Orbitals interaction energy

AiiEohc ntgcao 137.036 Thus, the spin-orbit interaction energy is about 5 X 10 times smaller than 1 11-... 

Since the spin-orbit interaction energy is small, the solution of equations (7.43) to obtain E is most easily accomplished by means of perturbation theory, a technique which is presented in Chapter 9. The evaluation of E is left as a problem at the end of Chapter 9. 

Using first-order perturbation theory, show that the spin-orbit interaction energy for a hydrogen atom is given by... 

This relative arrangement of cation vacant orbitals corresponds to the drop in orbital interaction energy in the series of complexes C2H2Li+ > C2H2K+ > C2H2Na+. 

Experimental log k2 values were correlated with Brown para-localization energies, Dewar reactivity numbers, Herndon structure count ratios, Hess-Schaad resonance energy differences, indices of free valence, and second-order perturbation stabilization energies. The latter are based on Fukui s frontier orbital theory [67] which classifies the Diels-Alder reaction of benzenoid hydrocarbons with maleic anhydride as mainly HOMO (aromatic hydrocarbon)-LUMO (maleic anhydride) controlled. However, the corresponding orbital interaction energy given by... 

It is possible to use as the basis functions symmetry-adapted combinations of primitive basis functions. This affords a decomposition of the orbital interaction energy of Eq. [21] according to irreducible representations of the point group... 

Orbital interaction energies are limited to a, ft and 0. The Fock matrix orbital interactions are limited to a, jl and 0, depending on whether the Htj interaction is, respectively i,i, adjacent, or further-removed. The value of ft does not vary smoothly with the separation of the orbitals, although logically it should decrease continuously to zero as the separation increases. 

Orbital interaction energies are calculated and vary smoothly with geometry. The EHM Fock matrix orbital interactions //, are calculated in a way that depends on the distance apart of the orbitals, so their values vary smoothly with orbital separation. 

The value in parentheses gives the percentage contribution to the total attractive interactions. The value in parentheses gives the percentage contribution to the total orbital interactions. Energy difference between the planar and perpendicular form. 

Apart from their asymptotic form for 0 < /3 < a%—a (Equations 2.20 and 2.23 in Section 2.1), the general expressions for the orbital interaction energies of the heteropolar MOs are rather complicated in the case of nonorthogonal AOs. For the sake of simplicity, we shall content ourselves with the simpler expressions occurring in the case of orthogonality between the interacting AOs Xi and 2 Under this assumption, Equations (2.58) simplify to ... 

First, we look at the term which is closely related to the term from the perturbational analysis. The trend predicted for the influence of the trans-ligand is reflected in the values. We also note that the orbital interaction energy for the tungsten nitrosyl hydride systems is essentially the same, independent of the number of phosphine ligands. For all three compounds, the steric contribution AP is stabilizing, and therefore dominated by attractive electrostatic interactions. Increasing the number of P-donors further increases the absolute value of A , and strengthens the TM-H bond. 

Figure 4.21. The orbital interaction energy scheme for sixfold coordinated titanium.

Increased a donation by itself cannot cause X-H bond rupture BD is responsible for this task. Sorting out these two components is important in understanding (T-bond activation, and the BD component, bd of the total M-Hj orbital interaction energy can be separated from [Pg.346]

Calculating the spin-orbit interaction energy 8.0 by finding the eigenfunctions and eigenvalues of the operator + Hs.o. where (u.i) is the Hamiltonian of Eq. (11.1), is difficult. One therefore usually estimates 8.0. by using perturbation theory. Except for heavy atoms, the effect of s.o. is small compared with the effect of (u.i), and first-order perturbation theory can be used to estimate 80.-... 

For a raany-electron atom, it can be shown (Bethe and Jackiw, p. 164) that the spin-orbit interaction energy is... 

We have based the discussion on a scheme in which we first added the individual electronic orbital angular momenta to form a total-orbital-angular-momentum vector and did the same for the spins L = S,- L, and S = 2i S,. We then combined L and S to get J. This scheme is called Russell-Saunders couplit (or L-S coupling) and is appropriate where the spin-orbit interaction energy is small compared with the interelec-tronic repulsion energy. The operators L and S commute with + W,ep, but when is included in the Hamiltonian, L and no longer commute with H. (J does commute with + //rep + Q ) If the spin-orbit interaction is small, then L and S almost commute with (t, and L-S coupling is valid. 

Hie spin-orbit interaction energy in diatomic molecules can be shown to be well approximated by A A2, where A depends on A and on the intemuclear distance R but not on 2. The spacing between levels of the multiplet is thus constant. When A is positive, the level with the lowest value of A + 2 lies lowest, and the multiplet is regular. When A is negative, the multiplet is inverted. Note that for A 0 the spin multiplicity 25+1 always equals the number of multiplet components. This is not always true for atoms. 

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