1. Calculation of Excitations of Superfluid Helium Nanodroplets Roman Schmied and Kevin K. Lehmann Department of Chemistry Princeton University 60 th Ohio State University International Symposium on Molecular Spectroscopy Columbus, June 23, 2005

2. Electronic HENDI spectra Phonon wings in 4 He nanodroplets: demonstrate superfluidity? Why is the ZPL split? How to estimate the phonon spectrum? glyoxal: from Stienkemeier and Vilesov JCP 115 (22), 2001, 10119 (HElium NanoDroplet Isolation)

3. First-Principles Approaches Quantum Monte Carlo techniques Only lowest excitations of each symmetry Only small droplets C 6 H 6 – He 14 excitations from DMC: from Huang and Whaley, PRB 67, 2003, 155419 Excitations localized around C 6 H 6 :

4. DFT to the rescue! !! Helium density, NOT electron density Bose symmetry included continuum theory Hydrodynamic description of flow Excitations as eigenmodes of oscillation

5. Spherical Simulations 1D simulations Excitations with any angular momentum No real-time dynamics, only phonons (normal mode analysis) No phonon-phonon interactions: linear theory DFT helium density around a 4 He atom density / nm-3

6. How good is DFT? (I) bulk Energy(P) bulk density(P) static response function (P=0), in particular the bulk compressibility bulk phonon spectrum (+pressure dependence) Calibration: momentum / nm -1 energy / cm -1 Orsay-Trento Density Functional (OTDF): Dalfovo et al., PRB 52(2), 1995, 1193

7. How good is DFT? (II) Energy: –358.8 cm –1 –DMC: –357.3(6) cm –1 Chemical potential: 3.2 cm –1 –DMC: 3.1(1) cm –1 Ag–He 100 : DFT from Mella, Colombo, Morosi, JCP 117 (21), 2002, 9695

8. procedure Input: –Pair potentials –Number of helium atoms Output: –Helium density –Phonons –Superfluid fraction 1.minimize energy 2.for each L: diagonalize dynamics matrix

9. Finite droplet excitations N=5000: angular momentum L phonon energy / cm -1 surface waves 10x bulk waves Compare to liquid-drop model:

10. Excitations around a dopant Some excitations are lowered “under” roton minimum Freezing: some phonons become unstable (imaginary frequency) DFT is (for now) unable to do freezing  =39cm -1 (5.5  He-He),  =0.2556nm momentum / nm -1 energy / cm -1

11. Split zero-phonon lines ZPL can split in 2 or 3 lines Peaks on phonon wing lowest L=5 excitation / cm -1

12. Superfluidity Thermally populated phonons induce normal fluid moment of inertia: Superfluid fraction:

13. “local superfluid fraction” Superfluidity is a global quantity We can define a local quantity Influence of dopant is minor unless frozen solvation shell local normal-fluid fraction

15. --> Q branches in spectra of (HCN) n, (HCCCN) n local normal-fluid fraction

16. Conclusions We can compute: –Density –Phonons –Superfluid fraction Large droplets Doped bulk Doped droplets ZPL splitting in electronic HENDI spectra Q branches

17. Acknowledgments Kevin Lehmann Charlotte Elizabeth Procter Fellowship

18. Freezing in solvation shell Very low energy phonons “under” roton minimum Localized in first solvation shell Such modes are few and far apart Explanation of ZPL splittings? 0.1x density / nm -3

19. Pair-correlation function Density around a helium atom in bulk DFT: Does not include Bose exchange of that atom with the fluid from Ceperley, RMP 67 (2), 1995, 279 DFT DMC, exp.