We create atoms with total spins satisfying Hund's rule. When optimized with eFF, hydrogen, helium, lithium, and beryllium all have nucleus-centered electrons, consistent with their electron configurations , , , and . In boron through neon, the valence electrons arrange themselves according to two rules: (1) same spin electrons form close-packed symmetric shells (i.e., nucleus centered point, line, triangle, or tetrahedron) and (2) shells of up-spin and down-spin electrons rotate relative to each other to minimize electron-electron repulsion. In neon, for example, the up and down spin electrons form two separate tetrahedral shells which interpenetrate each other to form a cubic lattice (Figure 5.3.
The new eFF properly reproduces periodic trends in the adiabatic ionization potential (Figure 5.4). It is remarkable that with only one set of parameters for the entire set of atoms, and with only spherical Gaussian functions, we are able to properly describe the balance between electron penetration and shielding, and the filling in of , , and shells, while also reproducing the special stability of half-filled subshells.
It is also clear why the old eFF worked well for the atoms hydrogen through carbon, but was not suitable for describing nitrogen, oxygen, fluorine and neon -- the ionization potentials starting from boron decreased rather than increased with increasing Z, which was acceptable for boron and carbon, but which led to incorrect IPs for larger Z atoms, culminating in neon being unstable. The old eFF had the incorrect Z dependence for ionization potential because it lacked stabilizing exchange interactions between valence electrons on the same atom.
In addition to ionization potentials, we computed atomic polarizabilities with eFF. This was done using a finite difference approach; the values plotted in Figure 5.5 are the averaged eigenvalues of the atomic polarizability tensor. Polarizabilities have units of volume, and can be taken as a measure of the size of the electrons in a system.
The simplest cases are the helium-like ions, which contain two nucleus-centered electrons, and a nucleus of variable charge. eFF values were compared to values computed using first-order coupled perturbed Hartree-Fock theory . We found that eFF gave polarizabilities that agreed well with theory over six orders of magnitude. This gave us confidence that eFF could properly describe polarizabilities over a wide range of electron sizes.
We found that first-row atomic polarizabilities were slightly too high (15%) for lithium through carbon, and too low (30-50%) for nitrogen and neon. Oxygen and fluorine have too-high polarizabilities because eFF gives those atoms a non-spherical charge distribution, which results in a permanent dipole moment that does not exist in the actual atom -- the atoms rotate in the presence of an electric field, which produces an artificially high polarizability. All in all though, we find that the general periodic trend is correctly reproduced. eFF may prove to be a useful way to obtain molecular polarizabilities as well, since dipole-dipole, atomic polarizability, and Pauli effects are taken into account in a self-consistent way.