Orbitals inactive

The electron density described by a core orbital space will of course strongly affect the nature of the active orbitals. The form of the inactive orbitals may be influenced by placing symmetry restrictions on them, or by invoking an initial orbital localization [11-15]. The localized orbitals that are not of interest for the VB description may then be placed in the core in the subsequent CASSCF or fully variational VB calculation and, if necessary, some or all of them may be frozen. 

There is one other step sometimes taken to make the CAS/RAS calculation more efficient, and that is to freeze the shapes of the core orbitals to those determined at the HF level. Thus, there may be four different types of orbitals in a particular MCSCF calculation frozen orbitals, inactive orbitals, RAS orbitals, and CAS orbitals. Figure 7.3 illustrates the situation in detail. Again, symmetry is the theoretician s friend in keeping the size of the system manageable in favorable cases. 

Before ending the discussion of the MCSCF Fock operator we shall transform it to a form which is more suitable in practical applications. Let us divide the occupied orbitals into two subsets the inactive orbitals, which are doubly occupied in all configurations, and the active orbitals, which are only partially occupied. The inactive orbitals we denote with indices ij,k,l..., and the active orbital with indices t,u,v,x... It is then possible to separate out the contribution from the inactive orbitals in the two-electron part of the Fock operator, by using the relation EipIO> = 28piIO>. The result of this separation can be written in the following form ... 

The inactive and active orbitals are occupied in the wave function, while the external (also called secondary or virtual) orbitals span the rest of the orbital space, defined from the basis set used to build the molecular orbitals. The inactive orbitals are kept doubly occupied in all configurations that are used to build the CASSCF wave function. The number of electrons occupying these orbitals is thus twice the number of inactive orbitals. The remaining electrons (called active electrons) occupy the active orbitals. 

However, there exist cases where it is advantageous to be able to use a larger set of active orbitals. The dimension of the CAS wave function can then become prohibitively large, and it may be of interest to look for other means of restricting the expansion length. This can be done in many ways, and results in a number of different types of MCSCF expansions. Assume that the number of inactive orbitals is and the number of active orbitals t, the number of active electrons being Na. We can then formally write the corresponding CAS wave function as ... 

This input specifies how many electrons are occupying the active orbitals. The number of electrons in inactive orbitals is of course twice the number of inactive orbitals. 

We may want to freeze some of the inactive orbitals, that is, not optimize... 

The number of inactive orbitals in each symmetry is given here. In our case they are all the doubly occupied a orbitals, which all belong to the first irreducible representation, alt of C2v. 

In CASSCF calculations only the rotation parameters Tit, T, and Tto have to be used, where i represents an inactive orbital, t an active orbital, and a an external orbital. Motivate ... 

Figure 1. Schematic representation of the determinants treated in the different DDCI approaches. The different classes of external determinants are labeled by the excitation operator, = uj,, that by acting on a determinant in the reference space generates a determinant of this class. The Following labels are used i, j for inactive orbitals t, u for active orbitals a, b for secondary orbitals. The open circles denote the creation of a hole in the inactive orbitals, whereas the crosses indicate the creation of an electron in the secondary space.

The BOVB method has several levels of accuracy. At the most basic level, referred to as L-BOVB, all orbitals are strictly localized on their respective fragments. One way of improving the energetics is to increase the number of degrees of freedom by permitting the inactive orbitals to be delocalized. This option, which does not alter the interpretability of the wave function, accounts better for the nonbonding interactions between the fragments and is referred to... 

This chapter considered molecules with high enough symmetry that can assist the distinction between active and inactive orbitals. Such facility is not always present in the general case, and this poses a danger that during the BOVB orbital optimization there will occur some flipping between the sets of active and inactive orbitals. This, however, depends on the BOVB level. 

Delocalization of the inactive orbitals (D-BOVB or SD-BOVB) is important for getting accurate energetics. Once again, it is important to make sure that the orbitals that are delocalized are the inactive ones, while the active set remains purely localized, which is the basic tenet of the BOVB method. To avoid a spurious exchange between the active and inactive spaces during the... 

BOVB Breathing orbital valence bond. A VB computational method. The BOVB wave function is a linear combination of VB structures that simultaneously optimizes the structural coefficients and the orbitals of the structures and allows different orbitals for different structures. The BOVB method must be used with strictly localized active orbitals (see HAOs). When all the orbitals are localized, the method is referred to as L-BOVB. There are other BOVB levels, which use delocalized MO-type inactive orbitals, if the latter have different symmetry than the active orbitals. (See Chapters 9 and 10.)... 

Here (p,

active orbitals of the left and right fragments, respectively, and the subscripts n and a stand for neutral and anionic fragments, respectively (recall that the cationic fragments have only inactive orbitals and no active ones). Note that the inactive orbitals (pi, (pi and (pi" of TVT11 are all different from each other, as are the active orbitals Ln, La, or Rn, Ra. These differences are pictorially represented in 9-11 by drawing orbitals with different sizes depending on the identity of the species as neutral, cationic or anionic. 

Several theoretical levels are conceivable within the BOVB framework. At first, the inactive orbitals may or may not be allowed to delocalize over the whole molecule (vide supra). To distinguish the two options, a calculation with localized inactive orbitals will be labeled "L", as opposed to the label "D" that will characterize delocalized inactive orbitals. The usefulness and physical meaning of this option will be discussed below using particular cases. 

Even though it is physically correct, the ROHF wave function suffers from the same defect as the GVB wave function for two-electron bonds. Thus, the active AOs are common for the two structures and are not adapted to their instantaneous occupancies, while the inactive orbitals are not adapted to the instantaneous charge of the fragments. Once again, this defect can be removed by use of the BOVB wave function that allows for different orbitals for different structures, as in eq 20 ... 

The computed equilibrium distance and bonding energy of F2- are displayed in Table 5. To appreciate better the sensitivity of active vs inactive orbitals to the breathing orbital effect, the latter has been introduced by steps In the first step no breathing orbitals are used (La = Lr, Ra = Rr, (Pi = cpf) this VBSCF calculation is nearly equivalent to the ROHF level. In the second step, only active orbitals are included in the breathing set (La 1- Lr, Ra Rr), while in the next step full breathing is permitted (La Lr, Ra Rr, cpi cpi ). The latter wave function, at the L-BOVB level, can be represented as in 26, 27 below. 

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