19: Multi-Determinant Wavefunctions
- Page ID
- 60576
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Corrections to the mean-field model are needed to describe the instantaneous Coulombic interactions among the electrons. This is achieved by including more than one Slater determinant in the wavefunction.
- 19.1: Introduction to Multi-Determinant Wavefunctions
- This page explores the limitations of single Slater determinant wavefunctions in unrestricted Hartree-Fock theory, highlighting issues like spin contamination. It stresses the need for multiconfigurational wavefunctions to manage strong electron correlations and ensure dynamic adjustments.
- 19.2: Different Methods
- This page covers key procedures for determining optimal wavefunctions in quantum chemistry, focusing on methods like MCSCF, CI, MPPT, and CC. It emphasizes the evaluation of electron integrals in molecular orbital theory, detailing efficient transformations from atomic orbital integrals and highlighting the need for multiple configurations in multiconfigurational approaches for accurate results.
- 19.3: Strengths and Weaknesses of Various Methods
- This page discusses variational methods (MCSCF, SCF, CI) that provide upper bounds for energy states but struggle with size-extensivity, posing issues for larger systems. Adjustable wavefunctions like CAS can help, but non-variational methods (MPPT, CC) achieve reliable size-extensive energies.
- 19.4: Further Details on Implementing Multiconfigurational Methods
- This page covers several advanced quantum chemistry methods, focusing on MCSCF, MPPT, MBPT, Coupled-Cluster, and DFT. It discusses the challenges of optimizing CSFs in MCSCF, the role of CSFs in perturbation theories, and the iterative nature of Coupled-Cluster methods. Key concepts like the Hohenberg-Kohn theorem in DFT are explained, alongside limitations of previous models like Thomas-Fermi.
Thumbnail: Schematic representation of the cluster-expansion-based classification. The full correlation is composed of singlets, doublets, triplets, and higher-order correlations, all uniquely defined by the cluster-expansion approach. Each blue sphere corresponds to one particle operator and yellow circles/ellipses to correlations. The number of spheres within a correlation identifies the cluster number. (CC SA-BY-3.0; Christoph N. Böttge, "Phonon-assistierte Lasertätigkeit in Mikroresonatoren").