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Lecture 18: Polyelectronic
Atoms
• Reading: Zumdahl 12.10-12.13
• Outline:
– Spin
– The Aufbau Principle
– Filling up orbitals and the Periodic Table

H-atom wavefunctions
• The Coulombic potential can be generalized:
e-
Ze 2
V (r) 
r
r
Z P+
• Z = atomic number (= 1 for hydrogen)
H-atom wavefunctions
• If we solve the Schrodinger equation using this
potential, we find that the energy levels are
quantized:
2 

Z 2  me4 
Z
E n   2  2 2  2.178x1018 J 2 
n 80 h 
n 
• n is the principle quantum number, and ranges
 from 1 to infinity.
Orbitals
Quantum Numbers and Orbitals
n
l
Orbital
1
2
0
0
1
0
1
2
1s
2s
2p
3s
3p
3d
3
ml
0
0
-1, 0, 1
0
-1, 0, 1
-2, -1, 0, 1, 2
# of Orb.
1
1
3
1
3
5
Spin
• Further experiments
demonstrated the need
for one more quantum
number.
• Specifically, some
particles (electrons in
particular)
demonstrated inherent
angular momentum.
Spin (cont.)
ms = 1/2
ms = -1/2
• The new quantum
number is ms
(analagous to ml).
• For the electron, ms
has two values:
+1/2 and -1/2
The Aufbau Principal
• For polyelectronic atoms, a direct solution of the
Schrodinger Eq. is not possible.
• When we construct polyelectronic atoms, we use
the hydrogen-atom orbital nomenclature to discuss
in which orbitals the electrons reside.
• This is an approximation (and it is surprising how
well it actually works).
The Aufbau Principal (cont.)
• When placing electrons into orbitals in the
construction of polyelectronic atoms, we use the
Aufbau Principle.
• This principle states that in addition to adding
protons and neutrons to the nucleus, one simply
adds electrons to the hydrogen-like atomic orbitals
• Pauli exclusion principle: No two electrons may
have the same quantum numbers. Therefore, only
two electrons can reside in an orbital
(differentiated by ms).
The Aufbau Principal (cont.)
• Finally, orbitals are filled starting from the lowest
energy.
• Example: Hydrogen
1s1
1s
2s
2p
• Example: Helium (Z = 2)
1s2
1s
2s
2p
The Aufbau Principal (cont.)
• Orbital configurations …
The Aufbau Principal (cont.)
• Lithium (Z = 3)
1s22s1
1s
2s
• Berillium (Z = 4)
2p
1s22s2
1s
2s
2p
• Boron (Z = 5)
1s22s22p1
1s
2s
2p
The Aufbau Principal (cont.)
• Carbon (Z = 6)
1s22s22p2
1s
2s
2p
Hund’s Rule: Lowest energy configuration is
the one in which the maximum number of unpaired electrons
are distributed amongst a set of degenerate orbitals.
• Nitrogen (Z = 7)
1s22s22p3
1s
2s
2p
The Aufbau Principal (cont.)
• Oxygen (Z = 8)
1s22s22p4
1s
• Fluorine (Z = 9)
2s
2p
1s22s22p5
1s
2s
2p
• Neon (Z = 10)
1s22s22p6
1s
2s
2p
full
The Aufbau Principal (cont.)
• Sodium (Z = 11)
1s22s22p63s1
Ne
[Ne]3s1
3s
• Argon (Z = 18)
[Ne] 3s23p6
Ne
3s
3p
The Aufbau Principal (cont.)
• We now have the orbital configurations for the first 18 elements.
• Elements in same column have the same # of valence electrons!
The Aufbau Principal (cont.)
The Aufbau Principal (cont.)
• What is the radial distribution for different orbitals?
The Aufbau Principal (cont.)
• Similar to Sodium, we begin the next row of the periodic
table by adding electrons to the 4s orbital.
• Why not 3d before 4s?
• 3d is closer to the nucleus
• 4s allows for closer
approach; therefore, is
energetically preferred.
The Aufbau Principal (cont.)
• Elements Z=19 and Z= 20:
Z= 19, Potassium: 1s22s22p63s23p64s1 = [Ar]4s1
Z= 20, Calcuim: 1s22s22p63s23p64s2 = [Ar]4s2
• Elements Z=21to Z=30 have occupied d orbitals:
Z= 21, Scandium: 1s22s22p63s23p64s23d1 = [Ar] 4s23d1
Z = 24, Chromium: [Ar] 4s13d5
exception
Z= 30, Zinc: 1s22s22p63s23p64s23d10 = [Ar] 4s23d10
The Aufbau Principal (cont.)
• This orbital filling scheme gives rise to the modern
periodic table.
The Aufbau Principal (cont.)
• After Lanthanum ([Xe]6s25d1), we start filling 4f.
The Aufbau Principal (cont.)
• After Actinium ([Rn]7s26d1), we start filling 5f.
The Aufbau Principal (cont.)
• Heading on column given total number of valence
electrons.
The Aufbau Principal (cont.)
Summary
• Electrons go into hydrogen-like orbitals to
construct polyelectronic atoms.