Drexel University
Department of Chemistry
3141 Chestnut St. Phila. PA 19104
A Reduced Dimensionality Model of Torsional
Vibrations in Branched Molecules
Evan Curtin and Karl Sohlberg
The Molecules
After finding the force constants for each molecule, the average
values were used to predict the frequency of the global torsional
mode.
Global Torsion Predictions
N=1
N=2
N=5
The Model
Frequency (cm-1)
Frequency (cm-1)
Reference
SHO
kl1
kl1
kl2
SHO
kt3
25
SHO*
0
1
0
50000
100000
Moment of Inertia (amu · Å2)
0
10000
20000
30000
Moment of Inertia (amu · Å2)
It has been shown1 that pseudo-diatomic fullerenes follow a
simple mass-spring model. We have found that when used to
model the torsional frequencies of branched alkanes, the
model diverges as the system size increases. This effect isn’t
seen in the alkyne case, due to their rigidity.
References
1. Superdiatomics and Picosprings: Cage−Cage Vibrations in C120O, C120O2, and
in Three Isomers of C130O. Hans-Jürgen Eisler, Frank H. Hennrich, Eva Werner,
Andreas Hertwig, Carsten Stoermer and Manfred M. Kappes. The Journal of Physical
Chemistry A 1998 102 (22), 3889-3897. DOI: 10.1021/jp980834s
2. Application of a Lumped-Inertia Technique to Vibrational Analysis of the TorsionalTwisting Modes of Low Molecular Weight Polyphenylenes and
Polyethynylphenylenes Xiange Zheng, Natalie Vedova-Brook, and Karl Sohlberg. The
Journal of Physical Chemistry A 2004 108 (13), 2499-2507 DOI: 10.1021/jp036659j
Inner 2 kt's
Only Inner kt
1
Force constants connecting concentric cylinders (kli) are used in tandem with force constants
connecting equivalent layers on either side of the molecule (kti). The coupled equations of motion
are solved by diagonalization of the inertia-weighted force constant matrix. Interactions between
layers beyond the nearest neighbor are neglected.
Numerical values for the force constants were found for each member of the series by minimizing
the relative sum of squares residuals between predicted and reference frequencies.
  kt1 , kt N , kl 1 ,..kl N 1 
Reference
Cylinder Model
SHO
10
1
0
20000
40000
60000
80000
Moment of Inertia (amu · Å2)
100000
The cylinder model reproduces the correct trend in the global
torsional frequencies, while a harmonic oscillator model
overestimates the frequencies.
Comparing SHO to Cylinders
SHO
10
Cylinder
SHO*
1
0.1
0.01
0
2000000
4000000
Moment of Inertia (amu · Å2)
6000000
Extending the models to larger systems shows the difference in
frequency between the presented model and the two SHO models.
SHO* was created using a decaying force constant.
Finding Force Constants
 0,i  i  kt1 , kt N , kl 1 ,..kl N 1  
 


i 1 

0,i


Frequency (cm-1)
kl2
kl1
 kl1  kt1 kt1

0
0
0
0
 I

I1L
I1L
1L


 kt1 kl1  kt1

kl1
0
0
0
0 

I1R
I1R
 I1R

 kl1

kl1  kl2  kt2
kt1
kl2
0
0
0 

I2L
I2L
I2L
 I2L



kl1
kt1
kl1  kl2  kt2
 0

I
I
I
2R
2R
2R




kl2
 0
0
0 
I3L



klN 1 
 0

0
0
0
I ( N 1) R 



klN 1  ktN
ktN 

0

I NL
I NL 



kl

kt
kl

kt
 0
N 1
N
N 1
N
0
0
0

I NR
I NR
I NR 

50
10
Perfect Match
Inner and Outer kt's
100
Alkyne Global Torsions
Reference
10
Training the Model
The Problem
100
100
3
150
The molecules used in the study constitute a set of branched hydrocarbons of increasing size.
Reference Frequency (cm-1)
Hydrogen atoms are omitted for clarity. N is the number of layers used in the model. Molecules
where N = 1 through N = 12 were used to benchmark the model.
Including the first two torsion constants reproduces the global
torsional frequencies with the highest accuracy.
kt1
Alkane Global Torsions
N = 12
Predicted Frequency (cm-1)
Dendrimers and star molecules are interesting classes of
molecules that show promise for application in polymer network
scaffolds and molecular drug delivery devices. Such molecules
also possess the feature that they can be constructed in a
controlled, stepwise manner. Owing to this feature, such
molecules are an ideal platform for the study of the properties of
dendrimer-based nanomaterials by extrapolating from a
systematic progression of smaller systems. Many properties of
interest depend on vibrational motion within the backbone of the
dendrimer and therefore there is interest in determining these
vibrational frequencies. Traditional vibrational analysis
techniques are well established, but become intractable for
large systems and are therefore impractical for application in
the nano-regime. Here we present a model to predict torsional
vibrational frequencies of branched molecules. The benchmark
against which the force constants were developed was full
normal coordinate analysis at the HFSCF/6-31G* level of
theory. The best agreement occurred when including force
constants between each concentric layer of the molecules, with
one innermost torsional force constant and one outermost
torsional constant. The numerical values of the optimized force
constants also give insight into which interactions dominate
torsional motions and the nature of torsional oscillations in
nanostructures.
Predicting Global Torsions
Frequency (cm-1)
Abstract
2
2N
The frequencies were assigned by counting radial and axial nodes in the corresponding
eigenvector.
Conclusion
The accuracy with which the proposed model recovers the variation
in global torsional frequencies with system size suggests that the
model can be used to extrapolate for systems far too large to subject
to traditional normal coordinate analysis.
The fact that the first two torsional constants best reproduced the
global torsional frequencies demonstrates the importance of nonbonded torsional interactions to vibrations of the system.
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