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Slide 1 of 60
Chapter 3 (continued)
Covalent Bonding: Orbitals
Valence bond theory
Molecular orbital theory
Slide 2 of 60
Bonding In H2
21
2
21
2
21
221 zyx ∂
∂+
∂∂
+∂∂
=∇
ψψ E=Η̂
abbaba RrrrrrU
111111
221112
+−−−−=
( )22211121
212
2
,,,,,)(8
ˆ zyxzyxUm
h+∇+∇−=Η
π
Slide 3 of 60
Atomic Orbital Overlap
• Valence Bond (VB) Theory states that a covalent bond is formed by the pairing of two electrons with opposing spins in the region of overlap of atomic orbitals between two atoms. This overlap region has a high electron charge density.
• The valence bond theory attempts to find the best approximation of optimal orbital overlap for all the bonds in a molecule.
Slide 4 of 60
0.7433.78+ effective nuclear charge
0.744.747Empirical
0.7494.02
0.8693.14
0.90.25
R (? )E (eV)wavefunction
)2()1( ba ϕϕψ =
)1()2()2()1( baba ϕϕϕϕψ +=
)1()2()2()1( baba ϕϕϕϕψ +=
)]1()2()2()1([
)1()2()2()1(
bbaa
baba
ϕϕϕϕλϕϕϕϕψ
+++=
Theoretical treatment
ionic
covalent
Slide 5 of 60
Bonding In H2S
Slide 6 of 60
Several Important Points
• Most of the electrons in a molecule remain in the same orbital locations that they occupied in the separated atoms.
• Bonding electrons are localized in the region of atomic orbital overlap.
• For orbitals with directional lobes, maximum overlap occurs when atomic orbitals overlap end to end.
• The molecular geometry depends on the geometric relationships among the atomic orbitals of the central atom that participate in bonding.
Slide 7 of 60
sp3 Hybridization Scheme
Hybridization
109.50
Slide 8 of 60
Bonding in Methane
Slide 9 of 60
Bonding in Ammonia
Slide 10 of 60
The sp2 Hybridization Scheme
Slide 11 of 60
The sp Hybridization Scheme
Slide 12 of 60
Hybrid Orbitals Involvingd Subshells
• This hybridization allows for expanded valence shellcompounds.
• A 3s electron can be promoted to a 3d subshell which gives rise to a set of five sp3d hybrid orbitals. These molecules have a trigonal bipyramidal molecular geometry.
• One 3s electron and one 3p electron can be promoted to two 3d subshells which gives rise to a set of six sp3d2
hybrid orbitals. These molecules have an octahedralmolecular geometry.
Slide 13 of 60
The sp3d and sp3d2 Hybrid Orbitals
sp3d
Slide 14 of 60
Hybrid Orbitals and TheirGeometric Orientations
Slide 15 of 60
Hybrid Orbitals andMultiple Covalent Bonds
• Covalent bonds formed by the end-to-end overlap of orbitals, regardless of orbital type, are called sigma (σ) bonds. All single bonds are sigma bonds.No nodal plane along inter-nuclear axis
• A bond formed by parallel, or side-by-side, orbital overlap is called a pi (π) bond.One nodal plane along inter-nuclear axis
sp2-hybrid
Slide 16 of 60
Descriptions of Ethylene
Rotational barrier for double bond
Slide 17 of 60
Valence Bond Theory of the Bonding in Acetylene
sp-hybrid
Slide 18 of 60
Characteristics ofMolecular Orbitals
• Molecular orbitals (MOs) are mathematical equations that describe the regions in a molecule where there is a high probability of finding electrons.
• Bonding molecular orbitals (σ, π) are at a lower energy level than the separate atomic orbitals and have a high electron probability, or electron charge density.
• Antibonding molecular orbitals (σ*, π*) are at a higher energy level than the separate atomic orbitalsand places a high electron probability away from the region between the bonded atoms.
Slide 19 of 60
The 1s Orbital
Ψ (r,θ,φ) = R(r) Υ0,0(θ,φ)
Υ0,0(θ,φ) = 1/2π1/2
Slide 20 of 60
+ +
Constructive Interference
ba ϕϕψ +=
ψ
2ψ
Slide 21 of 60
+
-
Destructive Interference
ba ϕϕψ −=
ψ
2ψ
Slide 22 of 60
Molecular Orbitals and Bondingin the H2 Molecule
++
+ -
)2()1()2()1()2()1()2()1(
)]2()2()][1()1([
)2()1(
baabbbaa
baba
total
ϕϕϕϕϕϕϕϕϕϕϕϕ
ψψ
+++=++=
=Ψ
ionic covalent
Slide 23 of 60
Bond order = ½ ( # of bonding electrons - # of anti-bonding electrons )
Electron configuration of H2 : (σ1s)2
B.O. of H2 = ½ (2 - 0) = 1
Bond energy = 435 kJ/mol
Bond length = 74 pm
ab
ab
SH
E−
=∆1
ab
ab
SH
E+
=∆1
Slide 24 of 60
H2-H2
+
He2
Slide 25 of 60
--0(σ1s)2(σ1s*)2He2
108238½(σ1s)2(σ1s*)1H2-
106269½(σ1s)1H2+
744351(σ1s)2H2
Bond length (pm)
Bond energy (kJ/mol)
B.O.Electron configuration
Species
Slide 26 of 60
Lewis Structure
Hetero-nuclear Diatomic Molecule
Slide 27 of 60
Slide 28 of 60
Electron configuration of Li2 : KK(σ1s)2
B.O. of Li2 = ½ (2 - 0) = 1
Bond length = 267 pm
2nd Period Homo-nuclear Diatomic Molecules
Slide 29 of 60
Molecular Orbitals Formed byCombining 2p Atomic Orbitals
++ +
____
_ + +__
1 node ⊥ inter-nuclear axis
+ +_ _
+_
1 node along inter-nuclear axis
_ +
+__
Slide 30 of 60
Diamagnetic Paramagnetic
Slide 31 of 60
Molecular Spectroscopy
UV-Vis
IRMicrowave
Slide 32 of 60
KE = h? - BE
The Photoelectric Effect
Slide 33 of 60
• X-ray Photoelectron Spectroscopy (XPS) - using soft x-ray (200-2000 eV) radiation to
examine core-levels.
• Ultraviolet Photoelectron Spectroscopy (UPS) - using vacuum UV (10-45 eV) radiation to examine
valence levels.
Photoelectron spectroscopy- a single photon in/ electron out process
Slide 34 of 60
Light sources: a Helium lamp emitting at 21.2 eV (He I radiation)or 40.8 eV (He II radiation)
Ultraviolet Photoelectron Spectroscopy(UPS)
Koopmans’Theorem
For a closed-shell molecule, the ionization energy of an electron in a particular orbital is approximately equal to the orbital energy.
I.E. = Eorbital = B.E.
such as H2 ? H2+ + e-
.... ,....
EKhvEIorEIhvEK−=
−=
Slide 35 of 60
Vibrational Fine Structure in PES
Slide 36 of 60
Vibrational energy states
)v'(....
)21
v()21
v()v( 2
+∆+=−⇒
+−+=
vib
eee
EEIEKhv
xhvhvE
where ΔEvib+(v’) = E+(v = v’) - E+(v = 0) is the extra (i.e., energy above
v = 0) vibrational energy of the ion.Lines corresponding to different vibrational energy levels of H2
+ , v = 0, 1, 2, 3, . . .,
Slide 37 of 60
Figure 4.4: He(I) UPS spectrum of HCl gas.
1. Loss of a bonding electron decreases the bond order, increasing the bond length in the resulting cation compared to the parent molecule.
2. Loss of a nonbonding electron has no effect on bond order or bond length.
3. Loss of an antibonding electron increases the bond order, decreasing the bond length of the cation compared to the parent molecule.
Slide 38 of 60
Slide 39 of 60
ΔN = N+ – J”where N+ and J”represent the rotational quantum numbers of the ion and the neutral, respectively.
Slide 40 of 60
Vibrational frequencies from UPS spectra of CO and N2
1706CO+ (B)
1549CO+ (A)
2200 ? weakly antibondingCO+ (X)2157COυ (cm-1)molecule
1936N2+ (A)
2331N2
Slide 41 of 60
Paramagnetic
Slide 42 of 60
Paramagnetism of Oxygen
Slide 43 of 60
Molecular Orbitals of Homo-nuclear Diatomic Molecules of 2nd Period
Slide 44 of 60
Bonding in Benzene
The σ-Bonding Framework
sp2
Slide 45 of 60
The π-Molecular Orbitals of Benzene
E
+
++ _
_
node
node
π-M.O. of benzene
Slide 46 of 60
3 nodes
2 nodes
Slide 47 of 60
Slide 48 of 60
Conjugated Double Bonds
E
π-M.O.
Bonding
Anti-bonding
Slide 49 of 60
Band Theory
• This is a quantum-mechanical treatment of bonding in metals.
• The spacing between energy levels is so minute in metals that the levels essentially merge into a band.
• When the band is occupied by valence electrons, it is called a valence band.
• A partially filled or low lying empty band of energy levels, which is required for electrical conductivity, is a conduction band.
• Band theory provides a good explanation of metallic luster and metallic colors.
Slide 50 of 60
bonding
Anti-bonding
Slide 51 of 60
The 2s Band in Lithium Metal
Bonding
Anti-bonding
e- e-Valence band
Conduction band
Slide 52 of 60
Band Overlap in Magnesium
Valence band
Conduction band
Slide 53 of 60
Band Structure of Insulatorsand Semiconductors
Slide 54 of 60
Density of states in (a) metal, (b) semimetal (e.g. graphite).
(a) (b)
Density of state= dn/dE
n = number of states
Slide 55 of 60
Conductivity of Graphite
insulator
e- -conductor
Slide 56 of 60
Fermi distribution (a) at T= 0, and (b) at T> 0. The population decays exponentially at energies well above the Fermi level.
Population,1
1/)( +
= − kTEeP µ
Fermi level - the highest occupied orbital at T= 0
(a) (b)
where, µ = chemical potential
When E= µ, P= 1/2
Slide 57 of 60
Fermi distribution at T> 0 for (a) Intrinsic semiconductor, (b)Fermi distribution and the band gap
(a) (b)population
Slide 58 of 60
Extrinsic semiconductor: (a) n-type, e.g. P doped Si(b) p-type, e.g. Ga doped Si.
Slide 59 of 60
Conductivity (σ) & Energy Gap (Eg)
kT
Ee
g
kTE g
2ln ln 0
2/0
−=
= −
σσ
σσ
Slide 60 of 60