The Configuration Interaction (CI) Method

The Hartree-Fock method produces an energy that is higher than the actual value (a consequence of the variational principle), due to the approximation of the wavefunction – the Schrödinger equation is not actually separable, and so the molecular orbital approximation introduces inaccuracy in this respect. It also treats coulombic repulsion between electrons in an average way only. For a more accurate picture, the instantaneous interaction between electrons must be considered – in the helium atom, for example, if one electron is near the nucleus at any given point in time, then it is energetically more favourable for the other electron to be further away from the nucleus. i.e. The probability density of finding another electron in the area immediately surrounding an electron is very small. In this way, the motion of the electrons is said to be correlated, and it is this instantaneous electron interaction (not just an average repulsion) that is referred to as electron correlation.
The correlation energy is the difference between the exact non-relativistic energy and the (non-relativistic) Hartree-Fock energy (equation (8.1)).

(8.1)

Configuration interaction (CI) is a method that includes instantaneous electron correlation. Also called configuration mixing (CM), this involves first- and higher-order corrections to the Hartree-Fock wavefunction that mix in elements of higher atomic orbitals, found in excited states. The exact wavefunction is represented as a linear combination of N-electron ‘trial’ functions, or configurations, and the linear variational method is used to optimise the coefficients of the different configurations (see Szabo and Ostlund[16]). In principle, the basis set of N-electron wavefunctions used could be complete, in which case an exact energy would be obtained. This is called full CI. It is, however, computationally extremely expensive and so generally the basis set is limited to a finite size.
For example, the ground state of beryllium would be represented by the Hartree-Fock SCF method as a linear combination of the 1s and 2s orbitals. In configuration interaction, contributions from excited states involving the 1s and 3s orbitals, for example, would be included in a linear variation function for the ground state.
Configuration interaction is useful for calculating excited states of molecules, where the Hartree-Fock method invariably fails.
For a much more detailed approach, the configuration interaction approach is described in a paper by G. Yan, et al.[23]

Figure 8.1 shows the potential energy curve for H₂ calculated using Hartree-Fock SCF, MCSCF and full CI methods:

Figure 8.1 - Potential energy curves for H2 (J.N. Harvey, “Molecular Electronic Structure”, University of Bristol, 2001)

8.1 The Multi-Reference Configuration Interaction (MRCI) Method

In conventional CI, the SCF wavefunction is used as a starting point (called the reference function) for obtaining the configuration state functions.
Multi-reference CI (MRCI) involves starting with an MCSCF wavefunction that is in itself a linear combination of configuration state functions (here termed reference CSFs), and using this to create further configuration state functions.
The MRCI approach is detailed in three papers by Knowles and Werner.[24]–[26]
A number of benchmark full CI and MRCI calculations on some small molecules have shown that certain MRCI calculations using CASSCF reference wavefunctions (singly and doubly excited; that is, with one and two electrons being excited from inactive to active orbitals) give potential energy functions very close to the full CI calculations.