2.1: Overview
- Page ID
- 541998
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)By the end of this chapter, you should know the following:
- How the Born-Oppenheimer approximation affects the molecular Hamiltonian
- The justification for making the Born-Oppenheimer approximation
- What a potential energy surface is and how it relates to the Born-Oppenheimer approximation
- How minima on a potential energy surface relate to molecular structure
- How molecular orbital theory is related to the single-electron approximation
- How to calculate the energy of a simple set of molecular orbitals, both from the variational principle and using matrix formalism
- How the outcome of a molecular orbital calculation depends on the choice of starting basis functions
- How the Pauli exclusion principle affects electronic structure
- Why Slater Determinants are used as wavefunctions in multielectron systems
- What the Self-Consistent Field approximation is
- What the Hartree-Fock limit is, and how it relates to electron correlation
- The differences between molecular orbital theory and valence bond theories
- The benefits and limitations of hybridized valence bond theory
- The use cases and approximations in Hückel theory
- General properties of Hückel orbitals and their energies
- The factors surrounding tradeoffs between computation time and accuracy in ab initio quantum chemistry
- How the choice of basis set affects ab initio chemistry
- What convergence is
- General properties and utility of STOs, GTOs, contracted functions, and split-valence basis sets
- The differences between and pros/cons of ab initio methods and model Hamiltonian methods
Skill goals (what you should be able to do/calculate):
- Identify terms affected by the Born-Oppenheimer and single-electron approximations
- Construct Hamiltonian matrices for molecules
- Interpret and populate MO diagrams
- Construct Slater Determinants
- Use MO diagrams and potential energy diagrams to explain molecular properties
- Calculate Hückel orbital energies
- Interpret simple basis set designations
- Determine whether a calculation is converged
The overall goal for this chapter is to understand the quantum mechanical nature of chemical bonds and the general structure of molecules in terms of molecular orbital theory. This chapter begins with a review of molecular orbital theory as applied to the simplest diatomic molecule, , and then applies the theory to polyatomic molecules with multiple electrons. These types of problems cannot be solved exactly, but variational methods and perturbation theory methods can be applied to yield approximate answers. Molecular orbital theory is one such approximation method that provides a useful qualitative description of bonding and makes many accurate predictions about energetics and structure. It also serves as the basis for many more advanced methods that are used in computational chemistry, some of which will be discussed in this chapter.