The central field approximation

Each electron moves independently of the other

in the electrostatic field from the nucleus and

the other N - 1 electrons.

This field is assumed to be spherically

symmetric.

1

1

,

22

1

1

( ) ( , )

( )2

( )ii i i isi

N

ii

N

i i i i ii

i i i in m

N

i i ii

N

ii

m

H

H

R r Y

H E

H V rm

E

sz

Leads to the concept of a configuration, i.e. a list

of the n and ℓ quantum numbers for all

electrons: 1 2

1 1 2 2( ) ( ) ....... ( ) mww wm mn n n

where w is the number of equivalent electrons in

the orbital

Unknown but

spherically symmetric

Determined numerically

Rörelsemängdsmoment

klassiskt

1: 0 bevarat/rörelsekonstant om:

2 : d.v.s. centrala krafter

d

dt

L r p

Lτ r F

FL

F r

Kvantmekaniskt

rörelsekonstant 0 [ , ] 0d

dtL L H L

Addition av 2 rörlesemängdsmoment

L = ℓ1 + ℓ2

Addition av 2 rörlesemängdsmoment

L = ℓ1 + ℓ2

Generell regel för att ta fram värdet av kvanttalet som

beskrives storleken av det totala momentet.

Vi använder beteckningarna ovan som exempel:

Kvanttalet L kan då anta följande värden

1 2 1 2 1 2, 1......

pd-configuration LS-coupling

Configuration Term

Central field Repulsion

Numerical example for 2p3d in O V, energies in cm

-1

E(2p3d) = 701810 Kinetic and central part of electrostatic

∆E (P - D) = 8980 Direct part of electrostatic repulsion

∆E (1F -

3F) = 15074 Exchange part of electrostatic repulsion

Permitted LS-terms with equivalent electrons

(Starting with d3 configuration there may appear more than one term with the same L and S

values. The number of such multiple terms is given by the subscripts in the table, and a new

quantum number – the seniority number – must be introduced)

pd-configuration LSJ-coupling

Configuration Term Level

Central field Repulsion Spin-orbit

Numerical example for 2p3d in O V, energies in cm

-1

E(2p3d) = 701810 Kinetic and central part of electrostatic

∆E (P - D) = 8980 Direct part of electrostatic repulsion

∆E (1F -

3F) = 15074 Exchange part of electrostatic repulsion

∆E (3F4-

3F3) = 235 Spin-orbit magnetic energy

Urvalsregler för E1 (elektisk dipol) strålning

J = 0, 1 ej 0 → 0

Endast en elektron får ändra orbital, n → n’’

= 1

I perfekt LS-koppling

S = 0

L = 0, 1 ej 0 → 0

Urvalsregeler och metastabila nivåer i He

No intersystem

lines ∆S = 0

Meta stable,

“nowhere to go!!

Skalning med Z längs en isoelektronisk sekvens.

2

2bindning 2

repulsion

4spinn-ban

ZT E R Z

n

E Z

E Z

I LS koppling började vi med att försumma spinn-ban

växelverkan, men för höga Z kan den bli stor p.g.a.

skalningen med Z4!

Ett annat extremfall är jj – koppling där vi börjar med

att försumma den elektrostatiska (icke-centrala)

växelverkan.

Departure from LS-coupling in Hg

Note the large fine structure in 6s6p 3P compared to the

1P -

3P separation. Not very good LS coupling.

Even stronger evidence for the departure from LS

coupling is the presence of the intense intercombination

line (∆S≠0) 6s2 1S0 – 6s6p

3P1 at 2536 Å.

pd configuration in jj-coupling

Configuration SO(p) SO(d) Coulomb rep.

Same number of energy levels and the same total J as in

LS-coupling. Only our names of the levels have changed.

LS to jj - coupling transition in a sp-configuration.

Relative energies as a function of the spin-orbit parameter, β.

Note how the 2 J = 1 states seems to “repel” each other

while the two unique J-values (0 and 2) just increase

linearly with the β-parameter.

3P0,1,2

1P1

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

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