6: Molecular Structure and Bonding
- Page ID
- 455291
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
- 6.1: The Born-Oppenheimer Approximation
- The Born-Oppenheimer approximation is one of the basic concepts underlying the description of the quantum states of molecules. This approximation makes it possible to separate the motion of the nuclei and the motion of the electrons.
- 6.2: The Orbital Approximation and Orbital Configurations
- 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 with each electron described by a product of a spin-orbitals Since electrons are fermions, the electronic wavefunction must be antisymmetric with respect to the permutation of any two electrons. A Slater determinant containing the molecular spin orbitals produces the antisymmetric wavefunction.
- 6.3: Linear Variational Method and the Secular Determinant
- A special type of variation widely used in the study of molecules is the so-called linear variation function, a linear combination of N linearly independent functions (often atomic orbitals). Quite often a trial wavefunction is expanded as a linear combination of other functions (not the eigenvalues of the Hamiltonian, since they are not known) .
- 6.4: Trial Functions Can Be Linear Combinations of Functions That Also Contain Variational Parameters
- An alternative approach to the general problem of introducing variational parameters into wavefunctions is the construction of a wavefunction as a linear combination of other functions each with one or multiple parameters that can be varied
- 6.5: 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.
- 6.6: 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.
- 6.7: 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.
- 6.8: 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.
- 6.9: 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.
- 6.10: 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.
- 6.11: 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.
- 6.13: 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.
- 6.17: Polar bonds
- The molecular orbital diagram of a heteronuclear diatomic molecule is approached in a similar way to that of homonuclear diatomic molecule. The orbital diagrams may also look similar. A major difference is that the more electronegative atom will have orbitals at a lower energy level. Two examples of heteronuclear diatomic molecules will be explored below as illustrative examples.
- 6.18: Molecular Term Symbols Describe Electronic States of Molecules
- Molecular term symbols specify molecular electronic energy levels. Term symbols for diatomic molecules are based on irreducible representations in linear symmetry groups, derived from spectroscopic notations. They usually consist of four parts: spin multiplicity, azimuthal angular momentum, total angular momentum and symmetry. All molecular term symbols discussed here are based on Russel-Saunders coupling.
- 6.19: The pi-Electron Approximation of Conjugation
- Molecular orbital theory has been very successfully applied to large conjugated systems, especially those containing chains of carbon atoms with alternating single and double bonds. An approximation introduced by Hückel in 1931 considers only the delocalized p electrons moving in a framework of \(\pi\)-bonds. This is, in fact, a more sophisticated version of a free-electron model.
- 6.20: Butadiene is Stabilized by a Delocalization Energy
- Delocalization energy is intrinsic to molecular orbital theory, since it results from breaking the two-center bond concept. This is intrinsic to molecular orbital theory with the molecular orbitals spreading further than just one pair of atoms. However, within the two-center theory of valence bond theory, the delocalization energy results from a stabilization energy attributed to resonance.
- 6.21: Benzene and Aromaticity
- The previous sections addressed the \(\pi\) orbitals of linear conjugated system. Here we address conjugated systems of cyclic conjugated hydrocarons with the general formula of \(C_nH_n\) where n is the number of carbon atoms in the ring. The molecule from this important class of organic molecule that you are most familiar with is benzene (\(C_6H_6\)) with n=6, although many other molecules exist like cyclobutadiene (\(C_4H_4\) with n=4).