Morse complex

Acyclic Matchings on Free Chain Complexes and the Morse Complex... 

Definition 11.23. Let (C, fi) be a free chain complex with a basis, and let A4 be an acyclic matching. The Morse complex... 

We remark that even if the chain complex C is infinite in both directions, one can still define the notions of the acyclic matching and of the Morse complex. Since each particular homology group is determined by a finite excerpt from C, we may still conclude that... 

Early experimental spectroscopic investigations on Rg- XY complexes resulted in contradictory information regarding the interactions within them and their preferred geometries. Rovibronic absorption and LIF spectra revealed T-shaped excited- and ground-state configurations, wherein the Rg atom is confined to a plane perpendicular to the X—Y bond [10, 19, 28-30]. While these results were supported by the prediction of T-shaped structures based on pairwise additive Lennard-Jones or Morse atom-atom potentials, they seemed to be at odds with results from microwave spectroscopy experiments that were consistent with linear ground-state geometries [31, 32]. Some attempts were made to justify the contradictory results of the microwave and optical spectroscopic studies, and... 

Since all photochemical reactions require the absorption of a photon, the result is that the reactant molecule is raised to a higher energy level. The outcome of this process depends on the nature of the upper and lower electronic states of the molecule. Four types of absorption behaviour are possible and we will first illustrate these by referring to Morse curves for the simple, diatomic, molecules. Although the potential energy of a complex molecule as a function of its molecular geometry is not a simple two-dimensional curve but a complex multidimensional surface, the conclusions arrived at by the use of Morse curves are instructive. 

Such considerations were extended to metal complexes in 1902 by Morse, who studied the distribution of divalent mercury between toluene and water at various Hg and CT concentrations. By taking complex formation in the aqueous phase into consideration Morse could determine the formation constants of HgCr and HgCb from distribution measurements, as well as the distribution constant of the neutral complex HgCl2. The overall extraction reaction can be written... 

The alkyl complexes mentioned above are very electron rich, and thus their susceptibility to reaction with O2 is not a great surprise, though the formation of stable organometallic oxidation products may be. Somewhat more unusual is the reaction of metal(III) hahdes with O2. Relatively recent results in this area begin with the report by Morse et al. of the oxidative addition of O2 to [Cp CrBr( ji- Br)]2, see Eq. 9 [23]. 

The corresponding EDFs for both potentials calculated for the same test set of the Fe(II) complexes are plotted in Figs. 5, 8. The systematic errors in all cases are close to zero, as can be seen from the parameters of the EDFs for the EHCF/MM with the Morse potential — r = —0.005, (7=0.056 with the NR potential — /r =0.002, (7=0.052. However, although somewhat improved the stiffness of the Morse potential again manifests itself the systematic errors for the separate LS and HS sets do exist and only approximately cancel each other in the total set, whereas for the NR... 

Figure 5. EDF for the difference between the experimental and the EHCF/MM calcn-lated bond lengths (A) over the entire test set of Fe(II) complexes. The Morse potential is used for the Fe-N bond. The scaled Racah parameters are used for the EHCF calculations (see text).

It has long been known that NO forms colored, paramagnetic complexes with many transition metals (Kon, 1968 Yonetoni et al., 1972 Chevion et al., 1977 Hilleetai., 1979 Morse and Chen, 1980 HorietaL, 1981 LoBrutto et al., 1983 Kon, 1975 Kon and Katakoa, 1969) the inorganic chemistry literature goes back at least to the early years of the twentienth century. Because NO has an odd... 

These potential energy terms and their attendant empirical parameters define the force field (FF). More complicated FFs which use different and/or more complex functional forms are also possible. For example, the simple harmonic oscillator expression for bond stretching can be replaced by a Morse function, Euorse (3), or additional FF terms may be added such as the stretch-bend cross terms, Estb, (4) used in the Merck molecular force field (MMFF) (7-10) which may be useful for better describing vibrations and conformational energies. 

The development of a consistent theory for a dissociative electron transfer is a recent challenge in the field of theoretical electrocatalysis. Progress in this field of electrochemistry has involved the use of an harmonic Morse curves [25] instead of harmonic approximations. Applying the principles of the theory of the activated complex to adiabatic dissociative electron transfer reactions, the work of Saveant resulted in the following expressions [24] for the Gibbs energy of activation... 

Any change in the dimensions of the activated complex along the reaction coordinate leads to a decrease in PE, behaviour not found in molecules. The Morse curve shown earlier (Figure 4.19) demonstrates that for a molecule displacement from the equilibrium position always leads to an increase in PE. The activated complex should be regarded as merely a critical configuration. Changes in dimensions in all other directions result in an increase in PE, behaviour typical of vibration about an equilibrium intemuclear distance. 

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