# 19: Multi-Determinant Wavefunctions

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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
To allow for a properly spin- and space- symmetry adapted trial wavefunction and to permit ΨΨ to contain more than a single CSF, methods which are more flexible than the single-determinant HF procedure are needed. In particular, it may be necessary to use a combination of determinants to describe such a proper symmetry function.
• 19.2: Different Methods
There are numerous procedures currently in use for determining the 'best' wavefunction that involve two fundamentally different kinds of parameters to be determined: the CI coefficients CI and the LCAO-MO coefficients. The most commonly employed methods used to determine these parameters include: (1) the multiconfigurational self-consistent field (MCSCF) method, (2) the configuration interaction (CI) method, (3) the Møller-Plesset perturbation method (MPPT), and (4) the Coupled-Cluster method.
• 19.3: Strengths and Weaknesses of Various Methods
Methods that are based on making the energy functional stationary (i.e., variational methods) yield upper bounds to the lowest energy of the symmetry which characterizes the CSFs.  All variational techniques suffer from at least one serious drawback; they are not size-extensive, which means that the energy computed using these tools can not be trusted to scale with the size of the system. This precludes these methods from use in extended systems (e.g., solids).
• 19.4: Further Details on Implementing Multiconfigurational Methods
In the CI method, one usually attempts to realize a high-level treatment of electron correlation. A set of orthonormal molecular orbitals are first obtained from an SCF or MCSCF calculation (usually involving a small to moderate list of CSFs). The LCAO-MO coefficients of these orbitals are no longer considered as variational parameters in the subsequent CI calculation; only the CI coefficients are to be further optimized.

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. Image used with permission (CC SA-BY-3.0; Christoph N. Böttge, "Phonon-assistierte Lasertätigkeit in Mikroresonatoren").