Multi-configurational self-consistent field

MCSCF wave functions are often used as reference states for multireference configuration interaction (MRCI) or multi-reference perturbation theories like complete 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.

Introduction

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 are atomic orbitals with small corrections) located at the two nuclei A and B:

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

where N_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 orbital φ₁ is nearly doubly occupied. The Hartree–Fock (HF) model assumes 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 case φ₁ are taken as sums of 1s atomic orbitals on both atoms, namely N₁(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)). 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 of ionic terms leads to an unphysical solution in this case.

Consequently, the HF model cannot be used to describe dissociation processes with open-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 the valence bond description of chemical bonds. With the coefficients C_ion and C_cov varying, the wave function will have the correct form, with C_ion = 0 for the separated limit, and C_ion comparable to C_cov at equilibrium. Such a description, however, uses non-orthogonal basis 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 and C₂ = 0 close to equilibrium, and C₁ will be comparable to C₂ for large separations.