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Next: Conclusion Up: Development of an electron Previous: Exchange and correlation functions   Contents

Performance of new functions on systems with nuclei

We start by testing the new exchange function on simple molecules with nuclei and s-like electrons (Figure 6.16). This includes the hydrides LiH, BeH, $ \mathrm{BeH_{2}}$ used to fit the exchange function, as well as the hydrogen systems $ \mathrm{H_{3}}$ (linear TS), $ \mathrm{H_{4}}$ (square TS), $ \mathrm{H_{3}^{+}}$, $ \mathrm{H_{4}^{2+}}$, and the lithium systems $ \mathrm{Li_{2}}$, $ \mathrm{Li_{2}^{+}}$, $ \mathrm{Li_{3}^{+}}$, and $ \mathrm{Li_{4}^{2+}}$. We find excellent agreement between eFF (exchange only) and unrestricted Hartree-Fock (6-311g** basis) for bond lengths (Figure 6.17), except for the lithium geometries, which are consistently too long by $ \mathrm{\approx 0.2 \AA}$.

Figure 6.16: Gallery of systems with nuclei and s-like electrons.

Table 6.2: Key to tested geometries and dissociation energies.
energy of relative to
$ \mathrm {H_{2}}$ $ \mathrm{H+H}$
LiH $ \mathrm{Li+H}$
BeH $ \mathrm{Be+H}$
$ \mathrm{BeH_{2}}$ $ \mathrm{BeH + H}$
$ \mathrm{H_{3}}$ (linear) $ \mathrm{H_{2} + H}$
$ \mathrm{H_{4}}$ (square) $ \mathrm{H_{2} + H_{2}}$
$ \mathrm{H_{3}^{+}}$ (triangle) $ \mathrm{2 H + H^{+}}$
$ \mathrm{H_{4}^{2+}}$ (tetrahedron) $ \mathrm{2 H + 2 H^{+}}$
$ \mathrm{Li_{2}}$ $ \mathrm{2 Li}$
$ \mathrm{Li_{2}^{+}}$ $ \mathrm{Li + Li^{+}}$
$ \mathrm{Li_{3}^{+}}$ (triangle) $ \mathrm{2 Li + Li^{+}}$
$ \mathrm{Li_{4}^{2+}}$ (tetrahedron) $ \mathrm{2 Li + 2 Li^{+}}$

Figure 6.17: eFF with exchange shows good agreement with Hartree-Fock for bond lengths and dissociation energies of s-electron systems.

There is good agreement on dissociation energies as well (Table 6.2, Figure 6.17, Table 6.7). For the first time, we obtain a reasonable energy for the $ \mathrm{H_{3}}$ linear transition state relative to $ \mathrm{H_{2} + H}$ (20.3 kcal/mol versus 24.3 kcal/mol UHF). We also obtain a reasonable energy for the forbidden $ \mathrm{H_{4}}$ square transition state (101.5 kcal/mol versus 121.4 kcal/mol UHF). Adding a proton to dihydrogen creates the two-electron triangular ion $ \mathrm{H_{3}^{+}}$, with a stability of -177.5 kcal/mol versus -187.6 kcal/mol UHF relative to separated atoms; adding another proton results in a tetrahedron that is barely stable relative to separated atoms (-8.3 kcal/mol versus -1.7 kcal/mol UHF).

We obtain correct dissociation energies for a variety of homonuclear lithium complexes, such as the dimer $ \mathrm{Li_{2}}$ (0.9 kcal/mol versus -4.0 kcal/mol UHF), the one-electron ion $ \mathrm{Li_{2}^{+}}$ which has a stronger bond (-31.4 kcal/mol versus -29.0 kcal/mol UHF), as well as the triangular cation $ \mathrm{Li_{3}^{+}}$ (-46.6 kcal/mol versus -46.0 kcal/mol UHF), and the tetrahedral cation $ \mathrm{Li_{4}^{2+}}$ (0.4 kcal/mol versus 1.2 kcal/mol UHF). That we are able to obtain correct dissociation energies for homonuclear lithium clusters is remarkable in light of the long length and weakness of the bond; it represents an extreme in bonding.

The molecules $ \mathrm {H_{2}}$, LiH, and BeH are underbound by 20-40 kcal/mol. We understand that $ \mathrm {H_{2}}$ is underbound because of deficiencies in the basis, but the weak bond in LiH and BeH is surprising, given that the ionization potentials of Li and Be match the Hartree-Fock values exactly (Li: 123 kcal/mol eFF versus 123 kcal/mol HF, 186 kcal/mol eFF versus 186 kcal/mol eFF). Previously we had believed that localized Gaussian functions were a good basis for representing LiH; we may have to reevaluate this notion. At least there is some consistency now between the dissociation energy of hydrides and the dissociation energy of $ \mathrm {H_{2}}$.

We assess the new eFF correlation function by comparing eFF correlation energies to B3LYP minus UHF in the above series of s-electron containing molecules (Figure 6.18). Adding correlation tends to decrease electron-electron repulsion and shrink bond lengths. For the most part, we obtain the correct change in bond length upon adding correlation, with eFF bond length differences systematically larger than B3LYP minus HF differences by -0.03 to -0.05 $ \mathrm{\AA}$. The bond length changes are especially large in the lithium clusters.

Figure 6.18: eFF reproduces some energy and bond length changes caused by adding correlation.

Where exact energies are available, the correlation energies agree well with exact minus UHF energies, though not as well with B3LYP minus UHF energies (Table 6.8, $ \mathrm {H_{2}}$: -16.5 kcal/mol eFF versus -20.8 kcal/mol exact, LiH: -22.8 kcal/mol versus -22.7 kcal/mol exact, $ \mathrm{BeH_{2}}$: -25.6 kcal/mol versus -23.8 kcal/mol exact, $ \mathrm{H_{3}}$ (linear): -11.1 kcal/mol versus -14.6 kcal/mol exact, and $ \mathrm{Li_{2}}$: -23.3 kcal/mol versus -20.5 kcal/mol exact).

We do less well for the geometries BeH (-16.9 kcal/mol versus -2.9 kcal/mol exact) and $ \mathrm{H_{3}^{+}}$ (-17.8 kcal/mol versus -36.4 kcal/mol exact) -- the reasons for these discrepancies are unknown. We tend to overestimate the correlation of lithium two-electron systems and underestimate the correlation of hydrogen two-electron systems, which suggests that there may be some effects of having nuclei nearby we should be including, or that core-valence correlation is not balanced as well as correlation between core-like electrons.

We turn to a simpler problem, finding the correlation energy of core electrons. Consider the effect of increasing the nuclear charge of a helium atom. As Z increases, the electrons are drawn more tightly inward, which causes the electron-electron repulsion to increase. We would expect that the correlation energy would increase as well; however, at the same time, it becomes more difficult to excite the electrons to virtual orbitals, which makes the electrons less mobile. The end result is that the correlation energy of core electrons remains virtually unchanged as Z increases. Density functional methods tend to overestimate the correlation energy of highly charged ions isoelectronic to helium [20]. With the eFF correlation function, we find that the correlation energy of the core-like electrons has the correct trend of remaining virtually unchanged as Z increases (Figure 6.19).

Figure 6.19: Atomic correlation energies, eFF reproduces major trends.

Encouraged by this success, we examined the total atomic correlations of the atoms helium through neon. We evaluated the correlation energy as a single point correction to atoms optimized with the eFF for p-like electrons discussed in the previous chapter. We reproduced the general trend and magnitude of correlation energies correctly (Figure 6.19), but (1) helium correlation energy is too small (2) lithium through carbon correlation energies are too small and increase too quickly; (3) oxygen through neon correlations are too large and increase too quickly. It will be interesting to discover what will happen once the correlation function is applied self-consistently to atoms.

Overall, we have remarkably good agreement between eFF (exchange only) and Hartree-Fock energies and bond lengths, suggesting that the new exchange function is a good candidate for further development. The correlation function slightly exaggerates the change in bond length that occurs when correlation is added, but reproduces overall trends correctly. Where exact energies are available, we observe reasonable agreement between eFF correlation energies and exact minus Hartree-Fock energies. Finally, we have shown some promising results in reproducing atomic correlation energies, but a definitive verdict will have to wait until we develop a system for handling p-like electrons compatible with the new exchange and correlation functions.

next up previous contents
Next: Conclusion Up: Development of an electron Previous: Exchange and correlation functions   Contents
Julius 2008-04-29