Exchange-type integrals

The Intermediate Neglect of Differential Overlap (INDO) method, originally developed by Pople and Beveridge [8] and Pople et al. [44], uses the ZDO approximation [Eq. (26)] only for two-center integrals. The elements of the INDO Fock operator are therefore modified with respect to CNDO mainly by the inclusion of one-center exchange-type integrals (pv pv). 

The second integral, which is an exchange-type integral (see Section 3.2.4), is much smaller than the first, and consequently the A-doubling difference of Eq. (3.4.35) is always positive (Horani et ai, 1967). 

Now, it is physically reasonable that the Coulomb-type integral is always greater than the associated exchange-type integral and, in fact, it has been proved that this is so, thus the bracketed repulsion term is always positive. 

Roothaan open-shell treatments involving a closed-shell subsystem and outer unpaired electrons interacting through two-index integrals of Coulomb and exchange type only [43],... 

These integrals are clearly of the exchange type because they involve the coulombic interaction of the 2s 2px y or z overlap charge density with itself. 

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

Mathematical models of cross-flow air-dryers and regenerators differ from those of the cross-flow reactors, since one must consider the fact that the sides of the channel walls are partly made of a reactive solid substance that is capable of exchanging a species with the process streams. These mass and heat balances result in nonlinear Volterra-type integral equations, which have been studied by Roy and Gidaspow [37,70]. 

The PEC reactor [power output 123 MW(th)/0 MW(el)] was of semi-integrated type (integrated intermediate heat exchangers and primary pumps) and was cooled by two sodiiun primary and secondary circuits. A test channel, hydraulically and thermally insulated from the core, was located at the core center, corresponding to the seven central positions. The maximum power of the experimental element inside the test channel was 3 MW(th). 

The major components of a shell and tube heat exchanger are a shell with tubes, a front end stationary head, and a rear end head. TBMA uses a letter to represent each different construction of these three components, see Figure 4, TEMA uses the combination of these three letters to describe the construction of a heat exchanger. For example, BEM is a heat exchanger with a bonnet type integral front cover, a shell, and a bonnet type integral end cover. 

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