Explanation of the energy expressions

What gives an electron $p$ character? Consider the electron configuration of neon, with its ten electrons. Two electrons pair up on top of the neon nucleus to form a helium-like core, while the other eight electrons -- four spin up and four spin down -- are valence electrons that surround the core. We assume that the core electrons form a high concentration of charge around the nucleus that is unperturbed by the presence of valence electrons. Then in order for valence and core electrons to have zero overlap and satisfy the Pauli principle, the valence electron must change sign over the region of the core electron.

For lithium and beryllium, the valence electrons are centered on top of the core electrons, and change sign over the core via a radial node contained in a $2s$ function. For neon, the eight valence electrons cannot all be centered on top of the core electrons, and they instead pack into a tetrahedral or cubic arrangement. In this case -- when the electron is shifted off-center -- a planar node is preferred, and is represented by a higher angular momentum $2p$ function with no radial nodes.

Imagine an electron approaching a nucleus with surrounding core electrons. Far from the core, the electron is spherical and has the characteristics of a $1s$ function. Closer to the core, the electron develops a planar node and attains p-character. Once on top of the core, the electron develops a radial node and becomes s-like again. In the eFF energy expressions, this $p$ character dependence is represented as a spline that is zero for r = 0, rises to a maximum when $x/s = 0.6$, then falls back to zero when $x/s = 1.5$ (Figure 5.1).

Figure 5.1: Pauli interaction between $p$-like versus $s$-like electrons.

The above explains why an electron's shape is modified by the presence of core electrons, which we assume to surround any nuclei with $Z>2$. We discover that bare nuclei, such as protons and helium nuclei, also affect the shape of nearby electrons, causing them to become more s-like.

Consider methane. The valence electrons that form the carbon-hydrogen bonds skew toward the proton and away from the nucleus, which make them more s-like than the valence electrons in neon. This is reflected in Pauling's hybridization model [3], which assigns neon valence electrons $p$ hybridization and methane valence electrons $sp^{3}$ hybridization. A proton's effect on hybridization is greatest when the proton is near the center of the electron; in the energy expression, this effect is represented by a spline which goes to 0.5 when $x/s$=0, then rises to become 1 when $x/s$ = 2.5.

We assume that these two factors -- electron proximity to cores, and electron proximity to bare nuclei -- are sufficient to determine an electron's hybridization/shape, which we represent with the scalar quantity $\chi_{ke}$; there is an analogous two-body quantity $\chi_{Pauli}$ we describe later as well, which includes angle effects. We now discuss how electron shape affects the components in the eFF energy expression.

First, p-like electrons have a higher kinetic energy than similarly sized s-like electrons, due to the presence of the planar node. This effect appears in the eFF energy expression as the factor $f_{ke}$, which scales between the two extremes $c_{s-ke}$, the multiplier for a pure $s$ function, and $c_{p-ke}$, the multiplier for a pure $p$ function. The hybridization variable $\chi_{ke}$ varies between one for a pure $s$ function and zero for a pure $p$ function, and is a multiplicative combination of contributions from all nearby protons and cores.

The end result is that the kinetic energy of electrons is raised slightly around cores, which is an effective two-body repulsion between cores and electrons. This interaction has the same character -- though the opposite sign -- as a conventional force field bond term.

Second, when electrons attain $p$ character, their Pauli repulsion can turn into Pauli attraction. At one extreme is the exchange interaction that stabilizes high-spin configurations of atoms, e.g., Hund's rule; at the other extreme is the repulsion between helium atoms that causes them to repel.

The physical origin of the Pauli attraction between valence electrons on the same atom is as follows. Recall from the last chapter that Pauli repulsion is the consequence of kinetic energy increase upon orbital orthogonalization. However, in an atom like neon, the $p$ electrons are already orthogonal to each other, due to the relative geometry of the planar nodes. Thus the kinetic energy repulsion vanishes, and instead a second-order attractive interaction -- two-body exchange energy -- becomes dominant. The exchange term arises because the Pauli principle causes same-spin electrons to avoid occupying the same region of space, which causes electron-electron repulsion to decrease. Hence we expect that neon valence electrons are stabilized not only by their attraction to the nucleus, but also by their Pauli attraction to each other.

In the electron force field, $E_{Pauli}$ is modified by two terms, $f_{repel}$, which ensures that electrons do not coalesce, and $f_{switch}$, which causes s-like electrons to repel and p-like electrons to attract. They occupy separate ranges: $f_{repel}$ dominates at high overlap, while $f_{switch}$ dominates at lower overlap.

The term $f_{switch}$ is a three-body term that modifies the Pauli interaction between two electrons and depends on the proximity of nearby cores and bare nuclei. It is a function not only of electron-nuclear distances, but also of the electron-core-electron angle; the attractive term reaches a maximum when the electrons are $\mathrm{90^{o}}$ apart from each other. The combined effects of $f_{repel}$ and $f_{switch}$ pushes the electrons apart to an angle larger than $\mathrm{90^{o}}$ in first-row atoms (Figure 5.1).

We consider the $f_{switch}$ modification to the Pauli interaction to be analogous to a conventional force field angle term, since it depends on the electron-core-electron angle. As with the bond term, the effect is limited in range with a strict spline cutoff, so it is not overly expensive to compute.

The $f_{repel}$ term is a two-body electron-electron term, but is new in this eFF; it addresses a problem found in the previous eFF, which is that same size electrons could coalesce under extreme conditions. In the new formulation, we have a singularity at $S=1$ so that coalescence of same size electrons is no longer possible. We have also added a term that encourages nearby electrons to have the same size. This helps to stabilize species such as methyl radical, where the radical electron would otherwise be too large. Figure 5.2 shows a comparison between old and new eFF versions of the Pauli repulsion between s-like electrons.

Figure 5.2: Pauli repulsion between $s$-like electrons is modified to make electron sizes more similar and prevent electron-electron coalescence.

We assume that electrostatics are unchanged between the old and new eFF, and we leave the terms $E_{nuc-nuc}$, $E_{nuc-elec}$, and $E_{elec-elec}$ the same as before. It is possible there could be some benefit to making these terms dependent on electron shape. However the electrostatics in our model are sensitive to small changes, and we have found that most modifications to them cause problems with atom and bond stabilities.