6.19.1 Introduction & Theory

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:

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.

The electronic energy is an exact functional of the 1-RDM and 2-RDM

Given the 1- and 2-PDMs, the generalized Fock matrices may be generated for this MCSCF. The derivation and further details are neatly described by Helgaker, Jorgensen, and Olsen, but the key results are summarized here. In the following, $m,n,p,q,\mathrm{\dots }$ are general indices, $i,j,k,\mathrm{\dots }$ are inactive indices, $t,u,v,w,\mathrm{\dots }$ are active indices, and $a,b,c,\mathrm{\dots }$ are virtual indices.

The generalized Fock matrix is defined as

where $h_{pq}$ are the 1-electron integrals and $g_{nqrs}$ are the 2-electron integrals and all indices run over all orbital classes (inactive, active, and virtual). This, generally non-symmetric, matrix can be simplified by taking advantage of the fact that the form of the density matrices when some indices are inactive or virtual are much simpler than when the indices are active. When the first index of the generalized Fock matrix is inactive and the second is general:

where the inactive and active Fock matrices are

In other words, the inactive Fock matrix is the Fock matrix formed from using only the inactive density and the active Fock matrix is sum of J and K matrices built from the active space 1-PDM. When the first index is active, and the second index is general, we have

where the auxiliary Q matrix is

and finally, if the first index is virtual then $F_{an}=0$. This formulation of the generalized Fock matrix is quite useful because it only requires density matrices with all indices active and two-electron integrals in the MO basis with three indices active and one general index, greatly reducing the storage and computational cost of the MO transformation. The orbital gradient is then given by