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, CanadaPage1

Open Questions for Pure Solid pH2

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. 223401, 92(2004)

Theoretical insights of vibrational frequency shift of pH2 in

solid pH2 are still unclear.

observed in pH2 clusters and solid

Page2

schematic diagram of

H. Li and co-workers, J. Chem. Phys. 139, 164315 (2013)M. E. Fajardo, J. Phys. Chem. A 117, 13504 (2013)

Open Questions for CO Doped Solid pH2

Theoretical investigation of of CO in doped solid pH2 is still left

undetermined.

of CO in pH2 clusters of CO isotopes in pH2 solid:

fcc structure: -2.961 cm-1

hcp structure: -2.974 cm-1

Page3

-1cm 013.0 fcchcp

fcc crystal hcp crystal

A

B

A

C

A

B

B

A

A

Part I - Pure Solid pH2-Methodologies

Algorithms:

Path Integral Monte-Carlo (PIMC)

Periodic Boundary Conditions

First order perturbation theory

Simulate at temperature:

T = 4.2 KIsaas 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, 070601, (2006)

R. J. Hinde, J. Chem. Phys., 128, 154308, (2008)H. Li and co-workers, J. Chem. Phys.,130, 144305, (2009)

N. Faruk and co-workers, (under revision)

pH2‐pH2 potential: recently obtained 1D potential

averaged from Hinde’s 6D H2‐H2 potential.

)()(totalHH 22

RVRGdRVv

)1( tonormalized is )( )),()(()( 01 33 NRGRVRVRV

Page4

Part I - Pure Solid pH2-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, 070601, (2006)

or

)0()(

/1/2

EA

aAA

PTP

Our choice:

P = 64

compared with:

extrapolated values ( )

Energy discrepancy: 6.4%

discrepancy: 1.5%

0

Extrapolation of E obtained with 144 atoms in hcp cell

)(E

Page5

Part I - Pure Solid pH2- 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 pH2 in hcp cell)

Observed R of hcp pH2 solid:

3.789 Å

Calculated R of both hcp and

fcc pH2 solid:

3.780 Å

Page6

Part I - Pure Solid pH2-

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))(N

First-order perturbation theory )()(totalHH 22

RVRGdRVv

NRRRGRRVRVRV /1)/1( ,)( ),/1())()(()( 32601 33

Page7

Part I - Pure Solid pH2- 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: 0.026 Å-3

Observed for 0 pressure hcp crystal: -11.38 cm-1

Calculated with

144 atoms in

the cell

Calculated with

144 atoms in

the cell

Page8

Part I - Pure Solid pH2- Summaries

H. P. Gush and co-workers, Can. J. Phys 176, 38 (1960)Isaas F. Silverra, Rev. Mod. Phys. 393, 52(1980)

hcpexperimenta

l

hcp calculated

fcc calculated

(Å)3.789

3.780with 144

atoms in cell

3.780with 108

atoms in cell

(cm-1)-11.38 -11.21

by extrapolate

-11.08by

extrapolate

(Å-3)0.026

0.0262with 144

atoms in cell

0.0262with 108

atoms in cell

22 HH R

pressure 0

Page9

fcc cell

hcp cell

Part II – CO Doped Solid pH2- Methodology

H. Li, P.-N. Roy, and R. J. Le Roy, J. Chem. Phys. 133, 104305 (2010) H. Li and co-workers, J. Chem. Phys.,139, 164315, (2013)

M. E. Fajardo, J. Phys. Chem. A 117, 13504 (2013)

pH2‐pH2 potential: same as the study of pure solid pH2.

pH2‐CO potential: obtained from Hui Li’s 4D H2‐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:

),(),(totalHCO 2

RVRddRVv

201 H ofnumber the tonormalized is ),( ),,(),(),( 33 pRRVRVRV

32 , 128 rottrans PP

Page10

Part II – CO Doped Solid pH2- Methodology

jiggling lattices in a rigid frame

P. Tao, thesis for master degree of science, University of Waterloo(2005)

pH2: Green ones: hold fixed

pH2: Blue ones: relaxing

CO: located in the

center, translating

and rotating R

all pH2 with R < Rrelax is relaxing,

and similar treatment when

choosing the total number of pH2

in the model.

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Rrelax

Part II – CO Doped Solid pH2- Structures

Studies of substitution site:

Nremove=0: pure pH2

Nremove=1: single substitution

Nremove=1: double substitution

Obtained by fcc structure

Nrelax = 42

Nfix = 822

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Classical MC

PIMC

Single substitution is

most stable.

Part II – CO Doped Solid pH2- Structures

minimal energy structure of single substitution in solid fcc pH2

icosahedral pH2 cage in (pH2)12‐CO cluster

S. Baroni and S. Moroni, Chem. Phys. Chem. 6, 1884 (2005)

Page13

Part II – CO Doped Solid pH2-

convergence studies of the number of relaxing pH2 (left) and the total number of pH2(right) in fcc pH2 matrix using of CO

Nrelax = 54, Ntotal = 1260: good approximation for pH2 matrix.

Nrelax is corresponding to Rrelax = 7.56 Å ≈ RH2-H2+ RH2-CO

Page14

Part II – CO Doped Solid pH2-

cm-1 cm-1 cm-1

experimental -2.961 -2.974 -0.013

calculated -3.244(1) -3.251(1) -0.007(2)

fcc hcp fcchcp

M. E. Fajardo, J. Phys. Chem. A 117, 13504 (2013)

of CO in solid pH2 of different structure

Page15

Part II – CO Doped Solid pH2- PIMC vs MC

Page16

distribution of pH2 around CO),( R

1-MC

1-PIMC

-1exp

cm 872.3

cm 244.3

cm 961.2

PIMC

Classical MC

Centre of mass of CO

R

θ

pH2

C O

Conclusion

Observed structures, densities and of both fcc and hcp pH2

crystal have been satisfyingly reproduced.

Single substitution site is most stable for CO doped pH2 matrix.

The obtained different values of in fcc and hcp pH2 matrix,

and the difference agree with observations very well.

Quantum mechanical treatment is critical to simulate pH2

matrix.

Page17

CO

Future works

Scaling pH2‐pH2 potential to provide a more realistic

solvation environment for doped CO in pH2 matrix .

Incorporating Worm Algorithm to handle the Bose

Exchange, thus to predict the rotational dynamics of doped

CO in pH2 matrix.

Scaling pH2‐CO potential to different isotopes of CO to study

isotope effect of .

Page18

CO

Page19

Acknowledgement

Supervisors: Prof. Robert J. Le Roy Prof. Pierre-Nicholas Roy

Prof. Marcel NooijenProf. Hui LiDr. Tao ZengNabil FarukMatthew Schmidt

Theoretical Chemistry Group, University of Waterloo

$$: NSERC and CFI Canada

Thank You!