Orbital relaxation

This lineshape analysis also implies tliat electron-transfer rates should be vibrational-state dependent, which has been observed experimentally [44]- Spin-orbit relaxation has also been identified as an important factor in controlling tire identity of botli electron and vibrational-state distributions in radiationless ET reactions. 

In this model, the Hartree-Foek orbitals of the parent moleeule are used to deseribe both the parent and the speeies generated by eleetron addition or removal. It is said that sueh a model negleets orbital relaxation whieh would aeeompany the eleetron addition or removal (i.e., the reoptimization of the spin-orbitals to allow them to beeome appropriate to the daughter speeies). 

That singly exeited CSFs allow for orbital relaxation ean be seen as follows. Consider a wavefunetion eonsisting of one CSF ( )i... ( )i... ( )n to whieh singly exeited CSFs of the form ( )i... ( )m... (1)n have been added with eoeffieients Ci m ... 

In summary, doubly exeited CSFs are often employed to permit polarized orbital pair formation and henee to allow for eleetron eorrelations. Singly exeited CSFs are ineluded to permit orbital relaxation (i.e., orbital reoptimization) to oeeur. 

CCSD Energy Second Derivatives. The number in the parenthesis is CCSD with orbital relaxation. 

Based on the foregoing discussion, one might suppose that the Fukui function is nothing more than a DFT-inspired restatement of frontier molecular orbital (FMO) theory. This is not quite true. Because DFT is, in principle, exact, the Fukui function includes effects—notably electron correlation and orbital relaxation—that are a priori neglected in an FMO approach. This is most clear when the electron density is expressed in terms of the occupied Kohn-Sham spin-orbitals [16],... 

Differentiating this expression with respect to the number of electrons gives a frontier orbital approximation in Equation 18.13 plus a correct due to orbital relaxation [17,18],... 

In most cases, the orbital relaxation contribution is negligible and the Fukui function and the FMO reactivity indicators give the same results. For example, the Fukui functions and the FMO densities both predict that electrophilic attack on propylene occurs on the double bond (Figure 18.1) and that nucleophilic attack on BF3 occurs at the Boron center (Figure 18.2). The rare cases where orbital relaxation effects are nonnegligible are precisely the cases where the Fukui functions should be preferred over the FMO reactivity indicators [19-22], In short, while FMO theory is based on orbitals from an independent electron approximation like Hartree-Fock or Kohn-Sham, the Fukui function is based on the true many-electron density. 

Figures 7 and 8 plot deviations of total energies from FCI results for the various methods. It is clear that the CASSCF/L-CTD theory performs best out of all the methods smdied. (We recall that although the canonical transformation operator exp A does not explicitly include single excitations, the main effects are already included via the orbital relaxation in the CASSCF reference.) The absolute error of the CASSCF/L-CTD theory at equilibrium—1.57 mS (6-31G), 2.26 m j (cc-pVDZ)—is slightly better than that of CCSD theory—1.66m j (6-31G), 3.84 m j (cc-pVDZ) but unlike for the CCSD and CCSDT theories, the CASSCF/L-CTD error stays quite constant as the molecule is pulled apart while the CC theories exhibit a nonphysical turnover and a qualitatively incorrect dissociation curve. The largest error for the CASSCF/L-CTD method occurs at the intermediate bond distance of 1.8/ with an error of —2.34m (6-3IG), —2.42 mE j (cc-pVDZ). Although the MRMP curve is qualitatively correct, it is not quantitatively correct especially in the equilibrium region, with an error of 6.79 mEfi (6-3IG), 14.78 mEk (cc-pVDZ). One measure of the quality of a dissociation curve is the nonparallelity error (NPE), the absolute difference between the maximum and minimum deviations from the FCI energy. For MRMP the NPE is 4mE (6-3IG), 9mE, (cc-pVDZ), whereas for CASSCF/ L-CTD the NPE is 5 mE , (6-3IG), 6 mE , (cc-pVDZ), showing that the CASSCF/L-CTD provides a quantitative description of the bond breaking with a nonparallelity error competitive with that of MRMP.

KT states that the IP is given by the HF orbital energy with opposite sign (— ,), calculated in the neutral system. It has long been recognized that in general there is an excellent agreement between the KT values and the experimental ones because of the fortuitous cancellation of the correlation and orbital relaxation effects. A survey of Table III reveals that KT consistently overestimates the hrst IPs except for P2. 

That singly excited CSFs allow for orbital relaxation can be seen as follows. Consider a wavefunction consisting of one CSF l(f>i... (f>j... (f l to which singly excited CSFs of the form l( ) ]... (f>m... ()xl have been added with coefficients Qjm ... 

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