8.6 DFT calculations

The structures were relaxed by performing highly converged density functional calculations. The geometries were obtained from all-electron calculations [157] using the B3LYP hybrid functional [154] and Dunning's cc-pVDZ basis set [158], which has been found an accurate and computationally affordable combination.[152,159,156] The B3LYP functional is a so-called hybrid functional because it includes a fraction of exact Hartree-Fock exchange.

All-electron, and not pseudopotential calculations were performed in order to facilitate comparison of the geometries obtained with those of previous studies. DFT calculations using the LDA, two gradient corrected functionals (PBE[160] and BLYP) and a hybrid functional (B3LYP) were then performed at the B3LYP optimised geometries for comparison with the DMC results.

The assertion that cc-pVDZ basis sets are sufficient [152,159,156] was tested by performing a series of calculations with Dunning's more extensive basis sets. The cc-pVDZ basis is an all-electron basis designed for correlated calculations consisting of two large $s$-type contracted sets of Gaussians for the $1s$ electrons, an additional uncontracted $s$-type Gaussian, a single $p$-type contraction, an additional $p$-type Gaussian and a single $d$-type Gaussian. Dunning [158] also defined an augmented basis set, aug-cc-pVDZ which augments the cc-pVDZ basis by an additional $s$,$p$ and $d$ Gaussians.

The ring and $\mathrm{D}_6$ fullerene isomers of $\mathrm {C}_{24}$ (see figure 8.3) which have significantly different coordination. The results of these calculations are given in table 8.1. Although the cc-pVDZ basis is ``incomplete'', the relative energies obtained are very similar to those obtained at the aug-cc-pVDZ level indicating the cc-pVDZ basis is probably sufficient for the structures tested.

Table 8.1: Basis set and geometry dependence of $\mathrm {C}_{24}$ isomers from all-electron density functional calculations. Energies are in a.u. per molecule. Sufficiently converged results were not obtained for the BLYP geometry, aug-cc-pVDZ basis.

	Geometry	Method	cc-pVDZ	aug-cc-pVDZ
Ring	B3LYP	B3LYP	-913.896423049	-913.923269541
		BLYP	-913.682193134	-913.716096665
	BLYP	B3LYP	-913.891887356
		BLYP	-913.687457613
$\protect\mathrm{D}_{6}$ Cage	B3LYP	B3LYP	-913.905987459	-913.931247631
		BLYP	-913.628848548	-913.660348572
	BLYP	B3LYP	-913.901588045	-913.926145351
		BLYP	-913.633002984	-913.663869638

The sensitivity of the total energies to changes in the geometries was assessed by comparing the energies of the fully relaxed ring and $\mathrm{D}_6$ fullerene isomers of $\mathrm {C}_{24}$ using the BLYP and B3LYP functionals. These functionals give significantly different energetic orderings for the isomers. The differences between the BLYP and B3LYP geometries were small - less than 0.03 angstroms in bond lengths and less than 0.4 degrees in bond angles. These differences in structure give rise to changes in the relative energies of the ring and fullerene of less than 0.27 eV for each of the density functionals investigated. The relative energies are therefore rather insensitive to the density functional used to obtain the geometries, but they are more sensitive to the density functional used to calculate the relative energies. Changes in the relative energies of less than 0.27 eV are small compared with the overall spread of energies, but some changes in the orderings of the isomers closest in energy could occur.