Perturbation selection

The effects of electron correlation are investigted through the CIPSI (Configuration Interaction with Perturbatively Selected Configurations) calculations (4) of the molecular states. 

Table 4-1. CPU time for the perturbation selection. Cyan Fluorescent Protein, C H l C (Crsymmetry), with DZP level basis sets. The Is core and corresponding virtual orbitals were frozen. Total number of active space is 290 (51 occ. 239 unocc.)...

C, O and N atoms were treated as the frozen orbitals. Perturbation selection [29] was carried out at the LevelTwo level of thresholds. 

The perturbation selection breaks the invariance of the SAC/SAC-Cl energies on the unitary transformation of orbitals among the occupied and/or unoccupied manifolds. This is the source of the discontinuity in the potential energy surface, if the external perturbation induces a large (sudden) mixing of orbitals within the occupied and unoccupied orbitals. Suppose a deformation of benzene from the Dgh geometry to less symmetric one, the degenerate set of orbitals in occupied or unoccupied manifold makes a sudden (discontinuous) deformation of MOs (see Ref. [44] for some details]. The MOD method [46] can solve this problem that may occur in the optimization... 

The SAC-CI method formally scales as O(N ), where Ais the mrmber of basis functions. This scaling property means that the computational requirement rapidly increases with the size of the system. Therefore, the SAC-CI program adopted the perturbation selection method. With this method, the scaling property can be relaxed by the suitable selections of the excitation operator. In this case, LMOs are a rational choice for the reference orbitals. However, the selection may cause the discontinuity in the potential energy surface. This problem can be solved by the MOD method [46,47], as introduced in the Section 2. 

Localized MOs were used for the reference orbitals. The perturbation selection was carried out for selecting the doubleexcitation operators. 

The SAC-Cl method is very effective for describing both (i) spin-polarization and (ii) electron correlation corrections [136,140], For calculating the HFSCs, we have to be careful about perturbation selection [44], since the energy and spin density are very different properties. In the SAC-CI method, we can avoid the perturbation selection, since the sizes of the matrices to be diagonalized are small without selection, in contrast to ordinary Cl methods. The perturbation selection method taking into account of the HFSCs is also possible [142]. 

For homonuclear molecules, the g or u symmetry is almost always conserved. Only external electric fields, hyperfine effects (Pique, et al., 1984), and collisions can induce perturbations between g and u states. See Reinhold, et al., (1998) who discuss how several terms that are neglected in the Born-Oppenheimer approximation can give rise to interactions between g and u states in hetero-isotopomers, as in the HD molecule. An additional symmetry will be discussed in Section 3.2.2 parity or, more usefully, the e and / symmetry character of the rotational levels remains well defined for both hetero- and homonuclear diatomic molecules. The matrix elements of Table 3.2 describe direct interactions between basis states. Indirect interactions can also occur and are discussed in Sections 4.2, 4.4.2 and 4.5.1. Even for indirect interactions the A J = 0 and e / perturbation selection rules remain valid (see Section 3.2.2). 

The and e/f labels are really two different bookkeeping devices for the same physical property, but the e/f labels are more convenient, mainly for optical transition and perturbation selection rules. 

This e/f degeneracy between the two same-TV components will be lifted by interaction with a 2II state. If the potential curves of the 2 + and 2II states are identical and the configurations of the 2 + and 2II states axe Perturbation selection rules for unsymmetrized basis functions require that the following interactions be considered. The 2n1//2 state experiences two types of Afl = 0 interactions with 2E]f/,2 spin-orbit,... 

Several other kinds of information are available from the information in Fig. 5.7. Two consecutive vibrational levels of A1 are crossed by the same vibrational level of e3 . The Q branch (F2) crossing occurs in va = 1 and 0 at J = 48.9 and 61.5. Since the deperturbed A1 term energies at these two J-values are known by interpolation, accurate values for B(e3 ) and F(e3 ) can respectively be determined from the slope and intercept of the straight line drawn through these va = 1, J = 48.9 and va == 0, J = 61.5 term values. Alternatively, the J-values of all three 1II 3 (Fi,F2,F3) crossings are determined accurately at many perturbations, for example, (56.5, 61.5, 66.0) in Si160 A1 v = 0. Each 3X TV-level consists of three near-degenerate J components. The perturbation selection rule is A J = 0, thus the F3 and F3... 

Callis PR, Scott TW and Albrecht AC (1983) Perturbation selection rules for multiphoton electronic spectroscopy of neutral alternant hydrocarbons. J Chem Phys 78 16-22... 

Nearly all the peaks in the calculated Nj spectrum have a number of basis operators that contribute significantly. This indicates that the simple molecular orbital picture of the shake-up process is insufficient. " The results emphasize the need for some selection technique, such as the perturbation theory approach employed here, in the choice of the primary operator space. The addition of several extra shake-up basis operators by the perturbation selection criterion lowers the peak from 17.12 to 16.79 eV. The most important of these extra operators involves removal of a electron and de-excitation of a second Itt electron to the lw level. The importance of this operator, which acts only on the correlation part of 0>, is not obvious by pure chemical intuition. 

Introduction of a heavy atom perturber into the vicinity of a chromo-phore such as Trp leads to a heavy atom effect (HAE). The HAE drops off very rapidly with distance beyond van der Waals contact and thus is an indicator of short-range interactions. The HAE depends not only on distance, but on the location of the heavy atom with respect to the coordinate axes of the molecule. For an aromatic (tt, tt ) state such as we find in Trp, a relatively small HAE is found if the perturber atom is located in the molecular plane. Large perturbations require overlap between perturber and the n orbitals of the molecule. The sublevel specificity of the HAE has been shown by both theory and experiment" to depend on the location of the heavy atom. If z defines the out-of-plane direction, then location of the perturber directly along z perturbs selectively, whereas displacement into the xz plane perturbs both and T, and displacement into the yz plane perturbs Ty and T. The theory is based only on symmetry arguments and does not consider the relative sizes of the HAE when more than one sublevel is involved. 

We also performed direct SAC-CI calculations for the n- and branched alkanes and five amides. The basis sets used in the SAC-CI calculations were aug-cc-pVDZ (C, N, and O) and cc-pVDZ (H). To reduce the computational cost of SAC-CI, the excitation operators were selected using the perturbation selection scheme [35]. Because we calculated several excited states, moderate accuracy with the level two threshold for the perturbation selection was used 5 x 10 and 1 x 10 a.u. for the ground and excited states, respectively. The direct algorithm was adopted in the SAC-CI calculations [36]. 

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