Quantum Monte Carlo Simulation of Vibrational Frequency Shifts in Pure and Doped Solid para-Hydrogen Lecheng Wang, Robert J. Le Roy and Pierre- Nicholas Roy Chemistry Department, University of Waterloo Waterloo, Ontario, Canada Page1

Open Questions for Pure Solid pH 2 H. P. Gush and co-workers, Can. J. Phys 176, 38 (1960), Isaas F. Silverra, Rev. Mod. Phys. 393, 52(1980) T. Oka, Annu. Rev. Phys. Chem. 299, 44(1993) G. Tejeda and co-workers, Phys. Rev. Lett , 92(2004) Theoretical insights of vibrational frequency shift of pH 2 in solid pH 2 are still unclear. observed in pH 2 clusters and solid Page2 schematic diagram of

H. Li and co-workers, J. Chem. Phys. 139, (2013) M. E. Fajardo, J. Phys. Chem. A 117, (2013) Open Questions for CO Doped Solid pH 2 Theoretical investigation of of CO in doped solid pH2 is still left undetermined. of CO in pH 2 clusters of CO isotopes in pH 2 solid: fcc structure: cm -1 hcp structure: cm -1 Page3 fcc crystalhcp crystal A B A C A B B A A

Part I - Pure Solid pH 2 -Methodologies Algorithms:  Path Integral Monte-Carlo (PIMC)  Periodic Boundary Conditions  First order perturbation theory Simulate at temperature: T = 4.2 K Isaas F. Silverra, Rev. Mod. Phys. 393, 52, (1980) N. Blinov and co-workers, J. Phys. Chem. A, 120, 5916, (2004) M. Boninsegni and co-workers, Phys. Rev. Lett. 96, , (2006) R. J. Hinde, J. Chem. Phys., 128, , (2008) H. Li and co-workers, J. Chem. Phys.,130, , (2009) N. Faruk and co-workers, (under revision) pH 2 ‐ pH 2 potential: recently obtained 1D potential averaged from Hinde’s 6D H 2 ‐ H 2 potential. Page4

Part I - Pure Solid pH 2 -Methodologies Fittings of numbers of beads P: N. Blinov and co-workers, J. Phys. Chem. A, 120, 5916, (2004) M. Boninsegni and co-workers, Phys. Rev. Lett. 96, , (2006) Our choice: P = 64 compared with: extrapolated values ( ) Energy discrepancy: 6.4% discrepancy: 1.5% Extrapolation of E obtained with 144 atoms in hcp cell Page5

Part I - Pure Solid pH 2 - Structures R: distance between nearest neighbors in solid. In periodic boundary conditions: R is determined by the size of the cell. Isaas F. Silverra, Rev. Mod. Phys. 393, 52(1980) E varies as a function of R (144 pH 2 in hcp cell) Observed R of hcp pH 2 solid: Å Calculated R of both hcp and fcc pH 2 solid: Å Page6

Part I - Pure Solid pH 2 - N: Number of atoms inside one cell in periodic boundary conditions. H. P. Gush and co-workers, Can. J. Phys 176, 38 (1960), varies linearly with 1/N: PIMC (left) and classical MC (right)(fcc) First-order perturbation theory Page7

Part I - Pure Solid pH 2 - and Densities H. P. Gush and co-workers, Can. J. Phys 176, 38 (1960) Isaas F. Silverra, Rev. Mod. Phys. 393, 52(1980) Energy (left) and (right) varies as a function of density (hcp) Observed densities for 0 pressure hcp crystal: Å -3 Observed for 0 pressure hcp crystal: cm -1 Calculated with 144 atoms in the cell Page8

Part I - Pure Solid pH 2 - Summaries H. P. Gush and co-workers, Can. J. Phys 176, 38 (1960) Isaas F. Silverra, Rev. Mod. Phys. 393, 52(1980) hcp experimental hcp calculated fcc calculated (Å) with 144 atoms in cell with 108 atoms in cell (cm -1 ) by extrapolate by extrapolate (Å -3 ) with 144 atoms in cell with 108 atoms in cell Page9 fcc cell hcp cell

Part II – CO Doped Solid pH 2 - Methodology H. Li, P.-N. Roy, and R. J. Le Roy, J. Chem. Phys. 133, (2010) H. Li and co-workers, J. Chem. Phys.,139, , (2013) M. E. Fajardo, J. Phys. Chem. A 117, (2013) pH 2 ‐ pH 2 potential: same as the study of pure solid pH 2. pH 2 ‐ CO potential: obtained from Hui Li’s 4D H 2 ‐ CO potential using Adiabatic Hindered Rotor approximation. Algorithms:  Path Integral Monte-Carlo (PIMC)  First order perturbation theory Simulate at temperature: T = 2.4 K Number of beads in PIMC: Page10

Part II – CO Doped Solid pH 2 - Methodology jiggling lattices in a rigid frame P. Tao, thesis for master degree of science, University of Waterloo(2005) pH 2 : Green ones: hold fixed pH 2 : Blue ones: relaxing CO: located in the center, translating and rotating R all pH 2 with R < R relax is relaxing, and similar treatment when choosing the total number of pH 2 in the model. Page11 R relax

Part II – CO Doped Solid pH 2 - Structures Studies of substitution site: N remove =0: pure pH 2 N remove =1: single substitution N remove =1: double substitution Obtained by fcc structure N relax = 42 N fix = 822 Page12 Classical MC PIMC Single substitution is most stable.

Part II – CO Doped Solid pH 2 - Structures minimal energy structure of single substitution in solid fcc pH 2 icosahedral pH 2 cage in (pH 2 ) 12 ‐ CO cluster S. Baroni and S. Moroni, Chem. Phys. Chem. 6, 1884 (2005) Page13

Part II – CO Doped Solid pH 2 - convergence studies of the number of relaxing pH 2 (left) and the total number of pH 2 (right) in fcc pH 2 matrix using of CO N relax = 54, N total = 1260: good approximation for pH 2 matrix. N relax is corresponding to R relax = 7.56 Å ≈ R H2-H2 + R H2-CO Page14

Part II – CO Doped Solid pH 2 - cm -1 experimental calculated-3.244(1)-3.251(1)-0.007(2) M. E. Fajardo, J. Phys. Chem. A 117, (2013) of CO in solid pH2 of different structure Page15

Part II – CO Doped Solid pH 2 - PIMC vs MC Page16 distribution of pH 2 around CO PIMC Classical MC Centre of mass of CO R θ pH2pH2 CO

Conclusion  Observed structures, densities and of both fcc and hcp pH 2 crystal have been satisfyingly reproduced.  Single substitution site is most stable for CO doped pH 2 matrix.  The obtained different values of in fcc and hcp pH 2 matrix, and the difference agree with observations very well.  Quantum mechanical treatment is critical to simulate pH 2 matrix. Page17

Future works  Scaling pH 2 ‐ pH 2 potential to provide a more realistic solvation environment for doped CO in pH 2 matrix.  Incorporating Worm Algorithm to handle the Bose Exchange, thus to predict the rotational dynamics of doped CO in pH 2 matrix.  Scaling pH 2 ‐ CO potential to different isotopes of CO to study isotope effect of. Page18

Page19 Acknowledgement Supervisors: Prof. Robert J. Le Roy Prof. Pierre-Nicholas Roy Prof. Marcel Nooijen Prof. Hui Li Dr. Tao Zeng Nabil Faruk Matthew Schmidt Theoretical Chemistry Group, University of Waterloo $$: NSERC and CFI Canada Thank You !