The central field approximation N 2 H 2 () - 1 electrons ... · Departure from LS-coupling in Hg...
Transcript of The central field approximation N 2 H 2 () - 1 electrons ... · Departure from LS-coupling in Hg...
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