2 electrons; the helium atom and its peculiarities

More than one electron: ÆPauli principle

Wolfgang Pauli Nobel 1945

Non-central forces due to electron repulsion

MNW-L4 The Helium atom; classical approximation

a Without repulsion: a/2 a/2 2 E = 2E1 = −2Z E0 = −8⋅()13.6 eV = −108.8eV

With repulsion; electrons at fixed distance a0, Repulsive energy: e2 = 2E0 = 27.2eV 4πε0a0

Calculated: Etot = −81.6eV

Experimental: Etot = −79eV

MNW-L4 The Helium atom: quantum mechanics

πε 2 2 2 2 h 2 2 2e πε2e e H = − (∇1 + ∇2 )− − + 2me 4 0r1 4 0r2 4πε0r12 +2e

Ekin attraction repulsion

r r r r Two-particle Schrödinger equation: HΨ(r1,r2 )= EΨ(r1,r2 )

(0) First approximation: unperturbed system (no repulsion) H = H1 + H2 2 2 Each: h 2 2e Hi = − ∇i − 2me 4πε0ri

r r r r Product wave function: Ψ(r1,r2 )=ψ nlm (r1)ψ nlm (r2 )=ψ1ψ 2 ψ ψ (0) r r ψ H Ψ(r1,r2 )(=ψHψ1 + H2 ψ1 )2 = (H1 1)()2 + 1 H2 2 ψ ψ = E1 1 2 + E2 1 2 = E 1ψ 2 ψ ψ MNW-L4 ψ The Helium atom: quantum mechanics

πε 2 2 2 2 h 2 2 2e πε2e e H = − (∇1 + ∇2 )− − + 2me 4 0r1 4 0r2 4πε0r12 +2e

Ekin attraction repulsion

2 2 2e2 Z 4 Each:H = − h ∇2 − has solution E = −R = −R i 2m i 4πε r i 2 2 e 0 i ni ni

⎛ 1 1 ⎞ Hence: E = −8R⎜ + ⎟ ⎜ 2 2 ⎟ ⎝ n1 n2 ⎠

MNW-L4 Intermezzo Æ electron-electron repulsion as a perturbation

2 ()1 e First order perturbation E = n1l1m1;n2l 2m2 n1l1m1;n2l 2m2 = J 4πε0r12

5 ⎛ Ze2 ⎞ This is a mathematical problem – with a solution: J = ⎜ ⎟ ⎜ ⎟ 4 ⎝ 4πε0a0 ⎠

So: ⎛ 5 ⎞ Etot = −2Eb + J = ⎜− 8 + ⎟R ⎝ 2 ⎠

Ecalc= -5.5 R ; Eexp = -5.8 R

MNW-L4 The orbital angular momentum of an electron in orbit

r e r Semiclassical approach Hence μ = − L 2me μr

So a magnetic moment -e is associated with +Ze an angular momentum 2πν

r L From classical electrodynamics: Angular velocity ω = 2πν What happens to a magnetic dipole in a B-field ? μ Magnetic moment π It rotates, and alines in the field = IA = r2eν

Angular momentum ω Magnetic interaction: 2 2 μ L = mvr = m r = 2πmvr r r VM = − ⋅ B = −μzBz Zeeman effect MNW-L4 Space quantisation

From classical electrodynamics: Origin of the spin-concept

-Stern-Gerlach experiment; -A magnetic dipole moment μ undergoes space quantization Otto Stern no force in a magnetic field B Nobel 1943

-A magnetic dipole undergoes a torque, is oriented in a B field

-A magnetic dipole μ undergoes a force in an inhomogeneous B-field, i.e. a field gradient (Stern-Gerlach experiment)

-Theory: the periodic system requires an additional two-valuedness

MNW-L4 Electron spin

No classical analogue for this phenomenon 1 s = 2 h 1 s = 2 h Spin is an angular momentum, so it should satisfy Pauli: 2 2 There is an additional “two-valuedness” S s,ms = h s()s +1 s,ms in the spectra of atoms, behaving like an angular momentum Sz s,ms = hms s,ms 1 1 s = ,m = ± Goudsmit and Uhlenbeck 2 s 2 This may be interpreted/represented as an angular momentum

Note: the spin of the electron cannot be explained from a classically “spinning” electronic charge Problem to be solved MNW-L4 The Helium atom and the Pauli principle

1) 2 electrons cannot occupy the same orbital (quantum state)

2) The 2 electrons must have different quantum numbers: α and β:

ψα 3) He must have an anti-syψ β mmetric wave function (under interchange of the 2 electrons) 1 ψ β Ψ()2,1 = []()2 1 ()− 2 ψα ()1 = −Ψ ()()1,2 2 If α = β then Ψ = 0

Two possibilities for having an anti-symmetric wave function: ψ A S A Ψ ()1,2 = r1,r (2 χ 1 ),2 () spatial ψ and spin χ ψ A A S Ψ ()1,2 = r1,r (2 χ 1 ),2 ()

MNW-L4 EXTRA Addition of spins in a 2-electron system

r r r M = m + m S = s1 + s2 S s1 s2 ; S= 0, 1 MS = -1, 0, 1

S =1, M S = 1 = ↑,↑ 1 A triplet of symmetric S =1, M S = 0 = (↑,↓ + ↓,↑ ) 2 spin wave functions S =1, M S = −1 = ↓,↓

1 A singlet of an anti-symmetric S = 0, M S = 0 = (↑,↓ − ↓,↑ ) 2 spin wave function

Two distinct families of quantum states in Helium:

A S Ortho-helium: triplet states ψ χ S A Para-helium: singlet states ψ χ

MNW-L4 Energy levels and Spectral lines in Helium

“Singly-excited” states in Helium, with one electron in (1s)

He (1s)(nl) configurations

Selection rules: similar as in H

ψ S χ A ψ Aχ S Δn = free

Δl = ±1 Δm = −1,0,+1 ΔS = 0

(1s)(2s) levels are metastable

MNW-L4 EXTRA Ordering of singlet and triplet states: qualitative ψ ψ Symmetric and Anti-symmetric spatial waveψ functions r r 1 r r r r Ψ± ()r1,r2 = []n l m ()r1 n l m r2 − ()n l m r1 ψ n l m ()r2 () 2 1 1 1 2 2 2 2 2 2 1 1 1 r r r r 2 If: r1 ≈ r2 density becomes small for Ψ− ()r1,r2

Fermi-hole

r r 2 While density of Ψ+ ()r1,r2 increases

Effect of “exchange”; a “force” related to the Pauli principle Æ quantum interference

MNW-L4 EXTRA Ordering of singlet and triplet states: qualitative

Fermi-hole in case of an anti-symmetric spatial wave function r r 2 Ψ− ()r1,r2 In this state the electrons tend to be distanced from each other Æ less repulsion Æ more binding For the triplet states (or symmetric spin states)

Conversely, in case of symmetric spatial function r r 2 Ψ+ ()r1,r2 Electrons tend to be close to each other Æ more repulsion Æ less binding energy

ÆTriplet states are lower in energy than corresponding singlets

MNW-L4