Download Conjugated Bonding in Cyanine Dyes: A "Particle In A Box" Model

CHEM 332L
Physical Chemistry Laboratory
Revision 2.1
Conjugated Bonding in Cyanine Dyes: A "Particle In A Box" Model
In this laboratory exercise we will examine the Visible Spectra of a series of Cyanine Dye
molecules and determine max for each compound. These results will then be compared with
max values obtained by treating the conjugated  electrons of the molecules as a free-electron
gas confined to a one-dimensional box whose length is that of the molecule.
The "Particle In A Box" problem is one that is ubiquitous in elementary courses on quantum
mechanics. This is because it illustrates a number of key principles concerning quantum
mechanics and because the solution of the Schrodinger Wave Equation for this system is
analytically solvable. In this model system a "particle" of mass m is confined to a onedimensional "box" along the x-axis between x=0 and x=L. Within the "box" the particle
experiences no potential, outside the "box" it is subject to an infinite potential. This model
results in a Wave Equation (  = E) within the "box" of:
(Eq. 1)
Solution of this 2nd Order Linear Homogeneous Differential Equation, after application of
appropriate boundary conditions and normalization, yields the following wave functions:
n = 1,2,3, …
0≤x≤L
(Eq. 2)
with associated energies:
(Eq. 3)
Absorption of a photon of appropriate energy:
E = Eph = h = hc/
(Eq. 4)
where  and  are the wavelength and frequency of the photon respectively, will lead to an
energy transition as pictured below.
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Here, nu and nl are quantum numbers associated with the upper and lower quantum states
involved in the transition, respectively. For a Particle In A Box, this transition involves an
energy change of:
E =
(Eq. 5)
Although seemingly abstract, Hans Kuhn has shown (J. Chem. Physics, 1949) that the Particle In
A Box model can be used to predict the wavelength of maximum absorbance (max) for a series
of organic dyes possessing highly conjugated bonds and James R. Arnold has used this model to
predict the energies of single covalent bonds (J. Chem. Physics, 1956). In these constructions
the molecular "bond" is modeled as a one dimensional box containing a free electron gas
consisting of the bonding electrons. At least qualitatively, both Kuhn and Arnold obtain results
which are correct.
We will follow Kuhn's lead and predict max for a series of Cyanine dyes, which will then be
compared with experimental determinations to judge the quality of the predictions.
Consider the dye 1,1'-diethyl-4,4'-cyanine iodide:
The conjugated series of double bonds 1-4 extends from the Nitrogen atom on one end of the
molecule to the Nitrogen atom on the opposite end. The N electrons (10 e- 's in this case; the
lone pair on the Nitrogen must be included in the count):
N = 2j + 2
where j = # of double bonds
(Eq. 6)
involved in this conjugated system can be thought of as occupying molecular electronic states
according to the Pauli Exclusion Principle.
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An electronic transition can occur from the Highest Occupied Molecular Orbital (HOMO) to the
Lowest Unoccupied Molecular Orbital (LUMO) when a photon is absorbed according to
Equation 4. For our dye molecules in solution, we will assume:
max = hc / E
(Eq. 7)
The HOMO-LUMO transition will then occur between the N/2 and the N/2+1 levels. Thus, we
can estimate E according to Equation 5 as:
E = EN/2+1 - EN/2
=
=
(Eq. 8)
This result can be used to predict the max for the compound's spectrum, if we know L, the length
of the molecular "box."
To a first approximation, these N electrons can be modeled as confined to a one-dimensional box
of length equal to that of the conjugated chain, designated L for our purposes. Based on the
length of the C=C bonds involved in the conjugated chain, L can be determined according to:
L = Nxl
where l = length of C=C bond = 139 pm
(Eq. 9)
According to Kuhn, the length of the conjugated chain should be taken as "the length of the
polymethine ziz-zag chain between the nitrogen atoms plus one bond distance to either side."
The purpose of including a "bond distance" on either side of the Nitrogen atoms is to include the
"distance" occupied by the lone pairs of the Nitrogen. Further, the aromatic rings at the ends of
the conjugated chain provide a soft rather than abrupt rise in the potential energy at the ends of
the "box." This softness can be accommodated using an appropriately chosen empirical fitting
parameter. We are only interested in the qualitative alignment of the model with the
experimental data, so we will not account for these details.
Based on the Particle In A Box model, we expect the electronic transition to produce a fairly
sharp spectral band. However, for solutions of these compounds, the spectral band is observed to
be fairly broad; on the order of tens of nanometers.
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This can be accounted for by considering molecular vibrations within the transitioning
molecules. If the electronic transitions from state to state occur within a time interval much
shorter than that of a molecular vibration, the absorption from many molecules shows up as a
broad band; those molecules that have their conjugated bond system stretched or compressed
during a transition, absorb at longer and shorter wavelengths than the absorption maximum, max.
So, in this study, we will measure the max for a series of cyanine dyes and compare the results
with those expected based on the Particle In A Box model. These results will then provide us
with some insight into the electronic structure of these compounds and the nature of a chemical
bond.
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Procedure
Prepare 10-4 M Methanol solutions of the following three cyanine dyes:
Take the Visible Spectrum of each solution and determine max (include an error estimate) for
each compound. If the absorbance maximum is much greater than unity, your solutions should
be diluted and the spectrum re-taken. This will minimize the occurrence of dimers which could
affect the determination of the max.
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Data Analysis
1.
Determine max for each compound whose spectrum was taken, using the Particle In A Box
model. Compare each result with the experimental value. Comment.
2.
Compare your experimentally determined max values with those of the literature.
Comment.
3.
We wish to examine James R. Arnold's approach to understanding covalent chemical
bonding between two atoms, A and B. Using this approach, each atom is modeled as a one
dimensional Particle In A Box. For atom A the "box" will have a width of 2rA, where rA is
the atomic radius. Similarly for atom B, a "box" of width 2rB is used. In each case, the
particle is a free electron. Once atoms A and B bond to form the molecule A-B, a "box"
containing two electrons of length rA + rB + R, where R is the bond distance, is used. This
is as pictured:
The bond energy can be approximated by the difference between the energy of the
molecule and that of the two atoms, assuming the each system is in the ground state (n =
1):
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Bond Energy = EA + EB - EA-B
(Eq. 10)
Each energy Ei can be determined via Equation 3. (Recall EA-B involves two electrons in
the ground state.)
a) Estimate the S-S bond energy using Arnold's approach as outlined above. According to
Huheey, the van der Waals radius (rvdw) of a Sulfur atom is 1.80Ǻ and the Sulfur-Sulfur
bond length in S8 is 2.05Ǻ. Experimentally this bond energy has been determined as
49 kcal/mol. Calculate the percentage error in your estimate.
b) Arnold sets the atomic radii ri as being proportional to an atom's non-bonding radius:
ri = k rvdw
where k is an empirical fitting parameter. Arnold finds a value of k = 1.185 provides a
"best fit to all the data." Use this modified approach to determine the S-S bond energy.
Again, calculate the percentage error in this estimate.
c) Assuming this model is a reasonable picture of reality, what is the physical source of
the bonding between the two atoms A and B?
d) In a real bond, what is the physical source of the bonding between the two atoms A and
B?
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References
Arnold, James R. "'Smoothed Potential' Theory of Chemical Binding" J. Chem. Physics 24
(1956) 181.
Brooker, L.G.S., Keyes, G.H. and Williams, W.W. "Color and Constitution. V. The Absoprtion
of Unsymmetrical Cyanines. Resonance as a Basis for a Classification of Dyes" J. Am.
Chem. Soc. 64 (1942) 199.
Fisher, Nellie I. and Hamer, Frances M. "A Comparison of the Absorption Spectra of Some
Typical Cyanine Dyes" Proc. Roy. Soc., Ser. A 154 (1936) 703.
Gerkin, Roger E. "A Molecular Spectral Corroboration of Elementary Operator Quantum
Mechanics" J. Chem. Ed. 42 (1965) 490.
Goodisman, J. Contemporary Quantum Chemistry: An Introduction Plenum Publishing, New
York, 1979.
Huheey, James E., Keiter, Ellen A. and Keiter, Richard L. Inorganic Chemistry: Principles of
Structure and Reactivity, 4th Ed. Harper Collins College Publishers, New York, 1993.
Kuhn, Hans "A Quantum-Mechanical Theory of Light Absorption of Organic Dyes and Similar
Compounds" J. Chem. Physics 17 (1949) 1198.
Sime, Rodney J. Physical Chemistry: Methods, Techniques, and Experiments Saunders College
Publishing, Philadelphia, 1990.
Shoemaker, David P.; Garland, Carl W. and Nibler, Joseph W. Experiments in Physical
Chemistry McGraw-Hill, New York, 1996.