Population Energy Increments

There are two additive terms to the energy, POP and TORS, that have not been mentioned yet because they are zero in minimal ethylene. The POP term comes from higher-energy conformers. If the energy at the global minimum is not too far removed from one or more higher conformational minima, molecules will be distributed over the conformers according to the Boltzmann distribution 

A molecule has three nondegenerate conformers (Fig. 5-8). One is 450 x 10 J above the ground state, and the second high-energy conformer is 900 x 10 J above the ground state. What are the percentages of each of the three conformers relative to the total number of molecules in a sample of the normal conformational mixture at 300 K  

Equal spacing between energy levels is not unusual. In the case of the harmonic oscillator, it is the rule. 

Suppose that the computed ground-state enthalpy of formation of a molecule is = 

The energies of the two higher-energy conformers described in Exercise 5-2 are 450 X 10-22j(6.022 x 1022mol- ) = 2.71 kJmol- and 

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The acid-catalyzed ester hydrolysis provides a good target for MM treatments. DeTar first used hydrocarbon models in which an ester was approximated by an isoalkane (74) and the intermediate (75) by a neoalkane (76). He assumed that if the rate of reaction truly is not influenced by polar effects but is governed only by steric effects of R, as has been generally postulated, the rate must be proportional to the energy difference (AAH ) between 74 and 76. The AAH f is mainly determined by the van der Waals strain in these branched alkanes. Nonsteric group increment terms were carefully adjusted, and statistical mechanical corrections for conformer populations... 

Here, (> and /> are initial and final states of the supermolecular complex, 11 is the induced incremental polarizability (a second-rank tensor), fio and fij are unit vectors in the direction of the electric polarization of incident and scattered waves, which are often specified in the form of subscripts V and H, for vertical and horizontal, (aif = (Ej- — Ei)lh is the energy difference of initial and final state in units of angular frequency, Pi T) is the population of the initial state (a function of temperature), <5 (to) is Dirac s delta function, and the summation is over all initial and final states of the collisional complex. 

Both the lack of a sharp definable mobility and threshold voltage are attributable to a gradual density of states profile at the valence or conduction band (HOMO/LUMO) frontiers of the material. Because carriers are relatively localized and hop from molecule to molecule, a continuum of states means that there is a distribution of energy barriers to conduction. As the Fermi level of the semiconductor moves closer to a higher density of states, states closer to the edge (which appear more mobile) are populated, and the incrementally added new carriers are significantly more mobile than deeper charges which were added earlier [127]. 

As indicated, the heat of formation (HFO) is taken as the sum of three terms. The first is the steric energy (E). The second is the bond enthalpy (BE), which is calculated from normal bond increments. The third is a partition function increment (PFC), which is itself the sum of three terms a conformational population increment (POP), a torsional contribution term (TOR), and a translation / rotation term (T / R). Since the molecular mechanics calculation has considered only one conformation, the POP and TOR terms are defined to be zero for this calculation (but see below). The T/R term is a molecular translational and rotational term that is always taken to be 2.4kcal/mol at room temperature, since even an anfi-butane molecule that is not rotating about the C2-C3 bond will still be undergoing translation and rotation of the entire molecule as a unit. Therefore, HFO is the total of the bond increment contributions (BE = —35.08) plus the steric energy (E = 2.17) plus the partition function increment (PFC = 2.4). The sum, -30.51 kcal/mol, is slightly lower than the literature value of -30.15 kcal/mol. ... 

Incremental market entry is facilitated because the proposed hierarchical hub-spoke nuclear assets - both reaetor plants and fuel cycle faeilities - are sealable and because the end user of energy services will find the substitution nearly transparent electricity, water, and chemical fuel (hydrogen) will eontinue to be produced - from a different source. Being scaleable, the infrastructure need not be deployed all at onee clusters of power plants can be built up incrementally at a eity s perimeter as population and/or energy use/capita increase. Additional... 

The unraveling of the averaging that goes into the definition of a thermal reaction rate constant shows that its temperature dependence is not quite a simple matter. The state-to-state reaction rate constants vary with temperature because the state-to-state cross-section depends on the collision energy. In addition, for a system in thermal equihhrium, the populations of the different initial states are themselves temperature-dependent. Problem F shows that if an increment in the collision energy is as effective in promoting reaction as an increment in the internal energy of the reactants, then the two sources of T-dependence will be equivalent. As we have noted, such equivalence is not necessarily the case. 

BE = bond increment GE = group increment HFE = Hartree-Fock energy MH = molar heat content MM2, MMP2, MM3, MM4 = molecular mechanics programs PFC = partition function contribution POP = population contribution SE = structural features increment TOR = torsional contribution. 

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