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Exchange and correlation functions for systems with nuclei

With the uniform electron gas exchange and correlation functions established, we now attempt to describe systems containing nuclei. Ideally, we would be able to apply the uniform electron gas functionals to these systems without modification. As a test case, we examined the Hartree-Fock repulsion between two helium atoms, and between a helium and a hydrogen atom. However, we found that the uniform electron gas exchange function by itself was not repulsive enough to reproduce the interaction (Figure 6.12).

Figure 6.12: We modify the exchange interaction to fit properly the interaction energy of $ \mathrm {He_{2}}$ and HeH.

We saw this as an opportunity to develop a new exchange functional that could draw from some of the insights gained from the uniform electron gas work, as well as from the previous incarnations of eFF, while correcting some past deficiencies. In the work on hydrogen plasma and uniform electron gas, we used a Pauli repulsion function of the form

$\displaystyle E_{Pauli} = \sum_{\sigma_{i} = \sigma_{j}} \left(\frac{S_{ij}^{2}...
..._{ij}^{2}} + (1 - \rho) \frac{S_{ij}^{2}}{1 + S_{ij}^{2}}\right) \Delta T_{ij}
$

with scaling factors added. It proved to be a capable function, able to correctly determine the relative energies of a wide range of hydrocarbon conformers. However, it was slightly too repulsive, giving an energy barrier for $ \mathrm{H_{2} + H \rightarrow H + H_{2}}$ as 42 kcal/mol rather than the exact 9 kcal/mol. It was also possible for same-spin electrons of the same size to coalesce.

In the work on p-like electrons, we used a Pauli repulsion function of the form

$\displaystyle E_{Pauli} = \sum_{\sigma_{i} = \sigma_{j}} \frac{1}{2} \frac{S_{i...
...\cdot \frac{S_{ij}}{1-S_{ij}^{2}} + c (1 - S_{ij}) \right) \cdot \Delta T_{ij}
$

which had a singularity when $ s_{i} = s_{j}$ and $ S \rightarrow 1$, eliminating the coalescence problem. However it was not repulsive enough, which caused structures like tetrahedral $ \mathrm{H_{4}}$ to be inappropriately stable.

We notice that the uniform electron gas function has a singularity as $ S \rightarrow 1$, since it is missing a $ t_{12}$ term. This led us to try puting the $ t_{12}$ term back into our exchange function, but with a scaling factor different from the one in front of the $ t_{11} + t_{22}$ term. We justify this procedure on the grounds that when nuclei are present, the kinetic energy at the electron centers is underestimated, since we are missing the proper nuclear-electron cusp, while the Gaussian description of electrons at the electron midpoint is a relatively better representation. Thus in computing the kinetic energy change upon orthogonalization, which moves electron density from the electron midpoint to the electron centers, we should multiply the $ t_{ii}$ terms by a larger factor than the $ t_{ij}$ terms. We write the final exchange function 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$. The parameters, adjusted to reproduce the $ \mathrm {He_{2}}$ repulsion and the correct bond length for LiH, are specified in the energy expression section.

The new exchange reproduces the repulsion of $ \mathrm {He_{2}}$ and HeH well, and has the desired anticoalescence singularity. Plotted against the previous Pauli repulsion functions, we see it is more repulsive than the p-like electron function, and slightly less repulsive than the original Pauli function, as desired (Figure 6.13).

Figure 6.13: Comparison of the new exchange potential to previous Pauli potentials, showing that the new potential has a reasonable amount of repulsion.

In order to obtain correct geometries for LiH and $ \mathrm{BeH_{2}}$, we need to include a relative-size-dependent term $ f_{size}$ in the function, as we did in the p-like electron eFF, but we find that the parameter multiplying it is smaller than it was previously (0.15 versus 3).

In developing a correlation function for systems containing nuclei, we use $ H_{2}$ bond breaking as a test case, which turns out to present some complications, since both unrestricted and restricted Hartree-Fock formalisms have problems describing the correct dissociation of hydrogen molecule. The unrestricted HF wavefunction for $ H_{2}$ is simply $ \phi_{1}(r_{1}) \phi_{2}(r_{2})$, where $ \phi_{1}$ and $ \phi_{2}$ are orbitals localized on different nuclei. However, since electrons are indistinguishable, the wavefunction $ \phi_{2}(r_{1}) \phi_{1}(r_{2})$ should be equally valid, and it turns out that the generalized valence bond wavefunction $ \phi_{1}(r_{1}) \phi_{2}(r_{2}) + \phi_{2}(r_{1}) \phi_{1}(r_{2})$ has a lower energy than the UHF wavefunction. This difference is termed ``static correlation,'' since it represents the interaction of electrons occupying orbitals that are far apart from each other. The exact energy is even lower than the GVB energy, and we assume that the energy difference arises from ``dynamic correlation,'' the stabilizing interaction of electrons that are near to each other.

We can thus take the difference between exact and GVB energies to be a measure of dynamic correlation, and the difference between exact and UHF energies to be a measure of dynamic and static correlation. When we apply the uniform electron gas correlation functional to $ \mathrm {H_{2}}$, we find that it falls off similarly to the exact minus GVB curve, suggesting that we account for dynamic but not static correlation (Figure 6.14). The exact minus UHF curve has a peculiar behavior, reaching a maximum near the point where the wavefunction transitions from a closed- to an open-shell form. Creating a function that reproduces this peak would be challenging, and we sidestep this issue by claiming that, like DFT, our eFF correlation functional reproduces dynamic and not static correlation effects.

Figure 6.14: We scale the correlation function to match the long range falloff of the GVB correlation energy in $ \mathrm {H_{2}}$.

To better match the falloff of the exact minus GVB curve, we multiply the uniform gas correlation energy by three. Future functionals would likely interpolate between these two extremes.

To clarify what effects eFF includes and what it does not, we plot the $ \mathrm {H_{2}}$ bond-breaking potential energy surface (Figure 6.15). We see that eFF with exchange is nearly 20 kcal/mol above the UHF curve, due to the deficiency inherent in the eFF single Gaussian basis. The total correlation energy at that point is nearly 27 kcal/mol, but we only account for 18 kcal/mol of it. We also see that both forms of eFF fall off like UHF, not GVB. We conclude that we are accounting for dynamic correlation properly, but not static correlation; and that a further future correction will be needed to account for deficiencies in the single Gaussian basis. We have not attempted to have either the exchange or correlation functional correct for basis set deficiencies, because we would like to keep the terms of the eFF force field as ``clean'' and focused on single tasks as possible.

Figure 6.15: Comparison of $ \mathrm {H_{2}}$ potential energy curves; we limit the correlation function to correcting correlation, not deficiencies in the basis.


next up previous contents
Next: Performance of new functions Up: Development of an electron Previous: Dynamic properties of the   Contents
Julius 2008-04-29