Spin, electronic, paired

Phthalocyanine complexes exhibit magnetic moments corresponding to low-spin, electron-paired species (57). One exception is lithium phthalocyanine manganate(O), a spin-free case which may occur for metals in their lowest oxidation state (57). On the basis of the low spin observed for phthalocyaninechromium(II) and phthalocyaninechloroiron(III), a metal-metal interaction which extends throughout the lattice has been postulated (55),... 

Fig. 11.8. A gear to test the power of the PauU exclusion principle fFerrai holeV fal The reference electron with the radius vector r and probe electron with the radius vector r + Cp (both with the same spin coordinate). From the sphere shown, an electron of the same spin as that of the reference one is expelled. Therefore, the same sphere is a residence for two electrons with opposite spins (electron pair) (b) Two parabolic Fermi holes result of the PauU exclusion principle. In each hole, the reference electrm is shown (represented by a small ball). A narrow well means a laige value of C(r) and therefore a small value of ELF, which means a small propensity to host an electron pair. In contrast to that, a wide well corresponds to a laige propensity to home there an election pair.

This indicates that the Hartree-Eock wavefunction has a hole for parallel-spin electron pairs. This hole is called the Fermi hole or exchange hole. In contrast, for anti-parallel spins (cti 02), the Hartree-Fock wavefunction approaches... 

This indicates that, for electron pairs of different spins, electronic motions are independent and have no correlation with each other. Obviously, it is not consistent with the above discussion that wavefunctions have Coulomb holes for close electrons. The Coulomb holes for anti-parallel-spin electron pairs are the main cause for electron correlation excluded in the HartrecF-Fock wavefunction. 

SCSN-MP2 is another parameterization of the spin scaling parameters which completely neglects the contribution from antiparallel-spin electron pairs to the MP2 energy while scaling the parallel contribution by 1.76 (HiU and Platts 2007). These spin-component scaled for nucleobases (SCSN) parameters were obtained by minimizing, with respect to SCS parameters. 

Soon after the appearance of the ELF, some interesting interpretations and remarks were given by Dobson [41]. He stated that the kernel of the ELF formula (cf. Eqs. (10) and (11)) is valid for states with zero Schrodinger current density, which explicitly means that time dependency would change the formula. Additionally, Dobson connected the Fermi hole curvature with the kinetic energy of the relative motion of same-spin electron pairs. 

It was claimed that this formulation follows Silvi s approach of spin-pair composition (cf. Sect. 2.10). Indeed, the Laplacian terms are connected with the number of (correlated) same-spin electron pairs in small arbitrary chosen region. However, the... 

As the 2-particle control functimi can serve, for instance, the same-spin electron pair density, />f"(ri,r2) normalized to the total number of spin electron pairs NaiNg— l)/2. The integral of the same-spin pair density over each micro-cell volume yields the number of same-spin pairs in the micro-cell. As will be shown later, the number of pairs U,- can be approximated by the integral of the pair density Laplacian (in which case 5c = 8/3 and tc(r,) is the Fermi hole curvature). The number of micro-cells with Uf" restricted to the value coo could be determined if the number of pairs formed between the micro-cells would be known as well. 

Fig. 10 ajRSP for Ne atom [72, 80] with micro-cells restricted to enclose 10 same-spin electron pairs. Left-, schematic diagram of section through the space partitioning. Right the height of the bars corresponds to the electron population in micro-cells... 

The volumes of the micro-cells are determined by the restriction < >, i.e., a change of the Wu value implies the change of the micro-cell volumes in accordance with Eq. (37). In case of micro-cells restricted to enclose a fixed amount Wo of same-spin electron pairs, the distribution of electron populations will scale with (Ou as follows (with 5c = 8/3, cf. Eq. (41) and Sect. 4.4) ... 

Because shows how alone an electron is, it could be inferred that the high ELI-D values around the hydrogen position describe a located opposite-spin electron pair (both a-spin and / -spin electrons are alone and there are 2 electrons in the H basin, which for independent particle would correspond to a pair). The inspection... 

It could be suspected that for hydrocarbons, the influence of correlation is not as much important as for other compounds. Then, possibly, the ELI-q for the singlet-coupled electrons would be of minor importance for the bonding analysis. To examine the extent of opposite-spin electron pairing in case of the C-C and C-H bonds, the C3H6 was analyzed. The CISD calculation of the cyclopropane molecule was performed with the triple-zeta basis CCT using 18 electrons excited into 75 orbitals. The correlated 2-matrix was used to compute the electron density as well as the ELI (and the spin-pair composition). 

Considering instead of ELI-q, the correlated spin-pair composition reveals that for this descriptor the extent of opposite-spin correlation is of minor importance. The spin-pair composition can be seen, similarly to ELI-q, as being based on the q-restricted space partitioning. However, the ratio between the opposite-spin and same-spin electron pairs behaves like the independent opposite-spin pairs would be taken into account (which in this view would in certain sense he proportional to the values of ELI-D see below). 

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