in atoms • Atomic units (a.u.) --> standard usage

– mass m e unit of mass – elementary charge e unit of charge – length and time such that numerical values of and 4 ⇥ 0 are unity – then atomic unit of length 2 4⇥0 11 a.u. (length) = a0 = 2 5.29177 10 m e me ⇥ a0 17 – and time a.u. (time) = 2.41888 10 s c ⇥ e2 1 – where = is the fine structure constant 4⇤⇥0c 137.036 2 – unit = Hartree EH = 2 27.2114 eV mea0

QMPT 540 Hamiltonian in a.u. • Most of atomic can be understood on the basis of

N 2 N N pi Z 1 1 HN = + + Vmag 2 ri 2 ri rj i=1 i=1 i=j | | | | • for most applications V V eff = s mag ⇥ so i i · i i • Relativistic description required for heavier atoms – binding sizable fraction of electron rest mass – binding of lowest s state generates high-momentum components

• Sensible calculations up to Kr without Vmag • Shell structure well established

QMPT 540 Shell structure • Simulate with N N H0 = H0(i) i=1 2 • with pi Z H0(i)= + U(ri) 2 ri

• even without auxiliary potential ⇒ shells (2 + 1) (2s + 1) – hydrogen-like: degeneracy Z2 = – but n 2 n 2 does not give correct shell structure (2,10,28...

– degeneracy must be lifted

– how?

QMPT 540 Effect of other electrons in neutral atoms • Consider effect of electrons in closed shells for neutral Na • large distances: nuclear charge screened to 1 • close to the nucleus: electron “sees” all 11 protons

0 • approximately:

!5

!10 V(r) [a.u.] Sodium

!15

!20 012345 r [a.u.]

• lifts H-like degeneracy: 2s < 2p

3s < 3p < 3d

• “Far away” orbits: still hydrogen-like! 523-14 Example: Na • Fill the lowest shells

• Use schematic potential H n⇤m m = n⇤m m 0 | s n | s • Ground state: fill lowest orbits according to Pauli

300m , 211 1 , 211 1 ,...,100 1 , 100 1 | s 2 2 2 2 i⌘ (Na) | 0 i • Excited states?

523-14 Closed-shell atoms • Neon

• Ground state

(Ne) = 211 1 , 211 1 ,...,100 1 , 100 1 | 0 i | 2 2 2 2 i

• Excited states

1 n` (2p) = a† a (Ne) | i n` 2p | 0 i • operators: see later

• Note the H-like states

• Splitting?

• Basic shell structure of atoms understood ⇒ IPM 523-14 Periodic table

HF F-RPA Σ(2)

523-14 Level sequence (approximately)

523-14 Helium • IPM description is very successful • Closed-shell configuration 1s2 • Reaction more complicated than for Hydrogen • DWIA (distorted wave impulse approximation)

N 1 N 2 S = dp a n | p | 0 ⇥ ⇥ agreement with IPM! → 1

Phys. Rev. A8, 2494 (1973)

523-14 Other closed-shell atoms • Spectroscopic factor becomes less than 1 • Neon 2 p removal: S = 0.92 with two fragments each 0.04 • IPM not the whole story: fragmentation of sp strength • Summed strength: like IPM • IPM wave functions still excellent • Example: Argon 3 p S = 0.95 a.u

3 2.0 Argon − • Rest in 3 small fragments 1.6 3p

1.2

0.8

0.4

0.0

Differential cross section (10 0123 p (a.u.) 523-14 Argon spectroscopic factors • s strength also in the continuum: Ar++ + e • note vertical scale • red bars: 3s fragments exhibit substantial fragmentation

8%

523-14 Fragmentation in atoms • ~All the strength remains below (above) the Fermi energy in closed-shell atoms • Fragmentation can be interpreted in terms of mixing between

a N | 0 • and N a a a† ⇥ ⇤ | 0

• with the same “global” quantum numbers • Example: Argon ground state N = (3s)2(3p)6(2s)2(2p)6(1s)2 | 0 | + 1 N 2 5 2 6 2 • Ar ground state (3p) = a = (3s) (3p) (2s) (2p) (1s) | 3p | 0 | 1 N 1 6 2 6 2 • excited state (3s) = a = (3s) (3p) (2s) (2p) (1s) | 3s | 0 | 2 N 1 2 4 2 6 2 • also (3p) 4s = a a a† = (4s) (3s) (3p) (2s) (2p) (1s) | 3p 3p 4s | 0 | 2 N 1 2 4 2 6 2 • and (3p) nd = a3pa3pand† 0 = (nd) (3s) (3p) (2s) (2p) (1s) | | | 523-14