Number of active orbitals

Maximum number of active orbitals in the Cl 15 Maximum number of determinants 350... 

Table 4. HF dependence of the polarizability tensors (in atomic units) on the number of active orbitals employed in the °°°CAS -MCSCF calculations using the daug-cc-pVQZ basis set at three internuclear distances...

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-... 

We shall not perform the somewhat elaborous calculation of the MC wave function in detail. A somewhat simpler example is the dissociation of a double bond and it is given as an exercise (exercise 2). Here we only note that the number of configuration state functions (CSF s) will increase very quickly with the number of active orbitals. In most cases we do not have to worry about the exact construction of the MC wave function that leads to correct dissociation. We simply use all CSFs that can be constructed by distributing the electrons among die active orbitals. This is the idea behind the Complete Active Space SCF (CASSCF) method. The total number of such CSFs is for N2 175 for a singlet wave function. A further reduction is obtained by imposing spatial symmetry. All these CSFs are not included in a wave... 

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 ... 

The CASSCF method itself is not very useful for anything else than systems with few electrons unless an effective method to treat dynamical correlation effects could be developed. The Multi-Reference Cl (MRCI) method was available but was limited due to the steep increase of the size of the Cl expansion as a function of the number of correlated electrons, the basis set, and the number of active orbitals in the reference function. The direct MRCI formulation by P. Siegbahn helped but the limits still prevented applications to larger systems with many valence electrons [20], The method is still used with some success due to recent technological developments [21], Another drawback with the MRCI approach is the lack of size-extensivity, even if methods are available that can approximately correct the energies. Multi-reference coupled-cluster methods are studied but have not yet reached a state where real applications are possible. 

This number increases quickly with the number of active orbitals and with todays computational technology the practical limit lies around 15 orbitals unless the number of electrons (or the number of holes) is much smaller than 15. If we want to... 

The major problem with the CASSCF method is the limited number of active orbitals that can be used. However, one notes in many applications that some of these orbitals will have occupation numbers rather close to two for the whole process one is studying, while others keep low occupations numbers. The restricted Active Space (RAS) SCF method was developed to handle such cases [15, 24], Here, the active space is partitioned into three subspaces RAS1, RAS2, and RAS3 with the following properties ... 

The first point is that A is normally read from disk in triangular form, and must be transcribed to full square form, a process which is thoroughly vectorisable. The amount of store allowed for each column of the square form of A would normally be N words. However in the case that N is divisable by 8, we allocate N+1 words, which prevents subsequent store conflicts. A similar store allocation algorithm is applied to Q, whose dimension is in general N rows by M columns (M[Pg.26]

Examples of photoreactions may be found among nearly all classes of organic compounds. From a synthetic point of view a classification by chromo-phore into the photochemistry of carbonyl compounds, enones, alkenes, aromatic compounds, etc., or by reaction type into photochemical oxidations and reductions, eliminations, additions, substitutions, etc., might be useful. However, photoreactions of quite different compounds can be based on a common reaction mechanism, and often the same theoretical model can be used to describe different reactions. Thus, theoretical arguments may imply a rather different classification, based, for instance, on the type of excited-state minimum responsible for the reaction, on the number and arrangement of centers in the reaction complex, or on the number of active orbitals per center. (Cf. Michl and BonaCid-Kouteck, 1990.)... 

One way to reduce the CSF expansion length is to allow only a subset of orbitals to be employed in generating the CSF list. In this case the wavefunction is invariant to transformations among the active orbitals (the orbitals potentially occupied in some CSF) and to transformations among the virtual orbitals (the orbitals that are not occupied in any CSF), but it depends on transformations that mix the orbitals of these subspaces with each other. The number of expansion terms in this subspace full Cl is given by Eq. (201) where the number of active orbitals is used instead of the total number of orbitals. 

Table IV shows the number of variational degrees of freedom for full Cl expansions and for RCI expansions for singlet wavefunctions for which the number of active electrons is equal to the number of active orbitals. The number of variational degrees of freedom is one less than the CSF expansion length. This number is used in Table IV for the direct comparison of the various non-linear expansion wavefunctions discussed in the following sections. Expansion space reductions due to spatial symmetry are neglected in this list. Only the variational degrees of freedom due to the expansion coefficient variations are given since the remaining ones, resulting from orbital variations, depend not on the number of active orbitals but instead on the total number of basis functions. Expansion spaces for the RCI wavefunction and the more general direct product full Cl expansions discussed previously, which explicitly include the effects of spatial symmetry, may be efficiently represented with the symmetry-dependent DRT.

As pointed out in the introduction, the CASSCF configuration space quickly becomes unmanageably large when the number of active orbitals is increased. While this does not create any serious problems in most applic-... 

Number of active orbitals. 357, 379 Numbering system of orbitals. 17 Nylon 6, 476... 

As can be computed from Table 20.3, the maximum differences in excitation energy for the largest three (two) numbers of active orbitals is 0.09 (0.05) eV. We can, therefore, consider that the excitation energies at the MC-SCF level are almost converged values for... 

The number of active orbitals that can be used in a CASSCF calculation is limited. Some problems needs larger active spaces than can be used today. It is then possible to use a less general way to construct the wave function, for example, the restricted active space (RAS) SCF method [79]. Such methods can use larger active spaces, but there exist no corresponding RASPT2 program yet. Such a code would be of great value. 

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