Filling of Electronic States - usual filling sequence: - consider ...

Filling of Electronic States
- usual filling sequence:
1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 2
4d 10 5p 6 6s 2 4f 14 5d 10 6p 6 7s 2 6d 10 5f 14
- consider potassium (K): the 4s shell is filled
before the 3d shell because of the stronger
binding of the 4s electrons
- the transition elements have electrons in the
3d or 4d subshells respectively
- the lanthanides (rare earths) have identical
5s 2 5p 6 6s 2 configurations and differ by the
number of electrons in the in the 4f shell
- the actinides have identical 6s 2 6p 6 7s 2 configurations and differ only in the 5f and 6d
electrons
- for both lanthanides and actinides the chemical properties are dominated by the outer s and
p electrons and thus result in very similar properties for all of them
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Hund's Rule
When filling electrons in any subshell spins first remain unpaired in every quantum state
and have parallel spin.
examples: Z configuration spin of p electrons
Boron 5 1s 2 2s 2 2p 1
Carbon 6 1s 2 2s 2 2p 2
Nitrogen 7 1s 2 2s 2 2p 3
Oxygen 8 1s 2 2s 2 2p 4
Fluorine 9 1s 2 2s 2 2p 5
Neon 10 1s 2 2s 2 2p 6
note:
- The Coulomb repulsion between electrons gives rise to Hund's rule. Electrons with the same
spin in the same subshell must have different magnetic quantum numbers m l which have
different spatial distributions minimizing the overlap and thus the Coulomb energy.
- ferromagnetism of the 3d metals iron (Fe 26 ), cobalt (Co 27 ) and nickel (Ni 28 ) is a
consequence of Hund's rule
Fe 26 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2
3d 6 phys4.10 Page 2

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Spin Orbit Coupling
The interaction between the spin magnetic moment of an electron with its orbital angular
momentum is called spin-orbit coupling.
- this effect gives rise to fine structure doubling of spectral lines
classical model:
- consider electron moving around
nucleus with charge +Ze
- in the reference frame of the electron
the positively charged nucleus
generates a magnetic field B
- the spin magnetic moment μ B of the electron interacts with that field giving rise to a
potential energy of the electron spin
- every electron state with orbital angular momentum will be split into two substates
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Spin orbit coupling in hydrogen
- nuclear field
- mean radius
- orbit frequency
- for an electron in a 2p state
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Total Angular Momentum
Each electron in an atom contributes with its orbital angular momentum L and its spin
angular momentum S to the total angular momentum J of the atom.
- consider the simple case of atoms with a single electron in the outer shell, i.e. all group I
elements (H, Li, Na, K, ... ) and ions such as He + , Be + , Mg + , B 2+ , Al 2+ etc.
- in these atoms the total angular momentum is given by
- with the magnitude
- and z-component
- the simultaneous quantization of L, S and J allows only for certain relative orientations to
be allowed
- for one electron atoms there are only two relative positions possible
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Relative orientation of L and S in one electron
atoms
- two possibilities:
quantization of total
angular momentum J for
the two possibilities of
coupling between L and S
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Precession of angular moments
without external field:
- the magnitude and direction of J is conserved
- L and S precess around J
with external magnetic field B:
- J precesses around B
- J z is quantized
- L and S precess around J
- this gives rise to the anomalous Zeeman
effect
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LS coupling
- in many electron atoms the total angular momentum J is the vector sum of the individual
orbital angular moments L i and spin angular moments S i
in most of the many electron atoms
- the individual orbital angular moments L i are observed to couple to a resultant orbital
angular moment L
- the individual spin angular moments S i couple to the resultant spin angular
momentum S
- the moments then interact via the spin orbit coupling to form the total angular
momentum J
the angular momentum magnitudes L, S, J and their z-components L z , S z , J z are quantized
in the usual way with their respective quantum numbers
phys4.10 Page 9
quantization rules:
note that the resultant angular
moment quantum numbers are
indicated by capital letters,
whereas the individual electron
quantum numbers are not
capitalized
note: - L and M L quantum numbers are always integer or 0
- all other quantum numbers are half integer for an odd number of
electrons
- and integer or 0 for an even number of electrons
- for L > S the total angular momentum J can have 2S + 1 different values
- for L < S the total angular momentum J can have 2L + 1 different values
phys4.10 Page 10

Hyperfine structure
- atomic nuclei also have intrinsic angular moments and magnetic moments
- these also contribute to the total angular moments
- the angular moments are small and thus make only small (but observable and also useful)
contributions to the total energy of electrons
- the three sources of angular momentum in an atom
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Term Symbols
- as for single electron states symbols are used to indicate the total orbital angular
momentum of a many electron atom
- in this case capitalized letters are used
- a superscript number before the letter indicates the multiplicity of the state, i.e. the number
of possibilities to combine the total L and total S to a total angular momentum J
- for L > S the total momentum ranges from L - S to L + S, i.e. the multiplicity is 2 S + 1
- in the situation S = 0 the multiplicity is 1, hence there is only one state with J = L that is
called a singlet
- for S = 1/2 there is a doublet state (L+ 1/2, L - 1/2)
- for S = 1 there is a triplet state (L+ 1, L, L - 1)
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- for S > L the multiplicity is given by 2 L + 1
- the total angular momentum of the state realized is then indicated by a subscript after
the letter
- example: 'doublet P three halves'
- in atoms with a single outer shell electron the principal quantum number is used as a
prefix
- e.g. in sodium (Na)
phys4.10 Page 13
Find the different possible values of J for two electrons in with orbital quantum numbers
l 1 = 1 and l 2 = 2 and spin quantum numbers s 1,2 = ± 1/2.
- three different possibilities to form L = |l 1 - l 2 | … |l 1 + l 2 | = 1, 2, 3
- two possibilities to form S = |s 1 - s 2 | … |s 1 + s 2 | = 0, 1
- orbital and spin angular
moments
- five possibilities to form J = |L - S| … |L + S| = 0, 1, 2, 3, 4
phys4.10 Page 14

Hydrogen (H, Z = 1) Spectrum:
- states labeled by n and l
- selection rule: Δl = ± 1
detailed structure of n = 2 and
n = 3 states:
- splitting of H α line into 7 sub
lines
- states with same n and different
j have different energies
- also states with same n and j but
different l have different energies
Lamb shift (discovered 1947)
between 2 2 S 1/2 and 2 2 P 1/2
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Sodium (Na, Z = 11) Spectrum:
- electron configuration 1s 2 2s 2 2p 6 3s 1
- single outer s electron in potential of nucleus
screened by closed inner shells
- spectrum similar to that of Hydrogen for large l
states (3d, 4f, 5f, 6f)
- lower angular momentum states have higher
probability densities close to the nucleus, resulting
in stronger binding and thus lower energies
single electron atom
radial electron
probability densities
P(r) dr in dependence
on angular quantum
number l
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Helium (He, Z = 2 ) spectrum:
selection rules:
- 2 electrons in the 1s ground state
- LS coupling is dominant
- in single electron transitions ΔL = 0 is
forbidden and Δl = ± 1 = ΔL is required
- for initial states J = 0, J has to change by
±1
- no radiative spin transitions between para
and ortho Helium (ΔS = 0), only
transitions by collisions
- no n = 1 triplet due to exclusion principle
- large binding energies
- largest ionization energy of any element
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Mercury (Hg, Z = 80) spectrum:
- 2 outer shell electrons, 78 closed subshell inner electrons
- spectrum shown for one of the two electrons in lowest energy state
- singlet (S=0) and triplet (S=1)
states of outer shell electrons are
observed
- violation of ΔS = 0 selection rule, e.g.
in 3 P 1 -> 1 S 0 transitions
- violation of ΔS = 0 selection rule and
ΔJ = 0 rules in 3 P 2 -> 1 S 0 and 3 P 0 ->
1
S 0 transitions
- break-down of LS coupling is
observable
- in high Z atoms JJ coupling
dominates
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Experiment Mercury (Hg) Spectrum:
* yellow (579.1 nm) visible line 6 1 D 2 -> 6 1 P 1
** green (546.1 nm) visible line 7 3 S 1 -> 6 3 P 0
*** blue (435.8 nm) visible line 7 3 S 1 -> 6 3 P 1
**** UV line(253.7 nm) visible line 6 3 P 1 -> 6 1 S 0
phys4.10 Page 19