Physical Chemistry 2 nd Edition

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

2. 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.

3. Outline • The Simplest One-Electron Molecule • The Molecular Wave Function for Ground-State • The Energy Corresponding to the Molecular Wave Functions • Closer Look at the Molecular Wave Functions • Combining Atomic Orbitals to form Molecular Orbitals

4. Outline • Molecular Orbitals for Homonuclear Diatomic Molecules • The Electronic Structure of Many-Electron Molecules • Bond Order, Bond Energy, and Bond Length • Heteronuclear Diatomic Molecules • The Molecular Electrostatic Potential

5. 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 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

6. 23.1 The Simplest One-Electron Molecule: H2+ • The quantities R, ra, and rbrepresent the distances between the charged particles.

7. 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 whereФ = atomic orbital (AO)ψ = molecularwave function σ = molecular orbital (MO)

8. 23.2 The Molecular Wave Function for Ground-State H2+ • For two MOs from the two AOs, where ψg = bonding orbitals wave functions ψu = antibonding orbitals wave functions

9. 23.3 The Energy Corresponding to the Molecular Wave Functions ψgand ψu • The differences ΔEg and ΔEubetween the energy of the molecule is as follow: where J = Coulomb integralK =resonance integral or the exchange integral

10. 23.3 The Energy Corresponding to the Molecular Wave Functions ψgand 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 playsa central role in the lowering of the energy that leads to the formation of a bond.

11. Example 23.1 Show that the change in energy resulting from bond formation, , can be expressed in terms of J, K, and Sab as

12. Solution Starting from we have

13. Solution Thus

14. 23.4 A Closer Look at the Molecular Wave Functions ψgand ψu • The values of ψg and ψualong the molecular axis are shown.

15. 23.4 A Closer Look at the Molecular Wave Functions ψgand ψu • The probability density of finding an electron at various points along the molecular axis is given by the square of the wave function.

16. 23.4 A Closer Look at the Molecular Wave Functions ψgand ψu • Virial theorem applies to atoms or molecules described either by exact wave functions or by approximate wave functions. • This theorem states that

17. 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

18. 23.5 Combining Atomic Orbitals to Form Molecular Orbitals • The two MO energies are given by where ε1 = bonding MOε2 = antibonding MO • Molecular orbital energy diagram:

19. Example 23.2 Show that substituting in gives the result c1 = c2.

20. Solution We have

21. 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 toeach of two symmetry operations: • Rotation about the molecularaxis • Inversion through the center of the molecule

22. 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.

23. 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.

24. 23.7 The Electronic Structure of Many-Electron Molecules • 2 remarks about the interpretation of MO energy diagrams: • Total energy of a many-electron molecule is not the sum of the MO orbital energies. • Bonding and antibonding give information about the relative signs of the AO coefficientsin the MO.

25. 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.

26. 23.8 Bond Order, Bond Energy, and Bond Length • Bond order is defined as • For a given atomic radius, the bond length is expected tovary inversely with the bond order.

27. Example 23.4 Arrange the following in terms of increasing bond energy and bond length on the basis of their bond order:

28. Solution The ground-state configurations for these species are

29. 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.

30. 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 σor πsymmetry.

31. 23.9 Heteronuclear Diatomic Molecules • The symbol * is usually added to the MOs for the heteronuclear molecule to indicate an anti-bonding MO.

32. 23.9 Heteronuclear Diatomic Molecules • The 3σ, 4σ and 1πMOs for HF are shown from left to right.

33. 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 valuesin the region around a molecule. where q = point charger = distance from thecharge

34. 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.