Multi-configurational self-consistent field (MCSCF) is a method inquantum chemistry used to generate qualitatively correct reference states of molecules in cases whereHartree–Fock anddensity functional theory are not adequate (e.g., for molecular ground states which are quasi-degenerate with low-lying excited states or in bond-breaking situations). It uses a linear combination ofconfiguration state functions (CSF), or configuration determinants, to approximate the exact electronicwavefunction of an atom or molecule. In an MCSCF calculation, the set of coefficients of both the CSFs or determinants and the basis functions in the molecular orbitals are varied to obtain the total electronic wavefunction with the lowest possible energy. This method can be considered a combination betweenconfiguration interaction (where the molecular orbitals are not varied but the expansion of the wave function is) and Hartree–Fock (where there is only one determinant, but the molecular orbitals are varied).

MCSCF wave functions are often used as reference states formultireference configuration interaction (MRCI) or multi-reference perturbation theories likecomplete active space perturbation theory (CASPT2). These methods can deal with extremely complex chemical situations and, if computing power permits, may be used to reliably calculate molecular ground and excited states if all other methods fail.

For the simplest single bond, found in the H₂ molecule,molecular orbitals can always be written in terms of two functions χ_iA and χ_iB (which areatomic orbitals with small corrections) located at the two nucleiA andB:

${\displaystyle {\phi }_i=N_i({\chi }_{iA}\pm {\chi }_{iB}),}$

whereN_i is a normalization constant. The ground-state wavefunction for H₂ at the equilibrium geometry is dominated by the configuration (φ₁)², which means that the molecular orbitalφ₁ is nearly doubly occupied. TheHartree–Fock (HF) modelassumes that it is doubly occupied, which leads to a total wavefunction

${\displaystyle {\mathrm{\Phi }}_1={\phi }_1({\mathbf{r}}_1){\phi }_1({\mathbf{r}}_2){\mathrm{\Theta }}_{2,0},}$

where${\displaystyle {\mathrm{\Theta }}_{2,0}}$ is the singlet (S = 0) spin function for two electrons. The molecular orbitals in this caseφ₁ are taken as sums of 1s atomic orbitals on both atoms, namelyN₁(1s_A + 1s_B). Expanding the above equation into atomic orbitals yields

${\displaystyle {\mathrm{\Phi }}_1=N_1^2\left[1s_A({\mathbf{r}}_1)1s_A({\mathbf{r}}_2)+1s_A({\mathbf{r}}_1)1s_B({\mathbf{r}}_2)+1s_B({\mathbf{r}}_1)1s_A({\mathbf{r}}_2)+1s_B({\mathbf{r}}_1)1s_B({\mathbf{r}}_2)\right]{\mathrm{\Theta }}_{2,0}.}$

This Hartree–Fock model gives a reasonable description of H₂ around the equilibrium geometry – about 0.735 Å for the bond length (compared to a 0.746 Å experimental value) and 350 kJ/mol (84 kcal/mol) for the bond energy (experimentally, 432 kJ/mol (103 kcal/mol)^[1]). This is typical for the HF model, which usually describes closed-shell systems around their equilibrium geometry quite well. At large separations, however, the terms describing both electrons located at one atom remain, which corresponds to dissociation to H⁺ + H⁻, which has a much larger energy than H· + H· (two hydrogen radicals). Therefore, the persisting presence ofionic terms leads to an unphysical solution in this case.

Consequently, the HF model cannot be used to describe dissociation processes withopen-shell products. The most straightforward solution to this problem is introducing coefficients in front of the different terms in Ψ₁:

${\displaystyle {\mathrm{\Psi }}_1=C_{\text{ion}}{\mathrm{\Phi }}_{\text{ion}}+C_{\text{cov}}{\mathrm{\Phi }}_{\text{cov}},}$

which forms the basis for thevalence bond description ofchemical bonds. With the coefficientsC_ion andC_cov varying, the wave function will have the correct form, withC_ion = 0 for the separated limit, andC_ion comparable toC_cov at equilibrium. Such a description, however, uses non-orthogonalbasis functions, which complicates its mathematical structure. Instead, multiconfiguration is achieved by using orthogonal molecular orbitals. After introducing an anti-bonding orbital

${\displaystyle {\phi }_2=N_2(1s_A-1s_B),}$

the total wave function of H₂ can be written as a linear combination of configurations built from bonding and anti-bonding orbitals:

${\displaystyle {\mathrm{\Psi }}_{\text{MC}}=C_1{\mathrm{\Phi }}_1+C_2{\mathrm{\Phi }}_2,}$

where Φ₂ is the electronic configuration (φ₂)². In this multiconfigurational description of the H₂ chemical bond,C₁ = 1 andC₂ = 0 close to equilibrium, andC₁ will be comparable toC₂ for large separations.^[2]

A particularly important MCSCF approach is thecomplete active space SCF method (CASSCF), where the linear combination ofCSFs includes all that arise from a particular number of electrons in a particular number of orbitals (also known asfull-optimized reaction space (FORS-MCSCF)). For example, one might define CASSCF(11,8) forNO, where the 11 valence electrons are distributed between all configurations that can be constructed from 8 molecular orbitals.^[3][4]

Since the number of CSFs quickly increases with the number of active orbitals, along with thecomputational cost, it may be desirable to use a smaller set of CSFs. One way to make this selection is to restrict the number of electrons in certain subspaces, done in therestricted active space SCF method (RASSCF). One could, for instance, allow only single and double excitations from some strongly occupied subset of active orbitals, or restrict the number of electrons to at most 2 in another subset of active orbitals.

• Charlotte Froese Fischer
• Douglas Hartree
• Vladimir Fock
• Yakov Frenkel
• Hartree–Fock method
• Quantum chemistry computer programs

• Cramer, Christopher J. (2002).Essentials of Computational Chemistry. Chichester: John Wiley and Sons.ISBN 0-471-48552-7.

• Electronic structure methods