Pseudo-effective operator

The pseudo-effective operator H up to second order in both cases is given by... 

Thus there is evidence from MO theory that a pseudo-co-operative effect is active in water hydrogen-bonding bonds become stronger when more molecules are added to the chain. This is not to say that all bonds are of equal strength in a chain in fact it is very likely that they are not equal. This strengthening of hydrogen-bonds is evidence that chains may be present in liquid water the higher the temperature, the shorter the chains. 

In this section, we introduce another interesting orbital model with the frustration effect, termed the honeycomb lattice orbital model. We consider the model where the doubly degenerate orbitals, e.g. the d i y7. and d y orbitals, which are described by the pseudo-spin operator T,, are located at each site in a two dimensional honeycomb lattice. The explicit form of the model Hamiltonian is given by... 

The pseudo-Liouville operator for hard-sphere interactions can be written directly in terms of the binary collision operator and this leads to considerable simplifications in the formalism. It is not difficult to treat more general interactions, but a considerable number of manipulations must be carried out to express the results in terms of generalized binary collision operators. To avoid these difficulties, we make use of hard-sphere interactions whenever no violence is done to the qualitative features of the effects we are studying. 

The pseudo-Liouville operator does couple these doublet fields to triplet fields such as 8 abs cds involving the solvent molecules. Thus one of the simplest forms for the pair kinetic equation can be obtained by explicitly including doublet and triplet fields in the generalized Langevin equation. This procedure yields a treatment of the effects of solvent dynamics on the motion of the reactive pair that is much more sophisticated than that given in the singlet kinetic equation discussed in the preceding... 

A pseudo-convective heat-transfer operation is one in which the heating gas (generally air) is passed over a bed of solids. Its nse is almost exchisively limited to drying operations (see Sec. 12, tray and shelf dryers). The operation, sometimes termed direct, is more aldu to the coudnctive mechanism. For this operation, Tsao and Wheelock [Chem. Eng., 74(13), 201 (1967)] predict the heat-transfer coefficient when radiative and conductive effects are absent by... 

Boys and Cook refer to these properties as primary properties because their electronic contributions can be obtained directly from the electronic wavefunction As a matter of interest, they also classified the electronic energy as a primary property. It can t be calculated as the expectation value of a sum of true one-electron operators, but the Hartree-Fock operator is sometimes written as a sum of pseudo one-electron operators, which include the average effects of the other electrons. 

The structure was refined with block diagonal least squares. In cases of pseudo-symmetry, least squares refinement is usually troublesome due to the high correlations between atoms related by false symmetry operations. Because of the poor quality of the data, only those reflections not suffering from the effects of decomposition were used in the refinement. With all non-hydrogen atoms refined with isotropic thermal parameters and hydrogen atoms included at fixed positions, the final R and R values were 0.142 and 0.190, respectively. Refinement with anisotropic thermal parameters resulted in slightly more attractive R values, but the much lower data to parameter ratio did not justify it. 

Conjugation of the 7t-electrons of the carbon-carbon double bond with the LUMO sulfur 3d-orbitals would be expected to stabilize the Hiickel 4n -I- 2 (n = 0) array of n-electrons in the thiirene dioxide system. No wonder, therefore, that the successful synthesis of the first member in this series (e.g. 19b) has initiated and stimulated several studies , the main objective of which was to determine whether or not thiirene dioxides should be considered to be aromatic (or pseudo-aromatic ) and/or to what extent conjugation effects, which require some sort of n-d bonding in the conjugatively unsaturated sulfones, are operative within these systems. The fact that the sulfur-oxygen bond lengths in thiirene dioxides were found to be similar to those of other 802-containing compounds, does not corroborate a Hiickel-type jr-delocalization... 

Figure 19 Schematic effect of the STAR operator on 2JCH and 3,/CH couplings. The vicinal component of magnetization in the long-range response that is two-bond coupled to a protonated carbon experiences modulation, which serves as a pseudo-evolution for this coupling. In contrast, the vicinal component of magnetization in the long-range response that is three-bond coupled to a protonated carbon does not exhibit a F, skew. Homonuclear modulation during the evolution period f, is still present, as the full experiment is not a constant-time experiment.

P. Durand and J.P. Malrieu Effective Hamiltonians and Pseudo-Operators as Tools for Rigorous Modelling., in K.P. Lawley (ed.) Ah Initio Methods in Quantum Chemistry., J. Wiley sons, New York, pp 321 12 (1987). 

As the conclusion, we may safely say that the operator of the Stark effect of the hydrogen-like atom possesses just the correct number of pseudo-eigenvalues and no... 

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