Inclusion of the electron spin

The Hartree method is a conceptionally important step for things to come, especially concerning density-functional theory (see Section 2.12), simply because of the idea of self-consistency. The systematic inclusion of the electronic spin and, therefore, electronic exchange, leads to yet another method, which is the most important method of molecular quantum chemistry. 

INCLUSION OF THE ELECTRON AND NUCLEAR SPINS 3.6.1 Magnetic Hamiltonian with electron and nuclear spins... 

An additional feature that must be incorporated into the perturbation theory treatment of the IIN problem is the inclusion of the in-plane (Ayy) hyperfine terms because of the small T, - energy difference. In this model, the energies of the electron spin only (Am, = 0) transitions of triplet IIN are given by... 

The inclusion of the correlation corrections to the spin-spin coupling calculation via electron propagator is quite straightforward since at third order it can be written as... 

The purpose of this review is to discuss the main conclusions for the electronic structure of benzenoid aromatic molecules of an approach which is much more general than either MO theory or classical VB theory. In particular, we describe some of the clear theoretical evidence which shows that the n electrons in such molecules are described well in terms of localized, non-orthogonal, singly-occupied orbitals. The characteristic properties of molecules such as benzene arise from a profoundly quantum mechanical phenomenon, namely the mode of coupling of the spins of the n electrons. This simple picture is furnished by spin-coupled theory, which incorporates from the start the most significant effects of electron correlation, but which retains a simple, clear-cut visuality. The spin-coupled representation of these systems is, to all intents and purposes, unaltered by the inclusion of additional electron correlation into the wavefunction. 

The electron density is expanded in a similar way (with the inclusion of the m = 0 term). We use 7 Fourier coefficients in the expansion of electron density, 6 Fourier coefficients in the expansions of and V (only the first four coefficients turned out to be significantly different from zero), and 8 coefficients (for each of the spin-parallel and spin-antiparallel cases) in the two-body term. 

There are two types of electron correlation static and dynamic. The static correlation is related to the behavior of HF method at the dissociation limit of the molecule and deals with the long range behavior of this approach. On the other hand dynamic electron correlation is related to the electron repulsion term and is the reciprocal function of a distance between two electrons and thus represents short range phenomena. However, it should be noted that the electron correlation in the HF method is included in the indirect manner by the consideration of an electronic motion in an effective potential field due to the nuclei and the rest of the electrons and due to the inclusion of electron spin. Therefore, despite the known shortcomings, HF method has been extensively used in chemical calculations and has been quite successful for systems which are not extensive for electron correlation. 

As noted in the introduction to this chapter, a very wide range of experimental methods is now available for the study of the electronic structures of earth materials. Many of the most important methods have been discussed in the preceding sections, and an attempt made in Appendix B to list all of the relevant methods, along with a very brief explanation of each technique and information on further reading. One major spectroscopic method was not discussed above but is worthy of inclusion in this chapter electron spin resonance. 

For inclusion of the target states it is useful to define a further reduction of the notation by separating the radial coordinate of the projectile from the projectile-spin—angle and target coordinates. The variables displayed in the state vectors indicate the context of the reduced notation. The coordinate—spin representation of an N + l)-electron state is given in the notation of (7.26) as... 

By overlapping each H Is orbital with each one of these hybrid orbitals of C, in a maimer similar to H2, four equivalent functions are defined each localized in each formal CH bond four localized functions., as graphically illustrated in Fig. 8.6. After eventual mixing of ionic contributions and the inclusion of the spin-functions, the global electronic wavefunction can be constructed. 

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