Quantitative Modelling of the Shifts and Splitting
in the Infrared Spectra of SF6 in an Ar Matrix ‡
Tao Peng and Robert J. Le Roy
Guelph-Waterloo Centre for Graduate Work in Chemistry and Biochemistry, University of Waterloo
60th International Symposium on Molecular Spectroscopy, June 22, 2005
‡ Research
supported by the Natural Sciences and Engineering Research Council of Canada
Introduction
 SF6: Octahedral molecule (Oh symmetry)
 15 (3N-6=15) vibrational modes

 3 normal mode:
asymmetric stretching vibration
triply degenerate
2
Introduction

Frequency shifts and splitting

SF6 (gas phase):  3 = 948 cm-1
3
SF6 in Ar matrix: unexpectedly
large amount of structure, with
9-11 observed peaks

Monte Carlo simulation
Experimental spectra for Ar/SF6=10,000; deposited at 10 K.

Lower : unannealed. Upper : annealed, 31K.1
Spectra Fitting
1B.
I. Swanson and L. H. Jones, J. Chem. Phys. 74, 3205 (1980)
3
Frequency Shifts
By 1st order perturbation theory:

E0  0 Vp 0

E1i (i=1-3) are the roots of
det[ 1i Vp 1k  E1i  i ,k ]  0
 i  (E1i  E0 ) / hc
4
Instantaneous-dipole/induced-dipole (IDID) Mechanism

The SF6 molecule has no permanent dipole moment,
but when displaced from equilibrium, an instantaneous electric
dipole moment arises …

2
 
3    

 
E 0  0 VP 0   V0 ( R j ) 
6 
2R j  Q3  
j 1 



2
 

     R j ,i R j ,k
 3
1i V p 1k   V0 ( R j ) i ,k 
 
 4 i ,k 
6
2


2R j  Q3  
Rj
j 1 


n
n

Conclude: for any particular arrangement of perturbers, we can
calculate the vibrational frequency shifts and splitting pattern.

Other models for the perturbation will give the same type of splitting
patterns.
5
Vacancy Sites
Yellow circles denote Ar atoms removed from FCC lattice to form vacancies for SF6.
Site 1
Site 2
Site 4a
Site 4b
Site 3a
Site 4c
Site3b
Site 5a
6
Flowchart of Simulation
Initial Configuration
Potential Calculation
MEQ: the maximum moves to
equilibrate the system. In our
simulation, the potential
energy doesn’t decrease
significantly after a relatively
short run of ~1,000
moves/atom. Therefore, MEQ
needn’t be very large.
MAV: the number of MC steps
used to obtain the frequency
shifts distribution. Tests
considered between 30,000
moves/atom and 1,000,000
moves/atom.
Choose Particle
No
Move Particle
Ignore Trial Move
Potential Calculation
No
ΔE ≤ 0 ?
No
Random No. x
0≤x≤ 1
x ≤ e-ΔE/kT ?
Yes
No
Accept Move
Yes
MEQ ≤ moves ?
Yes
Calculate Shifts
MAV=moves ?
Yes
Results
7
Some Details of the Simulation

Ar atoms are initially in perfect FCC lattice positions. SF6 is initially at the center
of the chosen vacancy site.

There are ~1,500 Ar atoms surrounding SF6.

SF6 is allowed to translate and rotate.

Inner Ar atoms are allowed to move during MC simulation. Outer ones are
frozen in perfect FCC configuration. Convergence tests varied No. of moving
atoms from ~55 to ~625.

The frequency shifts depend only upon the relative positions of Ar atoms and
their distances from the center of mass of SF6.

The overall potential energy depends both on the orientation and the position of
the SF6 molecule.

The overall potential energy is required for performing the MC simulation. It also
helps determine the relative importance of the different possible lattice vacancy
types.
8
Simulated Spectra

After the initial equilibration,
frequency shifts are calculated at
every move, i.e., every MC
configuration gives us three sticks in
the frequency shift axis.

After MAV ( ~108 ) moves, we count
the No. of sticks within the same shift
interval and obtain a distribution
curve for this particular site.
9
Simulated Spectra

After the initial equilibration,
frequency shifts are calculated at
every move, i.e., every MC
configuration gives us three sticks in
the frequency shift axis.

After MAV ( ~108 ) moves, we count
the No. of sticks within the same shift
interval and obtain a distribution
curve for this particular site.

The frequency shifts data were first
fitted to a sum of Gaussians, using
standard non-linear least-squares
techniques.
10

No constraints
Peak Type
Parameter
Value
Uncertainty
------------------------------------------------------------------1 Gaussian position
-6.7357
0.0003
width
0.3205
0.0008
height
12.21
0.02
2 Gaussian position
-6.2308
0.0002
width
0.3478
0.0005
height
21.35
0.02
DSE= 0.1322

Fix Peak Area Ratio 1:2
Peak Type
Parameter
Value Uncertainty
-------------------------------------------------------------------1 Gaussian position
-6.7379
0.0003
width
0.3100
0.0006
height
12.16
0.02
2 Gaussian position
-6.2323
0.0002
width
0.3536
0.0004
height
21.32
DSE= 0.1500
11
12
Site 2 Presented a Problem

The expected area ratios of 1:1:1 or
1:2 or 2:1 or 3 are not found at first.

Two sub spectra were then obtained.
13
Results of Simulation
14

Because the calculated frequency shifts are smaller than
experimental ones, the model clearly is not perfect. Some
scaling correction will be necessary, such as
 exp  A   sim  B

The total spectrum must be the linear combination of different
sites:
Stotal  p1  S1 ( ,  )  p2 a  S2 a ( ,  )  p2b  S2b ( ,  )  

A least-squares fit to the experimental spectrum was used to
determine these parameters (A, B, p1 , p2a , p2b , … etc.).
15
Fit to the Experimental
Spectrum
A
0.9597
B
-4.237
DSE
21.02
Site
Relative population
2a
0.46
2b
3.97
3a
7.25
3b
2.04
4a/b
0.63
4c
14.64
5a
4.37
16
Conclusion

Our fit shows that the observed spectrum is a linear
combination of spectra for SF6 trapped in different types of
lattice sites and the its peaks are assigned to different types
of lattice sites.

Use of the IDID mechanism in a Monte Carlo averaging
procedure successfully reproduces the experimental IR
spectra for the  3 vibrational band of SF6 in an Ar matrix.

The need for the ad hoc scaling and shifting of frequency
shifts showing shortcomings of our frequency shift model
should be addressed.

The effect of uncertainty in the Ar-SF6 potential function
should be examined.
17
Thank You!
18
19

We sorted the frequency shifts into two groups based on the maximum
peak separations, and we obtained the two plots:

The area ratios are found to be 2:1 and 1:2 , respectively, which is
reasonable.
20