Configuration interaction

Configuration interaction (CI) is a post Hartree-Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born-Oppenheimer approximation for a quantum chemical multi-electron system. Two meanings are connected to the term configuration interaction in this context. Mathematically, configuration simply describes the linear combination of Slater determinants used for the wave function. In terms of a specification of orbital occupation (for instance, (1s)²(2s)²(2p)¹...), interaction means the mixing (interaction) of different electronic configurations (states). Due to the long CPU time and immense hardware required for CI calculations, the method is limited to relatively small systems.

In contrast to the Hartree-Fock method, in order to account for electron correlation, CI uses a variational wave function that is a linear combination of determinant built from spin orbitals (denoted by the superscript SD),

$\Psi =\sum _{I=0}c_{I}\Phi _{I}^{SD}=c_{0}\Phi _{0}^{SD}+c_{1}\Phi _{1}^{SD}+{...}$

where Ψ is usually the electronic ground state of the system. If the expansion includes all possible Slater determinants of the appropriate symmetry, then this is a full configuration interaction procedure which exactly solves the electronic Schrödinger equation within the space spanned by the one-particle basis set. When solving the CI equations, approximations to excited states are also obtained, which differ in the values of their coefficients c_I.

Two meanings are connected to the term configuration interaction in this context. Mathematically, configuration simply describes the linear combination used for the wave function. In terms of a specification of orbital occupation (for instance, (1s)²(2s)²(2p)¹...), interaction means the mixing (interaction) of different electronic configurations (states).

The CI procedure leads to a general matrix eigenvalue equation:

$\mathbb {H} \mathbf {c} =\mathbf {e} \mathbb {S} \mathbf {c} ,$

where c is the coefficient vector, e is the eigenvalue matrix, and the elements of the hamiltonian and overlap matrices are, respectively,

$\mathbb {H} _{ij}=\left\langle \Phi _{i}^{SD}|\mathbf {H} ^{el}|\Phi _{j}^{SD}\right\rangle$,

$\mathbb {S} _{ij}=\left\langle \Phi _{i}^{SD}|\Phi _{j}^{SD}\right\rangle$.

Slater determinants are constructed from sets of orthonormal spin orbitals, so that $\left\langle \Phi _{i}^{SD}|\Phi _{j}^{SD}\right\rangle =\delta _{ij}$, making $\mathbb {S}$ the identity matrix and simplifying the above matrix equation.

See also

• Quantum chemistry
• Post Hartree-Fock
• Electron correlation
• Multireference configuration interaction