Antisymmetrization implications

In the case of ET, the reaction path starts with the electron in LUMO of the donor. Along the reaction path, the orbital shifts character, from donor to acceptor orbital. At the transition state LUMO and LUMO+1 for the full system are symmetric and antisymmetric combinations of the two local LUMO s (fig.2). Since for any of the two orbitals, the negative of the orbital energy approximates the electron affinity according to Koopmans theorem, the following implication holds ... 

Figure 4.3 Comparision of EDA frozen energies in kcal/mol along an angle for water dimer [upper] and formamide dimer [lower] with DEDA [blue curves] and MO-EDA [red curves] The MO-EDA employs the Heitler-London [HL] antisymmetrization of two fragments wave functions to represent the frozen density state. Reprinted with permission from Lu, Z., Zhou, N., Wu, Q. and Zhang, Y. Directional dependence of hydrogen bonds A density-based energy decomposition analysis and its implications on force field development./Chem Theory Comput 7, 4038-4049 [2011]. Copyright [2011] American Chemical Society.

Note that the complete wavefunction as written in Eq. (2.47) changes sign if the labels of the electrons (1 and 2) are interchanged. W. PauU pointed out that the wavefunctions of all multielectronic systems have this property. The overall wavefunction invariably is antisymmetric for an interchange of the coordinates (both positional and spin) of any two electrons. This assertion rests on experimental measurements of atomic and molecular absorption spectra absorption bands predicted on the basis of antisymmetric electrOTiic wavefunctirais are seen experimentally, whereas bands predicted on the basis of symmetric electronic wave-functions are not observed. Its most important implication is the Pauli exclusion principle, which says that a given spatial wavefunction can hold no more than two electrons. This follows if an electron can be described completely by specifying its spatial and spin wavefunctions and electrons have only two possible spin wave-functions (a and fi). 

For azoalkanes, the orbital from which excitation occurs is the antisymmetric combination of two s-p-hybrid nitrogen lone pairs, i.e., an in-plane n molecular orbital [105]. A special characteristic of azoalkanes is that their configuration can be cis or Irans (Scheme 3.1), which has implications for the spatial distribution of lone-pair... 

There are two important properties of the antisymmetrizer. First, if it is applied to a wavefunction that is already properly antisymmetric, it will make no change. The implication of this statement is that if the antisymmetrizer is applied twice to an arbitrary function, the same result will be achieved as if it is applied only once the second application does not do anything. This is the condition of idempotency, and Aj is an idempotent operator. The operator equation that expresses this fact is... 

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