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Quantum Physics Lecture 13
Multi electron atoms Z>1
Pauli exclusion principle
Filling orbitals
The Periodic table
Moseley’s Law
Interatomic (chemical) bonding
More than 1 electron atoms…
Simple picture: can add electrons to hydrogen-like quantum states
assumes the nucleus increasing charge is shielded by inner electrons
and ignores electron – electron interaction energies….
Try it anyway….
But: Pauli exclusion principle – no two electrons can occupy
the same quantum state (fermions, see Lecture 11)
Two spins states possible – ‘up’ and ‘down’
↑ or ↓ ±

2
So maximum of two electrons in each state one ↑ and the other↓
Many electron atoms
For Z>1, fill quantum states with max. 2 electrons each
First quantum number n (c.f.Bohr energy level)
Second quantum number l (angular momentum state) where 0 ≤ l ≤ (n-1)
Third quantum number ml where |ml| ≤ l
Many electron atoms (cont.)
Examples: Helium atom
Contains 2 electrons, both can be in 1s state (lowest energy)
provided one is spin up the other spin down
Notation for the ground state 1s2
Lithium (Z=3)
1s shell filled (like He)
Extra electron goes into 2s shell
Notation 1s2 2s1
2s orbital further out…
Nuclear charge screened by 1s shell, effective
charge more like H but further out
So less well bound
2s electron can be lost in bonding (ionicity)
Z>1 The periodic table of elements
n=1 n=2 ……
…….
l=0
l=1
l=2
l=3
s state – 2 electrons
p state – 6 electrons
d state – 10 electrons
f state – 14 electrons
Gives the basic structure of the
Periodic Table of the elements
Periodic Table of Elements
1-7s
4-5f
3-6d
2-7p
For Hydrogen, s,p,d,f, states have same energy for given n (c.f. Bohr)
This Degeneracy of states is broken for Z >1 (by e-e interaction potentials)
So s fills before p, before d etc., the gap increasing
as Z becomes larger.
X-Ray spectra
X-Rays are emitted by impact of high
energy electrons on elements
λmin =
hc
eV
Continuous spectrum due Bremsstrahlung & other
scattering processes
Molybdenum spectrum shown
Impacting electrons cause electrons in core (lowest energy) states to be
knocked out. For high Z atoms, these are very tightly bound states (K shells),
so require high energies (many keV) to eject them
Spectrum shows sharp peaks, due to emission of photons by outer electrons
falling to vacated core states. Energy (frequency) is characteristic of element.
N.B. Lower energy spectroscopy shows energies which often have little to do with the Z
number of the atom – a problem for early atom models!
Moseley’s Law
Moseley found that
(
)
f ∝ Z −1
2
The first time Z was
spectroscopically
determined…
One other electron in K-shell, so nuclear charge screened by 1e, i.e. reduced to Z-1
Transition from n=2 to n=1 gives (Bohr model)
2
3
ΔE =
4
(
)
Ryd Z − 1
Which agrees very closely with Moseley’s experiment.
Actually the most important early evidence for nuclear model of atom!
Bonds between atoms
Isolated atom in ground state Ψ
e.g. H atom 1s state
Probability of finding electron is ∝ ⎮ψ⎮2
Note: Wavefunctions can be +ψ or –ψ
What happens when two atoms
approach each other?
Wavefunctions of adjacent atoms 1 & 2 combine,
so two possibilities: ψ1+ ψ2 or ψ1- ψ2
Bonds between atoms (cont.)
OR
Diatomic Molecule = interference of electron ‘waves’
(i.e. adding/subtracting)
Antibonding
Bonding
Electron more likely to be
between nuclei compared to
isolated atom - saves
electrostatic energy
⇒ Bonding state
Electron is removed from region
between nuclei compared to
isolated atom
Costs energy.
Anti-Bonding state
Overall energy saving (= bonding) if electrons go into bonding state
e.g. OK for H2+ or H2 molecules.
Electrons are ‘shared’ – covalent bond
Bonds between atoms (cont.)
Antibonding
Bonding
Note for He2 (4 electrons), Pauli principle means two e’s in
antibonding state as well as bonding state
so no overall energy saving
(inert gases – no bond - no He2)
Mid-periodic table elements (half-filled orbitals) tend to have strongest
bonds (e.g. melting points. etc.)
ψ is ‘periodic’ inside atom & decaying outside – ‘barrier’ between atoms
but electrons move between atoms by tunnelling.
➞ Exponential variation of energy of interaction with separation
– Interatomic forces