Quantum Physics Lecture 13 - Trinity College, Dublin...Quantum Physics Lecture 13 Multi electron atoms Z>1 Pauli exclusion principle Filling orbitals The Periodic table Moseley’s

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 ↓

So maximum of two electrons in each state one ↑ and the other↓

±

2

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)

n=1 - l=0 s state – 2 electrons n=2 - l=1 p state – 6 electrons …… l=2 d state – 10 electrons ……. l=3 f state – 14 electrons

Z>1 The periodic table of elements Gives the basic structure of the Periodic Table of the elements

Periodic Table of Elements

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.

1-7s 4-5f 3-6d 2-7p

X-Ray spectra

X-Rays are emitted by impact of high energy electrons on elements

Continuous spectrum due Bremsstrahlung & other scattering processes

Molybdenum spectrum shown

λmin =

hceV

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

The first time Z was spectroscopically determined…

f ∝ Z −1( )2

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)

Which agrees very closely with Moseley’s experiment.

Actually the most important early evidence for nuclear model of atom!

ΔE =

34

Ryd Z −1( )2

Bonds between atoms

Wavefunctions of adjacent atoms 1 & 2 combine, so two possibilities: ψ1+ ψ2 or ψ1- ψ2

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?

Diatomic Molecule = interference of electron ‘waves’ (i.e. adding/subtracting)

OR

Bonds between atoms (cont.)

Bonding

Antibonding

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

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

Bonding

AntibondingBonds between atoms (cont.)