Introduction

The configuration interaction expansion of an approximate solution to the electronic Schrödinger equation is typically written

$$\vert \Psi \rangle = c_0 \vert \Phi_0 \rangle + \sum_{ia} c_i... ...k,a<b<c} c_{ijk}^{abc} \vert \Phi_{ijk}^{abc} \rangle + \ldots$$ (1)

$\vert \Phi_0 \rangle$ is the so-called ``reference,'' typically obtained from a Hartree-Fock self-consistent-field (SCF) procedure as the best single Slater determinant (or configuration state function, CSF) which describes the electronic state of interest. $\vert \Phi_i^a \rangle$ is the determinant formed by replacing spin-orbital i in $\vert \Phi_0 \rangle$ with spin orbital a, etc. These notes follow the convention that i,j,k denote orbitals occupied in the reference, a,b,c denote orbitals unoccupied in the reference, and p,q,r are general indices. The widely-employed CI singles and doubles (CISD)  wavefunction includes only those N-electron basis functions which represent single or double substitutions relative to the reference state and typically accounts for about 95% of the correlation energy for small molecules near their equilibrium geometries.

Head-Gordon, Pople, and others have advocated the use of configuration interaction with only single substitutions (CIS) as the starting point for investigations of excited electronic states. In their 1992 paper, Foresman, Head-Gordon, Pople, and Frisch [1] list the following desirable properties of CIS: well defined (and differentiable), applicable to large systems, size-consistent, variational, and providing directly comparable (i.e. orthogonal) electronic state solutions. They go on to present equations for the CIS energy and gradient when the reference is a single determinant obtained from an SCF procedure. These notes present a derivation of the CIS energy for general and several specific types of single-determinant references. Later, we discuss some extensions which are necessary for reliable treatments of open-shell systems.