## Electronic Structure Problem

For the present purposes, we define the modern electronic structure problem as finding the ground-state energy of nonrelativistic electrons for arbitrary positions of nuclei within the Born-Oppenheimer approximation.1 If this can be done sufficiently accurately and rapidly on a modern computer, many properties can be predicted, such as bond energies and bond lengths of molecules, and lattice structures and parameters of solids.

Consider a diatomic molecule, whose binding energy curve is illustrated in Figure 1. The binding energy is given by

where *E*_{0}(*R*) is the ground-state energy of the electrons with nuclei separated by *R*, and *E*_{A} and *Z*_{A} are the atomic energy and charge of atom *A* and similarly for *B*. The minimum tells us the bond length (*R*_{0}) and the well-depth (*D*_{e}), corrected by zero-point energy ( ), gives us the dissociation energy (D_{0}).

The Hamiltonian for the *N* electrons is

where the kinetic energy operator is

the electron–electron repulsion operator is

and the one-body operator is

For instance, in a diatomic molecule, *v*(**r**) = −*Z*_{A}/*r* − *Z*_{B}/|**r** − **R**|. We use atomic units unless otherwise stated, setting , so energies are in Hartrees (1 Ha = 27.2 eV or 628 kcal/mol) and distances in Bohr radii (1 *a*_{0} = 0.529 Å). The ground-state energy satisfies the variational principle:

where the minimization is over all antisymmetric *N*-particle wavefunctions. This *E* was called *E*_{0}(*R*) in Eq. (1).*

Many traditional approaches to solving this difficult many-body problem begin with the Hartree–Fock (HF) approximation, in which Ψ is approximated by a single Slater determinant (an antisymmetrized product) of orbitals (single-particle wavefunctions)2 and the energy is minimized.3 These include configuration interaction, coupled cluster, and Møller-Plesset perturbation theory, and are mostly used for finite systems, such as molecules in the gas phase.4 Other approaches use reduced descriptions, such as the density matrix or Green's function, but leading to an infinite set of coupled equations that must somehow be truncated, and these are more common in applications to solids.5

More accurate methods usually require more sophisticated calculation, which takes longer on a computer. Thus, there is a compelling need to solve ground-state electronic structure problems reasonably accurately, but with a cost in computer time that does not become prohibitive as the number of atoms (and therefore electrons) becomes large.