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Atoms and Nuclei
PA 322
Lecture 8
Unit 2: Many-electron atoms
(Reminder: http://www.star.le.ac.uk/~nrt3/322)
Spectra of many-electron atoms
•  Topics
–  L-S (= Russell-Saunders) coupling framework
–  terms for non-equivalent electrons
–  terms for equivalent electrons
–  Hund’s rules for L-S coupling: predicting the ground state
configuration
–  j-j coupling
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Lecture 8
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Many-electron atoms
General considerations
– 
typically atoms have more than one active electron
– 
exact solutions to Schrödinger equation formally not possible,
but approximate treatments give essentially correct picture
– 
L-S coupling (= Russell-Saunders coupling) provides
framework for discussing such atoms in wide variety of
circumstances
– 
valid unless individual electron spin-orbit coupling dominates
(⇒ j-j coupling)
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Lecture 8
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L-S coupling in many-electron atoms
•  L-S coupling issues
–  Effects which produced corrections to energy levels:
1. Spin correlation: electron-electron interaction as for helium atom
2. Orbital angular momentum coupling
3. Spin-orbit coupling
–  size of effects varies from atom to atom, cannot be trivially predicted
–  for low Z atoms (and low n): effects order 1 > 2 »3
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Lecture 8
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L-S coupling framework
for many-electron atoms
•  Spin angular momenta of the electrons couple:
–  Total spin angular momentum S = Σ si (vector sum)
–  Related total spin angular momentum quantum number S
•  Orbital angular momenta of electrons couple:
–  Total orbital angular momentum L = Σ li (vector sum)
–  Related total orbital angular momentum quantum number L
•  Resultant total spin and orbital angular momenta S and L couple to
produce total angular momentum vector J = L + S
•  Valid quantum numbers are now:
[s1, s2 ( … sN)], l1, l2 (…lN), L, S, J but not ml1, ms1 … as coupling of l vectors
and s vectors destroys their space quantisation but now have analogous
new projection quantum number for J called mJ .
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Terms for non-equivalent electrons
in many-electron atoms
•  Non-equivalent electrons: those which belong to different (n,l) subshells (e.g. orthohelium 1s2s)
•  Any (s, l) combination allowed without violating Pauli Exclusion
Principle
•  Terms can be worked out using relationships for L, S and J applying
rule for vector addition:
–  for two active electrons with spin quantum numbers s1 and s2,
orbital angular momentum numbers l1 and l2:
l2
•  S = |s1-s2| … (s1+s2) integer steps
•  L = |l1-l2| … (l1+l2)
integer steps
L
•  J = |L-S| … (L+S)
integer steps
l1
–  equivalent formulae for > 2 electrons (but not considered here)
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Lecture 8
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For
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examp
• 
• 
• 
Terms for non-equivalent electrons
in many-electron atoms
S=0 L=1 1P
S=0 L=2 1D
S=0 L=3 1F
S=1 L=1 3P
S=1 L=2 3D
S=1 L=3 3F Examples for non-equivalent electrons:
–  npnd, i.e., p & d optically active electrons:
–  L = |l1-l2| … (l1+l2) but l1=1, l2=2 ⇒ L = 1, 2, 3
1P –  S = |s1-s2| … (s1+s2) ⇒ S = 0, 1 (s1=s2=½)
S=0 L=1 J=1
1
1
S=0 L=2 J=2
D2
–  produces 6 terms: 1P, 1D, 1F, 3P, 3D, 3F
1
S=0 L=3 J=3
F3
3
taking J into account in each case
S=1 L=1 J=0,1,2
P0, 3P1, 3P2 3D , 3D , 3D S=1 L=2 J=1,2,3
–  J = |L-S| … (L+S)
1
2
3
3F , 3F , 3F S=1
L=3
J=2,3,4
2
3
4
–  e.g., for 3F, L=3, S=1 ⇒ J = 2, 3, 4 thus full levels are 3F2, 3F3, 3F4 –  in total there are 12 levels for the 6 terms 1P, 1D, 1F, 3P, 3D, 3F
for each J there are 2J+1 degenerate mJ values
–  e.g., npnd configuration: 60 microstates (see next slide)
–  same number as expected in decoupled representation, i.e., if angular momentum
coupling and spin correlation were ignored
•  e.g., npnd in decoupled representation:
(2l1+1) x (2s1+1) x (2l2+1) x (2s2+1) = 3 x 2 x 5 x 2 = 60 ‘microstates’
•  same number of microstates as computed from mJ values
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For
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Σ = 60
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Terms for equivalent electrons
in many-electron atoms
• 
• 
• 
Equivalent electrons: those which belong to same (n,l) sub-shells
Allowed combinations must of course be consistent with Pauli
Exclusion Principle (all quantum numbers cannot be same)
Simple application of L-S coupling rules used for non-equivalent
electrons produces some terms that are forbidden
–  working out valid terms has to be done the hard way
•  E.g. simple application for 2p2:
•  l1 = l2 = 1 ⇒ L = 0, 1, 2 & S = 0, 1
⇒ apparent terms 1S, 3S, 1P, 3P, 1D, 3D •  But this would result in too many states!
1S
3S 1P
3P
1D
3D
1 + 3 + 3 + 9 + 5 + 15 = 36
• 
Total number of apparently allowed microstates (36) ≠ number
for decoupled electrons (15) - see next table
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Lecture 8
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Possible quantum numbers for np2 configuration
number
ml
ms
ml
ms
mS
mL
mJ
1
+1
+½
+1
-½
0
+2
+2
2
+1
+½
0
+½
+1
+1
+2
3
+1
+½
0
-½
0
+1
+1
4
+1
+½
-1
+½
+1
0
+1
5
+1
+½
-1
-½
0
0
0
6
+1
-½
0
-½
-1
+1
0
7
+1
-½
-1
+½
0
0
0
8
+1
-½
-1
-½
-1
0
-1
9
0
+½
+1
-½
0
+1
+1
10
0
+½
0
-½
0
0
0
11
0
+½
-1
+½
+1
-1
0
12
0
+½
-1
-½
0
-1
-1
13
-1
+½
0
-½
0
-1
-1
14
-1
+½
-1
-½
0
-2
-2
15
-1
-½
0
-½
-1
-1
-2
decoupled
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coupled
Lecture 8
Note, also avoided
configurations that are
same due to
indistinguishability of
electrons.
must be equivalence
between degeneracies
(=number of allowed
microstates) in the L-S
coupled and decoupled
representations
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Terms for equivalent electrons
in many-electron atoms
• 
How to decide which terms are allowed and which terms are forbidden?
• 
Staying with application for 2p2:
–  apparent (i.e. possible w.r.t. s, l values) terms were
•  3D, 1D, 3P, 1P, 3S, 1S
•  in fact only 1S, 1D & 3P are allowed, others are forbidden by
application of PEP to decoupled representation
• 
Apply rule: for two equivalent
electrons, states with odd values of
L+S are not allowed.
Easy to see problem with 3D
–  2p2 & 3D means
•  s1= ½, s2= ½, S=1 and thus ms1 = ms2 = ½ (ie spins parallel)
•  l1=1, l2=1, L = 2 and thus ml1 = ml2 = +1 (to get L = 2 requires l1 and
l2 to be parallel)
•  this is not allowed by PEP in decoupled representation (all qn the
same) and is thus not allowed
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Terms for equivalent electrons
in many-electron atoms: full treatment
•  Working rule to sort out valid terms:
–  must be equivalence between degeneracies (=number of
allowed microstates) in the L-S coupled and decoupled
representations
•  In practice (see previous table for example)
–  tabulate all the possible (ml, ms) values for each electron, i.e. for
decoupled representation
–  compute the possible values of mS, mL, mJ –  compute apparent terms from s and l values in L-S coupling
scheme (i.e., S, L, J values)
–  check that terms are valid by looking to see if appropriate mS, mL,
mJ occur in table
•  if not, term is not allowed!
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Terms for equivalent electrons
in many-electron atoms
•  Further issues for terms of
equivalent electrons
–  closed (full) shells have term
1S since
0
•  all electrons must be
paired, ie. S = Σ si = 0
•  L = Σ li = 0 since
orientations of li must be
such as to give Σmli = 0
–  valid terms for subshells of q
electrons are the same as for
subshells with N-q electrons
where N is the closed (full)
subshell complement,
•  e.g., p2 has same terms as
p4 as N=6 for p subshells
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Possible terms for equivalent electrons
ns0
1S
ns1
2S
ns2
1S
np0
1S
np1
np2
2P
1S, 1D
np3
np4
2P, 2D
1S, 1D
np5
np6
Lecture 8
3P
4S
3P
2P
1S
13
Hund’s rules
• 
Now know which terms are possible, but which correspond to
ground configuration?
• 
Hund’s rules give the answer:
– 
rules give lowest energy term and level for many-electron
atoms
– 
for L-S coupling
– 
for ground state configuration of atom only
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Hund’s rules
1.  state with highest spin multiplicity has lowest energy
⇒ Smax
2.  if more than 1 term with highest spin multiplicity then term with
highest L has lowest energy
⇒ Lmax
3.  for terms giving more than 1 level:
•  lowest J has lowest energy if outermost subshell is less than half
full
⇒ Jmin if q < N/2
•  highest J has lowest energy if outermost subshell is more than
half full
⇒ Jmax if q > N/2
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Examples of application of Hund’s rules
For
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examp
• 
• 
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Carbon atom: 1s22s22p2
– 
terms: 1S 3P0, 1, 2 1D
– 
rule 1 selects 3P?
– 
rule 3 selects 3P0
Oxygen atom: 1s22s22p4
– 
same terms: 1S 3P0, 1, 2 1D
– 
rule 1 selects 3P?
– 
rule 3 selects 3P2
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Examples of application of Hund’s rules
For
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examp
Note, this is excited
state. Hund’s rules
usually (but not always)
work in such cases.
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(1983), (1985)
Application of Hund’s rules
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Atom
Terms
Configuration
H (Z=1) 2S
1s1 2S1/2 He (Z=2) 1S
1s2 1S0 Li (Z=3) 2S
1s2 2s1 2S1/2 Be (Z=4) 1S
1s2 2s2 1S0 B (Z=5) 2P
1s2 2s2 2p1 2P1/2 C (Z=6) 1D, 3P, 1S
1s2 2s2 2p2 3P0 N (Z=7) 4S, 2D, 2P
1s2 2s2 2p3 4S3/2 O (Z=8) 1D, 3P, 1S
1s2 2s2 2p4 3P2 Lecture 8
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j-j coupling
• 
• 
• 
• 
For high Z (and high n at lower Z) the spin-orbit coupling effects
for individual electrons become large
L-S coupling no longer valid, instead j-j coupling:
–  j1 = l1 + s1
for electron 1
–  j2 = l2 + s2
for electron 2 ….
… and finally J = Σ ji
Energy levels different, transition rules different, notation different!
j-j coupling example:
–  6s6p configuration l1= 0 s1 = ½ ⇒ j1 = ½; l2= 1 s2 = ½ ⇒ j2 = 1/2,
3/2 –  notation [j1 j2]J ⇒ [1/2 3/2]2, [1/2 3/2]1, [1/2 1/2]1, [1/2 1/2]0
–  if j-j coupling transition 6s6p → 6s2: selection rule ΔJ = 0, ±1
(not 0 → 0)
–  if L-S coupling: ΔS=0 ΔL=0, ±1 ΔJ=0, ±1 (not 0 → 0)
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Reading
•  Softley Chapter 5, sections 5.1 - 5.3
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