eFF, a method to simulate large scale excited electron dynamics

Understand delocalization and correlation

In an effort to describe better delocalized electrons, such as the ones that appear in aromatic compounds, we fit a new eFF, called eFF3, to the energy of a uniform electron gas whose density is allowed to vary. The jellium system serves as an extreme case of a flat potential where electrons mix freely and delocalize.

Two reference energies are available, Hartree-Fock, which corresponds to electrons moving independently of each other in a mean-field approximation [50]; and an exact energy obtained from a parameterization [26] of a quantum Monte Carlo calculation [25]. By taking the difference between these two energies, we compute a correlation energy, which the original eFF made no effort to incorporate, but which is necessary to include to achieve our ultimate goal of chemical accuracy.

Fit exchange and correlation to a uniform electron gas model

Our preliminary efforts, which we call eFF3, fits exchange and correlation energies separately to the uniform electron gas references. We represent the uniform electron gas as a periodic lattice of localized electrons, similar to the packings proposed by Wigner, but at much higher densities.

We define an exchange energy as a pairwise sum over same-spin electrons, and a correlation energy as a pairwise sum over opposite-spin electrons. For the uniform electron gas, we use exchange and correlation functions defined as:

    $\displaystyle E_{exch} = \sum_{\sigma_{i} = \sigma_{j}} \frac{1}{2}\ \frac{a_{exch} S_{ij}^{2}}{1 - S_{ij}^{2}} \cdot (t_{11} + t_{22})$  
    $\displaystyle E_{corr} = \sum_{\sigma_{i} \neq \sigma_{j}} \frac{-a_{corr}}{1 + b_{corr} s_{avg}} \cdot S_{ij}^{1/2}$  
where the parameters are $ a_{exch} = 0.5$, $ a_{corr} = 0.111283$ hartrees, $ b_{corr} = 0.110253\ \mathrm{bohr^{-1}}$; and the kinetic energy sum $ t_{ii} + t_{jj}$ and average electron size $ s_{avg}$ are defined as
    $\displaystyle t_{ii} + t_{jj} = \frac{3}{2}\left(\frac{1}{s_{i}^{2}}+\frac{1}{s_{j}^{2}}\right)$  
    $\displaystyle s_{avg} = \sqrt{(s_{i}^{2} + s_{j}^{2})/2}.$  
where $ s_{i}$ and $ s_{j}$ are the sizes of electrons $ i$ and $ j$.

Using these simple pairwise functions, we are able to reproduce the equation of state curves of both the Hartree-Fock and exact uniform electron gas systems (Figure 35).

Figure 35: Uniform electron gas energy versus density. eFF with exchange matches Hartree-Fock, while eFF with exchange and correlation matches exact quantum Monte Carlo energies.

We find the energies of different crystalline packings to be similar ($ <0.005$ hartrees/electron), suggesting that the uniform electron gas under eFF is highly fluxional (Figure 36).

Figure 36: Uniform electron gas represented as different packings of localized electrons.

An exchange-only functional for systems with nuclei

Having considered the uniform electron gas, we move on to systems containing nuclei, such as $ \mathrm {He_{2}}$ dimer. We find that the potential obtained in the previous section from the uniform electron gas is not sufficiently repulsive.

To address this discrepancy, we add on an extra term that is dependent on the one particle exchange energy $ t_{12} = \left<\phi_{1}\right\vert-\frac{1}{2}\nabla^{2}\left\vert\phi_{2}\right>$, and adjust its weighting to reproduce the $ \mathrm {He_{2}}$ dimer potential energy surface.

We also add on a term that depends on the ratio of electron sizes, which is zero in the uniform electron gas, but may make a contribution in molecules containing dissimilar nuclei. We adjust its weighting to reproduce the correct bond length of lithium hydride.

The final exchange functional for systems with nuclei is as follows:

$\displaystyle E_{exch} = \sum_{\sigma_{i} = \sigma_{j}} \frac{1}{2}\ \frac{S_{i... ...c_{exch} \right) (t_{11} + t_{22}) - c_{exch} \ \frac{2 t_{12}}{S_{ij}}\right) $

where $ f_{size} = s_{1}/s_{2} + s_{2}/s_{1} - 2$, and and $ a_{exch} = 0.4$, $ b_{exch} = 0.15$, and $ c_{exch} = 1$. We find it reproduces the potential energy surfaces of $ \mathrm {He_{2}}$ dimer and HeH well (Figure 37).

Figure 37: We modify the exchange interaction to fit properly the interaction energy of $ \mathrm {He_{2}}$ and HeH

With this functional we are able to reproduce with high accuracy the Hartree-Fock geometries and energies of simple compounds containing atoms from Z=1-6 (Figure 38, 39). No effort has been made yet to combine eFF2 and eFF3 into a single force field that could handle s and p electrons with high accuracy, but it would be a natural next step.

Figure 38: Simple compounds with s-like electrons and nuclei from Z=1-6.

Figure 39: eFF with exchange only shows good agreement with Hartree-Fock for bond lengths and dissociation energies of s-electron systems.

An exchange + correlation functional for systems with nuclei

To translate the correlation function from the uniform electron gas to atoms and molecules, we simply multiply the correlation energy by a factor of three. This gives reasonable trends for atomic correlation energies [51,52] (Figure 40).

Figure 40: Atomic correlation energy comparison.

However, more work is needed, as seen when we try to reproduce correlation energies for the systems studied in the previous section (Figure 41).

Figure 41: eFF reproduces some energy and bond length changes caused by adding correlation.

We note that this method of computing correlation in principle carries some advantages over density functional theory, as it is an inherently non-local description [53]. It may be possible, for example, to achieve a good description of van der Waals forces by adding additional pairwise terms to the eFF correlation functional.