### Lecture 20 (Slides) October 10

```H Atom Wave Functions
• Last day we mentioned that H atom wave
functions can be factored into radial and
angular parts. We’ll directly use the radial part
most in this course. (The “sigma” introduced
for H wave functions in the text is for printing
convenience only.)
General Chemistry: Chapter 8
Slide 2 of 50
Probability Plots for the H Atom
• We often describe the probability of finding
the electron in the H atom as a function of its
position in three dimensional space. This
requires an evaluation of Ψ2 and three
dimensional plots. Due to the wave like
properties of electrons the maximum value of
r that should be used in such plots is not
obvious (there is a small likelihood that the
electron will be found far from the nucleus).
Final Note on Coordinates
• For the motion of a particle described using
spherical polar coordinates the r, θ and φ can
vary over the ranges:
• 0 ˂ θ ˂ ; 0 ˂ φ ˂ 2 and 0 ˂ r ˂ ∞
• For electron motion in an atom or molecule we
expect the most likely r values to be very
small. Why? (Aside: Using Cartesian
coordinates we’d have -∞ ˂ x ˂ +∞ , etc.)
Orbitals and Electron Density
• In practice it is customary to draw a boundary
surface enclosing the smallest volume which
has, say, a 95% probability of containing the
electron. Chemists also speak in using these
plots of electron density. The s orbitals are
again a special case. The wave functions for s
orbitals, the Ψ(r,ϴ,φ), have in this case no
angle dependence – the probability of finding
the electron somewhere in space depends
“only” on the r value.
Orbitals have Different Shapes
• It follows from the previous slide that, for s
orbitals, the smallest volume that will have a
95% probability (say) of containing the
electron will necessarily always be a sphere.
Other orbitals have associated wave functions
which show dependence on all of r, ϴ and φ.
As a result, these orbitals have more complex
shapes. Nodal planes are seen for p orbitals as
seen on the next slides (d orbitals later).
s orbitals
FIGURE 8-24
•Three representations of the electron probability density for the 1s orbital
General Chemistry: Chapter 8
Slide 7 of 50
2s orbitals
FIGURE 8-24
•Three-dimensional representations of the 95% electron probability density
for the 1s, 2s and 3s orbitals
General Chemistry: Chapter 8
Slide 8 of 50
FIGURE 8-27
Three representations of electron probability for a 2p orbital
General Chemistry: Chapter 8
Slide 9 of 50
FIGURE 8-28
The three 2p orbitals
General Chemistry: Chapter 8
Slide 10 of 50
Electron Spin – Another Quantum
Number
• Charged particles in motion can generate
magnetic fields or act as magnets. Electrons in
an atom can move rapidly around the nucleus
(orbital angular momentum) and can also have
spin angular momentum. Experiments show
that spin angular momentum is also quantized.
For electrons we introduce a fourth and final
quantum number, ms, which can have values of
+½ or – ½.
8-9 Electron Spin: A Fourth Quantum Number
Number
FIGURE 8-32
•Electron spin visualized
General Chemistry: Chapter 8
Slide 12 of 50
Stern Gerlach Experiment - Spin
• In light of the previous slide one might be
tempted to pass a beam of H atoms through a
magnetic filed and see if the beam splits into
two parts corresponding to the two possible ms
values. In fact, an experiment analogous to this
was first done with a beam of silver (Ag)
atoms which also have a single unpaired
electron. The result is shown on the next slide.
General Chemistry: Chapter 8
Slide 13 of 50
FIGURE 8-33
The Stern-Gerlach Experiment
General Chemistry: Chapter 8
Slide 14 of 50
8-10 Multi-electron Atoms
• Schrödinger equation was for only one e-.
• Electron-electron repulsion in multielectron atoms.
• Hydrogen-like orbitals (by approximation).
General Chemistry: Chapter 8
Slide 15 of 50
• A final plot, the radial probability distribution,
is used to gain insight into an electrons likely
location in space. Here we are describing not
the likelihood that an electron will be found at
a particular point in space but rather at a
particular distance from the nucleus. One can
imagine constructing spheres of differing sizes
inside the H atom. Imagine that each of these
spheres is covered with very small “boxes”
(volume elements).
(cont’d)
• It follows that, if the electron distribution within
the atom is uniform, then the “boxes” on a large
sphere should be more likely to contain the
electron than the (necessarily fewer) boxes on a
small sphere. The number of boxes on a sphere
should be proportional to the surface area. Asphere
= 4πr2. The radial distribution function, P(r) then
takes the form
• P(r) = 4πr2R(r)2
[We should write Rn,l(r) !]
• FIGURE 8-35
Zeff is the effective nuclear charge.
General Chemistry: Chapter 8
Slide 18 of 50
• Experiments show that for H, He+, Li2+ and
other one electron species there are many
degenerate energy levels. For example, the 2s
and 2p subshells have the same energy (3s, 3p
and 3d subshells also have the same energy).
In many electron atoms the new electronelectron interactions cause the degeneracy of
subshells to disappear. The result is portrayed
(qualitatively) on the next slide.
FIGURE 8-36
Orbital energy-level diagram for the first three electronic
shells
General Chemistry: Chapter 8
Slide 20 of 50
Electron Configurations
• Electrons can be distributed amongst the
subshells/orbitals of an atom in different ways
–producing different electron configurations.
The most stable configuration has the lowest
energy – corresponding to the situation where
electrons get as close to the nucleus as possible
while staying as far away from each other as
possible.
Electron Configurations (cont’d)
• We will deal with familiar (?) material for the rest
of this lecture. The relevant concepts – Aufbau
Principle, Pauli Exclusion Principle and Hund’s
Rule are summarized on the next slide. Hund’s
rule tells us that electrons occupy equivalent
orbitals singly when possible and with their spins
parallel. Do the (initially at least) singly occupied
orbitals make sense in terms of the coulombic
interactions between electrons? (Think, for
example, of the 3 distinct p orbitals).
Electron Configurations
• Aufbau process
– Electrons occupy orbitals in a way that
minimizes the energy of the atom.
• Pauli exclusion principle
– No two electrons can have all four quantum
numbers alike.
• Hund’s rule
• When orbitals of identical energy (degenerate
orbitals) are available, electrons initially occupy
these orbitals singly.
General Chemistry: Chapter 8
Slide 23 of 50
Allowed Quantum Numbers
• Class example: Write (a) two possible sets of
the four quantum numbers (n, l, ml and ms) for
the H atom (b) the four quantum numbers for
each of the two electrons in He and (c) two
possible sets of the four quantum numbers for
the Li atom.
Populating Orbitals
• If we continued with the previous exercise
we’d see that for each value of n there is one s
orbital (unique ml value), three p orbitals (three
ml values) and five d orbitals (five ml values: 2, -1, 0, +1, +2). This means that s, p, d
subshells can contain (at most) 2, 6 and 10
electrons respectively. Relative subshell
energies comes from experiment – next slide.
FIGURE -37
The order of filling of electronic subshells
General Chemistry: Chapter 8
Slide 26 of 50
Representing Electron Configurations
spdf notation (condensed)
1s22s22p2
spdf notation (expanded)
1s22s22px12py1
spdf notation – Orbital Diagram
General Chemistry: Chapter 8
Slide 27 of 50
Writing Electron Configurations
• Class examples: Use the Aufbau principle to
write condensed configurations and orbital
diagrams for F, F-, P, Na, Na+ and Bill Gates
favourite atom. Which atoms have unpaired
electrons? Can an atom with an even number
of electrons have unpaired electrons?
The Aufbau process
General Chemistry: Chapter 8
Slide 30 of 50
The Aufbau Process – Sc through Zn
General Chemistry: Chapter 8
Slide 31 of 50
Electron Configurations and the
Periodic Table
FIGURE 8-38
General Chemistry: Chapter 8
Slide 32 of 50
Valence Shell Configurations
• The occupied shell with the highest value of n
is called the valence shell. When atoms
undergo chemical change electrons in the
valence shell can be lost or shared with other
atoms. The valence shell can also pick up
electrons. Atoms with similar chemical
properties often have the “same” valence
shell electron configuration. For example, Li,
Na, K, Rb, Cs and Fr have an ns1 valence shell
configuration.