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Lecture notes for CHEM 1000
The Hydrogen Atom
Explaining the periodic table
by understanding the
hydrogen atom
Presented by Dr. Maggie Austen
Monday, May 9th, 2005
@ York University
Understanding atoms
by understanding electrons
Some history of atomic theory…
In the mid 1800’s Faraday
discovered cathode rays.
In 1897 Thomson determined that
the rays were composed of small
particles; electrons.
In 1911 Rutherford proposed the
nuclear model of atoms.
He discovered protons in 1919.
Understanding electrons
by studying atomic spectra
• Each atom has a
characteristic spectrum of
emission and absorption.
• The wavelengths of the
hydrogen atom are
described by a simple
mathematical formula.
1 ~ 1 1 
= R  2 − 2 
λ
 n1 n2 
The Bohr model of the atom,
1913
• Assumed that electrons
orbit the nucleus with
discrete (‘quantized’)
values of angular
momentum, l = nh/2π.
• Explained how electrons
could occupy space
around the nucleus
without crashing into it.
• Reproduced the observed
energy spectrum for
hydrogen atoms.
• Did not reproduce the
energy spectra for multielectron atoms.
• Did not explain the effects
of magnetic fields on
atomic spectra.
• Gave no reason why the
angular momentum should
be quantized.
• One other problem…
Electrons as matter waves
• In 1905, to explain the photoelectric
effect, Einstein said that light can be
particle-like.
• De Broglie suggested in 1924 that
matter might also be wave-like,
λp = h.
• Confirmed by electron diffraction
patterns in 1927.
• Standing waves with l = nh/2π needed
to avoid destructive interference.
Explains quantization, but …
Matter waves and wavefunctions
The probability, or electron density,
is the square of the wavefunction.
• The wave-like properties of
electrons are described by their
wavefunction.
• A wavefunction is a mathematical
function of 3D space (x,y,z) that
contains all the measurable
information about the system,
such as:
– energy
– momentum
– spatial probability distribution.
At the nodes the probability is zero
Why do we need to talk about
probabilities?
• Schrodinger’s equation
H Ψ = EΨ relates the
energy of the particle to its
wavefunction.
• For a one-dimensional
box-like potential well the
wavefunctions are sine
waves with wavelength =
2L/n and E = n2h2/8mL2.
Particle in a box
• As the box gets smaller, the
wavelengths get shorter and the
energy increases.
En = n2h2 / 8mL2 = p2/2m
• As the particle gets more
localized its momentum, p = h/λ,
increases.
• If there are no walls, we can
localize an electron by summing
many sine waves of different
wavelengths, i.e. many different
momenta.
See also Heisenberg’s Uncertainty Principle
Probability distributions
• A single sine wave extends to
infinite distance.
• Particles of known energy
(momentum) cannot be
localized.
• Electrons in atoms are therefore
not limited to circular orbits.
• The electron is distributed over
the (infinite) volume of the atom.
• The electron density is
determined by the potential
energy field, centered at the
nucleus.
Orbitals of the hydrogen atom
• Solutions to Schrodinger’s equation for the H atom
reproduce the observed energy spectrum.
• There are n2 different wavefunctions with each energy, En.
A wavefunction
for one electron
is called an
orbital.
Shells, Subshells & Orbitals
For atoms with many
electrons we often talk
about shells and subshells.
E.g. the ‘valence shell’
contains the chemically
active electrons.
• This figure shows 3 energy levels, or shells, n.
• Each shell has n subshells, or ‘types’ of orbitals, l,
and a total of n2 orbitals.
• Each subshell consists of 2l+1 orbitals.
Increasing nodes
• Like the particle in a
box wavefunctions,
the number of nodes
increases as the energy
and principal quantum
number, n, increase.
• Other quantum
numbers describe
other properties of the
wavefunction.
Radial nodes
Spherical s orbitals
• The simplest orbitals are
the spherical ‘s’ orbitals.
• The highest probability (or
electron density) is at the
nucleus, then it falls off
exponentially.
• The density falls off more
slowly as n increases Î
larger ‘size’ of orbital;
1s < 2s < 3s
Ψ2(r)
ρ(r)
90 %
Summary of Some Quantum Numbers of Electrons in Atoms
Name
Symbol
Permitted Values
Property
principal
n
positive integers
(1,2,3,…)
orbital energy & size
total nodes = n-1
angular
momentum
l
integers from 0 to n-1
orbital shape
planar nodes = l
(The l values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.)
magnetic
ml
integers from -l to 0 to +l
orbital orientation
(The ml values do not directly correspond to x,y, and z.)
Each electron is described by a set of quantum numbers, (n, l, ml).
Each quantum number is restricted to certain values, depending on
the previous quantum number.
Orbital shapes, described by l.
l = # of planar nodes
s
Visualizing p orbitals
l = 1; ml = +1, 0, -1
0
R2
Ψ2 = Y2R2
p
1
d
2
Y
Visualizing d orbitals
l = 2; ml = +2, +1, 0, -1, -2
• Only exist for n > 2; 3d, 4d, etc.
• There are 2l + 1 = 5 possible values of ml, so there
are 5 different orientations for the d orbitals.
• The dz2 orbital has cone-shaped nodes, rather than
nodal planes.
Y2
Orbits versus Orbitals
• We have modified the Bohr model to account for waveparticle duality and the uncertainty principle.
• An additional quantum number, ms, is required to explain
magnetic field effects.
• Schrödinger’s equation can be written for any electron
system, but cannot be solved exactly for more than one
electron.
• We must make some assumptions / approximations.
• Start by modifying the H-atom orbitals to fit the multielectron atom picture.
How does this help with
explaining the periodic table?
• The periodic table has
rows of 2, 8, 18 & 32
{= 2n2} elements.
• It has blocks of 2, 6,
10 & 14 {= 2(2l+1)}
elements.
• This suggests that
orbitals are being
filled sequentially with
2 electrons per orbital.
Questions remaining:
The spin quantum number
•
The effects of a magnetic field
can be explained by postulating
the electrons have intrinsic
magnetic moments, associated
with the charged particle
‘spinning’ about an axis.
•
Pauli was able to use the idea of
a spin quantum number, ms,
having two possible values: ±½,
to explain why each orbital can
accept no more than 2 electrons.
The Pauli Exclusion Principle
Why 2 electrons / orbital?
Why s before p before d?
•
Why 4s before 3d?
Energy level splitting in multielectron atoms
There must be something
about the electron-electron
repulsion in multi-electron
atoms that makes s orbitals
lower in energy than p & d
orbitals.
If this effect become very
extreme, the 4s might be lower
in energy than the 3d orbitals.
These effects are best understood by looking at radial distribution functions.
Radial Probability Distributions
• What is the probability
of finding the electron
in a given ring, or
rather on a given
sphere?
• P(r, dr) = N / Ntot
P(r , dr ) =
ρ (r ) ⋅ V (r , dr ) ρ (r ) ⋅ 4π ⋅ r 2 dr
=
N tot
N tot
P(r , dr ) ∝ r 2 ⋅ R 2 (r )dr
Radial distribution functions
When the hydrogen atom is in the n = 1 state (1s orbital), the
most probable distance from the nucleus is the Bohr radius, ao.
R2(r)
The most probable distance increases with n but decreases with l.
Note also that the radial nodes decrease with l.
Notice that the 2s orbital has a small region of electron density in
close to the nucleus, while the 2p orbital does not.
r2R2(r)
r2R2(r)
Radial distribution functions
The 2s orbital penetrates the n = 1 shell more than the 2p does.
Thus the 2p orbital is more shielded from the nuclear charge.
The surface area of a sphere is 4πr2.
The probability of finding the electron at a given distance,
summed (integrated) over all angles is 4πr2R2(r).
There are more points at larger distances; no points at r = 0.
Similarly, the 3s orbital penetrates more and is less shielded than
3p, which is less shielded than 3d.
Therefore 3s is least repelled by the core (1s, 2s, 2p) electrons,
thus most attracted to the nucleus, and lowest in energy.
Summary of H-atom orbitals
Summary of Electron Quantum Numbers in Hydrogen Atoms
Name
principal
angular
momentum
magnetic
Symbol
n
l
ml
spin
s
spin
ms
Permitted Values
Property
positive integers(1,2,3,…) orbital energy & size
integers from 0 to n -1
integers from -l to 0 to +l
‘type’
l
min.
n
s
0
1
#
ml
planar
nodes
#
‘lobes’
1
0
1
1
2
2
4
3
8
l
2l
0
orbital shape
p
1
2
orbital orientation
d
e-
½ for all electrons
magnitude of
+½ or -½
direction of e- spin
2
3
5
0, ±1, ±2
spin
For multi-electron atoms, the other quantum numbers also
affect the energy, and thus the order of orbital filling.
3
-1, 0, +1
f
3
4
7
0, ±1, ±2, ±3
summary
l
l +1
2l +1
0, ±1… ± l
shape
Uncertainty Principle, 1920s
• Particles interact with
photons of a similar
wavelength.
• Measurements are affected
by the measuring device.
• We cannot know the exact
location and momentum
simultaneously.
• Heisenberg found that
∆x∆p > h/4π
Dmitri
Mendeleev
1871
The hydrogen atom orbitals, YR
The Schrödinger Equation
HΨ = EΨ
Ψ = Yl,m(θ,φ) Rn,l (r)
h2  d 2
d2
d 2  Ze 2
Hˆ = − 2  2 + 2 + 2  −
8π m  dx
dy
dz  r
Mosley, 1913: X-ray spectra &
Bohr model reveal atomic numbers