Chapter 3.

Many-Electron Atoms

Reading: Bransden & Joachain, Chapter 8

Central Field ApproximationAssumption: each of the atomic electrons moves in an effective spherically symmetric potential V(r) created by the nucleus and all the other electrons.

Hamiltonian of the N-electron atom

∑∑∑

∑∑∑

> ==

> ==

+⎟⎟⎠

⎞⎜⎜⎝

⎛−∇−=

+⎟⎟⎠

⎞⎜⎜⎝

⎛−∇−=

N

ji

N

j ij

N

i ir

N

ji

N

j ij

N

i ir

rrZ

re

rZe

mH

i

i

11

2

1 0

2

1 0

22

2

121

442

r

rh

πεπε

in a.u.

No spin-orbit interactions, spin-spin interactions, relativistic effects, nuclear corrections etc.

Schrödinger equation

( ) ( ) ),(,,...,,,...,,121

212111

2iiiNN

N

ji

N

j ij

N

i ir srqqqqEqqq

rrZ

i

rr =Ψ=Ψ

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−∇− ∑∑∑

> ==

( ) ( ) ( ),..,...,...,,..,...,...,,..,...,..., jiijjiij qqqqqqP Ψ−=Ψ=Ψ

Total wave function: antisymmetric

Schrödinger equation for spatial wave function

( ) ( )NN

N

ji

N

j ij

N

i ir rrrErrr

rrZ

i

rrrrrrr ,...,,,...,,1

21

212111

2 ψψ =⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−∇− ∑∑∑

> ==

Central field approximation (independent particle model): Each electron moves in an effective spherically symmetric potential V(r) created by the nucleus and all the other electrons.

∑∑∑> ==

≈+−=N

ji

N

j ij

N

ii r

rSrSrZrV

11

1)(),()(

∞→−−

−→

→−→

rrNZrV

rrZrV

,)1()(

0,)(Screening

?)( →rV at intermediate distance

Thomas-Fermi and Hartree-Fock methods

Unperturbed Hamiltonian

∑∑∑∑∑∑∑∑

∑ ∑

> ==> ==> =

= =

<<−=⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

+∇−==⎟⎠⎞

⎜⎝⎛ +∇−=

+=

N

ji

N

j ij

N

ii

N

ji

N

j ij

N

ii

i

N

ji

N

j ij

iri

N

i

N

iiirc

c

rrS

rrV

rZ

rH

rVhhrVH

HHH

ii

111111

2

1 1

2

1

1)(1)(1

)(21,)(

21

rr

Schrödinger equation for central field wave function

)()()(21

)()...()(

)(21

2

21

1

2

21

ruErurV

rururu

ErVH

ll

N

i

nlmnlnlmr

Naaac

N

icccircc

rrr

r

=⎥⎦⎤

⎢⎣⎡ +∇−

=

=⎥⎦⎤

⎢⎣⎡ +∇−= ∑

=

ψ

ψψψ separable into N equations

one-electron orcentral field orbitals

)(

),()()(

r

YrRru

nlm

lmnlnlm llr

r

ψ

φθ

≠

=

: hydrogen wave function

• V(r) spherically symmetric → Enl are independent of ml• Enl depend on both n and l.

Spin-Orbitals and Slater Determinants

slslsl mlmnlmnlmmnlm YrRruqu ,2/1,2/1 ),()()()( χφθχ ==r

Spin-orbitals:

slsl mnlmnlmnlmr uEurV =⎥⎦⎤

⎢⎣⎡ +∇− )(

21 2

r

Slater determinantsTotal N-electron (central field) wave function: antisymmetric (Pauli exclusion principle).

),...,,( 21 Nc qqqΨ

)()()(

)()()()()()(

!1),...,,( 222

111

21

NNN

Nc

quququ

ququququququ

Nqqq

νβα

νβα

νβα

L

MOMM

L

L

=Ψ),,,(

...,,,

sl mmln→νβα

Quantum numbers

describing an atom in which one electron is in sate α, another in state β, and so on.

νβα EEEEc +++= L

α

γβ

ν:

1

23N

…

Eigenstates ElectronsEigenvalue:

( ) ( ),..,...,...,,..,...,..., jiij qqqq Ψ−=Ψ

0=Ψ→= βα Pauli exclusion principle

The Slater determinant will vanish if two electrons have the same values of the four quantum numbers n, l, ml and ms (two columns are equal in the determinant).

Pauli exclusion principle: no two electrons in an atom can have the same set of the four quantum numbers n, l, ml and ms.

The Slater determinant includes N! permutations of q1 ,q2,…, qN.

∑ −=ΨP

NP

Nc ququqPuN

qqq )()()()1(!

1),...,,( 2121 νβα K

Permutation of the electron coordinates

+1: even permutation, −1: odd permutation

Sum over all permutations

↓−↑ == χχ )(,)( 1002/1,1001002/1,100 ruuruue.g. He ground state (1s)2 1S :

[ ] [ ]↓↑−↑↓=−=

==Ψ

↓↑↓↑

↓↑

↓↑

21)()()1()2()2()1(

21)()(

)2()()2()()1()()1()(

21

)()()()(

!21),(

2100110021001100

21002100

11001100

22

1121

rurururu

rurururu

ququququ

qqc

χχχχ

χχχχ

βα

βα

Electron States in a Central Field

∑ −=ΨP

NP

Nc ququqPuN

qqq )()()()1(!

1),...,,( 2121 νβα K

νβα EEEEc +++= L

Eigenstate

Eigenvalue

nlmlmnlmnlm EEYrRquslsl

== ,),()()( ,2/1χφθ

)()()()()1(221

22

2

rRErRrVrRrll

drd

rdrd

nlnlnlnl =+⎥⎦

⎤⎢⎣

⎡ +−+−

Ordering of individual energy levels Enl

n=1 n=2 n=3 n=4 n=5

4f4d4p4s

5p5s

:

3d3p3s

2p2s

1sl = 0, 1, 2, 3, …

Spectroscopic notation s p d f …

nlE

lnE

nl

nl

given for

given for

↑←↑

↑←↑

more screening because ofthe larger angular momentum

↑+←↑ )( lnEnl

ds EE 34 <)12(2 += lgl

Degeneracy

1,,1,0 −= nl L

E

different from H atom(En+1>En)

Shells and subshells

0

2

~ aZnr

nlmAverage radius of electron orbit in a hydrogenic atom

Electrons which have the same value of the principal quantum number n are said to belong to the same shell.- Maximum number of electrons in a shell = 2n2 (closed or filled shell)

Electrons having the same value of n and l are said to belong to the same subshell.- Maximum number of electrons in a subshell, degeneracy = 2(2l+1)- Closed (or filled) subshell: an assembly of 2(2l+1) equivalent electrons

Degeneracy

)!(!!νδν

δ−

=dν : number of electrons occupying a Enl levelδ= 2(2l+1) : degeneracy of the level

c.f. d=1 for a closed subshell.

e.g. Ground state of the carbon atom (1s2 2s22p2)

15,6,2:21,2,2:21,2,2:1

=========

dpdsds

δνδνδν

g = 15

Periodic System of the Elements

Ground state of neutral atoms (Pauli exclusion principle):Z electrons (Z: atomic number, nuclear charge) occupy the lowest individual energy levels.

Valence electrons: electrons are in the the subshell of highest energy• least tightly bound electrons• in insufficient numbers to form another closed subshell

Russell-Saunders notationJ

S L12 +

T. Spin AMT. AM

T. orbital AM e.g. H2/1

2S

01SHe

SLJrrr

+=

Electron configuration: filling up shell and subshell structure

n =2 (L-Shell)n =1 (K-Shell)Z Element El. Config.3 Li [He]2s4 Be [He] 2s2

5 B [He] 2s22p… … …

10 Ne [He] 2s22p6

Z Element El. Config.1 H 1s2 He 1s2

n=1 n=2 n=3 n=4 n=5

4f4d4p4s

5p5s

:

3d3p3s

2p2s

1sl = 0, 1, 2, 3, …

Spectroscopic notation s p d f …

n =3 (M-Shell)Z Element El. Config.11 Na [Ne] 3s… … …

18 Ar [Ne] 3s23p6

Since E4s < E3d (screening by [Ar]), the added electrons in K (Z=19) and Ca (Z=20) go into the 4s rather than the 3d subshell.

n =4 (N-Shell)Z Element El. Config.19 K [Ar]4s20 Ca [Ar] 4s2

… … …

… … …

29 Cu [Ar] 4s3d10

30 Zn [Ar] 4s23d10

31 Ga [Ar] 4s23d104p

24 Cr [Ar] 4s3d5

25 Mn [Ar] 4s23d5

21 Sc [Ar] 4s23d

… … …

36 Kr [Ar] 4s23d104p6

First transition (iron) group, 3d

E4s ~ E3d

Second transition (palladium) group, 4d(from Z=39 to Z=48)Third transition (platinum) group, 5d(from Z=71 to Z=80)

Rare-earth elements (lanthanides), 4f ~ 5dfrom Z=57 to Z=70

Actinides, 5f ~ 6dfrom Z=89 to Z=102

For large Z, relativistic effects (e.g., spin-orbit coupling) become important and prevent the simple decoupling of the space and spin parts of the wave functions.

noble (inert) gases: closed shell

alkali metals: ns1

[He]

Li (Z=3)

2s1

[He]

Ne (Z=10)

2s12p6

Ionization Potentials

[He]

Na (Z=11)3s1

L-Shell

K-ShellZ

0

2

~ aZnr

nlm

Chemical Properties and Periodic Table

alkali metal

inert gashalogens

Pauli’s exclusion principle → diversity of chemical and material properties

Models for Central Field Approximation

Central field approximation (independent particle model): Each electron moves in an effective spherically symmetric potential V(r) created by the nucleus and all the other electrons.

∑∑∑> ==

≈+−=N

ji

N

j ij

N

ii r

rSrSrZrV

11

1)(),()(

∞→−−

−→

→−→

rrNZrV

rrZrV

,)1()(

0,)(Screening

?)( →rV at intermediate distance

Thomas-Fermi and Hartree-Fock methods

Fermi Electron GasFermi electron gas: A system consisting of a large number N of free electrons confined to a certain region of space (large cube of side L: 3D infinite potential well).

Free particle Schrödinger equation: )()(2 2

2

2

2

2

22

rErzyxm

rrh ψψ =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛= z

Lny

Ln

xL

nL

r zyxnnn zyx

πππψ sinsinsin8)(21

3v

Spatial Eigenfunctions

L,3,2,1,, =zyx nnnvanishing at the boundary

(i) degenerated

2

1L

E ∝∆(ii)( ) 2

2

22222

2

22

22n

mLnnn

mLE zyxnnn zyx

hh ππ=++=Energy eigenvalues

szyxszyx mnnnmnnn rq ,2/1)()( χψψ v=Total wave functions

Density of states D(E) : Number of electron quantum states (spin-orbitals) per unit energy range

dEED )( - Number of electron states between E and E+dEnz

ny

n

222zyx nnnn ++=

Total number of states for energies up to E= volume of the piece of the sphere × 2

22

2

22 22 h

h mELnnmL

Eπ

π=→=

2/32/3

22

2/3

2

3

33

23

1

231

31

34

812

VEm

mEL

nnNs

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

==

h

h

π

ππ

ππ

dEVEmdEEDdNs2/1

2/3

22

22

1)( ⎟⎠⎞

⎜⎝⎛==hπ

2/12/3

22

22

1)( VEmED ⎟⎠⎞

⎜⎝⎛=hπ

nx

2/12/3

22

22

1)( VEmED ⎟⎠⎞

⎜⎝⎛=hπ

Fermi energy EF

ground state at T=0

EFfilled vacant

2/32/3

22

0

2/12/3

220

23

1

22

1)(

F

EE

VEm

dEEVmdEEDN FF

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛== ∫∫

h

h

π

π

( )

( ) 3/1222

3/222

3,2

,32

ρπ

ρρπ

==

==

FF

F

kmk

VN

mE

h

h

FF

EE

tot NEVEmdEEVmdEEEDE FF

532

512

21)( 2/5

2/3

220

2/32/3

220=⎟

⎠⎞

⎜⎝⎛=⎟

⎠⎞

⎜⎝⎛== ∫∫ hh ππ

Thomas-Fermi theory

Ftot E

NEE

53

==

Thomas-Fermi TheoryN-electron atom: Fermi electron gas in the ground state, confined to a region of space by a central potential V(r) (V → 0, r → ∞). ?)(?,)( == rrV ρ

)(,0)(2 max

2

rVEErVm

pE F +=<+=

independent of r

Total energy of an electron

[ ] ( ) 3/12max2

222

max 3,)(2)()(2

ρπ=−=→+= FFF krVEmrkrV

mkE

h

h

)( 0max rVE = at r=r0 : boundary

[ ]

)(0

)(23

13

)(

max

2/3max

2/3

222

3

EV

rVEmkr F

>=

−⎟⎠⎞

⎜⎝⎛==hππ

ρ

)(rV

rEmax

r0

EF(r)

Electrostatic potential )(1)( rVe

r −=φ

[ ]⎪⎩

⎪⎨⎧

<Φ

≥ΦΦ⎟⎠⎞

⎜⎝⎛

=→−=

−=Φ

0,0

0,)(23

1)(

)()( 2/32/3

22max

0

0 remr

eErr

hπρφ

φφ

Poisson’s equation [ ] )()(1)(0

2

22 rerr

drd

rr ρ

ε=Φ=Φ∇

[ ] [ ]⎪⎩

⎪⎨⎧

<Φ

≥ΦΦ⎟⎠⎞

⎜⎝⎛

=Φ0,0

0,)(23)(1 2/3

2/3

20

22

2 remerr

drd

r hεπ

00 4

)(limπεZerr

r=Φ

→Boundary condition

∫ =0

0

2)(4r

NdrrrρπNormalization

)(4

)(,0

xZerrbxr χπε

=Φ= where 3/103/7

3/2

2)3( −= Zab π

Dimensionless variable x and function χ

⎩⎨⎧

<≥

=−

0,00),()( 2/32/1

2

2

χχχχ xx

dxxd

,1)0(

Thomas-Fermi equation

=χBoundary condition for neutral atoms : 0)( =∞χ

Universal function χ(x) for all neutral atoms ← numerical integration

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−+−=⎟

⎠⎞

⎜⎝⎛−=

br

rZe

rZe

br

rZerV χ

πεπεχ

πε1

444)(

0

2

0

2

0

2

Central potential

Screening: monotonically decreasing as r increases

neutral atom

Hartree-Fock Method: Self-Consistent Field

Trial function for N-electron atom: ),...,,( 21)(

Nk

c qqqΨ

Spherically symmetric electrostatic potential energy:

),...,,()()( 21)()()(

Nk

ckk qqqrS

rZrV Ψ←+−=

One electron Schrödinger equation: )1()1()1()(2 )(21 +++ =⎥⎦

⎤⎢⎣⎡ +∇− k

mnlmk

nlk

mnlmk

r slsluEurVr

?1 EEE kk δ<−+ No: k= k+1

Yes

One electron Schrödinger equation: )1()1()1()0(2 )(21

slsl mnlmnlmnlmr uEurV =⎥⎦⎤

⎢⎣⎡ +∇− r

∑=Ψln

fnl

fN

fc EEqqq

,

)()(21

)( ),,...,,(f= k+1

k=1

self-consistent field

iteration

21 HHH +=Non-relativistic Hamiltonian for N-electron atom

∑∑

∑ ∑

> =

= =

=

−∇−==⎟⎟⎠

⎞⎜⎜⎝

⎛−∇−=

N

ji

N

j ij

iri

N

i

N

ii

ir

rH

rZhh

rZH

ii

12

2

1 1

21

1

21,

21

rr

Variational method: trial function for N-electron atom = Slater determinant

Expectation value of Hamiltonian (energy) for a trial wave function, φ[ ] 1,0 =ΦΦΦΦ=Φ≤ HEE

)()()(

)()()()()()(

!1),...,,( 222

111

21

NNN

N

quququ

ququququququ

Nqqq

νβα

νβα

νβα

L

MOMM

L

L

=Φ),,,(

...,,,

sl mmln→νβα

Quantum numbers

∫ == µλλµλµ δdqququuu )()(*Orthonormal spin-orbitals

HP

NP

N ANquququPN

qqq Φ=−=Φ ∑ !)()()()1(!

1),...,,( 2121 νβα K

)()()(),...,,( 2121 NNH quququqqq νβα K=Φ∑ −=P

P PN

A )1(!

1

Hermitian and projection operator

where and

( )AA =2

H1 and H2 are invariant under permutations of the electron coordinates

[ ] [ ] 0,, 21 == AHAH∑∑∑∑> ===

==⎟⎟⎠

⎞⎜⎜⎝

⎛−∇−=

N

ji

N

j ij

N

ii

N

i ir r

HhrZH

i1

211

21

1,21

r

Energy expectation value for the trial wave function, Φ

[ ] ΦΦ+ΦΦ=ΦΦ=Φ 21 HHHE

)()(,)()(

)1(

!!!

11

12

111

iiiiii

N

iHiHHiH

N

i P

P

HHHHHH

quhquIIquhqu

hPh

AHNAHNAAHNH

λλλλ

λλ

λλ ===

ΦΦ=ΦΦ−=

ΦΦ=ΦΦ=ΦΦ=ΦΦ

∑∑

∑∑∑==

µλλµ δ=uu

( )

( )

[ ]νβαµλ

µλ µλµλµ

λµµλ

µλ µµλµλ

µλλµµλµλµλ

,,,,21

)()(1)()(

)()(1)()(21

)()(1)()()()(1)()(

111)1(

!!!

,

11

22

222

K=

−=

⎥⎥⎦

⎤−

⎢⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−=

Φ−Φ=ΦΦ−=

ΦΦ=ΦΦ=ΦΦ=ΦΦ

∑∑

∑∑

∑

∑∑∑∑∑

≠

≠

> => =

KJ

ququr

ququ

ququr

ququ

ququr

ququququr

ququ

Pr

Pr

AHNAHNAAHNH

jiij

ji

jiij

ji

jiij

jijiij

ji

N

ji

N

jHij

ijH

N

jiH

ijH

N

j P

P

HHHHHH

)()(1)()(

)()(1)()(

jiij

ji

jiij

ji

ququr

ququK

ququr

ququJ

λµµλλµ

µλµλλµ

=

= Directterm

Exchangeterm

[ ] [ ]∑∑∑ −+=ΦΦ+ΦΦ=Φλ µ

λµλµλ

λ KJIHHE21

21

Energy expectation value for a trial wave function, Φ

E[Φ] is stationary with respect to variations of the spin-orbitals uλ (λ=α,β,…,ν ).

Variational equation (p130-136):

00 =−→=− ∑∑∑λ

λλλλ µ

λµλµ δεδδεδ uuEuuE

Lagrange multipliers diagonalization

Hartree-Fock equations

νβαµλλλµµ

λµ

λµ

µµλ

,,,,),()()(1)(

)()(1)()(21

*

*2

K

r

==⎥⎥⎦

⎤

⎢⎢⎣

⎡−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−∇−

∑ ∫

∑∫

iijjij

j

ijjij

jii

r

quEqudqqur

qu

qudqqur

ququrZ

i

νβαµλδ λλµµ

λµ

λµ

µµλ

µλ ,,,,),()()(1)(

)()(1)()(21

*,

*2

Krrrrr

rrrrrr

==⎥⎥⎦

⎤

⎢⎢⎣

⎡−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛−∇−

∑ ∫

∑∫

iijjij

jmm

ijjij

jii

r

ruErurdrur

ru

rurdrur

rururZ

ss

i

λSpin-orbitals: χλλsmii ruqu

,2/1)()( r

=

)( id rV r

µ= : direct operator

)()( iiex rurV rr

λµ=: exchange operator

[ ] )()()()(21 )1()1(

,2

ik

ik

iex

mmid

ir ruErurVrV

rZ

ssi

rrrrr

++ =⎭⎬⎫

⎩⎨⎧

−+⎟⎟⎠

⎞⎜⎜⎝

⎛−∇− ∑ λλλ

µµµ µλδ

)()(i

k ru rµ

)()()()(21 2

iiiex

id

ir ruErurVrV

rZ

i

rrr λλλ =

⎭⎬⎫

⎩⎨⎧

−+⎟⎟⎠

⎞⎜⎜⎝

⎛−∇−

Corrections to the Central Field Approximation

∑=

=+=N

iHFHFHF ihHHHH

11 )(,

Correlation effects

0<−= HFexactcorr EEE[ ] ΦΦ=Φ=≤ HEEE HF0because

Spin-orbit interactions

i

i

ii

iiii dr

rdVrcm

rSLrH )(12

1)(,)( 222 =⋅= ∑ ξξrr

relativistic correction

Total Hamiltonian

21 HHHH HF ++=

L-S (or Russell-Saunders) coupling case: small and intermediate Z

21 HH >>(i)

j-j coupling case: large Z21 HH <<(ii)

L-S Coupling

1HHH HF += 21 HH >>Unperturbed Hamiltonian where

(γ : level configuration of Hc) is degenerated.γEnergy eigenstate of HHF ,e.g. carbon atom, C, 1s2 2s22p2, g=15

H1 lifts part of the degeneracy, but not completely: 3 states. PDS 311 ,,

[ ] [ ] 0,, == SHLHrr

,, ∑∑ ==i

ii

i SSLLrrrr

Since where

→ energy eigenstates can be written as SLMLSMγ

Terms: energy levels corresponding to definite values of L and S, LS 12 +

Multiplicity: 2S+1 = 1(singlet), 2(doublet), 3(triplet), 4, …

Closed subshell: L=0, S=0, S1

→ Energy eigenvalues are independent of ML and MS . LSγ

L

L

,,,,,3,2,1,0

FDPSL =

Incomplete subshells: optically active electrons

1. Non-equivalent electrons: electrons belonging to different subshells

Pauli exclusion principle is automatically satisfied.

21 SSSrrr

+=21 LLLrrr

+=Addition of angular momentum and

212121 ,,1, llllllL ++−−= L

212121 ,,1, ssssssS ++−−= L

(i) Configuration np n’p

1,0,2,1,02/1,1 2121 ==⇒==== SLssll

Terms: DPSDPS 333111 ,,,,,(ii) Configuration np n’d

1,0,3,2,12/1,2,1 2121 ==⇒==== SLssll

Terms: FDPFDP 333111 ,,,,,

(iii) Configuration np n’p n”d

configuration np n’p 1,0,2,1,0 11 == sl

2/1,2 22 == sl+ configuration n”d

GFDPS

GFDPSSLslD

FDPFDPSLslP

DDSLslS

GFDPSSLslD

FDPSLslP

DSLslS

44444

2222211

3

44422211

3

4211

3

2222211

1

22211

1

211

1

,,,,

,,,,,,2/3,2/1,4,3,2,1)1,2(

,,,,,,2/3,2/1,3,2,1)1,1(

,,2/3,2/1,2)1,0(

,,,,,2/1,4,3,2,1,0)0,2(

,,,2/1,3,2,1)0,1(

,2/1,2)0,0(

==→==

==→==

==→==

==→==

==→==

==→==

Terms:

Number of identical terms: 23224642,,,,,,,,, 4444422222 GFDPSGFDPS

2. Equivalent electrons: electrons belonging to same subshells

Pauli exclusion principle rules out certain values of L and S.

(i) Configuration ns2

Closed subshell: ML , MS = 0 → L=0, S=0, term S1

1,0,02/1,0 2121 ==⇒==== SLssll

SS 31 ,Terms: Not allowed because of the Pauli exclusion principle

(ii) Configuration np2, g = 15

1,0,2,1,02/1,1 2121 ==⇒==== SLssll

DPSDPS 333111 ,,,,,

Not allowed because of the Pauli exclusion principle

Terms:

Number ml1 ms1 ml2 ms2 ML

= ml1 + ml2

MS

= ms1 + ms2

1 1 ½ 0 ½ 1 12 1 ½ -1 ½ 0 13 0 ½ -1 ½ -1 14 1 -½ 0 -½ 1 -15 1 -½ -1 -½ 0 -16 0 -½ -1 -½ -1 -17 1 ½ 1 -½ 2 08 1 ½ 0 -½ 1 09 1 ½ -1 -½ 0 010 0 ½ 1 -½ 1 011 0 ½ 0 -½ 0 012 0 ½ -1 -½ -1 013 -1 ½ 1 -½ 0 014 -1 ½ 0 -½ -1 015 -1 ½ -1 -½ -2 0

1,0,1 −=lm

D1

D1

D1

D1

D1

P3

S1

P3

P3P3

P3

P3

P3

P3P3

DPSDPS 333111 ,,,,,Terms:

is missing : no ( )1,2,3 == SLD)1,2( == SL MM

: term with L=2 must exist.

→

)0,2()0,2( =−=== SLSL MMMM

( )0,2,1 == SLDdegeneracy = 5

: term with L=1, S=1 must exist.

→

...)1,1()1,1( −=−=== SLSL MMMM( )1,1,3 == SLP

degeneracy = 9

5+9=14 : eliminate P1 S3and

)0,0( == SL MM D1

P3

degeneracy = 1S1

Fine Structure of terms in L-S coupling

Total Hamiltonian

21 HHHH HF ++= 1H Correlation effects

LS 12 +

Spin-orbit interactionsi

i

ii

iiii dr

rdVrcm

rSLrH )(12

1)(,)( 222 =⋅= ∑ ξξrr

Energy eigenstates

L-S (or Russell-Saunders) coupling case21 HH >>

[ ] [ ] 0,,0, 22 ≠≠ SHLHrr [ ] 0,2 =JH

rbut where SLJ

rrr+=

( )2222 2

1 SLJASLAH −−=⋅=rr

H2 lifts part of the degeneracy of a state represented by a termJ

SS LL 1212 ++ →SLSLSLJ ++−−= ,...,1|||,|

DegeneracyJ

SS LL 1212 ++ →

)12)(12( ++= SLg )12)(12()12(||

++=+= ∑+

−=

SLJgSL

SLJ

Configuration np n’p

1,0,2,1,02/1,1 2121

==⇒====SL

ssll

DPSDPS 333111 ,,,,,Terms:

Hund’s rules

1. For given term, largest S → lowest energy

2. For given S,largest L → lowest energy

j-j Coupling

i

i

ii

iiii dr

rdVrcm

rSLrH )(12

1)(,)( 222 =⋅= ∑ ξξrr

21 HH <<

( )

( )222

112

)(21~

)(~~

iiiiii

N

iiiii

N

iiHF

SLJrhh

SLrhhHHH

−−+=

⋅+==+= ∑∑==

ξ

ξrr

Hamiltonian

[ ] iiiii jlnJh →r

,

∑=

=N

ijln iii

EE1

~Energy