Orbital corrections

Although the Hiickel method has now been supplanted by more complete treatments for theoretical analysis of organic reactions, the pictures of the n orbitals of both linear and cyclic conjugated polyene systems that it provides are correct as to symmetry and the relative energy of the orbitals. In many reactions where the n system is the primary site of reactivity, these orbitals correctly describe the behavior of the systems. For that reason, the reader should develop a familiarity with the qualitative description of the n orbitals of typical linear polyenes and conjugated cyclic hydrocarbons. These orbitals will be the basis for further discussion in Chapters 9 and 11. 

It is well-known that the hyperfine interaction for a given nucleus A consists of three contributions (a) the isotropic Fermi contact term, (b) the spin-dipolar interaction, and (c) the spin-orbit correction. One finds for the three parts of the magnetic hyperfine coupling (HFC), the following expressions [3, 9] ... 

In this way it was shown that the opt values derived from data for MCl, MFi, and MFg species gave excellent correlations with the occupation number, q, and that the n -<-75 (ys) peak positions could be well reproduced using Eqs. 5 (7) and 5 (2), with spin-orbit corrections. In all cases the correlations were significantly better when the relativistic terms were included than when they were omitted, and in Table 30 we list the xopt and oPt values derived from the 5 d data for MFg, MFg, and MF6 complexes. In the Table we also show the observed and the calculated band positions using the corrected forms of Eqs. 5 (7) and 5 (2). Once again the xopt vs. q plots yield slopes in excellent agreement with the ( —ri) values deduced from these equations. Finally, in Figs. 16... 

For the 5 d series therefore the inclusion of the spin-orbit corrections leads to improved x0pt vs. q plots, satisfactory predictions of the charge-transfer band via Eqs 5 (/) and 5 (2), and a strong correlation between... 

Thus, DFT-IGLO calculations of the 31P chemical shift without spin-orbital corrections yielded a value of S = 274, whereas additional spin-... 

In these expressions, n,m> is the number (0,1, or 2) of electrons occupying spatial orbital 0,-(O) in F0), and is a variational trial function (orthogonal to 0,-(O)) for the first-order orbital correction The expressions (1.20) allow us to treat the perturbative effects on an orbital-by-orbital basis, isolating the corrections associated with each HF orbital ,. Equations (1.18)-(1.20) involve only s/ng/e-electron operators and integrations, and are therefore considerably simpler than (1.5c) and (1.5d). 

Spin-orbit correction Eso is included for atomic species only. The spin-orbit correction is taken from experiment [49] where available and accurate theoretical calculations [50] in other cases. These corrections are particularly important for halide-containing systems [24], Molecular spin-orbit corrections are not included in G3 theory. 

On the other hand, high-level computational methods are limited, for obvious reasons, to very simple systems.122 Calculations are likely to have limited accuracy due to basis set effects, relativistic contributions, and spin orbit corrections, especially in the case of tin hydrides, but these concerns can be addressed. Given the computational economy of density functional theories and the excellent behavior of the hybrid-DFT B3LYP123 already demonstrated for calculations of radical energies,124 we anticipate good progress in the theoretical approach. We hope that this collection serves as a reference for computational work that we are certain will be forthcoming. 

The above is based on the calculation of a collective r for the whole molecule. This value changes the HOMO of either the diene or dienophile, as is necessary. This equation is accurate to about 0.5 eV on either side of the known values [15]. The value of ttotal is inserted into the HOMO-LUMO calculation as the parameter r Y), Note that in its pure form, this equation only yields values for the HOMO orbitals. Corrections are used for the calculation of the LUMO values. Table 1 contains examples of the Wiswesser Line Notation and the raw r values used in the computation of orbital energies. 

After this step one may already identify the paramagnetic and diamagnetic terms as arising from the pp and pn first order orbital corrections to the zeroth order wave function. The contribution is then further approximated by neglecting all off-diagonal terms in the pn-pn part of the Hessian... 

Includes 5s5p core-valence and spin-orbit corrections These values were cited in Ref (13). 

For atoms, G3 energies are defined to include a spin-orbit correction taken either from experiment or other high-level calculations. In addition, different coefficients are used in step (8). 

Table 10.2 Experimental values and spin- -orbit corrections...

A DFT-based third order perturbation theory approach includes the FC term by FPT. Based on the perturbed nonrelativistic Kohn-Sham orbitals spin polarized by the FC operator, a sum over states treatment (SOS-DFPT) calculates the spin orbit corrections (35-37). This approach, in contrast to that of Nakatsuji et al., includes both electron correlation and local origins in the calculations of spin orbit effects on chemical shifts. In contrast to these approaches that employed the finite perturbation method the SO corrections to NMR properties can be calculated analytically from... 

The MP4/6-31G(d) energy and the four corrections from step 4 are combined in an additive manner along with a spin-orbit correction AE(SO) for atoms only (hydrogen through chlorine) ... 

The atomic spin-orbit correction is taken from experiment96 where available and accurate theoretical calculations in other cases. The values are listed in Table 7. 

Table 7 Atomic Spin-Orbit Corrections (millihartrees) ...

Spin-orbit corrections are from Ref. 96 except where noted. Calculated value, Ref. 30. 

Atomic spin-orbital corrections, sometimes empirical, and a higher-level correction , HLC, to (hopefully) take any remaining inadequacies into account... 

These results imply that the electron density on the central carbon is higher than that on the terminal carbon.6 This makes no chemical sense and also disagrees with exact calculations. The error arises because we have ignored distortions of the molecular electron clouds. When the three-orbital correction term given in Equation (3.10) is added, VP2 becomes... 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

At B3LYP, including ZPE and atomic spin-orbit corrections from Reference 153. 

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