1. The gerade Rydberg states of molecular hydrogen Daniel Sprecher, 1 Christian Jungen, 2 and Frédéric Merkt 1 1 Laboratory of Physical Chemistry, ETH Zurich, Switzerland 2 Laboratoire Aimé Cotton du CNRS, Campus d‘Orsay, France International Symposium on Molecular Spectroscopy Ohio State University, June 20-24, 2011 TH07, June 21, 3:45pm, 1000 McPherson Lab

2. Motivation Measurement of rotational, vibrational, spin-rotation, and hyperfine splittings of H 2 + :  Test of ab initio calculations in one-electron systems  Metrology of fundamental constants (proton-to-electron mass ratio)  rotational, vibrational and electronic channel interactions  s-d and p-f mixing  fine and hyperfine splittings  singlet-triplet mixing and other uncoupling phenomena  shifts from the interaction with dissociation continua Problem: essentially no electric dipole spectrum Solution: spectroscopy of high-n Rydberg states of H 2 Requirement: model to extrapolate Rydberg series Expected perturbations: role of nuclear spin in photoionization

3. Introduction to molecular Rydberg states Properties of Rydberg states: highly excited electronic states energies follow the Rydberg formula:

4. Introduction to molecular Rydberg states para-H 2 : I = 0 and N + = 0, 2, 4, … ortho-H 2 : I = 1 and N + = 1, 3, 5, … gerade states: ℓ = 0, 2, … (s, d, …) ungerade states: ℓ = 1, 3, … (p, f, …) Symmetries:

5. Multichannel quantum-defect theory (MQDT) Born-Oppenheimer approximation Coulomb potential quantization condition for bound states: Energy levels of H 2 + Eigenquantum-defect matrix [V. I. Korobov, Phys. Rev. A 77, 022509 (2008)] [V. I. Korobov et al., Phys. Rev. A 79, 012501 (2009)] calculated ab initio

6. The eigenquantum-defect functions gerade manifoldungerade manifold 2pσB p 1 Σ d 1 Δ d 1 Π all f p 1 Π s 1 Σ d 1 Σ sd 1 Σ 4pσB”B 3pσB' H 2 + 1sσ H 2 + 2pσ 2sσEF 3dσGK 4dσP H 2 + 1sσ H 2 + 2pσ (2pσ) 2

7. I=0 for para-H 2, no hyperfine structure s Rydberg series (ℓ = 0) N + = 0 N = 0 (N = N + + ℓ), no fine structure S = 0 or 1, singlet-triplet splitting present MQDT calculations of para-H 2 S = 1 S = 0 (neglecting s-d interaction and v + >0 channels)

8. MQDT calculations of para-H 2 p Rydberg series (ℓ = 1), S = 0 N + = 0 N = 1 (N = N + + ℓ), no fine structure N + = 2 N = 3, 2, 1 p, ℓ = 1, N = 1 d, ℓ = 2, N = 2 f, ℓ = 3, N = 3 N + = 2 N + = 0 rotational channel interaction

9. MQDT calculations of para-H 2 s Rydberg series (ℓ = 0) N + = 0 N = 0, no fine structure neglect s-d interaction neglect v + >0 channels include s-d interaction neglect v + >0 channels include s-d interaction include v + =1 channels S = 0 S = 1 N + = 2 only N = 2 for ℓ = 0, but N = 4, 3, 2, 1, 0 for ℓ = 2

10. include hyperfine interaction Millimeter-wave data of ortho-H 2 G + =1/2 G + =3/2 S = 0 S = 1 transitions between high-n Rydberg states with an accuracy of 50 kHz Observed gerade states are: d, N + =1, N=1 neglect hyperfine interaction

11. Millimeter-wave data of ortho-H 2 For the G + =3/2 hyperfine components: observedab initio MQDT calculation

12. The gerade d (N=1) Rydberg series G + =3/2 G + =1/2 G + =3/2 rms deviation: 400 kHz

13. Laser spectra of the d Rydberg series [H. J. Wörner, PhD thesis (2007)]

14. Laser spectra of the d Rydberg series [H. J. Wörner, PhD thesis (2007)] N = 1 N = 3 N = 2

15. Laser spectra of the d Rydberg series N = 1 N = 3 N = 2 n=18, N + =3

16. Laser spectra of the d Rydberg series N = 1

17. Laser spectra of the d Rydberg series N = 1 n=7, ℓ=0, S=1, N + =1, v + =1

18. Conclusions Multichannel quantum-defect theory can reproduce the energies of high-n Rydberg states with an accuracy of better than 1 MHz spectroscopy of H 2 + Energy levels of H 2 + Eigenquantum-defect matrix