chap2 2 aufbau+principle
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Topics to be learned
(1) Aufbau
Principle ‐
Pauli exclusion principle
‐
Hund’s
rule‐
Effective nuclear charge, Slater’s rule
Aufbau
Principle
In the last lecture, we have learned(1) Schrödinger equation → to describe the electron wave function(2) Atomic wave functions‐
radial distribution function: 4πr2
R(r)2
‐
radial wave function, angular function (Ψn,l,ml = Rn,l (r)Yl,ml (θ,φ))‐
nodes for radial part and angular part
Bohr energyEn
= (‐13.6 /n2) eV Multielectron
Atoms
Wolfgang Pauli (1900‐1958), Austrian physicist who won the Physics Nobel prize for his idea of the exclusion principle
Fourth Quantum Number
Ex) 2pz (2,1,0,1/2) (2,1,0,-1/2)
No two electrons in an atom can have the same set of four quantum numbers (n, l, ml, ms)
For your information
Aufbau
principle (from the German
Aufbau meaning "building up, construction“):
electrons fill
the atomic orbitals
from the lowest energy up
Introduction to effective nuclear charge
2+
He, Z=2
For your information
Ionization Energy (IE): energy required to remove an electron from the gaseous atom
(just Z=2)
the effective nuclear chargeZeff
2 =24.6/13.6
Zeff = ~ 1.34
Estimation of Zeff
Zeff = Z - σ
provide numerical values for the effective nuclear charge
concept Slater's rules
1. The electrons are arranged in to a sequence of groups that keep s‐and p‐
orbitals
with the same principal quantum number
together[1s] [2s,2p] [3s,3p] [3d] [4s,4p] [4d] [4f] [5s, 5p] [5d] etc.2. If the electron in an (ns, np) group:a) Electrons in groups to the right of the (ns, np) group contribute
nothing to σ.b) Each of the other electrons in the (ns, np) group contributes 0.35
to σ. (i.e., the shielding is 35% effective)c) Each electron in the n‐1 shell contributes 0.85 to σ.d) Each electron in the n‐2 shell or below contribute 1.00 to σ3. If the electron in an (nd) or (nf) group:a) Electrons in groups to the right of the (nd) or (nf) group
contribute nothing to σ.b) Each of the other electrons in the (nd) or (nf) group contributes
0.35 to σ. c) Each electron in groups to the left of the (nd) or (nf) group
contributes 1.00 to σ
provide numerical values for the effective nuclear charge
concept Slater's rules
Group Other electrons in the same group
Electrons in group(s) with principal quantum number n-1
Electrons in all group(s) with principal quantum number < n-1
[1s] 0.3 N/A N/A
[ns, np] 0.35 0.85 1
[nd] or [nf] 0.35 1 1
(1s2) (2s22p6) (3s23p6) (3d6) (4s2)For Fe, Z=26
For a 3 d electron and a 4 s electron, estimate effective nuclear charge
3.454.553.45
Calculate Zeff
?* recall, RDF: radial Distribution Function
Homework!
Atom Zeff [1s] Zeff [2s] Zeff [2p]H 1.0He 1.7Li 2.7 1.3Be 3.7 1.95B 4.7 2.6 2.6C 5.7 3.25 3.25N 6.7 3.9 3.9O 7.7 4.55 4.55F 8.7 5.2 5.2
Ne 9.7 5.85 5.85
Hund’s
rule: For any set of orbitals
of equal energy (degenerated orbitals), the electron configuration with the maximum number of parallel spin is the ground state
Hund’s
rule
For instance, px , py , pz are degenerated orbitals
: how to fill electrons in degenerated orbitals
Coulombic
energy of repulsion: Πc
(+ energy)Exchange energy: Πe
(‐
energy)
EX on page 37
Total paring energy Π
= Πc
+ Πe
* For transition metal elements with 4s and 3d, this breaks down because the paring energy of 4s and 3d are almost the same
(Z=19)
(Z=20)
E(3d8) = 131.84 eV, E(3d64s2) = 92.16 eV (see next page)
3Πc +13Πe
2Πc +10Πe → hard to distinguish
Stronger shielding for 4s, smaller effective Z:4s electron will be first removedGround state for Sc+ : [Ar]3d14s1
Fe Fe2+
4s3d
4s3d
Please see the figure 2‐12 in p 43
Removal of electrons reduce the overall electron repulsion
Table 2‐7, p38, * elements
For your information
What we have learned
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