A New Potential Energy Surface for N 2 O-He, and PIMC Simulations Probing Infrared Spectra and Superfluidity How precise need the PES and simulations be? Lecheng Wang, Daiqian Xie, Hui Li, Robert J. Le Roy and Piere-Nicholas Roy

Open Questions for He-N 2 O Considering the Q 3 asymmetry stretch of N 2 O alone gives a poor prediction of the vibrational shift for the He-N 2 O dimer. Predictions for large clusters require a PES which is accurate at long range, which ab initio calculation could not yield. nano droplet limit A.R. W. McKellar, J. Chem. Phys 127, (2007) Yanzi Zhou, Daiqian Xie, Donghui Zhang, J. Chem. Phys 124, (2006) There have been no quantitative theoretical studies of the evolution of ν 3 the band origin shift (Δ ν 3 ). N=1: theoretical estimate: 0.17 cm -1 experimental value: 0.25 cm -1

Open Questions for He-N 2 O In 4 He nanodroplets the rotational constant of N 2 O is just half that of CO 2 In the gas phase, B is similar for both. Both CO 2 and N 2 O have 22 electrons, and have similar interactions with He. The only difference is monomer symmetry. The reason for the difference in the size dependence of B for N 2 O and CO 2 clusters is unclear! K. Nauta and R.E. Miller, J. Chem. Phys 115, (2001)

How to address these ‘tough’ problem? 1. Get a highly precise N 2 O-vibration-dependent PES 2. Determine the band origin shift 3. Predict the superfluidity Part I: PES Part II: Δ ν 3 Part III: f s

Part I: The PES – How to describe stretching Hence, our ab-initio He-N 2 O PES must be 4D, depending on r NO, r NN, R He-N 2 O and θ The strong coupling of Q 1 and Q 3 requires the symmetric stretch to be considered, and not just held fixed! A simple approach might be explicitly consider the asymmetric stretch coordinate while fixing the symmetric stretch coordinate How to incorporate asymmetric vibration of N 2 O? But … Δ ν for the dimer obtained in this way is only 0.189cm -1 while the observed value is 0.253cm -1 r NN r NO R He-N 2 O θ Centre of mass of N 2 O

Part I: The PES – Ab initio details Ab initio calculation details: Perform ab-initio calculations at the CCSD(T)/aug-cc-pVQZ level 713 different He-N 2 O geometries: R from 2 to10 Å, θ in steps of 10 ° He-N 2 O energy minimum(R,θ) at different (r NN,r NO ), in cm -1 At each of these geometries generate 30 {r NN,r NO } points Total of configurations The random error of the super-molecular approach introduces noise, especially at long range. How to smooth it out?

Part I: The PES – Smoothing over the noise Use long-range (LR) coefficients based on the polarizability of N 2 O to smooth PES at long range PES at long range has form: Vibrational averaged Experimental Leading term of LR PES is Main contribution to is

Part I: The PES – The 4D MLR form In the 4D MLR form, expand each radial parameter A= D e, R e, β i in terms of powers of r NO and r NN. 2D MLR: 4D MLR: Expansions in r NN and r NO using powers up to i = j =5 require only 598 non-zero parameters to represent the ab initio points with an RMS error of only cm -1.

Part I: The PES – From 4D to 2D Use the asymmetric-stretch wave functions for vibrational states to average to yield 2D Morse/Long-Range analytical functions were used to fit the averaged 2D PES. Only 59 fitting parameters are required to yield a standard error 0.027cm -1 for points at energies below 1000 cm -1.

How to address these ‘tough’ problem? 1. Get a highly precise N 2 O-vibration-dependent PES 2. Determine the band origin shift 3. Predict the superfluidity Part I: PES Part II: Δ ν 3 Part III: f s

Part II: Δν 3 – Simplify PIMC sampling Standard PIMC algorithm: The small value of compared to makes conventional PIMC simulations very difficult to converge for large clusters. Solution: use first-order perturbation theory; treat the small difference between as perturbation H. Li and R. J. Le Roy, Phys. Chem. Chem. Phys. 10, 4128 (2008) Strategy: avoiding sampling over kinetic energy operators Effect: less fluctuation and faster convergence MAKES SIMULATIONS FOR LARGE CLUSTERS POSSIBLE!

Part II: Δν 3 and the PES from surfaces with fixed LR PES has systematic discrepancy PDS = Potential Difference Surface Fixed LR Fitted LR exp Systematic discrepancy Discrepancy come from systematic error in finite temperature approximation of PIMC For He-N 2 O dimer

Part II: Δν 3 and the solvent distribution The distribution of solvent He atoms determines the evolution of the vibrational frequency shifts

Part II: Δν 3 – Is bending negligible? Contribution of the bending mode to the He-N 2 O interaction from the bending dipole of N 2 O. Given, the contribution of bending stretch to is of order cm -1 : very tiny → negligible. : the angle between and : perpendicular to linear N 2 O Obtain the vibrational average for in the same way as PES. Molecule moment and intermolecular faces

How to address these ‘tough’ problem? 1. Get a highly precise N 2 O-vibration-dependent PES 2. Determine the band origin shift 3. Predict the superfluidity Part I: PES Part II: Δ ν 3 Part III: f s

Part III: f s – The rotation constant B Classical Exp. Two-fluid Correlation Rotational Constant Both the two-fluid model and correlation function calculations reproduce the evolution of B Comparison with B values calculated using classical statistics (ignoring exchange) shows that the turn-around stems from Bose exchange effects

Part III: f s – The superfluid fraction Superfluid Fraction Exp. Calc f s obtained by boson PIMC simulations correspond to the experimental observations The large discrepancies for small clusters arise mainly because of deviations from the symmetric- top model assumed by the two-fluid theory

Part III: f s – How CO 2 and N 2 O differ Two fluid model: Classical Rotational Constant He-N 2 O He-CO 2 Two quantities determine the effective B: Total inertia Superfluidity

Part III: f s – How CO 2 and N 2 O differ f s for different PES Isotropic He-N 2 O He-CO 2 He-N 2 O Scaled He-N 2 O Symmetrized-He-N 2 O 3 trial PESs: Symmetrized Scaled to He-CO 2 Isotropic Both strong solvent-solute interaction and the symmetry lead to the differences between the B values of N 2 O-(He) N and CO 2 -(He) N.

Conclusions To explain the Δν 3 vibrational shifts requires a 4D ab-initio He-N 2 O PES, including symmetric stretching, rather than a PES depending only on R, θ and Q 3. Use of theoretical polarizability results for N 2 O to define the Q 3 stretching dependence of the long-range interaction improves the long-range PES behaviour and gives better predicted shifts Our simulation predictions of superfluid fraction and B constant evolution with N agree well with experiment Simulations performed using ‘artificial’ scaled PESs provide insight into the reason for the differences in rotation constant behaviour between CO 2 -(He) N and N 2 O-(He) N clusters.

… and thank you! This work was funded by NSERC Canada and relied on the computational resources of SHARECNET. Also thank Prof Guo for his thought-invoking discuss.

How do we determine ‘experimental’ f s