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Chapter 23
The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
Thomas Engel, Philip Reid
Objectives
• Usefulness of H2+ as qualitative model in
chemical bonding.
• Understanding of molecular orbitals (MOs) in
terms of atomic orbitals (AOs),
• Discuss molecular orbital energy diagram.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Outline
1. The Simplest One-Electron Molecule
2. The Molecular Wave Function for Ground-State
3. The Energy Corresponding to the Molecular
Wave Functions
4. Closer Look at the Molecular Wave Functions
5. Combining Atomic Orbitals to form Molecular
Orbitals
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Outline
6. Molecular Orbitals for Homonuclear Diatomic
Molecules
7. The Electronic Structure of Many-Electron
Molecules
8. Bond Order, Bond Energy, and Bond Length
9. Heteronuclear Diatomic Molecules
10. The Molecular Electrostatic Potential
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.1 The Simplest One-Electron Molecule: H2+
•
•
Schrödinger equation cannot be solved exactly
for any molecule containing more than one
electron.
We approach H2+ using an approximate model,
thus the total energy operator has the form
2
2
2
2


h
h
e
1
1
e
1
2
2
2
ˆ


H 
 a  b 
e 
 

2m p
2me
40  ra rb  40 R


where 1st term = kinetic energy operator nuclei a and b
2nd term = electron kinetic energy
3rd term = attractive Coulombic interaction
4th term = nuclear–nuclear repulsion
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.1 The Simplest One-Electron Molecule: H2+
•
The quantities R, ra, and rb represent the
distances between the charged particles.
2
2
2
2


h
h
e
1
1
e
1
2
2
2
ˆ
   
H 
 a  b 
e 
2m p
2me
40  ra rb  40 R


Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.2 The Molecular Wave Function for Ground-State H2+
•
•
For chemical bonds the bond energy is a small
fraction of the total energy of the widely
separated electrons and nuclei.
An approximate molecular wave function for
H2+ is
  caH 1sa  cbH 1sb
where Ф = atomic orbital (AO)
ψ = molecular wave function
σ = molecular orbital (MO)
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.2 The Molecular Wave Function for Ground-State H2+
•
For two MOs from the two AOs,
 g  cg H 1s  H 1s
 u  cu H 1s  H 1s
a
a
b
b


where ψg = bonding orbitals wave functions
ψu = antibonding orbitals wave functions
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.3 The Energy Corresponding to the Molecular Wave
Functions ψg and ψu
•
The differences ΔEg and ΔEu between the
energy of the molecule is as follow:
E g  E g  H aa
 K  S ab J
K  S ab J

and Eu  Eu  H aa 
1  S ab
1  S ab
where J = Coulomb integral
K = resonance integral or the exchange integral
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.3 The Energy Corresponding to the Molecular
Wave Functions ψg and ψu
•
•
J represents the energy of interaction of the
electron viewed as a negative diffuse charge
cloud on atom a with the positively charged
nucleus b.
K plays a central role in the lowering of the
energy that leads to the formation of a bond.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 23.1
Show that the change in energy resulting from
bond formation, E  E  H and E  E  H
, can be
expressed in terms of J, K, and Sab as
g
E g  E g  H aa 
g
aa
u
u
aa
-K  S ab J
K  S ab J
and Eu  Eu  H aa 
1  S ab
1  S ab
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
Starting from
H aa  H ab
Eg 
1  S ab
we have
H aa  H ab
H aa  H ab  1  S ab H aa
H aa  S ab H aa
Eg 
 H aa 
 H aa 
1  S ab
1  S ab
1  S ab
E g  E g  H aa 
H ab  S ab H aa
1  S ab



e2 
e2
  K  S ab  E1s 
S ab  E1s 
 J 
40 R 
40 R


   K  S ab J
E g 
1  S ab
1  S ab
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
Thus
Eu 
H aa  H ab
H  H ab  1  S ab H aa
 H ab  S ab H aa
 H aa  aa
 H aa 
1  S ab
1  S ab
1  S ab



e2 
e2



 S ab  E1s 
 K  S ab  E1s 
 J 

40 R 
40 R
K  S ab J



Eu

1  S ab
1  S ab
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.4 A Closer Look at the Molecular Wave Functions ψg
and ψu
•
The values of ψg and ψu along the molecular
axis are shown.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.4 A Closer Look at the Molecular Wave Functions ψg
and ψu
•
The probability density of finding an electron at
various points along the molecular axis is given
by the square of the wave function.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.4 A Closer Look at the Molecular Wave Functions ψg
and ψu
•
•
Virial theorem applies to atoms or molecules
described either by exact wave functions or by
approximate wave functions.
This theorem states that
E potential  2 Ekinetic
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.5 Combining Atomic Orbitals to Form Molecular
Orbitals
•
•
Combining two localized atomic orbitals gave
rise to two delocalized molecular wave
functions, called molecular orbitals (MOs)
2 MOs with different energies:
 b  c1b1  c2b2
 a  c1a1  c2 a2
•
Secular equations has the expression of
H11  
H12  S12
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
H12  S12
0
H 22  
23.5 Combining Atomic Orbitals to Form Molecular
Orbitals
•
The two MO energies are given by
H11  H12
H11  H12
b 
and  b 
1  S12
1  S12
•
where ε1 = bonding MO
ε2 = antibonding MO
Molecular orbital
energy diagram:
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 23.2
Show that substituting  b  H11  H12 in
c1 H11     c2 H12  S12   0
c1 H12  S12   c2 H 22     0
1  S12
gives the result c1 = c2.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
We have



H11  H12 
H11  H12
  c2  H12 
c1  H11 
S12   0
1  S12 
1  S12



c1 H11S12  H12   c2 H11S12  H12   0
c1  c2
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.6 Molecular Orbitals for Homonuclear Diatomic
Molecules
•
It is useful to have a qualitative picture of the
shape and spatial extent of molecular orbitals
for diatomic molecules.
• All MOs for homonuclear diatomics can be
divided into two groups with regard to each of
two symmetry operations:
1. Rotation about the molecular axis
2. Inversion through the center of the molecule
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.6 Molecular Orbitals for Homonuclear Diatomic
Molecules
•
The MOs used to describe chemical bonding in
first and second row homonuclear diatomic
molecules are shown in table form.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.7 The Electronic Structure of Many-Electron
Molecules
•
The MO diagrams show the number and spin of
the electrons rather than the magnitude and
sign of the AO coefficients.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.7 The Electronic Structure of Many-Electron
Molecules
•
2 remarks about the interpretation of MO
energy diagrams:
1. Total energy of a many-electron molecule is not
the sum of the MO orbital energies.
2. Bonding and antibonding give information
about the relative signs of the AO coefficients in
the MO.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.8 Bond Order, Bond Energy, and Bond Length
•
For the series H2→Ne2, the relationship between
Bond Order, Bond Energy, and Bond Length is
shown.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.8 Bond Order, Bond Energy, and Bond Length
•
Bond order is defined as
•
For a given atomic radius, the bond length is
expected to vary inversely with the bond order.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Example 23.4
Arrange the following in terms of increasing bond
energy and bond length on the basis of their bond
order: N 2 , N 2 , N 2 and N 22
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
Solution
The ground-state configurations for these species
are
  2  2  1  3 
: 1  1  2  2  1  1  3 
: 1  1  2  2  3  1  1 
: 1  1  2  2  3  1  1  1  1 
N : 1 g  1

2
N2
N

2
N
2
2
2
* 2
u
2
g
* 2
u
2
g
2
g
* 2
u
* 2
u
* 2
u
2
g
* 2
u
2
g
* 2
u
2
g
2
g
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
* 2
u
2
1
u
g
2
2
u
2
u
g
2
* 1
g
2
g
u
2
g
2
u
2
u
* 1
g
* 1
g
Solution
In this series, the bond order is 2.5, 3, 2.5, and 2.
Therefore, the bond energy is predicted to follow the

2
N

N
,
N

N
order 2
2
2
2 using the bond order alone.
However, because of the extra electron in the
antibonding 1 g* MO, the bond energy in N -2 will be
less than that in N 2 . Because bond lengths decrease
as the bond strength increases, the bond length will
follow the opposite order.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.9 Heteronuclear Diatomic Molecules
•
The MOs on a heteronuclear diatomic molecule
are numbered differently for the order in energy
exhibited in the molecules Li2N2:
•
The MOs will still have either σ or π symmetry.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.9 Heteronuclear Diatomic Molecules
•
The symbol * is usually added to the MOs for
the heteronuclear molecule to indicate an antibonding MO.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.9 Heteronuclear Diatomic Molecules
•
The 3σ, 4σ and 1π MOs for HF are shown from
left to right.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.10 The Molecular Electrostatic Potential
•
•
The charge on an atom in a molecule is not a
quantum mechanical and atomic charges
cannot be assigned uniquely.
Molecular electrostatic potential (Фr) can
be calculated from molecular wave function and
has well-defined values in the region around a
molecule.
where q = point charge
r = distance from the charge
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd
23.10 The Molecular Electrostatic Potential
•
It is convenient to display a contour of constant
electron density around the molecule and the
values of the molecular electrostatic potential
on the density contour using a color scale.
Chapter 23: The Chemical Bond in Diatomic Molecules
Physical Chemistry 2nd Edition
© 2010 Pearson Education South Asia Pte Ltd