Spin-orbit energy corrections

First-order spin-orbit energy corrections A (SO) are included in the G2 energies for the third row including 2P and 3P atoms and 2n molecules. Values for the these corrections are obtained from spin-orbit configuration interaction calculations.88 91... 

So, within the limitations of the single-detenninant, frozen-orbital model, the ionization potentials (IPs) and electron affinities (EAs) are given as the negative of the occupied and virtual spin-orbital energies, respectively. This statement is referred to as Koopmans theorem [47] it is used extensively in quantum chemical calculations as a means for estimating IPs and EAs and often yields results drat are qualitatively correct (i.e., 0.5 eV). 

For all results in this paper, spin-orbit coupling corrections have been added to open-shell calculations from a compendium given elsewhere I0) we note that this consistent treatment sometimes differs from the original methods employed by other workers, e.g., standard G3 calculations include spin-orbit contributions only for atoms. In the SAC and MCCM calculations presented here, core correlation energy and relativistic effects are not explicitly included but are implicit in the parameters (i.e., we use parameters called versions 2s and 3s in the notation of previous papers 11,16,18)). 

Goudsmit and Uhlenbeck electron spin. Thomas spin-orbit energy. Heisenberg and Jordan relativistic correction to energy. Net result recovery of Sommerfeld energy levels, different quantum numbers. 

Figure 3. Corrections to 2p Koopmans ionization energies when including (R) relaxation and gsm (Q) relaxation and qed plus nuc terms (G) relaxation and gsm plus qed, and nuc terms. Upper left 2pj/2 corrections lower left 2p3/2 corrections upper right spin-orbit splitting corrections to Koopmans values lower right spin-orbit splitting discrepancies with respect to experimental values.

The lowest order correction to the so-called Koopmans theorem, which equates the electron binding energies with spin orbital energies, can then be expressed as... 

Aev is a core-valence correction obtained as the difference between ae-CCSD(T)/cc-pCVQZ and fc-CCSD(T)/cc-pCVQZ energies. Azpve is the harmonic zero-point vibrational correction obtained at the ae-CCSD(T)/cc-pCVTZ level, AAnh. is the correction due to anharmonic effects, calculated at the fc-MP2/cc-pVDZ level. Amvd is the correction for scalar-relativistic effects (one electron Darwin and mass-velocity terms) obtained at the ae-CCSD(T)/cc-pCVTZ level [101, 102], Aso is a spin-orbit coupling correction, which may be non-zero only for open-shell species. For the C, O and F atoms, Aso amounts to —0.35599, —0.93278 and —1.61153 kJ/mol, respectively [103]. The remaining contributions take care of the correction to the full triple excitations and perturbative treatment of quadruples Ax = ccsdt/cc-pvtz - ccsd(t)/cc-pvtz, A(q) = E CCSDT(Q)/cc-pVDZ—-E ccsDT/cc-pVDz- The final atomization energies are obtained by adding all the incremental contributions... 

This actually overestimates the spin-orbital energy by a factor of 2, because we have neglected the fact that an electron in a circular or elliptical orbit does not travel at a uniform velocity V, but experiences acceleration. The effect of correcting for this is to cancel [5] (or nearly cancel, according to Schwinger) the g factor g, and we write... 

These methods can give us useful information on radicals in a manner similar to that for closed-shell systems, provided the exploitation is correct. Of course, in expressions for total energy, bond orders, etc., a singly occupied orbital must be taken into account. One should be aware of areas where the simple methods give qualitatively incorrect pictures. The HMO method, for example, cannot estimate negative spin densities or disproportionation equilibria. On the other hand, esr spectra of thousands of radicals and radical ions have been interpreted successfully with HMO. On the basis of HMO orbital energies and MO symmetry... 

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