Determination of any molecular property is accomplished by making a measurement in the lab. Due to recent development of high-speed computers, quantum mechanics has been made much more feasible, and as a result, can be incorporated into the lab. As a result, approximate values for properties such as dipole moment can be found before doing any experiment.
At the most basic level, quantum mechanics is a set of mathematical equations used to position all of the atoms and electrons so that the molecule has the lowest possible energy. In contrast with molecular mechanics, quantum mechanics does not find a geometry that spaces atoms equally. Instead, quantum mechanics will find a geometry resulting in the lowest energy. The process of minimizing the energy of the molecule is actually the boring, repetitive part of quantum mechanics; it is accomplished by initially choosing a reasonable structure (the result of molecular mechanics) and then moving one particle at a time until the energy of the system is as low as possible.
All of the following calculations have been made using quantum mechanics. Because these computations are very time consuming and very repetitive, a software package called CACHE Workshop was used to do the mathematics. Included in this package are three programs named after the respective calculation it performs: Extended Huckel, ZINDO, and MOPAC. Although they all perform quantum mechanics, they each make different assumptions. For Extended Huckel, it assumes that the structure of the molecule is determined by molecular mechanics. This assumption, among others, works well for aromatic or conjugated systems. Both ZINDO and MOPAC are semi-empirical in that they incorporate results from spectroscopy. ZINDO calculations are good for predicting electronic spectra, and MOPAC works the best for any general molecule since it makes the fewest approximations in doing the math.
The classic ball-and-stick model for each level of theory is given below. Feel free to click on these molecules and move them around as you please. After examination, you should find the structures to be very similar; this should not be a surprise as these are relatively simple molecules.
The structure on the left was determined without quantum mechanics. It was determined by treating each bond as a very stiff spring and each atom as a ball. Using this model, the lowest energy structure is the one in which all of the atoms are spaced as far away from each other as possible.
The middle structure and the right structure were determined after quantum mechanics. For these structures, atoms and bonds were not treated as balls and sticks. Rather, they were treated as charged particles. The subatomic particles were tweaked a little at a time until the lowest possible energy was found.
Bonds are usually thought of as shared electrons between atoms.
This idea can be deceptive if you limit yourself to just s-orbitals,
p-orbitals and etc. What you see below are actually a bunch of
orbitals added or subtracted together. It turns out that lower
energy--and more stable--bonds can be formed using this method; below
are a just a few of the bonding orbitals calculated by MOPAC. In
examining these molecules, it is refreshing to see a couple of
familiar orbitals predicted by quantum mechanics. For instance, in
the first model one can see some s-orbital overlap. However, as the
the third diagram illustrates, strange bonding and even anti-bonding
can also occur. More complicated molecular orbitals are very
difficult to interpret and have been excluded from this web page.
These orbitals are the most involved in chemical reactions since
they can most easily lose or gain electrons. The HOMO is the orbital
in which there are "valence shell electrons." If this orbital is not
filled, an additional electron from its surroundings is allowed to
join the molecule in this orbital. Otherwise, if the HOMO is full,
the next place that the electron can enter is in the orbital
specified in the LUMO. All of the molecular orbitals shown on this
page are for the lowest energy geometry of the molecule. The
HOMO is determined by putting pairs of electrons in the molecular
orbitals starting with the lowest energy orbital, just as the ground
state configuration of an atom is determined.
These are the HOMO and LUMO calculated using Extended Huckel. |
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These are the HOMO and LUMO calculated using ZINDO. |
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These are the HOMO and LUMO calculated using MOPAC. |
Electron denisty reveals where electrons are in the molecule.
Taking all of the orbitals into account, the three different methods
reveal the amount of electrons located in any given place at any
given moment. Lighter colors indicate a higher electron density while
the purple and the dark grey indicate low electron density.
Interestingly, each model gives a different description. As one can
see, all of the models show a high density around the oxygen atom. In
any case, if there are asymmetric places of high and low density of
electrons within the same molecule, then the molecule is polar. This
makes sense since electrons, which have a charge, are not spread
evenly throughout the molecule; therefore, there must be two "poles,"
or two different regions of charge.
Extended Huckel |
ZINDO |
MOPAC |
Without focusing on any particular atom, it is possible to see how
charge is spread throughout the molecule. In the diagrams below, blue
depicts negative charge and red depicts positive charge. It is
comforting to see that oxygen atoms are negatively charged as one
would expect without doing quantum mechanics. Additionally, notice
the correlation between the electron density and charge distribution.
A high electron density about the oxygen correlates to a negative
charge around the oxygen. When this occurs and there is an unequal
distribution of charge in a molecule, the molecule is polar. The more
separated, distinct, and unbalanced this distribution is, the greater
the polarity.
Extended Huckel |
ZINDO |
MOPAC |
Considering each atom of the molecule, the partial atomic charge is the charge that one atom experiences. Graphically, this model makes sense since oxygen is more electronegative than carbon and hydrogen (note that the same color coding applies as above). Again, the charge on these atoms create an unsymmetric and uneven charge distribution which means that this molecule is polar.
MOPAC
The mathematical derived quantity used to determine the direction and magnitude of the overall charge separtion (a vector) is called the dipole moment. Essentially, this is the weighted average of the atomic charges in 3 dimensional space. This is the essential quantity used to determine the polarity of some molecule. The greater the dipole moment, the more polar the molecule.
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Extended Huckel |
2.130 |
ZINDO |
1.537 |
MOPAC |
2.021 |
Actual--Experimentally Derived |
1.15 --found in CRC |