Discussion

The familiar overestimation of the LSDA functional is demonstrated by the total correlation energies in Table 3.4. These results are close to previously published LSDA correlation energies [155]. Correct total atomic correlation energies (relative to the HF reference) are known to range from 42 mhartrees for helium to 392 mhartrees for neon [156]. Thus, LSDA is overestimating by roughly a factor of two.

The work of Stoll et al. [152,153] closely resembles Table 3.3. They partitioned the LSDA correlation energy in this manner, but then compared just the $\alpha \beta$ component to the total correlation energy from conventional theory. However, as can be seen from the first two columns of Table 3.3, the $\alpha \alpha$ and $\beta \beta$ components are too large to ignore. The spin-parallel contributions are partly spurious, as indicated by the significant nonzero values for H, He and H₂. These represent a self-correlation for single electrons. These effects are undoubtedly present in the larger systems, contributing to their large $\alpha \alpha$ and $\beta \beta$ values -- far greater than their corresponding conventional results, even accounting for the poor treatment of core electrons with MP2 for this basis. The conventional results do show that spin-parallel effects are significant, beyond that accounted for by the exchange term.

The spin-antiparallel components, shown in Table 3.3, are still significantly larger for LSDA than MP2. For the helium atom, where there is only $\alpha \beta$ correlation, the LSDA value of 58 mhartrees is considerably larger than the known accurate value of 42 mhartrees, even after accounting for the 1.7 mhartree correction of Table 3.2, but no longer by a factor of two. The larger systems show even greater overestimation, but note that the MP2 results will be underestimates, due to the crude description of the inner-shell conventional correlation.

If the LSDA error was roughly constant for each atom, a convenient cancellation of errors would occur when examining most chemical properties. To see if this is the case the contributions of parallel and antiparallel correlation to chemical binding energies are listed in Table 3.6, along with their conventional counterparts.

  

Table 3.6: Spin Components of the Correlation Binding Energy (mhartrees)

	DFT(LSDA)				conventional(MP2)
	$\alpha\alpha+\beta\beta$	$\alpha \beta$	total		$\alpha\alpha+\beta\beta$	$\alpha \beta$	total
H2	1.88	48.85	50.74		0	29.87	29.87
N2	4.64	82.23	86.87		46.37	169.64	216.01
F2	3.37	19.05	22.41		40.74	105.44	146.18
FH	5.10	36.60	41.70		23.09	54.77	77.86
OH2	8.69	76.89	85.58		30.96	93.89	124.85
NH3	10.83	120.59	131.42		25.46	119.91	145.37
CH4	15.65	131.91	147.56		27.13	108.03	135.16

The conventional total correlation contribution shows that electron correlation does play a major role in binding. The DFT totals are mostly lower (with the exception of H₂ and CH₄), with F₂ being remarkably lower. When broken down into the spin components, the contributions of parallel spins are quite small by LSDA, even though the individual parallel values of Table 3.3 are large. The MP2 results indicate that LSDA underestimates the parallel contribution (except for H₂, where there is none). The LSDA $\alpha \beta$ correlation contributions to binding show improvement when compared with the general overestimation of total correlation energies by a factor of two. However, there are wide variations with the type of bond. LSDA describes moderately well the $\alpha \beta$ contributions to bonds involving hydrogen. Yet the triple bond in N₂, where antiparallel correlation in the three pairs is a major stabilizing factor, shows an LSDA underestimation by more than a factor of two. This is particularly disturbing, remembering that the total correlation energies are overestimated by about this factor. The LSDA functional also badly underestimates the $\alpha \beta$ correlation contribution to binding in the F₂ molecule. F₂ is bound by LSDA (just), however 99 mhartrees of the 122 mhartrees come from the exchange part, when conventional HF does not bind F₂.

Turning to the core-valence separations of Table 3.4, the first thing to note is that the inter core-valence correlation energies are small. This is not surprising as the respective orbitals are principally located in different spatial regions. Comparison with good conventional numbers is not really possible, due to the deficiencies of the basis set. On the whole, the LSDA functional shows a good separation of correlation energy into the core and valence regions.

Although good MP2 core-core correlation numbers are unavailable, the atomic LSDA values should be close to those for the corresponding two-electron ions (He, Li⁺, Be²⁺, B³⁺, $\ldots$). It is known that these remain fairly constant, approaching a limit of about 46 mhartrees [156]. The LSDA numbers here are larger by up to a factor of four, and show a pattern of increasing steadily. This is obviously a major contribution to the overestimation of total correlation energies by the LSDA functional.

The valence-valence LSDA correlation energies are also too large, by roughly a factor of two. Again, the inadequate MP2 basis set provides a worrying underestimation, yet the results are similar for the more sophisticated QCISD(T) numbers.

The LSDA functional provides roughly a constant error for the core-core correlation components in moving from atom to molecule. Thus the failure to accurately describe the inner-shell electrons is not related to any failures in the description of chemical bonding. This is also true of the core-valence component. This suggests that the valence-only behaviour of the LSDA functional is more important. The spin-component analysis of the valence correlation energies is summarized in Table 3.5. The orbital basis is of better quality in the valence region, allowing a more satisfactory comparison with conventional MP2 results.

The spin-parallel components of valence correlation energies are still too large, and again show spurious self-correlation effects (for example Li and Be, where the valence correlation should be zero). The LSDA $\alpha \beta$ terms, however, do show a far better agreement with MP2 than the all-electron results. LSDA is too large again, but only by a factor of about 1.5, not two as before.

The partitions can be continued to examine the spin components of the binding energy using only the valence density, which are listed in Table 3.7.

  

Table 3.7: Spin Components of the Valence Correlation Binding Energy (mhartrees)

	DFT(LSDA)				conventional(MP2)
	$\alpha\alpha+\beta\beta$	$\alpha \beta$	total		$\alpha\alpha+\beta\beta$	$\alpha \beta$	total
N2	3.85	90.89	94.74		44.89	167.90	212.79
F2	3.28	20.49	23.77		40.45	105.04	145.48
FH	5.02	37.29	42.31		22.76	54.44	77.20
OH2	8.47	78.99	87.46		30.36	93.38	123.73
NH3	10.39	124.88	135.27		24.49	118.95	143.44
CH4	15.20	133.97	149.17		25.80	106.23	132.02

As expected, the performance of the valence-only theory (for binding) is similar to the all-electron results of Table 3.6. The correlation bindings are mostly too small and there is incorrect division between the spin-parallel and antiparallel components. This is consistent with a good cancellation of errors with the inner-shell contributions in moving from atoms to molecules.

Finally, in Table 3.8, the spin components of the correlation energy contributions to the ionization energies are presented. The total LSDA contributions of the five molecules studied are all about 60 mhartrees. This bears no resemblance to the large variations seen for the conventional values. The LSDA parallel and antiparallel components are roughly constant as well, with antiparallel providing the majority of the total. This, again is in contrast to the MP2 results, which show a roughly equal parallel/antiparallel contribution, with the components changing in similar ways to the total. Table 3.8 demonstrates a failure of the LSDA functional.

  

Table 3.8: Spin Components of the Correlation Ionization Potentials (mhartrees)

	DFT(LSDA)				conventional(MP2)
	$\alpha\alpha+\beta\beta$	$\alpha \beta$	total		$\alpha\alpha+\beta\beta$	$\alpha \beta$	total
N2	24.92	32.25	57.17		-12.32	-8.86	-21.18
FH	26.00	36.08	62.08		24.02	21.22	45.24
OH2	24.27	34.31	58.58		21.43	19.78	41.21
NH3	22.43	32.58	55.01		17.27	17.14	34.41
CH4	23.33	32.20	55.53		9.45	11.49	20.93