Inactive subspaces

Just as in correlated descriptions based on MO theory, it is neither practical nor desirable to include electron correlation for all of the electrons in large systems. In common with other strategies, the orbital space is partitioned into inactive , active and unoccupied or virtual subspaces. Electron correlation is incorporated only for the active space, which corresponds to that part of the electronic structure which interests us most. A convenient representation of a general spin-coupled wavefunction takes the form ... 

The solution to the active orbital choice and the accompanying expansion length problems may be addressed in many cases by extending the direct product definition of the orbital space. The orbitals in the full Cl expansion were partitioned into three types in the definition of the FORS/CASSCF wavefunction inactive, active and virtual. Within each orbital type, the expansion may be considered to be full. That is, subject to the occupation restrictions on each subspace, all possible orbital occupations and spin... 

A CASSCF study starts with the selection of the inactive and active orbital subspaces. This must be done with great care, as will be discussed in detail later. The major problem is to construct a wavefunction which gives a balanced description of near-degeneracy and dynamical correlation effects for the section of the energy surface studied. 

Normally these molecular orbitals are obtained as expansions in a set of atom-centred basis functions (the linear combination of atomic orbitals (LCAO) method), m being the number of such functions. Recently, two-dimensional numerical integration methods have been developed to solve the MCSCF equations for linear molecules. The dimension m is then, in principle, infinite (practice, it is determined by the size of the grid used in the numerical integration). The molecular-orbital space is further divided into three subspaces the inactive, the active and the external orbitals. The inactive and active subspaces constitute the internal (occupied) orbital subspace, while the external orbitals are unoccupied. The CASSCF wavefunction is formed as a linear combination of configuration state functions (CSFs) generated from these orbitals in the following way. 

For rotations in the inactive and external subspaces we trivially obtain ... 

We did some preliminary calculations for BCu (which still could be treated with rather large active spaces) with different partitioning of the orbital space in CAS calculations. The notation is (frozen(inactive active n el) for orbital subspaces and n correlated electrons in the active space. The C2v symmetry was used in all computations. For most distances the wave function has definitively a two configuration form. The smallest active space considered is (0000 9331 2000 2 el) in the CASSCF calculation while in the subsequent CASPT2 calculation we used the (6220 3111 2000 2 el) space. The best would be to choose as the active space the valence orbitals of boron (2110) and the 3d,4s and the correlating 4d shell for Cu (5222). 

In the 1970s, a stream of nonempirical VB methods began to appear and were followed by many applications of accurate calculations. All these methods divide the orbitals in a molecule into inactive and active subspaces, treating the former as a closed-shell and the latter by a VB formalism. The programs optimize the orbitals, and the coefficients of the VB structures, but they differ in the manners by which the VB orbitals are defined. Goddard et developed the generalized VB (GVB) method, which uses semilocalized... 

Figure 5J Partitioning of the molecular orbital space into inactive, active, and external subspaces. The number of active subspaces differs between CASSCF, RASSCF, and GASSCF calculations...

---