# 9: The Chemical Bond: Diatomic Molecules

- Page ID
- 143024

- 9.1: The Born-Oppenheimer Approximation Simplifies the Molecular Schrödinger Equation
- The Born-Oppenheimer approximate is one of the most important and fundamental approximations in molecular quantum mechanics. This approximation separates the Schrödinger equation into two components, resulting in separate electronic and nuclear part of the wavefunction components. This separation is not exact, but approximate based on the separation of electronic and nuclear degrees of freedom we assume can be made based on the mass differential.

- 9.2: The H₂⁺ Prototypical Species
- The simplest conceivable molecule would be made of two protons and one electron, namely \(H_{2}^{+}\). This species actually has a transient existence in electrical discharges through hydrogen gas and has been detected by mass spectrometry and it also has been detected in outer space. The Schrödinger equation for \(H_{2}^{+}\) can be solved exactly within the Born-Oppenheimer approximation. This ion consists of two protons held together by the electrostatic force of a single electron.

- 9.3: Solving the H₂⁺ System Exactly (Optional)
- The \(H_2^+\) molecular system can be solved exactly (within the Born-Oppenheimer approximation) to generate wavefunctions called Molecular Orbitals. This involves recasting the system in elliptical coordinates.

- 9.4: The Overlap Integral
- Overlap integrals quantify the concentration of orbitals (often) on adjacent atoms in the same regions of space. Orbital overlap is a critical component in bond formation.

- 9.5: Evaluating the Overlap Integral (Optional)
- Evaluating the overlap integral for the 1s-1s atomic orbital basis functions is not simple and involves integration in elliptical coordinates.

- 9.6: Chemical Bond Stability
- From this LCAO-MO approach arises the Coulomb, Exchange (similar to HF calculations of atoms), and Overlap integrals. The concept of bonding and anti-bonding orbitals results.The application of LCAO toward molecular orbitals is demonstrated including linear variational theory and secular equations.

- 9.7: Bonding and Antibonding Orbitals
- The spatial structure of the bonding and antibonding molecular orbitals are contrasted demonstrating features such as a node between the nuclei. The expansion of the LCAO MOs using a greater basis set than just the 1s atomic orbitals is discussed.

- 9.8: A Simple Molecular-Orbital Treatment of H₂ Places Both Electrons in a Bonding Orbital
- To describe the electronic states of molecules, we construct wavefunctions for the electronic states by using molecular orbitals. These wavefunctions are approximate solutions to the Schrödinger equation. A mathematical function for a molecular orbital is constructed, \(\psi _i\), as a linear combination of other functions, \(\varphi _j\), which are called basis functions because they provide the basis for representing the molecular orbital.

- 9.9: Molecular Orbitals Can Be Ordered According to Their Energies
- The linear combination of atomic orbitals always gives back the same number of molecular orbitals. So if we start with two atomic orbitals, we end up with two molecular orbitals. When atomic orbitals add in phase, we get constructive interference and a lower energy orbital. When they add out of phase, we get a node and the resulting orbital has higher energy. The lower energy MOs are bonding and higher energy MOs are antibonding.

- 9.10: Molecular-Orbital Theory Does not Predict a Stable Diatomic Helium Molecule
- The occupied molecular orbitals (i.e., orbitals with electrons) are represented via an electron configuration like with atoms. For diatomics, these configurations are reflected at a "bond order" that is used to describe the strength and lengths of the bonds. They predict that stable molecules (i.e., observable) have bond orders that are > 0. For molecular orbitals consisting of only the 1s atomic orbitals, that suggests certain molecules will not exist. The typical example is the helium dimer.

- 9.11: Electrons Populate Molecular Orbitals According to the Pauli Exclusion Principle
- The Pauli exclusion principle plays as important a role in the understanding of the electronic structure of molecules as it does in the case of atoms. We are now in a position to build up and determine the electronic configurations of the homonuclear diatomic molecules by adding electrons two at a time to the molecular orbitals with the spins of the electrons paired, always filling the orbitals of lowest energy first.

- 9.12: Molecular Orbital Theory Predicts that Molecular Oxygen is Paramagnetic
- The molecular orbital configuration dictates the bond order of the bond. This in turns dictates the strength of the bond and the bond length with stronger bonds exhibiting small bond lengths. The molecular orbital configuration of molecular oxygen demonstrates that the ground-state neutral species has two unpaired electrons and hence is paramagnetic (attractive to external magnetic fields). This is a feature of MO theory that other theories do not predict.