Uncertainty in Molecular Photoionization!

Transcription

1 Uncertainty in Molecular Photoionization! Robert R. Lucchese! Department of Chemistry! Texas A&M University Collaborators:! At Texas A&M: R. Carey, J. Lopez, J. Jose! At ISMO, Orsay, France: D. Dowek and coworkers! At Tohoku Univ: K. Ueda and coworkers! At LBNL: T. Rescigno and W. C. McCurdy!! Funding: Welch Foundation, US Dept of Energy, NSF, JSPS! 1

2 Application and Validity Range! Very few experimental electron-molecular ion scattering cross sections! Molecular Photoionization! Total Cross Sections! Asymmetry Parameters! Molecular Frame Angular Distributions! Adiabatic Nuclei Approximation! Fixed Nuclei! Small Amplitude Motion! One electron in the continuum! 2

3 Method Description! Single Center Expansion! One electron in the continuum represented on numerical grid! Bound orbitals from standard Quantum Chemistry Code! Both linear molecules and non-linear polyatomic systems! Correlation in Target Wave Functions! Frozen-Core Hartree Fock! Modest CI expansion, up to 10,000 CSF! Selection of penetration terms to avoid pseudo-resonances! Coupled Channel Expansion! One channel! Coupled channels, no more than ~20 channels! 3

4 Schwinger Variational Expression! Variational approximation to M! t t t t! M R ψ S + χ R S χ R ( 1 G 0 V )ψ S! χ R ( 1 G 0 V ) = R ψ S t χ R t Expand and in basis sets and respectively:! u i v j M 0 = D ij = i,j R u i D 1 i,j v j S v i ( 1 G 0 V )u j 4

5 Φ i = j c ij ψ j ( ) Scattering Wave Function Ψ = A Φ i, χ i = c ij A ψ j, χ i i ( ) i, j ( ) Target! Scattering! Where A ψ j, χ i is an N electron spin-adapted state function constructed from CSF ψ j and continuum function χ i. The target states generally contain ,000 CSFs! Penetration terms occur when χ i is not orthogonal to the bound orbitals used to construct! ψ j Over-completeness and spurious resonances are avoided by selective removal of penetration terms!!! 5

6 Single-Center Expansion! Single-center expansion! l! r β = max f β lm ( r)y lm ( r ˆ )!! l=0!typical values: l max = 60 for N 2 and l max = 80 for CO 2! The matrix for photoionization is not symmetric n=0 φ i µ ψ lm = φ i µ GV ( ) n φ lm We compute a sequence of terms φ i µ ( GV ) n for the one initial state and take simple overlaps with the many homogeneous scattering states φ lm. 6

7 Verification! Consider verification within a given correlation model, i.e. number of channels and representation of targets! Convergence with respect to radial and angular grids! Convergence with respect to one-electron basis set size, complete basis set extrapolation! Depends on range of photo-electron kinetic energies! 7

8 Convergence of Partial-Wave Expansion! Position of! Resonance! N 2 Photoionization! The energy of a shape resonance converges as 1/l 3 8

9 Comparison With Complex Kohn Method Rescigno and McCurdy! Single-Center Expansion Complex Kohn 8 (3σ u ) 1 CO 2 Cross Section (Mb) (t 1g ) 1 Cross Section (Mbarn) (4σ g ) Photon Energy (ev) Photon Energy (ev) SF 6 One Channel! Hartree-Fock Target! CO 2 Two Channel! CASSCF Targets! 9

10 Validation! Validation possible for each level of theory (or model)! Fixed nuclei! Adiabatic nuclei! Different choices for channel coupling and target correlation! Comparison to experimental data! Total cross sections! Laboratory frame differential cross sections! Molecular frame differential cross sections! Consistency! Length vs Velocity! Convergence with respect to hierarchy of models! 10

11 SF 6 (1s) 1 Photoionization Single channel Hartree Fock! Shape resonance at a kinetic energy of 65 ev! Due to a high angular momentum barrier! 11

12 C 60 Photoionization! (6h u ) 1! 12

13 Multi-Channel Frozen Core Hartree Fock! Exp Coupled Channels Separated Channels σ (Mb) SF 6 (4t 1u ) 1 +(1t 2g ) 1 +(3e g ) 1 +(5t 1u ) 1 +(1t 2u ) 1 +(1t 1g ) Photon Energy (ev) Vibrational effects can explain some of the errors! 13

14 Bending in Core Ionization from Hot CO 2 and NO 2! Probe of the effects of bending on shape resonance! Just above threshold core ionization from hot molecules! Primary excitation is the bending! Angle resolved ion yield! Tanaka et al. New J. Phys. 12, (2010)! 14

15 Geometry Dependence! 0.7 Cross Section (Mbarn) I(90 ) 430 I(0 ) 440 Photon Energy (ev) N 2 O N c (1s) -1 θ = 180 θ = 172 θ = 164 θ = Cross Section (Mbarn) I(0 ) I(90 ) N 2 O N t (1s) -1 θ = 180 θ = 172 θ = 164 θ = Photon Energy (ev) CO 2! N 2 O! 15

16 Comparison to Experiment! Cross Section (Mbarn) CO 2 C (1s) -1 (0,0,0) I(0 ) (0,0,0) I(90 ) (0,1,0) I(0 ) (0,1,0) I(90 ) Cross Section (Mbarn) N 2 O N c + N t (1s) -1 (0,0,0) I(0 ) (0,0,0) I(90 ) (0,1,0) I(0 ) (0,1,0) I(90 ) Cross Section (Mbarn) Photon Energy (ev) CO 2 O (1s) -1 (0,0,0) I(0 ) (0,0,0) I(90 ) (0,1,0) I(0 ) (0,1,0) I(90 ) 550 Photon Energy (ev) Cross Section (Mbarn) Photon Enegy (ev) N 2 O O (1s) (0,0,0) I(0 ) (0,0,0) I(90 ) (0,1,0) I(0 ) (0,1,0) I(90 ) Photon Energy (ev)

17 CO Photoionization! 13 valence coupled channels, CASSCF targets! Cross Secion (Mbarns) X 2 Σ + A 2 Π B 2 Σ + Hamnett Plummer Plummer Cross Secion (Mbarns) D 2 Π 3 2 Σ Δ D 2 Π Σ Δ Hamnett "C" State Photon Energy hν (ev) Photon Energy hν (ev)

18 CO Photoelectron Asymmetry Parameters! Photoelectron Asymmetry Parameter Photon Energy hν (ev) 35 X 2 Σ + A 2 Π B 2 Σ + Marr Holmes 40 Photoelectron Asymmetry Parameter D 2 Π 3 2 Σ Δ 34 Photon Energy hν (ev)

19 CO Photoionization! Cross Secion (Mbarns) X 2 Σ + Mixed Length Velocity Hamnett Plummer Plummer Cross Secion (Mbarns) A 2 Π Mixed Length Velocity Hamnett Plummer Photon Energy hν (ev) Photon Energy hν (ev) Consistency Check using Length vs Velocity! 19

20 Molecular Frame Photoelectron Angular Distributions (MFPADs)! In linear molecules, this general expression simplifies to! I ( θ k,φ k,θ n,φ n ) = H L L N Y ( LN ˆk )Y L N ( ˆn ) L L =0,2 ( ) 0 N min L, L The MFPAD can be alternatively written using only real-valued functions for linearly polarized light as:!! ( ) = F 00 ( θ k ) + F 20 ( θ k )P 0 2 ( cosθ n ) +F 21 ( θ k )P 1 2 ( cosθ n ) cos ( φ k φ n ) +F 22 ( θ k )P 2 2 ( cosθ n ) cos 2[ φ k φ n ] I θ k,φ k,θ n,φ n ( ) Lucchese et al. Phys. Rev. A 65, (2002); Cherepkov and Raseev, J. Chem. Phys 103, 8238 (1995)! 20

21 Dissociative Ionization of NO! MFPADs measured by coincidence measurements of fragments and photoelectrons! The ion fragmentation threshold is just above 21 ev! The number of nearby states makes it important that correlation is included in the description of the ion states! Results will be presented for ionization leading to the c 3 Π state of NO +.! 21

22 Photoionization Leading to the c 3 Π State of NO +! 22

23 NO + States! Initial NO configuration! (1σ) 2 (2σ) 2 (3σ) 2 (4σ) 2 (5σ) 2 (1σ) 4 (2σ) 1 1 channel calculation! 5 channel calculation! 17 channel calculation! Exp.: Albritton et al. 1979!! CASCI Exp State (ev) (ev) Main Configuration X 1 Σ (2π) -1 a 3 Σ (1π) -1 b 3 Π (5σ) -1 w 3 Δ (1π) -1 b' 3 Σ (1π) -1 A' 1 Σ (1π) -1 A 1 Π (5σ) -1 W 1 Δ (1π) -1 c 3 Π (4σ) -1 B 1 Π (4σ) -1 (3) 3 Π (5σ) -1 (1π) -1 (2π) +1 B' 1 Σ (1π) -1 (4) 3 Π (5σ) -1 (1π) -1 (2π) +1 (3) 1 Π (5σ) -1 (1π) -1 (2π) +1 (3) 1 Σ (4σ) -1 (5σ) -1 (2π) +1 (5) 1 Δ (4σ) -2 (2π) +1 (7) 3 Σ (4σ) -1 (5σ) -1 (1π) -1 (2π) +2 23

24 Correlation in NO Photoionization Leading to the c 3 Π State of NO +! The 4σ and 5σ hole states included in the 5- channel calculation provide polarization when the electron is ejected in the axial direction! The 17-channel calculation includes 1π hole states that provide polarization of the target when the electron is ejected in the perpendicular direction! 24

25 MFPAD in NO Photoionization Leading to the c 3 Π State of NO +! Exp. - Lucchese, et al., Phys. Rev. A 65, (2002)! 25

26 Recoil Frame Photoelectron Angular Distribution (RFPAD)! Experimentally, MFPAD can be measured with angle resolved electron-ion fragment coincidence using the axial recoil approximation! In linear molecules when the axial recoil approximation fails one can only measure the RFPAD! Rotation before dissociation! Polyatomic linear molecules can also bend in the dissociation process! The RFPAD is all that can be obtained in nonlinear molecules when only two fragments are produced! 26

27 Effects of Rotation on RFPAD for Ionization Leading to the B 2 Σ g State of O 2+! θ k e θ R θ k e hν Ionization Rotation Dissociation 27

28 Valence-Shell Photoionization of O 2! 28

29 Photoionization of O 2 Leading to the B 2 Σ g State of O 2 + A. C. E. Brion, et al., J. Electron Spectrosc. Relat. Phenom. 17, 101 (1979).! B. J. A. R. Samson, et al., J. Electron Spectrosc. Relat. Phenom. 12, 281 (1977).! C. T. Gustafsson, Chem. Phys. Lett. 75, 505 (1980).! D. M. Braunstein and V. McKoy, J. Chem. Phys. 90, 150 (1989).! E. P. W. Langhoff, et al., Int. J. Quantum Chem. Symp. 13, 645 (1979).! 29

30 Effects of Rotation on the RFPAD! τ= 0.45 ps! about 50! rotation Lafosse et al., J. Chem. Phys. 117, 8368 (2002). 30

31 Conclusions! Verification! Grids can be converged so that quantities of interest are converged to within ~5%! One-electron basis sets could be converged, with appropriate extrapolation, not done yet! Validation and Uncertainties! Semi-quantitative agreement with experiment possible with lower level correlation treatments! More differential quantities require higher level calculations! Quantitative agreement possible away from narrow resonances! Nuclear motion can be important with narrow resonances or when resonances are strongly geometry dependent! Consequences of lack of consistency between electronic structure needed to obtain geometric structure and vibrational motion and the description of the electron scattering! 31

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