Active orbitals occupied

MMCC(2,3)/CI calculations. This alone leads to considerable savings in the computer effort, since we only have to construct NoNuri n moments where No (Af ) is the number of active orbitals occupied (unoc-... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Figure 51. 1,3-cyclohexadiene (a) and all-cw-hexatriene (b) isomers of CeHg with the respective third active orbitals (highest occupied n-orbital) and breaking C—C distances in angetroms. The bold line indicates the C2 rotation axis. Taken from Ref. [48]. 

As the contribution of the 5s and 5p orbitals of Ba to Wd is quite significant [43] for the BaF molecule, we have included these orbitals of Ba in our Cl active space for the calculation of Wd and p(, for the ground state of the BaF molecule. The occupied orbitals above the 25th are also included in the RASCI space from energy consideration. Thus altogether 17 active electrons (9a and 8(3) are included in the Cl space. The present calculations for Wd consider nine sets of RASCI space, which are constructed from 17 active electrons and 16, 21, 26, 31, 36, 46, 56, 66, and 76 active orbitals to analyze the convergence of Wd. 

With such a formulation, fpp = — IPp (Ionization Potential) when the orbital p is doubly occupied and fpp = —EAp (Electron Affinity) when the orbital is empty. The value of fpp will be somewhere between these two extremes for active orbitals. Thus, for orbitals with occupation number one, fpp = —j(IPp + EAp). This formulation is somewhat unbalanced and will... 

Table 5. Post-HF activation barriers for the insertion reaction of ethene into the Zr-CH3 bond of the HjSifCpEZrCH species. All the reported insertion barriers were obtained through single point calculations on the MP2 geometries of Tables 3 and 4 (corresponding to run 3 in this Table). In the valence calculations the Is orbitals on the C atoms, the orbitals up to 2p on the Si atom and up to the 3d on the Zr atom where not included in the active orbitals space. In the full MP2 calculations all occupied orbitals were correlated.

Firstly, inclusion of polarization functions on the C and H atoms of the reactive groups (CH3 and C2H4) reduces considerably the insertion barrier (compare runs 1 and 2 as well as runs 6 and 7 ) and seems to be mandatory. Instead, inclusion of polarization functions on the ancillary H2Si(Cp)2 ligand has a negligible effect on the calculated insertion barrier (compare runs 2 and 3 as well as runs 7 and 8). Extension of the basis set on the reactive groups lowers further the insertion barrier (compare runs 7 and 9). Both the MIDI basis set on Zr, and the SVP basis set on the remaining atoms decrease the insertion barrier (compare runs 3, 5 and 8). Finally, the extension of the active orbitals space to include all the occupied orbitals reduces sensibly the insertion barrier (compare runs 3 and 4). 

The generic chemical problem involving both dynamic and nondynamic correlation is illustrated in Fig. 1. The orbitals are divided into two sets the active orbitals, usually the valence orbitals, which display partial occupancies (assuming spin orbitals) very different from 0 or 1 for the state of interest, and the external orbitals, which are divided into the core (largely occupied in the target state) or virtual (largely unoccupied in the target state) orbitals. The asymmetry between... 

In active-space calculations, the total orbital space is usually partitioned into external core orbitals (c), active orbitals (a), and unoccupied virtual (external) orbitals (v). (There can additionally be some frozen core orbitals that remain doubly occupied throughout the calculation.)... 

The fact that the lowest two orbitals of the reactants, which are those occupied by the four 7t electrons of the reactant, do not correlate to the lowest two orbitals of the products, which are the orbitals occupied by the two o and two 7t electrons of the products, will be shown later in Chapter 12 to be the origin of the activation barrier for the thermal disrotatory rearrangement (in which the four active electrons occupy these lowest two orbitals) of 1,3-butadiene to produce cyclobutene. 

Because a particular active orbital may be occupied by zero, one, or two electrons in any given determinant, these MCSCF orbitals do not have unique eigenvalues associated with them, i.e., one cannot discuss the energy of the orbital. Instead, one can describe the occupation number of each such orbital i as... 

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. 

The inactive and external orbital spaces have the same properties as for CAS wave functions. The RAS 1 space consists of orbitals in which a certain number of holes may be created. One could for example allow single and double excitations out of this orbital space. Normally, all these orbitals would be doubly occupied in a CAS calculation. The RAS 2 space has the same properties as the active orbital space in a CAS wave function all possible occupations and spin couplings are allowed. Finally the RAS 3 space is allowed to be occupied with up to a given number of electrons. A variety of wave functions can be created using the RAS concept. For example, by making the RAS 2 space empty, one arrives at a conventional singles and doubles (triples,... 

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. 

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