# 6.17.1 Introduction

The complete active space (CAS) methods are a family of methods for dealing with strongly correlated systems. In this method, a subset of a system’s orbitals and electrons are denoted as active and the full configuration interaction (FCI) problem is solved exactly in this small active space. The remaining occupied orbitals are denoted inactive and are treated in a mean-field manner, while the remaining unoccupied orbitals are denoted virtual. In CAS-CI, this is the end of the matter. In CASSCF, the orbitals spanning these three spaces (inactive, active, and virtual) are then optimized to obtain the lowest possible energy. In other words, the CASSCF problem is to find the optimal (by energy) partitioning of the orbital Hilbert space. This allows moderately sized systems to be studied as long as the active space is relatively small, due to combinatorial growth in the number of possible Slater determinants that encompass all possible configurations within the active space. Indeed, the total number of possible Slater determinants for an active space with $M$ spatial orbitals, $N_{\uparrow}$ up spins and $N_{\downarrow}$ down spins is:

 $\displaystyle N_{Total}$ $\displaystyle=\dfrac{M!}{N_{\uparrow}!\left(M-N_{\uparrow}\right)!}\dfrac{M!}{% N_{\downarrow}!\left(M-N_{\downarrow}\right)!}$ (6.47)

Modern computing architectures can handle active spaces of approximately 18 electrons in 18 orbitals ($\approx 2\times 10^{9}$ determinants), though we do not recommend using such a large active-space for routine calculations.

Nuclear gradients for CASSCF calculation are also available in Q-Chem. In addition to full CAS calculations, arbitrary order truncated CI (CIS, CISD, CISDT, etc.) may also be carried out in the requested active space and orbitally optimized.