potential energy surface (kcal/mol); eFF properly dissociates , but the simplicity of the basis, as well as the neglect of electron correlation, leads to underbinding.
Pauli repulsion comes from the kinetic energy increase upon making orbitals orthogonal to each other. This effect increases with the overlap between orbitals, and is the dominant contribution to the Pauli energy when the interacting electrons are nodeless and spherical.
Comparison of Pauli repulsion and electrostatic repulsion between two wave functions with . The Pauli repulsion rises more sharply with increasing electron overlap than electrostatic repulsion, acting almost as a hard-sphere potential. This behavior gives rise to the basic rules of Lewis bonding and hybridization, which we discuss in greater depth in Sec. III B.
(a) Timings on a 2.33 GHz Xeon for the Auger fragmentation of , plasma etching of a diamond surface, and proton stopping in beryllium. (b) Linear scaling of calculation time using a 10 Å force cutoff; shown is the time spent on an energy/force evaluation on bulk lithium solid. The memory required also scales linearly with the number of electrons and nuclei.
eFF geometries of simple substituted hydrocarbons. The valence electrons spin pair into matching spatial orbitals, shown here as gray spheres. The basic rules of Lewis bonding and hybridization arise as a natural result of minimizing the sum of electrostatic potential, kinetic energy pressure, and Pauli repulsion. Geometries are in Table I.
eFF geometries of larger hydrocarbons. Bond lengths are in angstrom, and DFT values are given in parentheses for comparison. Overall, the bond lengths and angles of the carbon-carbon framework are correct, although carbon-carbon bonds attached to quaternary carbons are uniformly too long.
Multiple bonds can split or form symmetric “banana” pairs. In eFF, bonding is strongly preferred, but the resulting -like function is polarized and too diffuse, and incapable of providing substantive additional bonding strength.
Geometry optimization of a series of cyclic hydrocarbons. eFF reproduces the curved bonds of cyclopropane and the puckered (nonplanar) geometry of cyclopentane and cyclohexane; it also reproduces the energy difference between twist boat and chair conformations of cyclohexane.
eFF reproduces steric repulsions within a series of acyclic, cyclic, and fused cyclic alkanes. Overall the agreement with DFT relative energies is good, although there is slightly too much repulsion between carbon-carbon bonds.
A single bond attached to a substituted double bond finds its rotation severely restricted due to a steric interaction called allylic 1,3-strain. eFF reproduces the magnitude of this effect well.
eFF properly distinguishes between (a) the allowed reaction and (b) the forbidden reaction. In the first case there is a low energy linear transition state with partial bonds to both hydrogens, while in the second case, both reactant bonds must be simultaneously broken in forming the transition state, making it prohibitively high in energy. Contour lines are separated by 10 kcal/mol in (a) and 20 kcal/mol in (b).
Lithium hydride exchange reaction proceeds via formation of a precomplex intermediate, then exchange of atoms via a low barrier transition state with ionic character. eFF correctly reproduces the trapezoidal geometry of the transition state, but somewhat overestimates the barrier height for the reaction.
Lithium, beryllium, and boron hydrides containing ionic and/or electron-deficient multicenter bonds. Bond lengths are in angstrom with DFT values in parentheses for comparison. eFF obtains correct geometries and dissociation energies for these systems—not surprising, as the electrons are -like and well represented by eFF.
Bonding in lithium bulk solid, showing single spin electrons nestled in octahedral interstices between fcc ions; the EOS is also shown and compared with DFT (Ref. 46).
Bonding in beryllium bulk solid, showing electron pairs forming strong bonds within and between layers in a hexagonal close packed array of ions; the EOS is also shown as a function of lattice parameters and .
Potential energy of a uniform electron gas with respect to the density parameter . eFF agrees with the binding energy from HF, but neglects the correlation energy which would cause it to agree with the exact curve.
Comparison of eFF EOS to the ones from experiments where hydrogen is compressed (a) statically or (b) dynamically. In (a), the eFF EOS of solid hydrogen at 300 K matches the data from diamond anvil experiments (Ref. 60). In (b) the eFF single shock Hugoniot curve for liquid (solid black line) agrees with the data from many experiments: gas gun (Ref. 62) (red dots), machine (Ref. 63) (green dots), convergence geometry (Ref. 64) (orange), and laser ablation (Ref. 55) (blue). Note that the eFF results were published in 2007, while the results from laser ablation experiments (which agree well with eFF but not path integral theory) were published in 2009.
Comparison of EOS from eFF to other theoretical methods for fixed densities (, corresponding to ) and varying temperatures, showing good agreement in the regimes where the other theories are expected to be the most accurate. Specifically, eFF (solid line) matches the EOS from the Saumon–Chabrier chemical model [dotted line (Ref. 59)] in the molecular regime, and the EOS from PIMC calculations [circles, where solid indicates the EOS with energy and pressure corrections to obtain the correct dissociation energy for hydrogen molecules (Refs. 65–67)] in the atomic regime.
Phenomena associated with Auger processes modeled using eFF: (a) bond breaking and desorption of surface fragments as a result of core-hole relaxation, which may occur via an intermediate state containing two valence holes [Knotek–Feibelman mechanism (Ref. 70)]; and (b) the surface selectivity of Auger spectroscopy, which exists because secondary Auger electrons arising from surface layers are ejected and subsequently detected, while those from bulk atoms are trapped in the solid.
eFF predicted Auger fragmentation of core-ionized adamantane, showing a single trajectory in which the molecule was equilibrated at 300 K for 500 fs (time step of 5 as), a core electron was instantaneously removed, and the dynamics was propagated for 100 fs (time step of 1 as).
Electron dynamics during the Auger process. The red valence electron fills the core hole after 7 fs, the green electron is ejected after 12 fs, and the blue and purple electrons remain excited but bound over 50 fs.
Mechanisms of hydrogen desorption summarized. (a) The analysis was based on 410 trajectories of core-ionized adamantane . We distinguish three mechanisms using the following decision tree: if the ejected hydrogen was bound to the core-ionized atom, classify the event as the result of a direct Auger process. If not, examine whether more than two electrons were ever associated with the ejected proton at any point in the trajectory. If yes, classify the event as the result of a secondary impact process, otherwise classify it as a thermal process. (b) Direct Auger processes involve a transient (10–20 fs) depletion of charge to form a two valence-hole state, followed by limited recombination to produce ions or H atoms. (c) Thermal processes involve a slow leakage of ions from the molecule. (d) Secondary impact processes involve the scattering of an excited electron off an adjacent pair of bonding electrons, resulting in ionization of the bonding electrons and prompt ejection of an ion.
Geometries of primary, secondary, and tertiary-substituted carbon. The deviations highlighted in boldface are caused by a too-strong repulsion between C–C bonding electrons.
Geometries of double and triple bonds with bond lengths in angstrom.
Energy differences between conformers examined. Overall, the agreement between eFF and DFT is excellent.
Lithium and beryllium parameters with a comparison of eFF and experimental values. Lattice constants are somewhat too large—we explain this after considering the energetics of a uniform electron gas—and cohesive energies are too high, because the bulk solid is represented more properly than the free atoms making up the metal. Elastic constants are nearly perfect.
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