Physical Chemistry 2 nd Edition Thomas Engel, Philip Reid Chapter 23 The Chemical Bond in Diatomic Molecules

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules Objectives Usefulness of H 2 + as qualitative model in chemical bonding. Understanding of molecular orbitals (MOs) in terms of atomic orbitals (AOs), Discuss molecular orbital energy diagram.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.1 The Simplest One-Electron Molecule: H 2 + Schrödinger equation cannot be solved exactly for any molecule containing more than one electron. We approach H 2 + using an approximate model, thus the total energy operator has the form where 1 st term = kinetic energy operator nuclei a and b 2 nd term = electron kinetic energy 3 rd term = attractive Coulombic interaction 4 th term = nuclear–nuclear repulsion

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.1 The Simplest One-Electron Molecule: H 2 + The quantities R, r a, and r b represent the distances between the charged particles.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.2 The Molecular Wave Function for Ground-State H 2 + 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 H 2 + is where Ф = atomic orbital (AO) ψ = molecular wave function σ = molecular orbital (MO)

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.2 The Molecular Wave Function for Ground-State H 2 + For two MOs from the two AOs, where ψ g = bonding orbitals wave functions ψ u = antibonding orbitals wave functions

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.3 The Energy Corresponding to the Molecular Wave Functions ψ g and ψ u The differences ΔE g and ΔE u between the energy of the molecule is as follow: where J = Coulomb integral K = resonance integral or the exchange integral

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.3 The Energy Corresponding to the Molecular Wave Functions ψ g and ψ u 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.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules Example 23.1 Show that the change in energy resulting from bond formation,, can be expressed in terms of J, K, and Sab as

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules Solution Starting from we have

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules Solution Thus

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.4 A Closer Look at the Molecular Wave Functions ψ g and ψ u The values of ψ g and ψ u along the molecular axis are shown.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.4 A Closer Look at the Molecular Wave Functions ψ g and ψ u The probability density of finding an electron at various points along the molecular axis is given by the square of the wave function.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.4 A Closer Look at the Molecular Wave Functions ψ g and ψ u Virial theorem applies to atoms or molecules described either by exact wave functions or by approximate wave functions. This theorem states that

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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: Secular equations has the expression of

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.5 Combining Atomic Orbitals to Form Molecular Orbitals The two MO energies are given by where ε 1 = bonding MO ε 2 = antibonding MO Molecular orbital energy diagram:

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules Example 23.2 Show that substituting in gives the result c 1 = c 2.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules Solution We have

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.8 Bond Order, Bond Energy, and Bond Length For the series H 2 → Ne 2, the relationship between Bond Order, Bond Energy, and Bond Length is shown.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules Example 23.4 Arrange the following in terms of increasing bond energy and bond length on the basis of their bond order:

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules Solution The ground-state configurations for these species are

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules Solution 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.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.9 Heteronuclear Diatomic Molecules The MOs on a heteronuclear diatomic molecule are numbered differently for the order in energy exhibited in the molecules Li 2  N 2 : The MOs will still have either σ or π symmetry.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.9 Heteronuclear Diatomic Molecules The symbol * is usually added to the MOs for the heteronuclear molecule to indicate an anti- bonding MO.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 23.9 Heteronuclear Diatomic Molecules The 3σ, 4σ and 1π MOs for HF are shown from left to right.

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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 (Ф r ) 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

© 2010 Pearson Education South Asia Pte Ltd Physical Chemistry 2 nd Edition Chapter 23: The Chemical Bond in Diatomic Molecules 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.