Properties of Atomic
Orbitals and Intro to
Molecular Orbital Theory
Chemistry 754
Solid State Chemistry
Dr. Patrick Woodward
Lecture #15
Atomic Orbitals
• Four quantum numbers define the properties of
each atomic orbital
–
–
–
–
Principle quantum number, n = 1,2,3,…
Azimuthal quantum number, l = 0(s), 1(p), 2(d), 3(f), n-1
Magnetic quantum number, ml = - l,, …, l (i.e
.e px, py, pz)
Spin quantum number, ms = +1/2 or -1/2
• Pauli Exclusion Principle
– No two electrons can have the same set of quantum
numbers (each orbital can hold 2 e-)
• Hund’s Rule
– For degenerate orbitals the lowest energy configuration
maximizes the electron spin (no pairing of electrons if
avoidable)
1
Radial Nodes
For a given atomic orbital there are n-
l -1 radial nodes.
p and d-orbitals: Nodal planes
For a given atomic orbital there are
l nodal planes.
2
Relative Sizes of Atomic Orbitals
Atom
Radius s
Radius p
C
64 pm
65 pm
Si
115 pm
95 pm
Ge
120 pm
96 pm
Sn
140 pm
114 pm
Pb
148 pm
121 pm
Values correspond to radial expectation
values based on HartreeHartree-Fock calculations,
taken from
http://www.webelements
.com/webelements
webelements
http://www.webelements.com/
Relative Sizes of Atomic Orbitals
Atom
Radius s
Radius d
Ti
162 pm
53 pm
Zr
179 pm
84 pm
Hf
178 pm
88 pm
Cu
137 pm
33 pm
Ag
153 pm
55 pm
Au
156 pm
64 pm
Values correspond to radial expectation
values based on HartreeHartree-Fock calculations,
taken from
http://www.webelements
.com/webelements
webelements
http://www.webelements.com/
3
Relative Sizes of Atomic Orbitals
Atom
Radius s
Radius d
Nd
222 pm
35 pm
Yb
199 pm
25 pm
U
226 pm
53 pm
No
215 pm
40 pm
Values correspond to radial expectation
values based on HartreeHartree-Fock calculations,
taken from
http://www.webelements
.com/webelements
webelements
http://www.webelements.com/
Periodic Trends
• As you move either up a group (i.e. Pb → C) or
left to right across a period (i.e. K → Kr).
– The effective nuclear charge increases.
– The orbitals become more contracted (smaller).
– The valence electrons become more tightly bound to the
nucleus (1st ionization energy increases).
– The electronegativity increases.
• Relative Orbital Sizes
– The spatial extent (size) of the valence shell orbitals
decrease from (n)s (largest) → (n)p → (n-1)d → (n-2)f
[on a given atom].
• Lanthanide Contraction
– Due to the ineffective shielding of the 4f electrons
there is only a very small increase in the spatial extent
of the orbitals as you go from a 2nd row transition metal
(i.e. Nb)
Nb) to a 3rd row transition metal (i.e. Ta)
4
Special Properties associated with
filling a subshell for the first time
• Filling the 2p orbitals (B → Ne)
Ne)
– Short internuclear separation allows for effective p overlap,
allows for strong π bonding to occur
– s and p orbitals have similar spatial extent which favors
bonding geometries well suited for both orbitals
(hybridization)
• Filling the 3d orbitals (Sc → Zn)
– Overlap of 3d orbitals with ligand orbitals is not large, gives
rise to high spin configurations and Curie-Weiss magnetism
• Filling the 4f orbitals (La → Lu)
– The spatial extent of the 4f orbitals leads to negligible
overlap with other orbitals.
orbitals. Thus the chemistry of the
lanthanides is fairly constant, and the magnetism of these
elements in compounds is similar to that of a free ion.
Relativistic Effects
• What are relativistic effects?
– For very heavy nuclei the electron velocities approach
the speed of light, which causes them to get heavier
according to the Theory of Relativity. The increased
mass leads to a contraction of the orbital, which in turn
lowers it’s energy.
• What elements are most affected by
relativistic effects?
– The heavier elements, particularly the 6th period and
beyond (Cs, Ba,
Ba, La, …).
• Are all orbitals affected equally?
– No. The s orbitals are most strongly affected, while the
p-orbitals
p-orbitals are affected to a lesser extent. Relativistic
effects have little direct influence on the d and f
orbitals.
orbitals.
5
Group 14 Orbital Energies
Element
C
Ep (eV)
non-Relat.
-5.42
Es (eV)
Ep (eV)
Es (eV)
non-Relat. Relativistic Relativistic
-13.63
-5.42
-13.64
Si
-4.17
-10.83
-4.16
-10.87
Ge
-4.08
-11.61
-4.05
-11.92
Sn
-3.93
-10.05
-3.87
-10.78
Pb
-3.86
-9.72
-3.70
-12.17
For Pb Relativistic Effects lower the energy of the
6s orbitals by 2.35 eV, while the 6p are only
stabilized by 0.16 eV!
Relativistic Effects - Implications
Standard Reduction
Potentials
Reaction
CO → CO2
‘SiO’ → SiO2
GeO → GeO2
Ge2+ → Ge4+
SnO → SnO2
Sn2+ → Sn4+
Pb2+ → Pb4+
Potential (V)
-0.11
-0.97
-0.37
0.00
-0.09
0.15
1.69
As you go from Ge → Sn → Pb
the +2 oxidation state becomes
more stable, as it requires more
energy to remove electrons from
this orbital. [CO is stabilized by
π-bonding.]
Structure & Properties of
the Group 14 Elements
Element
C
Si
Ge
α-Sn
β-Sn
Pb
Structure
Diamond
Diamond
Diamond
Diamond
Tetragonal
FCC
Eg(eV)
eV)
5.5
1.1
0.7
0.1
Metal
Metal
As you go proceed down the group
the tendency for the s-orbitals
s-orbitals
to become involved in bonding
diminishes. This destabilizes
tetrahedral coordination and
semiconducting/insulating
semiconducting/insulating
behavior.
6
Relativistic Effects: Bonding Implications
• Contraction of the s-orbitals
s-orbitals
– decreases their spatial overlap with other atoms
– decreases their contribution to bonding
• Energetic stabilization of the s-orbitals
s-orbitals
– Stabilizes the +2 oxidation state for the heavier
elements
• Combined effects
– Inert pair effect (distorted coordination environment
for ions which retain their valence s-electrons, due to
formation of an sp-hybrid). This effect is pronounced for
the following ions Tl+, Pb2+, Bi3+, Sn2+, Sb3+, Te4+.
– The tendency for “sp3 hybridization” is reduced, which
impacts coordination preferences
• Oxidation state = +4: tetrahedral → octahedral
• Oxidation state = 0: tetrahedral → close packed
Orbital Overlap:
Molecular Orbital (MO) Theory
Antibonding
Orbital, σ*
Bonding
Orbital, σ
7
Overlap of p-orbitals
MO Diagram for 2nd Row Diatomic
Molecule
8
Orbital Interactions: Key Points
• The overlap of two atomic orbitals is dependent upon:
–
–
–
–
symmetry of the orbitals
distance between the orbitals
spatial extent of the orbitals
the energy difference between orbitals
• Increasing the overlap (spatial and energetic) leads
to the following:
– Stabilization of the bonding MO
– Destabilization of the antibonding MO
– The antibonding MO is destabilized to a greater extent than the
bonding MO is stabilized
• The spatial overlap in a bond depends upon symmetry
– It decreases as the number of nodal planes increases, σ > π > δ
– π and particularly δ bonds are more sensitive to changes in bond
angle
9