Quantum mechanics exchange energy

As explained in Section 2-2-3, the energy of pairing two electrons depends on the Coulombic energy of repulsion between two electrons in the same region of space, II., and the purely quantum mechanical exchange energy,. The relationship between the... 

In the equation above, the functional /ixc p is the exchange-correlation (XC) energy functional and is the sum of the conventional quantum mechanical XC energy... 

The operator J corresponds to the classical Coulomb repulsion, while the quantum mechanical exchange operator, K, contains the permutation operator P(12), which has the effect of interchanging the coordinates of electron 1 and 2. This causes the exchange operator to be non-local and difficult to plot, unlike the local, multiplicative Coulomb operator, J. However, the exchange energy is... 

Exchange interactions only occur if the donor and acceptor ions are close enough for direct overlap of their electronic wavefunctions. Consequently, energy transfer due to quantum mechanical exchange interactions between the D and A ions is only important at very short distances (nearest neighbor positions). In fact, the transfer probability varies similarly to the overlap of the wavefunctions Pt a where... 

Slater determinants have a number of interesting properties. First, note that every electron appears in every spin orbital somewhere in the expansion. This is a manifestation of the indistinguishability of quantum particles (which is violated in the Hartree-product wave functions). A more subtle feature is so-called quantum mechanical exchange. Consider the energy of interelectronic repulsion for the wave function of Eq. (4.43). We evaluate this as... 

A notable exception are chemisorbed complexes in zeolites, which have been characterized both structurally and spectroscopically, and for which the interpretation of electronic spectra has met with a considerable success. The reason for the former is the well-defined, although complex, structure of the zeolite framework in which the cations are distributed among a few types of available sites the fortunate circumstance of the latter is that the interaction between the cations, which act as selective chemisorption centers, and the zeolite framework is primarily only electrostatic. The theory that applies for this case is the ligand field theory of the ion-molecule complexes usually placed in trigonal fields of the zeolite cation sites (29). Quantum mechanical exchange interactions with the zeolite framework are justifiably neglected except for very small effects in resonance energy transfer (J30). ... 

The quantum-mechanical exchange-correlation energy EXCD>] and the Pauli-Coulomb component Wxc(r) of the potential can be expressed in terms of a field c(r). This field is derived via Coulomb s law from the quantum-mechanical Fermi-Coulomb hole charge distribution pxc(r, r/) at r7 for an electron at r as... 

It is thus apparent that there are two different types of bond, the homopolar or covalent and the ionic. In the former the main contribution to the bond energy is due to the quantum mechanical exchange phenomena... 

Similar mechanisms can be provided for other reactions, such as parahydrogen conversion, hydrogen exchange reactions, dehydrations, etc. Reactions of the exchange type, where the energy levels in the two species are close, if not identical, can proceed in the van der Waals layer at low temperatures by virtue of the quantum mechanical exchange interaction. However, other classes of reactions are unlikely to proceed exclusively in other than the chemisorbed state. 

The second possibility of energy transfer is known as exchange type or Dexter energy transfer. Dexter ET is based on quantum mechanical exchange interactions, therefore it needs strong spatial overlap of the involved wavefunctions of D and A. Since the overlap of electronic wavefunctions decays exponentially with distance, it is expected that the rate constant koA decreases even more rapidly with distance R than observed in the case of singlet transfer. A schematic presentation of Dexter ET is shown in Fig. 21. Dexter ET occurs typically over distances which are similar to the van-der-Waals distance, i.e. R = 0.5 - Inm. The rate constant drops exponentially with the distance Rda between D and A ... 

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