Second-order estimates

Sigma- and pi-type donor acceptor interactions Further details of the leading hp —hM donor-acceptor interactions are gathered in Table 2.5 and Figure 2.21. For each such interaction the table shows the hybrid form of the donor (hp)23 and acceptor (hM) orbitals, the occupancy of the acceptor, and the second-order estimate (cf. Eq. (1.35) or (2.7)) of the donor-acceptor stabilization energy. Let us now discuss some of the trends displayed in Table 2.5. 

The relative weakness of donor-acceptor stabilization in the cr-7t case is... 

The CASSCF wavefiinction is used as reference function in a second-order estimate of the remaining dynamical correlation effects. All valence electrons were correlated in this step and also the 3s and 3p shells on copper. Relativistic corrections (the Darwin and mass-velocity terms) were added to all CASPT2 energies. They were obtained at the CASSCF level using first-order perturbation theory. A level-shift (typically 0.3 Hartree) was added to the zeroth order Hamiltonian in order to remove intruder states [30]. Transition moments were conputed with the CAS state-interaction method [31] at the CASSCF level. They were... 

Fig.3.1. Estimates of the logarithmic-derivative function for free s electrons compared with the exact result D(x) = x cot x - 1, x = S/E explained in Sect. 4.4. The curve labelled to is the second-order estimate (3.51), E(D) is the third-order estimate (3.50). while Lau is the Laurent expansion (3.30) valid to third order in (E - EV)S2. The potential parameters used in the three estimates are derived in Sect.4.4 and listed in Table 4.4. The two open circles in the figure refer to the points (EVS2,DV) and (EVS2,D ), where EVS2 is K2S2 of Table 4.4...

Fig.4.2. Second-order estimate (3.51) of the function E(D) inverse to the logarithmic-derivative function D(E) for the 3d band in chromium. The second-order estimates of energies are denoted by a A...

Fig.4.3. Third-order (variational) estimate (3.50) of the energy function E(D) for the 3d band in chromium. The unphysical range of positive slope is indicated by a broken line. For comparison, the second-order estimate io(D) given in the previous figure is also included...

Fig.4.4. Estimates of the 3d potential function p(E) for chromium. The heavy line is the third-order estimate obtained from (2.9,3.50), the thin line is the second-order estimate obtained from (2.9,3.51), and the broken line is the linear approximation (3.41)...

From the second-order estimates (4.2,9) and the entries in Table 4.1 we... 

The non-Lewis /I-NBOs of vinoxy consist of the 1-c ric, hole (labeled LP in NBO output) as well as the usual valence antibonds ( ch> co co> co)and Rydberg orbitals. Figure 7.4d depicts the form of the most important /1-acceptor he, whose orbital energy (0.0874 a.u.) lies significantly below that of other non-Lewis NBOs (e.g., 0.0046 a.u. for the next-lowest Pco ) According to the second-order estimates... 

Kinetic exchange by second-order perturbation theory. Make a second-order estimate of the singlet energy taking into account the interaction between 5 = - ( ab + ba ) and the ionic states h = - ( ao + and h =... 

If Fdeie represents the effect of deleting a single <7 -+ cr NBO interaction, AEdeie may be directly compared (usually, within 10% accuracy) with the simple second-order estimate equation (16). However, for multiple dele-... 

---