18: Multiconfiguration Wavefunctions
- Page ID
- 60575
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept.
- 18.1: Optimization of the Energy for a Multiconfiguration Wavefunction
- This page covers the optimization of molecular orbitals through multi-electron wavefunctions represented as linear combinations of configuration state functions. It details the energy expectation value's dependency on CI coefficients and orthonormality constraints, introducing the variational method and deriving the CI-secular and Fock equations.
- 18.2: The Single-Determinant Wavefunction
- This page explores the single Slater determinant trial function in quantum many-body theory, focusing on its significance in Hartree-Fock (HF) theory. It details the simplification of density matrices for occupied spin-orbitals and the importance of the Fock operator, which accounts for Coulomb and exchange interactions.
- 18.3: The Unrestricted Hartree-Fock Spin Impurity Problem
- This page discusses the application of Hartree-Fock (HF) equations to open-shell systems, highlighting issues with spin symmetry and orbital energies for \(\alpha\) and \(\beta\) spins, which can lead to spin contamination. The unrestricted Hartree-Fock (UHF) method does not guarantee pure spin symmetry, requiring corrections like spin-projection. An alternative spin-adapted HF approach, while starting with a pure spin wavefunction, results in non-orthogonal orbitals that complicate calculations.
- 18.4: Atomic Orbital Basis Sets
- This page explores the types of basis orbitals in the LCAO-MO-SCF process, focusing on Slater-type orbitals (STOs) and Gaussian-type orbitals (GTOs), with an emphasis on the latter's computational advantages. It also details the construction of Gaussian basis sets like 4-31G and 6-31G, highlighting the significance of polarization and diffuse functions for enhancing electronic structure calculations.
- 18.5: The LCAO-MO Expansion
- This page discusses the Hartree-Fock (HF) equations, which are integro-differential equations used to describe quantum systems. It highlights the iterative nature of their solutions due to dependency on orbitals, emphasizing the self-consistent field (SCF) procedure that begins with an initial guess. For non-linear molecules, the Linear Combination of Atomic Orbitals-Molecular Orbital (LCAO-MO) methods convert the problem into a matrix eigenvalue equation, requiring further iterations.
- 18.6: The Roothaan Matrix SCF Process
- This page covers the SCF equation-solving process for quantum chemistry, detailing the formation of the Fock operator and the importance of one- and two-electron integrals. It highlights strategies for generating initial orbital coefficient guesses and the iterative nature of the SCF process, which stops when energy and coefficients converge. Additionally, the page mentions semi-empirical methods that leverage approximations to expedite calculations.
- 18.7: Observations on Orbitals and Orbital Energies
- This page covers key concepts in quantum chemistry regarding Hartree-Fock orbital energies and their interpretations. It explains how occupied orbitals correspond to N electrons and virtual orbitals to N+1 electrons, introduces Koopmans' theorem linking orbital energies with ionization and electron affinity, and details the total SCF electronic energy formula while highlighting double counting issues.