachment cross sections calculated from the oscillator strength density:
Steps in cross section calculations (Stieltjes imaging):
1) EOMEE-CCSD calculations of oscillator strengths for pseudostates
2) Quadrature calculations utilizing the relation between moments
and orthogonal polynomials
3)
Differentiation to obtain the oscillator strength density
Second approach: Electronic excitation to pseudostates
Result 1: Photodetachment from H anion
Dyson (dipole length)
▲: Stieltjes imaging
EOMEE-CCSD
aug-pVQZ + diffuse functions
Stieltjes imaging gives an excellent match with the experimental results.
Experiment (dots): Branscomb and Smith, Phys. Rev. 98, 1028 (1955)
Smith and Burch, Phys. Rev. 116, 1125 (1959)
Second approach: Electronic excitation to pseudostates
Result 2: Photodetachment from Li anion
▲: Stieltjes imaging
EOMEE-CCSD
aug-pVQZ + diffuse functions
Dyson/full orthogonalization
(dipole length)
Stieltjes imaging gives an excellent match with the experimental results.
Experiment (dots) : Kaiser et al., Z. Phys. 270, 259 (1974)
Second approach: Electronic excitation to pseudostates
Result 3: Photodetachment from O radical anion
Dyson/full orthogonalization
(dipole length)
▲: Stieltjes imaging
EOMEE-CCSD
aug-pVQZ + diffuse functions
Stieltjes imaging gives a good match with the experimental results.
Experiment (dots):
Branscomb et al., Phys. Rev. 111, 504 (1958); J. Chem. Phys. 43, 2906 (1965)
Second approach: Electronic excitation to pseudostates
Result 4: Photodetachment from F anion
Dyson/full orthogonalization
(dipole length)
▲: Stieltjes imaging
EOMEE-CCSD
aug-pVQZ + diffuse functions
Stieltjes imaging gives cross sections consistent with those from
Dyson orbital calculations.
Experiment (dots): Mandl, Phys. Rev. A 3, 251 (1971)
Conclusions
•
•
•
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.
Thanks to:
Professor John F. Stanton
NSF, DOE, The Welch Foundation
Feynman-Dyson amplitude in EOMIP
(ex) EOMIP-CCSD
Stanton and Gauss, J. Chem. Phys. 101, 8938 (1994)
Oana and Krylov, J. Chem. Phys. 127, 234106 (2007); 131, 124114 (2009)
Brauman’s treatment
Koopmans’ theorem
reference orbital from which
an electron is detached
continuum orbital
Partially orthogonalized plane wave
Slater functions
Reed et al., J. Chem. Phys. 64, 1368 (1976)
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)