Topological populations

The benefits of the reduced computational method are not achieved without some loss of information. The surfaces between atoms in the projected method are curtains that follow the projeaed density valley between atoms and extend perpendicular from the (x,z) plane. This curtain is schematically drawn in Figure 7a, with the comparable zero-flux surface drawn below in Figure 7b. The more electronegative atom is on the left. Relative to the topological populations, IPP will generally overestimate the population of the less electronegative atom and underestimate the population of the more electronegative atom. 

In Tables 5 and 6 we present two studies of the basis set dependence of the topological population. Both studies examined small organic molecules. The major cause for the differences in the populations is the change in the relative distance between the nucleus and the zero-flux surface. We comment further on this effect later on. 

Table 5 Effect of Basis Set on Valence Topological Population ... 

On the other hand, the topological populations for alkanes predict a reversed polarity — H. Wiberg and Wendoloski favored this polarity... 

Ritchie and Bachrach have examined the relative acidity of strained and unstrained hydrocarbons. The topological population of hydrogen correlates very well with the deprotonation energy of a series of hydrocarbons, better than the C—H distance or the value of the electron density at the C—H bond critical point (Table 13). 

The computational advantage of the projected populations over the topological populations can be substantial. However, the volume assigned to each atom in the projected method has no quantum mechanical significance. Furthermore, the choice of projection plane can affect the projected populations. The major advantage of the projected method is the ability to present contour plots of the electron density that reflect the total elearon density. Visualization of four-dimensional plots is very difficult, and the projection method reduces the dimensionality. 

Molecular descriptors must then be computed. Any numerical value that describes the molecule could be used. Many descriptors are obtained from molecular mechanics or semiempirical calculations. Energies, population analysis, and vibrational frequency analysis with its associated thermodynamic quantities are often obtained this way. Ah initio results can be used reliably, but are often avoided due to the large amount of computation necessary. The largest percentage of descriptors are easily determined values, such as molecular weights, topological indexes, moments of inertia, and so on. Table 30.1 lists some of the descriptors that have been found to be useful in previous studies. These are discussed in more detail in the review articles listed in the bibliography. 

Finally, an alchemical free energy simulation is needed to obtain the free energy difference between any one substate of system A and any one substate of system B, e.g., Ai- In practice, one chooses two substates that resemble each other as much as possible. In the alchemical simulation, it is necessary to restrain appropriate parts of the system to remain in the chosen substate. Thus, for the present hybrid Asp/Asn molecule, the Asp side chain should be confined to the Asp substate I and the Asn side chain confined to its substate I. Flat-bottomed dihedral restraints can achieve this very conveniently [38], in such a way that the most populated configurations (near the energy minimum) are hardly perturbed by the restraints. Note that if the substates AI and BI differ substantially, the transfomnation will be difficult to perform with a single-topology approach. 

Fig. 3.45 Time evolution of rule T12 on (a) r — 2 lattice, (b,c) intermediate lattices, defined by populating an r=2 lattice with a fraction p of vertices that have 6 nearest-neighbors, with p6 0.15, pc 0.30, and (d) r = 3. We see that the class-3 behavior on the pure range-r graphs in (a) and (b) can become effectively class-2 on certain intermediate (or hybrid) topologies.

It has been recently shown [12] that the ELF topological analysis can also be used in the framework of a distributed moments analysis as was done for Atoms in Molecules (AIM) by Popelier and Bader [32, 33], That way, the Mo( 2) monopole term corresponds to the opposite of the population (denoted N) ... 

On enthalpic grounds alone, under ambient conditions the Boltzmann factor for the chain pentamer would be reduced by about 1010 relative to that for the cyclic pentamer, showing that the former topology contributes negligibly to the equilibrium population distribution. 

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