These web pages are part of an end of term project for Math 308 (Euclidean
Geometry) at the University of British Columbia. The course is an introduction to
Euclidean Geometry, with this particular section focused on computer-drawn
proofs. The course extensively utilized PostScript and some limited html coding,
and was taught by Professor Bill Casselman.
The goal of this project is to illustrate the key role that geometry and 3D
representations play in understanding chemical reactivity as described through
atomic and molecular orbitals. To my dismay, I was unable to complete the
project as I had planned, and many of the details have the lost. As a result, the
treatment of the subject material is VERY brief, and generally qualitative. Many of
the important mathematical and physical elements of Quantum Chemistry have
been omitted. In spite of this, I hope that this can provide a gentle introduction in
one of the main results of Quantum Chemistry without being too mysterious. The
last section is a brief digression of Molecular Orbital Theory that uses the angular
elements that arise from Quantum Mechanics.
A Brief Introduction to Quantum Chemistry
Quantum Mechanics arose from deficiencies in classical physical mechanics
when applied to small particles, such as electrons and protons.
The first major breakthrough in Quantum Chemistry was the notion that energy
was quantized or discrete. Classical mechanics had regarded energy, like all
physical observables, as being continuous, and not discrete. In the late 1800’s
and 1900’s, major challenges to limitations of classical mechanics were
presented.
Blackbody Radiation
Imagine that a block of metal being heated to very, very high temperatures. While
metals in the real world eventually melt, let us assume that there is object called
a blackbody that can absorb and emit back energy of any frequency in a similar
fashion to a metal. As most metals are heated, they slowly turn from red, to
white, and then finally blue. The same effect is observed when an stove element
or a roaring campfire becomes increasingly hot. As the energy used to heat it
increases, the box begins to release energy of increasing frequency. The colours
that we see can be approximated by the “sum” of the energy being released. As
the energy increases, the band moves to the left, and becomes a mix of different
colours.
The classical mechanical interpretation yielded a result known as the RayleighJeans Law. The equation was directly proportional to the temperature, and the
square of the frequency. As such, if energy of increasing frequency was used,
the energy released should become more intense. The Rayleigh-Jeans
approximated the intensity at lower frequencies, such as in range of visible
radiation. After a certain frequency, the intensity would then decrease towards
zero.
In 1900, Max Planck offered an interesting interpretation. As in classical
mechanics, Planck assumed that the energy absorbed and then emitted back
from the blackbody was due to the vibrations of electrons. However, Planck
made the contradictory assumption that energy of the vibrations/oscillations had
to be integral multiples of the frequency, and therefore quantized. Classical
mechanics assumed that energy was continuous and could have any value.
Planck derived an equation (Planck’s Distribution Law) that was directly
proportional to a constant h (now known as Planck’s constant). The Distribution
Law was found to be in strong agreement with intensity distributions, and began
the notion that energy was quantized.
Photoelectric Effect
The photoelectric effect is observed when high energy light is directly towards a
metal surface. If the light is sufficiently high in energy, an electron is ejected from
the surface of the metal. Two results were found that were incongruent to
classical mechanics.
The first was that the kinetic energy of electron was not determined by the
intensity of the light hitting it. Classical mechanics viewed electromagnetic
radiation as an electric field that propagated perpendicular to its direction of
motion. As its intensity increases, so does the amplitude of the electric field.
When the light hits the electron on the metallic surface, the electron begins to
vibrate more and more, until it has enough energy to break free, and becomes
ejected.
An example in the real world would be pushing people on a swing. Let’s say that
a young child sitting in a swing wants to be pushed to a certain height (like the
electron being ejected). The child can be pushed in two ways: either in one
strong push, or a series of smaller ones. Pushing the child in one strong push is
analogous to one very high energy burst of low intensity. Alternatively, one can
slowly push child over a duration of time, and the child swings higher and higher
each time. This is like light of very low frequency (low energy) but very high
intensity.
As such, Classical Mechanics predicted that energy of any frequency should be
able to eject an electron, so long as it was intense enough. However, it was
experimentally discovered that there was a threshold frequency. When light of
lower frequency was used, no electrons were ejected, regardless of the intensity
of light used. If the frequency of light used was greater than the threshold
frequency, electrons would be ejected. As well, the kinetic energy of those
electrons was indirectly proportional to the frequency of light used. Not only did
this result contradict Classical Mechanics, it also lead to Albert Einstein’s
formulation that radiation existed in “small packets of energy” now known as
photons.
Einstein found a relation that expressed the kinetic energy of the ejected
electrons to be proportional to the frequency of light and h, Planck’s constant,
minus a constant (for a given metal), Φ, known as the work function. The reappearance of Planck’s constant was truly remarkable. The exact same number
had appeared in two totally different experiments. Both dealt with the
quantification of energy, and were in agreement with experiment results.
These two results, along with a third that Einstein found concerning molar heat
capacities at constant volume, helped strengthen the concept that energy could
be quantized in certain systems. This concept of quantization is fundamental for
the concepts used in the Schrodinger Wave Equation.
Wave-Matter Duality of Light
In the early 1900’s, the fundamental nature of light was still a mystery. Light
exhibited many properties that were consistent with both matter and
waves/energy. For example, when light enters a prism, it divides into many
colours, which are visual cues of electromagnetic radiation. Electromagnetic
radiation is a form of energy, like UV rays and microwaves. However, light acts
like matter when it is drawn into black holes. Attractive gravitational forces
require that objects have mass, as matter does. As such, light remains a
confusing amalgam of both being a wave and matter, but truly being neither.
In 1924, Louis deBroglie suggested an interesting claim. If light can show the
characteristics of both matter and wave, why couldn’t matter show wave-like
characteristics? deBroglie claimed that the wavelength of a moving object was
Planck’s constant divided by p, its momentum. Later, this result was “proven”
when both X-rays (a wave) and electrons (matter) were fired into a thin sheet of
aluminum. The pattern that was observed from both was very similar, and wavelike in nature.
Heisenberg Uncertainty Principle
From deBroglie’s results, it could be seen that any moving object traveled in a
wave-like path. While the result it generally inconsequential for large objects, the
problem remained for smaller particles, such as electrons.
In order us to measure the location of a moving electron, at a single moment in
time, light must be used to determine its location. Just as one sees by receiving
light that has bounced off objects, one can “see” an electron by measuring light
that has bounced off an electron. The light needs to interact with the electron in
someway, or there would be no detectable change, as it would be the same as
light going through a vacuum. However, once the light hits the electron, it gives it
an instantaneous boost of kinetic energy, which increases it momentum.
Conversely, it we know an electron’s exact position, there is no way we can learn
its momentum without colliding with light. At that moment, the electron becomes
“displaced” from its original position. As such, it is impossible to know both the
position and momentum of an object exactly with perfect accuracy.
The Heisenberg Uncertainty Principle is a key result used in Quantum Mechanics
because it provides a boundary for the accuracy and limitations of measurement.
In order for a measurement to be made, it requires interactions between the
various objects. It is this fundamental interaction that causes this variability.
Given the Uncertainty Principle, many results are given probabilistic
interpretations where surfaces and ranges are often best representations of
electron behaviour.
Schrodinger Equation
The Schrodinger Equation is a postulate, and is regarded as an axiom of
Quantum Chemistry. While no proof is possible, many of the postulates are
“reasonable” based on their congruency to experimental results.
The time-independent Schrondinger Equation (for the one dimension case)
utilizes the classical one-dimensional wave equation, deBroglie’s Equation, and
energy relationships. The final equation is:
(h/4π) ● (d2Ψ/dx2) + U(x) ● Ψ(x) = E ● Ψ(x)
where:
h is Planck’s Constant,
U(x) is the potential energy,
Ψ(x) is the wavefunction that describes the system,
and E is the overall energy of the system.
Particle in One-Dimension Box or Particle on a Line
For pedagogic purposes, the concept of the one-dimension box or singular line is
often used to generate some of complex concepts of wave equations.
Assume there is a particle that exists between the region 0 and A. At both 0 and
A, there is an infinite potential, which forces the particle to stay inside these
bounds. The probability of find the particle outside of these bounds is zero.
The particle experiences zero potential within these bounds, so it may freely
move around. When we apply this to the Schrodinger Equation, the second term
drops out, so the net result is:
(h/4π) ● (d2Ψ/dx2) = E ● Ψ(x)
or
(d2Ψ/dx2) + (8π2mE/h2) ● Ψ(x) = 0
where:
h is Planck’s Constant,
Ψ(x) is the wavefunction that describes the system,
E is the overall energy of the system,
And m is the mass of the particle
It can also be written in a more familiar way as:
(d2f/dx2) + (k) ● f(x) = 0
where k is a constant
Many mathematicians will recognize that the solution probably uses cosines and
sine. Hence, Ψ(x) behaves in a harmonic or sinusoidal wave. The energy of the
system is also quantized, since it can only take on certain, discrete values, which
happen to be solutions of the system. The presence of the 8π2 forces the system
to have a phase that has π in its denominator. The net result is a system which
takes on integral values of cosines and sines.
A general solution to this equation is:
Ψ(x) = A cos kx + B sin kx
where A, B, are constants,
and k = [(2mE)0.5]/h.
While this result may seem fairly abstract it provides two useful results. The first
is that the energy of the system is quantized given the boundary conditions.
Since the particle can only travel in sinusoidal waves, there are only discrete
energy levels it can take on. This should make some intuitive sense. If the
standing wave was allowed to take any shape or form, it would eventually cancel
itself out as it traveled from boundary point to boundary point. However, if it
travels in a “controlled” sinusoidal path, then it will be able to neatly fit into the
box without destroying itself.
The second key result is that the function can be square normalized. The term
square normalized simply means that the function, times its complement
(including imaginary terms) must equal one when integrated over the entire
region of interest. The region of interest is variable. In this particle in the box
example, we integrate over the entire line from 0 to A. in the real world, we would
integrate over all of x, y, and z.
Rigid Rotator
In order to approximate atomic systems, we can think of the atom as containing a
very heavy nucleus (with mass m1) with a relatively light electron (m2) that rotates
around the nucleus as a satellite. The distance of the nucleus and electron from
the center of mass is given by r1 and r2 respectively. The actual derivation of the
solution is quite lengthy, and involves the use of operators, and solving a second
order differential equation. The result is simply stated here:
The solution to the Schrodinger Equation for the Rigid Rotator:
EL = (h2/8π2I) ● L(L+1)
where I is the moment of inertia for the system (calculated for m1, m2, r1, r2)
and L is a non-negative integer
Notice that the energy is also discrete, and can only take on certain values (as
governed by the L term). It turns out that the wave functions of the rigid rotator
are spherical harmonics. A spherical harmonic is analogous to the sinusoidal
wave from particle-on-a-line example. A spherical harmonic can be though of as
a 3D-path that a particle can travel without “destroying” itself energetically.
However, this 3D-path is NOT fixed, and can take on many different shapes,
even for one energy level.
Orbitals in general
With the spherical harmonics, we can describe the 3D motion of a electron
around a nucleus. As such, the Schrodinger wave equation is decomposed into
two separate parts: the radial (r) and angular elements (Φ,Θ) which arise from
spherical harmonics. While the radial parts are relatively easy to describe,
however angular parts are quite subtle. The common decomposition for the
wavefunction is written as:
Ψ(r, Φ, Θ) = R(r) ● Y(Φ,Θ)
where R(r) is the radial portion, and Y(Φ,Θ) is the angular portion.
While wavefunctions specify information about the behaviour of a particle, we
often integrate over a limited range of r, Φ, or Θ in order to get the probability that
the particle lies in a particular volume, at any one given time.
The Periodic Table
This coloured Periodic Table is supposed to covey the major valence shell
groups. They are s-orbitals (red), p-orbitals (green), d-orbitals (blue), and f-
orbitals (purple). There is also one light blue square for lanthanum. While
lanthanum is formally a d-block metal, its chemistry and reactivity is very similar
to f-block metals. Furthermore, the first row of f-block metals (rare earth metals)
are known as lanthanoids, due to their similar chemistry. As such, lanthanum is
given a special box to denote its borderline behaviour. Also, the last six elements
in the seventh row [Rutherfordium (104) – Meitnerium (109)] have not been given
any formal designation.
This project will cover the s-orbitals, p-orbitals d-orbitals, and their applications in
Molecular Orbital Theory.
s-Orbitals
The s-orbitals are fairly easy to describe. The angular parts of the wavefunctions
turn out to be constants, so the orbital has a 3D representation that is
independent of angle; the net result is a sphere.
Due to the simplicity of the s-orbitals, now would be a good idea to take a
qualitative look at the concept of valence shells. Valence shells are the
interpretation of the energy levels of the Particle in a Box or Rigid Rotator
Solutions. Both systems describe discrete energy levels.
About the Radial Wavefunctions -- R(r)
The graph on the left is of the radial portions for the 1s (blue), 2s (green), and 3s
(red) orbitals. Each of the graphs is on the same x-axis (radius), but varying yaxis values. Notice that as the radius goes to infinity, that each of three functions
goes to zero. This fits in with our notion that each of the wavefunctions must be
normalized, and square integrate to 1.
While the 1s orbital is always positive, the 2s orbital changes sign once, and the
3s orbital changes sign twice. At every point where the radius becomes zero,
there will be no chance of finding the electron. These are radii called nodes.
The Radial Wavefunctions and their Probability Distributions – r2 ● R(r)2
The graph on the right is of the radial portions of the curved integrated with
respect to radius (as in standard integration in spherical co-ordinates). They are
graphed on the same x-axis, but variable y-axes. Some of the y-scales have
been exaggerated so their features are easier to see. The curve of the 1s orbital
reaches one maximum in its curve. The distance is known as a Bohr radius, and
is the same average distance that Niels Bohr obtained when he approximately
the behaviour of the Hydrogen Atom (which is 1s1). As mentioned previously, at
each x-intercept, there is zero probability chance of finding the electron. In the 2s
and 3s cases, the probability varies, going from regions of high and low. Because
the angular elements are spherical, one can imagine this as a set of nested or
concentric spheres. At each of the maxima, there is a solid layer, that alternates
with a nodal sphere or an empty volume.
p-Orbitals
p-Orbitals are governed by their radial functions and two angular elements. Many
of the principles of radial orbitals for the p-Orbitals are same as in the s-Orbitals.
As such, only their graphs are presented here:
The three equations for the 2p orbitals are given by:
Y+ = (3/4π)^1/2 sin Θ cos Φ
Y- = (3/4π)^1/2 sin Θ sin Φ
Y0 = (3/4π)^1/2 cos Θ
Each of the three orbitals has the different radial elements, since two of the
functions depend on Φ and Θ. It will turn out that the three orbitals are each
aligned along a different axis in the Cartesian plane. Using the CartesianSpherical conversion, we have:
The conversion formulas:
x = sin Θ cos Φ
y = sin Θ sin Φ
z= cos Θ
we obtain:
Ypx = (3/4π)^1/2 sin Θ cos Φ = (3/4π)^1/2 x/r
Ypy = (3/4π)^1/2 sin Θ sin Φ = (3/4π)^1/2 y/r
Ypz = (3/4π)^1/2 cos Θ = (3/4π)^1/2 z/r
This means that we can graph one function with respect to an axis, and then
rotate 90° to the other two axes, and we have a full set of p orbitals.
The angular element is given by:
Since there is no dependency on Φ, the graph is sketched as follows:
d-Orbitals
d-Orbitals are more complex wavefunctions than either s or p orbitals. Each of
the five has a dependency on all three variables. However, they can be
converted to Cartesian equivalents as in the p-Orbital case.
Radial Elements
The graph on the left is of the radial function of the 3d orbital. It contains no
phase changes. The graph on the right is of the radial function, integrated with
respect to the radius in spherical co-ordinates. The maxima occurs roughly at 9
Bohr radii.
Angular Elements
As in the p-Orbital case, the conversion of the angular portions of the
wavefunctions to Cartesian co-ordinates has reduced the unique angular
elements shapes to two. The first is given by the graph of.
The net result is a set of four lobes that
Y = (5/16π)1/2 (3cos2θ-1) = 3z2-r2/r2
Y= (5/4π)1/2 cos Θ sin Θ cos Φ = xz/r2
Y= (5/4π)1/2 cos Θ sin Θ sin Φ = yz/r2
Y= (5/4π)1/2 sin2 Θ cos Φ = xy/r2
Y= (5/16π)1/2 sin2Θ(cos2 Φ – sin 2 Φ) = x2 – y2 / r2
The second angular function turns out to have this strange shape. In 3D, has a porbital type center, along with a “ring” of opposite phase that goes around the
middle.
Molecular Orbital Theory
Molecular Orbital Theory or MO Theory utilizes concepts of atomic orbitals to
rationalize general behaviour of chemicals. MO Theory is based on the “mixing”
or combining of orbitals. As two atoms form a successful covalent bond, their
valence electrons become shared. In the case of atomic orbitals, only major
interactions between the nucleus and electrons are considered; no electronelectron interactions are considered. However, when electrons on shared, their
behaviour changes drastically.
The simple wavefunctions that defined electron motion with respect to one
nucleus is no longer applicable. The complications that arise from the
interactions of a second nucleus are very complex, and difficult to accurately
model. The resulting mathematical equations are very complex. MO Theory
takes a more general approach. Instead of combining the wavefunctions and
adding various correction factors, only the visible representations are combined.
The resulting linear combinations of the atomic orbitals give rise to a variety of
molecular orbitals.
In order for orbitals to mix, they need to be of comparable energy. If the energy
levels are two high or low, they have no net interaction with one another. Each
suitable pairing has a bonding orbital and an anti-bonding combination. The
bonding and antibonding orbitals between two s-orbitals is denoted denoted σ
and σ* respectively. Bonding and antibonding between a s-orbital and p-orbital or
two p orbitals is denoted by π and π* respectively. This notation will be left off the
MO diagrams produced due to space restrictions.
When drawing up the angular portions of the orbitals, different colours were used
to denote the various phases. While there exist no real differences between the
two phases, there are theoretical considerations. When two phases of the same
type align, they generate a mutual or constructive field that leads to bonding.
When two phases of the opposite type align, they generate a “destructive” field
that leads to anti-bonding.
There will be three examples presented: Hydrogen Fluoride, Carbon Monoxide,
Silicon Fluoride.
Hydrogen Fluoride (HF)
Hydrogen Fluoride (HF) is an excellent example to introduce the primary
concepts of MO Theory.
Background
Hydrogen has only one single valence electron in an s-orbital. It has an electron
configuration of 1s1.
The 1s refers to the valence shell. The regular “1” refers to its general energy
level (1st row element), and the s denotes its shape type (a spherical shape). The
superscript “1” means that it has one electron in this shell.
Fluorine has seven valence electrons available for bonding. It has an electron
configuration of [He]2s22p5.
The [He] symbol means that it has base electron configuration that is the same
as helium (1s2). The square bracket notation is used for both convenience and
clarity. Any electrons that are included in base element (in this case He) are
considered non-reactive. The two electrons in the 1s shell do not participate in
bonding.
The 2s22p5 means that there are seven electrons available for bonding: two
reside in a 2s orbital, and five are in a 2p orbital. This following is a picture of
their relative energies.
In total, the combination of the single electron of hydrogen and seven of fluorine
means that there are eight electrons available for bonding. The axis is shown on
the right as a reference point.
The first thing to note is that the 2s orbital of the F atom is too low to participate
in bonding with the 1s of the H. Let’s try some of the combinations of the 1s and
the various 2p orbitals.
Carbon Monoxide (CO)
Carbon Monoxide (CO) will build upon the general principles that the hydrogen
fluoride example introduced, but will be slightly more involved.
Background
Carbon has an electron configuration of [He]2s22p2, so it has four valence
electrons available for bonding. Oxygen has six valence electrons from its
electronic configuration of [He]2s22p4. This is a total of 10 electrons.
Unlike in the case of hydrogen fluoride, both atoms have s and p orbitals that are
available for bonding. As a result, there are more possible interactions.
The s - orbitals of both atoms can overlap to a high degree, so they generate a
very low energy bonding molecular orbital. However, if they are of opposite
phase, they generate a very high energy anti-bonding orbital.
The p - orbitals generate a total of nine combinations among themselves.
However, when two p-orbitals of different axes orientation align, there is little
overlap as shown in the various diagrams. As such, only orbitals of same axes
alignment are of any major consideration.
The pairing with the most bonding character lie on the axis that runs through both
atoms. One lobe of the p from each of the atoms can mix together. This creates a
moderate bonding orbital.
In the case of the last two, orbital overlap is a possibility, but it requires that the
two atoms be very close together, and the overlap is very minimal. As such, the
energy reduction from the orbital mixing is minimal. However, the resulting antibonding combination is only slightly higher in energy.
This is the simplified MO diagram.
One of the interesting effects that occurs is a phenomenon known as s-p mixing.
The s-s combination yields a large elliptical bonding orbital. A bonding
combination of two on axis orbitals yields a molecular orbital with a large central
lobe of one phase, with smaller lobes on the large axis. These orbitals are of
similar energy such that these can mix further.
References and Acknowledgements
McQuarrie, Donald A., Quantum Chemistry, University Science Books, Sausalito,
California, 1983.
Housecroft, Catherine E, Inorganic Chemistry, Prentice Hall, Essex England,
2001,1st Edition.
McMurry, John, Organic Chemistry, Brooks/Cole, Pacific Grove, California, 2000,
5th Edition.
Laidler, Keith J., Physical Chemistry, Houghton Mifflin, New York, 1999, 3rd
Edition.
I’d also like to take this opportunity to apologize for the (likely many) technical
mistakes in this project. My class notes from both classes are incredibly sketchy,
and the erroneous content of these web pages (especially the Quantum
Chemistry parts) should not be viewed as a reflection of these instructors!
Thanks to Dr. Bill Casselman (UBC) for his assistance on the project, and for
instructing this course.