Pairing theorem, description

Proof. We shall use the description of (C2) in terms of matrices given in Theorem 1.14. Suppose Z is a T-invariant O-dimensional subscheme in (C2), and corresponds to a triple of matrices (Bi, B2, i). Recall that it is given as follows Define a iV-dimensional vector space V as H°(Oz), and a 1-dimensional vector space W. Then the multiplications of coordinate functions z, z2 6 C define endomorphisms Bi, B2. The natural map Oc2 —> Oz defines a linear map i W V. Prom this construction, V is a T-module, and W is the trivial T-module. The pair (Bi,B2) is T-equivariant, if it is considered as an element in Hom(V, Q V), where Q is 2-dimensional representation given by the inclusion T C SU(2). (This follows from that (Zi,z2) is an element in Q.) And i is also a T-equi variant homomorphism W —> V. 

Thus the spectrum which arises when Eq. (8) is Fourier transformed consists of a set of -functions at the energies corresponding to the stationary states of the ion (which via the theorem of Koopmans) are the one-electron eigenvalues of the Hartree-Fock equations). The valence bond description of photoelectron spectroscopy provides a novel perspective of the origin of the canonical molecular orbitals of a molecule. Tlie CMOs are seen to arise as a linear combination of LMOs (which can be considered as imcorrelated VB pairs) and coefficients in this combination are the probability amplitudes for a hole to be found in the various LMOs of the molecule. 

The large maxima of the electron density are expected and are found at the nuclear positions Ra. These points are m-limits for the trajectories of Vp(r), in this sense they are attractors of the gradient field although they are not critical points for the exact density because the nuclear cusp condition makes Vp(Ra) not defined. The stable manifold of the nuclear attractors are the atomic basins. The non-nuclear attractors occur in metal clusters [59-62], bulk metals [63] and between homonu-clear groups at intemuclear distances far away from the equilibrium geometry [64]. In the Quantum Theory of Atoms in Molecules (QTAIM) an atom is defined as the union of a nucleus and of the electron density of its atomic basin. It is an open quantum system for which a Lagrangian formulation of quantum mechanics [65-70] enables the derivation of many theorems such as the virial and hypervirial theorems [71]. As the QTAIM atoms are not overlapping, they cannot share electron pairs and therefore the Lewis s model is not consistent with the description of the matter provided by QTAIM. 

The LDM codes for more than one aspect of the electron distribution in the molecule (one-electron density and pair density) as described in Sect. 3.1 above. Equations (3.5-3.10) show that an LDM contains information on atomic populations, atomic charges, the total number of electrons in the molecule (and their localized and delocalized subpopulations), and also two-electron information derived from the pair density, that is, the full atom-atom delocalization matrix of the system. It is thus expected that the LDM codes strongly for aromaticity by virtue of the first Hohenberg-Kohn theorem. The core question is how to get from LDMs to a description of aromaticity ... 

The approximate treatment of the bonding constraints in Eq. 3 may be motivated by recourse to the Flory theorem [3,4], which states that in polymer melts it is impossible to discern whether a pair of nonbonded nearest-neighbor united atom groups belongs to different polymer chains or to distant portions of the same polymer molecule. Thus, in the lattice model description of polymer systems, the excluded volume prohibition of multiple occupancy of a site is more important than the consequences of long-range chain connectivity. Based on the Flory theorem, we introduce the zeroth-order mean-field average... 

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