6: Understanding Molecules with QM

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6.1: Prelude to Covalent Bonding
Yet oxygen demonstrates very different magnetic behavior than nitrogen. We can pour liquid nitrogen through a magnetic field with no visible interactions, while liquid oxygen is attracted to the magnet and floats in the magnetic field. We need to understand the additional concepts of valence bond theory, orbital hybridization, and molecular orbital theory to understand these observations.
6.2: The Born-Oppenheimer Approximation
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.
6.3: 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.
6.4: 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.
6.5: 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.
6.6: 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.
6.7: 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.
6.8: 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.
6.9: Summary of Molecular Orbital Theory
Molecular orbital (MO) theory describes the behavior of electrons in a molecule in terms of combinations of the atomic wavefunctions. The resulting molecular orbitals may extend over all the atoms in the molecule. Bonding molecular orbitals are formed by in-phase combinations of atomic wavefunctions, and electrons in these orbitals stabilize a molecule. Antibonding molecular orbitals result from out-of-phase combinations and electrons in these orbitals make a molecule less stable.
6.10: Valence Bond Theory
A more sophisticated treatment of bonding is needed for systems such as these. In this section, we present a quantum mechanical description of bonding, in which bonding electrons are viewed as being localized between the nuclei of the bonded atoms. The overlap of bonding orbitals is substantially increased through a process called hybridization, which results in the formation of stronger bonds.
6.E: Advanced Theories of Covalent Bonding (Exercises)
These are homework exercises to accompany the Textmap created for "Chemistry" by OpenStax. Complementary General Chemistry question banks can be found for other Textmaps and can be accessed here. In addition to these publicly available questions, access to private problems bank for use in exams and homework is available to faculty only on an individual basis; please contact Delmar Larsen for an account with access permission.