Spin-orbit exchange

In order to proceed comfortably, the notation should again be simplified. Let us drop the superscript (0) denoting the zeroth-order functions (other functions will not appear in our considerations). Additionally, we omit the subscript k from the determinants containing the virtual spin-orbitals. In return, we explicitly specify the pattern of spin-orbital exchange. Using O Eq. 3.108, the Slater determinant can be written as... 

From the Slater rules it is easy to estimate that the only non-vanishing terms will arise from the double spin-orbital exchange in the wave function. The intuitive statement that the low-order corrections should have the larger impact and the above considerations lead to the most important accomplishment of this section the largest contribution to the corrections of the Hartree-Fock function arises from the functions with the doubly exchanged spin-orbitals. 

It is no longer a block diagonal matrix - all blocks contribute to its eigenvalues and one can count on some improvement. An interesting observation, however, is that here the functions with the single spin-orbital exchange also have influence on the energy via the (S f/ D) and (D H S) blocks. 

Limiting ourselves to the terms corresponding to not more than four spin-orbital exchanges and writing it in the ordered way according to the number of exchanges, one gets... 

Further terms are not necessary in such a case the functions on both sides of the integral would differ with four and more spin-orbital exchanges (and Hamiltonian is still a sum of one-and two-electron operators). Similarly, the expansion in the integral ( Eo ff o) will also be truncated on the second term ... 

The HF [31] equations = e.cj). possess solutions for the spin orbitals in T (the occupied spin orbitals) as well as for orbitals not occupied in F (the virtual spin orbitals) because the operator is Flennitian. Only the ( ). occupied in F appear in the Coulomb and exchange potentials of the Fock operator. 

Spin orbitals of a and p type do not experience the same exchange potential in this model because contains two a spin orbitals and only one p spin orbital. A consequence is that the optimal Isa and IsP spin orbitals, which are themselves solutions of p([). = .([)., do not have identical orbital energies (i.e. E p) and are... 

Thus E. is the average value of the kinetic energy plus the Coulombic attraction to the nuclei for an electron in ( ). plus the sum over all of the spin orbitals occupied in of the Coulomb minus exchange interactions. If is an occupied spin orbital, the temi [J.. - K..] disappears and the latter sum represents the Coulomb minus exchange interaction of ( ). with all of the 1 other occupied spin orbitals. If is a virtual spin orbital, this cancellation does not occur, and one obtains the Coulomb minus exchange interaction of cji. with all N of the occupied spin orbitals. 

This is an example of a Mobius reaction system—a node along the reaction coordinate is introduced by the placement of a phase inverting orbital. As in the H - - H2 system, a single spin-pair exchange takes place. Thus, the reaction is phase preserving. Mobius reaction systems are quite common when p orbitals (or hybrid orbitals containing p orbitals) participate in the reaction, as further discussed in Section ni.B.2. 

A I lai lcee-Fock calculation provides a set of orbital energies, e,. What is the significance oi these The energy of an electron in a spin orbital is calculated by adding the core inleraclion to the Coulomb and exchange interactions with the other electrons in the svstein ... 

Spin-orbitals of a and P type do not experience the same exchange potential in this model, which is clearly due to the fact that P contains two a spin-orbitals and only one P spin-orbital. 

The second term in equation (1-26) is the exchange contribution to the HF potential. The exchange operator K has no classical interpretation and can only be defined through its effect when operating on a spin orbital ... 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

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