Active subspaces

In the related work of Kim and Hynes [50], Equations (3.107) and (3.112) have been designated, respectively, by the labels SC (self-consistent or mean field) and BO (where Born-Oppenheimer here refers to timescale separation of solvent and solute electrons). More general timescale analysis has also been reported [50,51], Equation (3.112) is similar in spirit to the so-called direct RF method (DRF) [54-56], The difference between the BO and SC results has been related to electronic fluctuations associated with dispersion interactions [55], Approximate means of separating the full solute electronic densities into an ET-active subspace and the remainder, treated, respectively, at the BO and SC levels, have also been explored [52],... 

As shown by Clark [38] in a comprehensive computational study of (HnX XHn)+ radical cations (X= Li to C, Na to Si), one-electron bonds are already rather well described by simple Hartree-Fock theory. This is because the active system contains a single electron, so that the breathing orbital effect is ineffective in the active subspace, where each orbital is either empty or singly occupied as illustrated in 32, 33 for the OC bond. 

In the following we divide the internal orbital space into three subspaces the core orbitals, which are doubly occupied and not correlated the closed-shell orbitals, which are correlated and the active orbitals, which are only partially filled in the MCSCF wavefunction. We will show that it is sufficient to evaluate coupling coefficients only for the active subspace. For the special case that the reference wavefunction is a single closed-shell determinant, the algorithm then reduces to the closed-shell SCEP method described by Meyer, and no coupling coefficients have to be calculated explicitly. ... 

The generalized valence bond (GVB) method developed by Gaddard and coworkers does not employ a full Cl in the valence shell, but includes only the configurations needed to describe proper dissociation of a chemical bond. As will be demonstrated later, such a wavefunction represents a restricted form of the CASSCF wavefunction, where the active subspace is partitioned into subsets with a fixed number of electrons occupying the orbitals in each subset (acutally the GVB function is further restricted by allowing only specific spin couplings within each subset). 

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. 

Thus, in many cases it is possible to reduce the computational effort by adding constraints to the CASSCF wavefunction in the ways discussed above. The simple structure of the model is then lost to some extent. The selection of an active orbital space is extended to include several active subspaces with a fixed number of electrons attributed to each of them. It is not difficult to proceed one step further and allow limited excitations between the different subspaces. The number of electrons is then not fixed, but is allowed to vary between given limits. 

The wavefunctions generated using the model discussed above will in the following be called constrained complete active space (CCAS) wavefunctions. In the examples given above the different active subspaces were of different symmetry. Rotations between active orbitals are redundant variables in the CASSCF orbital optimization process. This is no longer the case when the active CSF space is reduced. In principle, it is necessary to introduce rotations... 

In calculations on larger systems it will, of course, not be possible to include all valence orbitals into the active subspace. It is also not necessary. Studies of an energy surface, i.e. for an chemical reaction, will normally be concentrated in regions where only one or two of the chemical bonds are broken. The calculation can then be performed by choosing as active orbitals only those taking part in the bond-breaking process. However, only semi-quantitative results can be expected from such calculations. Additional dynamical correlation effects have to be include via configuration-interaction calculations. 

Using the criteria for selecting the active subspace which were discussed above, it should not be necessary to include this orbital, since 2ag is correlated... 

The above occupations correspond to a constrained CASSCF wavefunction with the active subspace divided accordingly. This description is, however, not valid for the entire potential curve. At internuclear distances larger than 3 a.u. (around 1.6 A), the bonding changes completely and the dominant configuration is now... 

Is it possible to improve the results for NiH on the CASSCF level of accuracy by extending the active subspace The answer to this question is most probably no . The next important feature to include would be the radial correlation effects in the 3d shell of the nickel atom. The active subspace then has to include two sets of 3d orbitals together with the NiH electrons distributed among 12 orbitals. Such a calculation is well within the limits of the present capabilities, but it is not at all certain that it would give a balanced description of the correlation effects of the entire potential curve ... 

It is clear that an enlargement of the active subspace in NiH, which tries to account for the 3d pair correlation effects, will run into balance problems similar to those experienced in FH. The conclusion seems rather clear structure-dependent dynamical correlation effects in systems with high electron density cannot be accounted for in an MCSCF treatment in a balanced way. Large Cl or MBPT treatments then become necessary and the calculations have to include a large fraction of the total correlation energy, in order to give reliable results for relative energies. 

Using the same kind of augmented correlation-consistent basis sets employed in the above coupled cluster methods, but working with the complete active subspace (CASSCF and CASSCF( +1, +2)) approximations to the Cl expansion, Lawson and Harrison101 have investigated the variation with interatomic distance and spatial distribution of the quadrupole moments of P2, S2 and CI2. The a and it contributions to the quadrupole are resolved and the poor results obtained at the SCF level are attributed to the inadequate representation of the it system in the SCF approximation for P2 and S2. 

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...

---