physical chemistry 2 nd edition

Physical Chemistry 2Physical Chemistry 2ndnd Edition EditionThomas Engel, Philip Reid

Chapter 23 Chapter 23 The Chemical Bond in Diatomic Molecules

© 2010 Pearson Education South Asia Pte Ltd

Physical Chemistry 2nd EditionChapter 23: The Chemical Bond in Diatomic Molecules

ObjectivesObjectives

• 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.



OutlineOutline

1. The Simplest One-Electron Molecule2. The Molecular Wave Function for Ground-

State3. The Energy Corresponding to the

Molecular Wave Functions4. Closer Look at the Molecular Wave

Functions5. Combining Atomic Orbitals to form

Molecular Orbitals



OutlineOutline

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 Molecules10.The Molecular Electrostatic Potential



23.1 23.1 The Simplest One-Electron Molecule: HThe Simplest One-Electron Molecule: H22++

• 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

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

R

e

rr

e

m

h

m

hH

bae

eba

p

1

4

11

422ˆ

0

2

0

22

222

2



23.1 23.1 The Simplest One-Electron Molecule: HThe Simplest One-Electron Molecule: H22++

• The quantities R, ra, and rb represent the distances between the charged particles.

R

e

rr

e

m

h

m

hH

bae

eba

p

1

4

11

422ˆ

0

2

0

22

222

2



23.2 23.2 The Molecular Wave Function for Ground-The Molecular Wave Function for Ground-State HState H22

++

• 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

where Ф = atomic orbital (AO) ψ = molecular wave function σ = molecular orbital (MO)

ba sHbsHa cc 11



23.2 23.2 The Molecular Wave Function for Ground-The Molecular Wave Function for Ground-State HState H22

++

• For two MOs from the two AOs,

where ψg = bonding orbitals wave functions ψu = antibonding orbitals wave functions

ba

ba

sHsHuu

sHsHgg

c

c

11

11



23.3 23.3 The Energy Corresponding to the Molecular The Energy Corresponding to the Molecular Wave Functions Wave Functions ψψgg and and ψψuu

• The differences ΔEg and ΔEu between the energy of the molecule is as follow:

where J = Coulomb integralK = resonance integral or the

exchange integral

ab

abaauu

ab

abaagg S

JSKHEE

S

JSKHEE

1

and 1



23.3 23.3 The Energy Corresponding to the The Energy Corresponding to the Molecular Wave Functions Molecular Wave Functions ψψgg and and ψψuu

• 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.



Example 23.1Example 23.1

Show that the change in energy resulting from bond formation, , can be expressed in terms of J, K, and Sab as

aauuaagg HEEHEE and

1

and 1

ab

abaauu

ab

abaagg S

JSKHEE

S

JS-KHEE



SolutionSolution

Starting from

we haveab

abaag S

HHE

1

ab

ab

ab

sabsab

g

ab

aaababaagg

ab

aaabaaaa

ab

aaababaaaa

ab

abaag

S

JSK

S

JR

eESK

Re

ES

E

S

HSHHEE

S

HSHH

S

HSHHH

S

HHE

11

44

1

11

1

1

0

2

10

2

1



SolutionSolution

Thus

ab

ab

ab

sabsab

u

ab

aaababaa

ab

aaababaaaa

ab

abaau

S

JSK

S

JR

eESK

Re

ES

E

S

HSHH

S

HSHHH

S

HHE

11

44

11

1

1

0

2

10

2

1



23.4 23.4 A Closer Look at the Molecular Wave A Closer Look at the Molecular Wave Functions Functions ψψgg and and ψψuu

• The values of ψg and ψu along the molecular axis are shown.




• The probability density of finding an electron at various points along the molecular axis is given by the square of the wave function.




• Virial theorem applies to atoms or molecules described either by exact wave functions or by approximate wave functions.

• This theorem states thatkineticpotential EE 2



23.5 23.5 Combining Atomic Orbitals to Form Combining Atomic Orbitals to Form Molecular OrbitalsMolecular Orbitals

• Combining two localized atomic orbitals gave rise to two delocalized molecular wave functions, called molecular orbitals (MOs)

• 2 MOs with different energies:

• Secular equations has the expression of

2211

2211

aaa

bbb

cc

cc

0221212

121211

HSH

SHH



23.5 23.5 Combining Atomic Orbitals to Form Combining Atomic Orbitals to Form Molecular OrbitalsMolecular Orbitals

• The two MO energies are given by

where ε1 = bonding MO ε2 = antibonding MO

• Molecular orbital energy diagram:

12

1211

12

1211

1 and

1 S

HH

S

HHbb




Show that substituting in gives the result c1 = c2. 0

0

22212121

12122111

HcSHc

SHcHc12

1211

1 S

HHb



SolutionSolution

We have

21

12121121212111

1212

1211122

12

1211111

0

011

cc

HSHcHSHc

SS

HHHc

S

HHHc



23.6 23.6 Molecular Orbitals for Homonuclear Molecular Orbitals for Homonuclear Diatomic Diatomic Molecules 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 axis2. Inversion through the center of the

molecule



23.6 23.6 Molecular Orbitals for Homonuclear Molecular Orbitals for Homonuclear Diatomic Diatomic Molecules Molecules

• The MOs used to describe chemical bonding in first and second row homonuclear diatomic molecules are shown in table form.



23.7 23.7 The Electronic Structure of Many-Electron The Electronic Structure of Many-Electron MoleculesMolecules

• The MO diagrams show the number and spin of the electrons rather than the magnitude and sign of the AO coefficients.



23.7 23.7 The Electronic Structure of Many-Electron The Electronic Structure of Many-Electron MoleculesMolecules

• 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.



23.8 23.8 Bond Order, Bond Energy, and Bond LengthBond Order, Bond Energy, and Bond Length

• For the series H2→Ne2, the relationship between Bond Order, Bond Energy, and Bond Length is shown.



23.8 23.8 Bond Order, Bond Energy, and Bond LengthBond 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.




Arrange the following in terms of increasing bond energy and bond length on the basis of their bond order:

22222 N and N,N ,N



SolutionSolution

The ground-state configurations for these species are

1*1*2222*22*22

2

1*222*22*22

2222*22*22

122*22*22

111132211:N

1132211:N

3112211:N

312211:N

gguugugug

gugugug

guuugug

guugug



SolutionSolution

In this series, the bond order is 2.5, 3, 2.5, and 2. Therefore, the bond energy is predicted to follow the order using the bond order alone. However, because of the extra electron in the antibonding MO, the bond energy in will be less than that in . Because bond lengths decrease as the bond strength increases, the bond length will follow the opposite order.

22

-222 NN,NN

*1 g -2N

2N



23.9 23.9 Heteronuclear Diatomic MoleculesHeteronuclear Diatomic Molecules

• The MOs on a heteronuclear diatomic molecule are numbered differently for the order in energy exhibited in the molecules Li2N2:

• The MOs will still have either σ or π symmetry.




• The symbol * is usually added to the MOs for the heteronuclear molecule to indicate an anti-bonding MO.




• The 3σ, 4σ and 1π MOs for HF are shown from left to right.



23.10 23.10 The Molecular Electrostatic PotentialThe 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



23.10 23.10 The Molecular Electrostatic PotentialThe 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.

physical chemistry 2 nd edition

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