9: Chemical Bonding in Diatomic Molecules
- Page ID
- 11786
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Our basis for understanding chemical bonding and the structures of molecules is the electron orbital description of the structure and valence of atoms, as provided by quantum mechanics. We assume an understanding of the periodicity of the elements based on the nuclear structure of the atom and our deductions concerning valence based on electron orbitals.
- 9.1: The Born-Oppenheimer Approximation Simplifies the Schrödinger Equation for Molecules
- This page covers the Born-Oppenheimer approximation in quantum chemistry, which simplifies molecular studies by separating the motions of nuclei and electrons. It describes how this approximation treats nuclei as stationary due to their greater mass, allowing for efficient computation of electronic states and molecular properties.
- 9.2: The H₂⁺ Prototypical Species
- This page discusses molecular orbital theory, beginning with the hydrogen molecular ion \(\ce{H2^{+}}\), which has two protons and one electron. Using the Schrödinger equation and the Born-Oppenheimer approximation, the theory employs the Linear Combination of Atomic Orbitals (LCAO) method to describe molecular orbitals.
- 9.3: The Overlap Integral
- This page explores the molecular orbital theory of \(\ce{H2^{+}}\) through linear combination of atomic orbitals, detailing bonding and antibonding orbitals affected by charge density. It presents calculations for charge density, finding \(3.7 \times 10^{-7}\;\text{pm}^{-3}\) for bonding and zero for antibonding due to a node. Stability and energy implications are also discussed, along with expressions for overlap integrals and a mention of exercises on integrals between functions.
- 9.4: Chemical Bond Stability
- This page details the calculation of molecular orbital energies for \(\ce{H_2^{+}}\) using the Linear Combination of Atomic Orbitals method. It explains key components like Coulomb, exchange, and overlap integrals, and their roles in energy equations. The text highlights the contributions of proton repulsion and electron distribution to total energy, along with defining the Coulomb and exchange integrals.
- 9.5: Bonding and Antibonding Orbitals
- This page explores the molecular orbitals of the \(\ce{H^{+}}\) ion using the LCAO method, which includes bonding and antibonding characteristics. The bonding \(\sigma_{1s}\) orbital stabilizes the system through constructive interference, while the antibonding \(\sigma_{1s}^*\) destabilizes it through destructive interference.
- 9.6: A Simple Molecular-Orbital Treatment of H₂ Places Both Electrons in a Bonding Orbital
- This page discusses constructing wavefunctions for molecular electronic states via molecular orbitals using linear combinations of basis functions. It highlights the variational method for optimizing these functions to minimize ground state energy, describes electron wavefunctions' antisymmetry, and aligns with the Aufbau Principle and symmetry.
- 9.7: Molecular Orbitals Can Be Ordered According to Their Energies
- This page discusses the LCAO-MO method for analyzing diatomic molecules, highlighting qualitative insights for homonuclear molecules and quantitative analysis for heteronuclear and polyatomic ones. It notes the formation of bonding and antibonding molecular orbitals from ns and np orbitals, affecting bond order and stability.
- 9.8: Molecular-Orbital Theory Does not Predict a Stable Diatomic Helium Molecule
- This page explains bond order through molecular orbital theory, emphasizing its role in molecular stability and strength. Bond order is defined as the difference between bonding and antibonding electrons, illustrated with examples of diatomic molecules such as \(H_2\) and \(\ce{He_2}\). While \(H_2^+\) and \(\ce{He_2^+}\) are stable but weaker, \(\ce{He_2}\) has a bond order of 0, indicating instability.
- 9.9: Electrons Populate Molecular Orbitals According to the Pauli Exclusion Principle
- This page explains the Pauli exclusion principle, which restricts electron occupancy in atomic and molecular orbitals and affects molecular electronic structures, preventing collapse toward nuclei. For stable bonding, electrons must have paired spins in the same orbital, as seen in diatomic hydrogen (H2). In contrast, helium (He2) is unstable due to antibonding orbitals, while charged species like He2+ demonstrate stability.
- 9.10: Molecular Orbital Theory Predicts that Molecular Oxygen is Paramagnetic
- This page explores the relationship between bond order, bond length, and bond energy in diatomic molecules, using molecular oxygen (\(\ce{O2}\)) as an example to illustrate its paramagnetic behavior linked to molecular orbital theory. It highlights the significance of unpaired electrons in \(\ce{O2}\) and how this affects reactions with organic compounds due to a spin barrier.
- 9.11: Photoelectron Spectra Support the Existence of Molecular Orbitals
- This page discusses Werner Heisenberg's uncertainty principle from the mid-1920s, which illustrates the inverse relationship between the uncertainties of position and momentum for particles. It highlights that confining a particle's position increases momentum uncertainty, showcasing the wave nature of particles. An exercise compares the uncertainties and wavelengths of a baseball and an electron, offering insights into quantum behaviors versus classical mechanics.
- 9.12: Molecular-Orbital Theory Also Applies to Heteronuclear Diatomic Molecules
- This page explains the formation and bonding characteristics of heteronuclear diatomic molecules through molecular orbital theory, emphasizing concepts like polar covalent bonds, resonance, and electron configurations. It highlights nitric oxide's unique properties due to its odd electron count and discusses the creation of molecular orbital diagrams, which account for electronegativity differences and predict magnetic properties in various molecules, including ions such as CN⁻ and OCl⁻.
- 9.14: Molecular Term Symbols Describe Electronic States of Molecules
- This page explains molecular term symbols for diatomic molecules within the Russell-Saunders coupling framework. It details the four components of these symbols—spin multiplicity, azimuthal angular momentum, total angular momentum, and symmetry—while highlighting differences between homonuclear and heteronuclear diatomics.
- 9.15: Molecular Term Symbols Designate Symmetry
- This page discusses the quantum numbers for diatomic molecules, highlighting their similarities and differences with atomic quantum numbers, particularly the angular momentum quantum number \(Λ\). It covers the impact of parity and reflection symmetries on molecular orbitals, including symmetric (g) and anti-symmetric (u) designations. The rules for determining overall orbital reflection apply mainly to Σ states.
- 9.E: Chemical Bond in Diatomic Molecules (Exercises)
- This page explores molecular orbital theory, focusing on diatomic molecules like H2, O2, and NO. It details the use of elliptic coordinates for overlap integrals, bond order calculations for various molecules, and how these relate to molecular stability.
Thumbnail: A covalent bond forming \(\ce{H2}\) where two hydrogen atoms share the two electrons. (CC BY-SA 3.0; Jacek FH via Wikipedia; modified by LibreTexts)