**Beta Carotene and Lycopene**

** **

Beta Carotene and lycopene have important biological activity as carotenoids (derivatives of carotene, often yellow to red colored pigments). Beta Carotene plays an important role as a provitamin of vitamin A (a provitamin is a precursor to a vitamin that is biologically processed by an organism). Lycopene, on the other hand, is a useful anti-oxidant that has important anti-cancer and anti-mutagenic properties. Spectroscopy is a useful tool in determining the composition of many foods because of the characteristic coloring property of cartenoid compounds. In particular, fruits and vegetables can be analyzed with spectroscopy to determine the amounts of nutrients they contain. These spectroscopic methods are a much cheaper alternative to mass spectroscopy or other compositional analysis as they don’t require expensive equipment or require a long tedious process. In one study, the researchers used these spectroscopic methods to track the growth of tomatoes through their life cycle; tracking the levels of chlorophyll and lycopene throughout the life cycle of a tomato plant (*1).*

* *

**Introduction to Particle in a Box Theory**

The Particle in a Box (PIB) model describes a particle moving in one dimension between two walls (the sides of the box). The potential energy between 0 and L vanishes and outside tends to infinity. In this way, the particle is free to move anywhere along the length of the box, but cannot move outside the walls of the box. This is because it is impossible for the particle to gain the infinite energy needed to overcome the potential energy barriers at either end of the box. The Schroedinger equation for this model gives the wavefunction for the particle which describes the behavior of the particle and is given by:

where *h* is Planck’s constant and *m* is the mass of the particle. Solving this equation for the total energy *E* of the particle gives:

where *h* is Planck’s constant, *n* is the discrete energy level, *m* is the mass of the particle, and *L* is the length of the box. H-bar is equal to h/(2π). There are several points to keep in mind when applying this model to pigments such as lycopene and beta-carotene. First, notice that the energy at each energy level is inversely proportional to L^{2}. This means increasing the length of the box *decreases* the energy of every energy level. Second, remember that the difference between energy states decreases for higher energy states.

**Application of the PIB Model to Conjugated Systems**

** **

A conjugated system is a system of alternating single and double bonds. The pi electrons in conjugated systems can be likened to the particles residing inside the box in our model and the length of the conjugated system can be likened to the length *L* of the box. The pi electrons of the conjugated system are essentially delocalized across the length of the conjugated system and are therefore free to move about the entire length of conjugated system. This freedom of mobility corresponds to the “zero” potential energy from 0 to L in our PIB model. The pi electrons cannot move outside of the conjugated system where there are only sigma bonds. Thus, at 0 and L (the ends of the conjugated system) there is infinite potential energy. In this way, the electrons are confined to the “box” that is the conjugated system by the infinite potential energy that is outside of the conjugated system.

The application of the particle in a box model to the conjugated systems of Beta- carotene and lycopene applies to other conjugated systems as well. As the level of conjugation increases, the “length” of the particle in a box approximation as well as the ground state energy level increases. This means that with a more conjugated compound, one can expect the energy levels to become closer and closer. Smaller energy jumps leads to longer wavelength absorptions. As is seen with the carotenoids, the conjugated systems of 9-11 conjugations absorb most wavelengths in the visible spectrum except for the longest ones. It has also been observed that systems with less than eight conjugations only absorb in the non-visible, higher-energy, ultraviolet region. With increasing conjugation, the compounds begin to absorb in the higher energy regions of the visible light spectra, starting with the violets, then blues, etc. up the visible spectra. Although the exact model breaks down with increased conjugation, the particle in a box model can be used as a very rough approximation for a compounds conjugation and can thus be used as a tool for compound identification.

**How do electronic transitions fit in to all this?**

When a system is conjugated, the electrons of the p orbitals are delocalized over the entire framework of the conjugated system because the p orbitals are parallel to one another and form a linear plane for the electrons to easily traverse through. Under the assumption that the pi electrons of the system are non-interacting, the energetics of this system can be approximated by the simple quantum particle in the box model. This assumption is not entirely outrageous, as if the pi electrons are delocalized over the conjugated framework, they will naturally tend to spread out in order to minimize the electron-electron repulsions. Solving Schrodinger’s equation yields both the wave function:

and the energy function:

Interestingly, we notice a relationship between the quantum number n and the number of pi electrons of the conjugated system. Take, for example, 1,3-butadiene. We see here that the number of each nodes in the different energy levels of the pi system correspond exactly to those seen in the PIAB model. Using the PIAB model also allows for the determination of the amount of energy required for the excitation of an electron of the conjugated system. Take for example, 1,3,5-hexatriene.

.

Here, we can calculate the change in energy between the highest and lowest occupied orbitals and set this equal to the amount of energy for a certain wavelength to solve for the wavelength needed for excitation of the electron. ** **

**How do we explain the observed colors and what explains the discrepancies between predicted and actual results?**

Conjugated systems have unique properties which give rise to their colors. Basically, when the energy of the incident light (hν) matches the energy differences between the ground state and the excited state is allowed under specific selection rules, the transition can take place. The electrons that absorb a photon of light of the correct wavelength can be promoted to a higher energy level. Most of these electronic transitions are of a p-orbital electron to a p-anti-bonding orbital (π to π*), but non-bonding electrons can also be promoted (n to π*). If the wavelength of the light absorbed falls into the ultraviolet and visible region, the compound will have visible color, and it will exhibit a color complementary to the one being absorbed. This relationship is demonstrated by the color wheel below:

β-Carotene contributes to the orange color of many different fruits and vegetables and lycopene has a dark red color. However, these observations do not totally agree with our predicted colors. One important factor we left out was the electron repulsion. In the particle in a box model, we assume that electrons are confined in a particular region. The repulsion will be minimized when the box is small enough and only when the box gets longer, the repulsion gets bigger and becomes a more important factor in determining the energy levels than the factor that they are confined in a box. As we can see, the Coulomb’s potential will be 1/L, contrasting with the PIB model which has energy levels proportional to 1/L^{2}, thus being a major factor when L increases. Therefore, we observe the trend that when the molecules become longer, the real value will have a smaller energy gap than those predicted.

**References**

Arias, R.; Lee, T.C.; Logendra, L.; Janes, H. *J. Agric. Food Chem.* **2000**, *48*, 1697–1702.

CHEM 260 Lecture Material

http://web.nmsu.edu/~snsm/classes/chem435/Lab5/

Oxtoby, David W.. *Principles of Modern Chemistry*. 6 ed. New York: Brooks Cole, 2007. Print.

http://dev.physicslab.org/Document.aspx?doctype=5&filename=ModernAtomicNuclear_EnergyLevelDiagrams.xmlhttp://www2.chemistry.msu.edu/faculty/reusch/VirtTxtJml/Spectrpy/UV-Vis/spectrum.htm

http://bouman.chem.georgetown.edu/S02/lect13/lect13.htm

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