Ab Initio Quantum Chemistry Methods - Accuracy and Scaling

Accuracy and Scaling

Ab initio electronic structure methods have the advantage that they can be made to converge to the exact solution, when all approximations are sufficiently small in magnitude and when the finite set of basis functions tends toward the limit of a complete set. In this case, configuration interaction, where all possible configurations are included (called "Full CI"), tends to the exact non-relativistic solution of the electronic Schrödinger equation (in the Born-Oppenheimer approximation). The convergence, however, is usually not monotonic, and sometimes the smallest calculation gives the best result for some properties.

The downside of ab initio methods is their computational cost. They often take enormous amounts of computer time, memory, and disk space. The HF method scales nominally as N4 (N being the number of basis functions) – i.e. a calculation twice as big takes 16 times as long to complete. However in practice it can scale closer to N3 as the program can identify zero and extremely small integrals and neglect them. Correlated calculations scale even less favorably: Møller–Plesset perturbation theory (MP2) as N5, MP4 as N6 and coupled cluster as N7. Density functional theory (DFT) methods using functionals which include Hartree–Fock exchange scale in a similar manner to Hartree–Fock but with a larger proportionality term and are thus more expensive than an equivalent Hartree–Fock calculation. DFT methods that do not include Hartree–Fock exchange can scale better than Hartree–Fock.