We have outlined a new eFF that accounts for changes in electron shape caused by the influence of nearby nuclei and core electrons. Previously we had assumed that the Pauli principle was manifested solely by electrons repelling each other. In this chapter, we demonstrate an important exception to the rule -- when electrons are orthogonal to each other, as they are when they attain character and are at a angle to each other, the Pauli repulsion is reduced and can become attractive. This is due to the same decrease in electron-electron repulsion between same spin electrons which causes the ground state of atoms to be high spin (Hund's rule).
By including electron shapes, we can with a single set of parameters obtain correct ionization potentials and polarizabilities for atoms from hydrogen through neon. Electrons arrange themselves into s-like and p-like shells naturally, and the special stability of s shells, p shells, and even half-filled p shells emerges naturally from the eFF energy expressions.
Conceptually it should be possible to extend eFF to higher-row atoms by parameterizing interactions between electrons and larger cores, such as neon, argon, and so on.
We obtain good geometries and bond dissociation energies for atom hydrides and hydrocarbons as well, in the process correcting many of the issues afflicting the old eFF, such as carbon-hydrogen bonds that were too variable in length, and double bonds that were too long. In most cases, hydrocarbon geometries containing single and double bonds optimized with eFF match B3LYP geometries well, even for flexible molecules such as cycloheptene and cyclooctatetraene.
The new version of eFF has clear limitations as well. It is too easy to turn double bonds into single bonds, and triple bonds are unstable relative to double bond diradicals, suggesting that banana bond electrons repel each other too strongly. We do not account for conjugation, so benzene appears as 1,3,5-cyclohexatriene. We also find that in making it easier for valence electrons to pack together to form atoms, we have degraded the accuracy of part of the old eFF -- the Pauli repulsion between s-like electrons is now underestimated in a variety of systems. This underestimation causes heteroatom bonds and hydrogen bonds to be too short and too strong, and causes hydrogen and boron clusters to arrange themselves in unphysical configurations.
To correct this problem, we have attempted to modify the Pauli potential to be more repulsive while still preserving a proper description of first row atoms, but it has not been straightforward to do. We may have overconstrained our energy expression by assuming that exchange attraction could be approximated by a functional form similar to that used to describe exchange repulsion. In the future, it may be useful to investigate separate functional forms for (1) exchange repulsion between s-like electrons, which arises from the increase of kinetic energy upon orbital orthogonalization; and (2) exchange attraction between p-like electrons, which arises from the decrease of electron-electron repulsion as a consequence of the Pauli principle.
Nonetheless, our results suggest strongly that (1) it is possible to describe systems with p-like electrons using only spherical Gaussian functions,(2) electron shape can be specified implicitly, by considering the position of the electron relative to the nuclei of the system, and (3) the dominant effects to consider for interactions of p-like electrons are changes in kinetic energy and Pauli interactions.
These terms take the form of two and three body terms involving nuclei and electrons, analogous to the bond and angle terms found in traditional force fields. It is more than likely that future eFFs will find it advantageous to include such terms in order to account for the diversity of electron shapes present in molecular systems while maintaining the simplicity of propagating spherical Gaussian functions.