Lecture 6
Electronic Structure of the H-Atom
Chapter 1 Sections 1-9
Matter: Atoms
Review Fundamentals-Sections A,B
The Nuclear Atom: electron e-, proton p+, neutron n0, unit of charge e
The Wave Nature of Light, Electromagnetic Radiation
Radiation: the motion of one system of matter that influences the motion of another system of
matter at a distance.
Think of a bar magnet spinning around its center and its influence on another magnet at a
distance away.
When electrons move they set up electromagnetic radiation
wavelength
frequency
speed of light, c, 3.00x108 m/s

c
   c

Self-test 1.1B
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c

Line spectra
When light is passed through a prism, a continuous spectrum of light results (figure a). When an
electrical discharge is sent through a gas such as neon, the gas glows. When this light is passed
through a prism, a line spectrum results. Each element has its own set of lines at different
wavelengths, its emission spectrum. When high current is passed through a low-pressure sample
of hydrogen gas, the hydrogen H2 dissociates into two H atoms. The single electron in an H atom
is excited to higher energy by the electrical discharge and then the electron releases this energy
in the form of light. The spectral lines for the H atom are given in figure b:
 1
1 


2 2 n 2 
  C
n  3, 4, 5, 6
(Visible region, Balmer series)
In general,
 = R[ 1/n12 – 1/n22]
n1 = 1,2, …, n2 = n1 + 1, n1 +2, …,
Self-test 1.2B
If we pass white light through a vapor composed of the atoms of an element, a series of dark
lines will appear on an otherwise continuous spectrum. This is called an absorption spectrum.
The lines are at the same as in the emission spectrum.
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The observation that there are discrete spectral lines suggests that the electron in an atom can
have only certain energy levels
Quantum Theory
Quantized Energy Levels
Max Planck proposed that the exchange of energy between matter through radiation occurs in
quanta or ‘packets’ of energy. If an oscillating atom releases an energy E into the surroundings,
then radiation of frequency  will be detected.
E  h
Planck’s constant,
h  6.626 10 34 J  s
Self-test 1.4B (1.3B not needed)
Albert Einstein’s interpretation of the Planck relation: the energy transferred between matter as
radiation is transferred as ‘particles’ or photons, each with energy = h∙ν
Photoelectron effect
E (energy of photon) = hν = (energy needed to eject e-) + KEe (kinetic energy of ejected e-)
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Bohr’s model
h∙ν = Eupper - E lower
 1 
En  RH  2
n 

E  2.18 10
18
n  1, 2, 3, 4, ...
J
12   0
E  E f  Ei  h
E
1 
RH  1
 

2  2 
 h 
h
ni n f 
Bohr’s Model of the Atom
Electrons travel in orbits of certain energy.
Explains line spectrum of H atom, but no other atoms.
The Wave-Particle Behavior of Matter and Light
The photoelectron effect demonstrates the particle nature of light, but there is another effect that
demonstrates its wave nature: diffraction:
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a) Constructive interference
b) Destructive interference
Diffraction patterns:
Light or surface of ‘ripple tank’
Diffraction pattern of X-rays on Ni surface
The pattern only appears if the two holes are placed n∙ apart ( wavelength, n integer) or
if X-ray  corresponds to internuclear distance in Ni crystal.
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Einstein: The  of light times its momentum, mc equals h.
Proved when light from a distant star was pulled by the gravity of our sun.
E  mc 2
E  h  h
c

mc 2 
hc


hc
h

mc 2 mc
DeBroglie proposal (1923): replace c (speed of light) by v (speed of particle).
Then every particle has an associated wavelength,  = h / m∙v
Proved when Davisson and Germer (1925) obtained a diffraction pattern when a beam of
electrons travelling at the proper velocity was diffracted off of Ni.
Uncertainty Principle
Particle: has a trajectory, a path on which location and linear momentum are specified at each
instant; it is localized.
Wave: spread out over the surface (wave on water) or the whole length (guitar string);
delocalized.
It is inherently impossible to know simultaneously both the exact momentum (p=mv) of the
electron and its exact location in space (x).

x ∙ p ≥ h/4
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Quantum Mechanics and Atomic Orbitals
What is the character of the “associated wave” of an object such as an electron, an atom, a
molecule, a red blood cell, a baseball, …?
I don’t know. I can’t see it; but if I know the mathematical function that describes it, I can
calculate any physical property of that object. (a pragmatic viewpoint)
Schrödinger's wave equation
Wavefunctions are usually represented by the symbol . Remember that a function depends on
something. Take the function y=2x. The equation can also be written f(x)=2x or (x)=2x.
The “particle-like” behavior of the object is to have a center of mass localized at a point in space
that follows a trajectory. The “wavelike-like” behavior is to have an associated wave that fills all
of space. The tie-in between the behaviors is the interpretation that the square of the
wavefunction at a point (x,y,z) in space gives the probability of finding the particle at that point
in space.
Probability density, (x,y,z)2 , at a given point in space represents the probability that the
particle will be found at that location (x,y,z).
Electron density: regions where there is a high probability of finding the electron are said to be
regions of high electron density.
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The success of the Schroedinger equation is that it can be used to calculate the energy levels for
the H atom to reproduce the frequencies of the line spectrum exactly, and of other atoms and
molecules to within certain numerical accuracies.
Before we explore the Schroedinger equation for the H atom in detail, let’s apply the equation to
a simple model so that we can familiarize ourselves with this new way of thinking of or ‘seeing’
nature
Particle in a Box
The H atom consists of a proton and an electron. No assumptions are made as to where the
electron is in relation to the proton. Now let us make an assumption so that we have a model. We
will calculate the energy levels of our model H atom and compare them to experiment (line
spectrum).
The model is: the atom is a box that holds the electron and
the wave-like character of the electron is like a guitar string
held at two points at a distance, L from each other. The form
of the wavefunction is the sine wave, f(x)=sin(∙x). As we
pluck the string, various ‘modes’ appear according to how
hard we pluck the string. Each one of these modes is
described by a ‘quantum number’, n. When n=1, the lowest
energy mode, there is only one ‘bulge’ vibrating up and
down. We picture it as one upward ‘bulge’. If the string is
plucked with enough energy, then another mode appears
where one half bulges up while the other half bulges down.
The mid-point does not move; its amplitude is zero, which is
a node. Notice that each mode has n-1 nodes (besides the
tied down end-points). There is only one string but in the
diagram we place it at the different energy levels. Each level
represents a different energy state, En. ΔE = Ef – Ei.
Each level is a sine function that fits the physical conditions. We have to adjust the function to
account for the fact that the string is attached at end-points, the ‘frets’, at a distance of L (called
boundary conditions). F(x) = sin(n/2L ∙x). The ½ comes from the fact that the first and
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following modes are based on half of a sine wave (one bulge) and that one full sine wave = 2π
(two bulges). So,
(x) = sin(n/2L ∙x)
(x) =
(The outside
∙ sin(n/2L ∙x)
term is used so that the areas under all the waves are equal.)
Now let’s apply the Schroedinger equation to the model. The equation resembles Hamilton’s
equation that was developed in the 1800’s to describe the mechanics of vibrations in structures.
H=E
The energy of the vibration is composed of KE and PE. H = KE + PE. The movement of the
vibrations contribute to the KE term. The PE comes from attractions such as gravity and opposite
charges. We neglect the PE in this crude model and just say that the electron is in a box with the
size of an atom, L= 150pm (150∙10-12m).
We now solve the equation for the energy E at each mode or level n, that is En. The KE = ½ mv2.
If we multiply the numerator and denominator of KE by m, we arrive at another expression,
KE = ½ mv2 = ½ (mv)2 /m = ½ p2 /m
This is an equivalent expression for KE in terms of the momentum, mv. We now ‘go quantum
mechanical’ and replace mv by deBroglie’s relation =h/mv
KE = ½ (mv)2 /m = ½ (h/)2 /m
(x) = (2/L)1/2 ∙ sin(n/2L ∙x)
KE ∙ (x) = En ∙ (x)
½ (h/)2 /m ∙ (2/L)1/2 ∙ sin(n/2L ∙x) = En ∙ (2/L)1/2 ∙ sin(n/2L ∙x)
but the wavelengths  correspond to the box length L by (2L/n)
½ (h/)2 /m = En
En = ½ (h/2L/n)2 /m
En = n2h2
8mL2
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In spectroscopy, ΔE = Efinal - Einitial
ΔE = En+1 - En = [(n+1)2h2 / 8mL2] – [n2h2 / 8mL2] = (2n+1) ∙ h2/8mL2
ΔE = h∙ν = h∙c/ = (2n+1) ∙ h2/8mL2
Doing Example 1.8 in the text, the  for a particle in a box that drops from level 2 to level 1 =
25nm. This is far from the spectroscopic value of 122nm for an H atom but considering the
crudeness of the model, it is surprising that the result is only one order of magnitude away from
the experimental value.
Self-test 1.8B
The Hydrogen Atom
Orbitals and Quantum Numbers
a) The complete solution to Schrodinger's equation for the hydrogen atom yields a set of
wave equations and corresponding energies. These wavefunctions are called orbitals.
An orbital is a mathematical function that describes the wavelike behavior of an
electron. (It is NOT a “region of space”!!!)
b) There is a quantum number for each dimension in space. The wavefunction for the
particle in a box resembles a guitar string, which is a line, a one dimensional object. If we
place the string along the x direction, then the quantum number, n is designated nx. F(x) =
(x) = sin(nx∙p∙x).
Instead of the “particle in a box with a string” model, let’s play music on a drum. Our model is a
“particle in a cylinder with a drum head”. In other words, we are looking at the vibrations of a
drum, which uses the two dimensions of a circular surface rather than one for a string. The up
and down ‘bulges’ along the string are replaced by up and down bulges on the surface of the
drum.
Ch 1 Electronic Structure H-atom
We now have two quantum numbers for the two dimensions of the drum-head vibrations. The
first one, n, called the principle quantum number, gives the energy levels. This is the same as in
the 1-D string. The second quantum number, l, is used to distinguish between the directions that
the nodal lines can take in all the upper levels. Notice that in the string there are nodal points
between the upward and downward bulges. This is where the value of the function is zero
(cos90˚ = 0). On the drum-head there are nodal lines between the bulges.
Ch 1 Electronic Structure H-atom
Before Schroedinger applied wave functions to the H atom, he
was studying the mathematics of vibrations on the surface of a
sphere. This is just the 3-D extension to the 1-D string and 2-D
drum vibrations.
We will look at his application to the H atom in two ways. First,
in looking at the quantum numbers and shapes of the vibrations.
Then, we will examine the mathematical functions themselves.
Ia) Quantum numbers
Quantum mechanics uses three quantum numbers n, l, and ml, to describe an atomic orbital.
1. principal quantum number, n, can have integral values of 1, 2, 3.and so forth. As n increases
the orbital becomes larger and increases in energy. The energy for the isolated H atom only
depends on this first quantum number, n. [!!Trick question!!]
For the hydrogen atom En = -h∙R/n2. When the Schroedinger
equation is solved for ions with only one electron in total,
He+, Li2+, Be3+, …, then
En = -Z2∙h∙R/n2., where Z is the atomic number of the
nucleus.
2. azimuthal quantum number, l, can have values from 0
to n–1 for each value of n. This quantum number gives the
orbital its shape, and is also known as the angular
momentum quantum number.
Value of l
0
1
2
3
Letter used
s
p
d
f
3. magnetic quantum number, ml, can have integral values between l and –l, including zero.
This quantum number describes the orientation of the orbital in space.
The collection of orbitals with the same value of n is called an electron shell.
Ch 1 Electronic Structure H-atom
The set of orbitals with the same n and l values is called a subshell.
Each subshell is designated by a number (the value of n) and a letter (s, p, d, or f, corresponding
to the value of l)
ex:
2s subshell
n = 2, l = 0
3f subshell
n = 3, l = 3.
Restrictions on the possible values of the quantum numbers.
1. The shell with the principal quantum number n will consist of exactly n subshells.
n=1
1 subshell
n=2
2 subshells
n=3
3 subshells
n=4
4 subshells
2. Each subshell consists of a specific number of orbitals. Each orbital corresponds to a different
allowed value of ml.
For a given value of l there are 2l + 1 allowed values for ml, ranging from l to –l.
each s (l = 0) subshell consists of one orbital
(2(0)+1)=1
each p (l = 1) subshell consists of three orbitals
(2(1)+1)=3
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each d (l = 2) subshell consists of five orbitals
(2(2)+1)=5
3. The total number of orbitals in a shell is n2, where n is the principle quantum number of the
shell.
n
1
2
3
4
n2
1
4
9
16
2
8
18
32
orbitals in a shell
elements in a row
Ib) Representations of Orbitals
The s orbitals

spherically symmetrical.

probability function approaches zero as the distance, r, from the nucleus increases.

2s orbital has a node. Regions where the probability,

3s orbital has two nodes.

As n increases, the electron is more likely to be farther from the nucleus. The size of the
 2 , goes to zero.
orbital increases as n increases.
Contour representations: give relative sizes of orbitals.
1s
2s
3s
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The p orbitals

not spherically symmetrical

all p orbitals have nodes at the nucleus

on either side of the node there are lobes

n = 2 has three p orbitals; all the same size, but one oriented along each axis: px, py, pz.

p orbitals increase in size as n increases.
z
y
x
pz
px
py
d orbitals

four of d (l = 2) orbitals have "four-leaf clover" shapes.

lie primarily in a plane: dxy, dxz, dyz, with lobes between axes. the
d x 2  y2 lies in the x-y
plane along the axes.

the
orbital has two lobes along the z axis and a "doughnut" in the xy plane. It has the
same energy as the other four d orbitals.
The f orbitals (l = 3) are more complicated than the d orbitals. Where the d orbitals have two
pairs of bi-lobes, f orbitals have three pairs.
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II) Mathematical Equations for the H atom
It is easier to solve the Schroedinger equation for an atom if polar coordinates (r, are used
instead of Cartesian (x,y,z) coordinates. The parameter r is just the distance between the electron
and the nucleus and the angles ( are the two remaining coordinates. We can split (r,
into the product of two functions, one the depends only on the distance r, called the Radial
Function, R(r), and another called the Angular Function, Y() because it depends on the angle
and  coordinates.
(x,y,z(r,R(r) ∙ Y(
The full set of equations are: (Don’t worry! We’ll make them so simple even a chemist can
understand them.)
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Remember: just look for the variables r and  and treat all the constants and parameters
as if they = 1. You can then graph simplified functions.
Examples:
Radial Functions
S functions (orbitals), l = 0
n=1: 1s orbital, n=2: 2s orbital, n=3: 3s orbital, etc.
(r,R(r) ∙ Y(
ns(r,R(r) ∙ Y(
ns(r,R(r) ∙ (1/4π)1/2 ≈ R(r)
1s = e-r
2s = (2-r)∙e-r/2
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3s = (3-2r+r2)∙e-r/3
Angular Functions
The three 2p orbitals (x,y,z) are each a product of the same radial
function R(r)≈e-r/2 times an angular function that places the orbital
along the x, y, or z axes and gives it a nodal plane.
It is important to know that the +and – signs on the orbital do NOT
refer to the charge of the electron but to the values of the function!!!
The fun of it, you might trace the signs and nodes of the 3pz orbital
(bonus material)
Ch 1 Electronic Structure H-atom