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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 40 ra rb 40 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 40 ra rb 40 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 caH 1sa cbH 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 40 R 40 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 40 R 40 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 c1b1 c2b2 a c1a1 c2 a2 • 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 Li2N2: • 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