Page 43 - 71 the abc of dft_opt

10.4. DISSOCIATION ENERGIES 85 86 CHAPTER 10. PROPERTIES
0 10.6 Transition metals
-1
-2 In solid state physics, an infamous failure is the prediction that the non-magnetic structure
v XC (r) -3 of iron is of slightly lower energy than the magnetic one.
In general, density functionals are much less reliable for transition metal complexes than
-4
-5 first- and second-row elements, with bond-length errors more likely in the 0.05 ˚ Arange.
-6
Ne atom
-7 exact 10.7 Weak bonds
LDA
-8
0 0.2 0.4 0.6 0.8 1
The remarks above apply to covalent, metallic, and ionic bonds. But many processes are
r determined by much weaker bonds, such as hydrogen bonds and dispersion forces. LDA
Figure 10.5: Exchange-correlation potential of Ne atom, both exactly and in LDA. significantly overbinds both hydrogen bonds, and van der Waals dimers.
Between two isolated pieces of matter (no significant density overlap) there are always van
take the mean absolute error (MAE), which is 0.25 eV. This is much better than HF, der Waals forces due to fluctuating dipoles. In absence of permanent dipoles, and ignoring
but not accurate enough for many purposes. relativistic effects, the energy between two such pieces behaves as
• Interestingly, if we compare LSDX with the exact exchange numbers, we find that LSDX C 6
E bind = − , R → ∞ (10.1)
almost always underestimates the ionization energy, and has an MAE of 0.4 eV. Thus R 6
LSDX is a poorer approximation to exact exchange than LSD is to the exact result. where R is their separation. The exchange energy is always repulsive, so the existence of C 6
Again, there is a cancellation of errors between exchange and correlation. is a purely correlation effect.
LDA cannot reproduce this asymptotic behavior, since any contribution to the energy
Electron affinities difference must come from a density difference. Thus the LDA correlation energy vanishes
Electronegativity and Hardness exponentially in this limit.

10.4 Dissociation energies
10.8 Gaps
We consider first atomization energies, being the energy difference between molecules or
On the other hand, for solids, these eigenvalues are often plotted as the band structure in
solids at equilibrium, and their constituent atoms. These are called cohesive energies of
solids. To calculate these, the zero-point energy of the molecule (or solid) must be added to solid-state texts. In these cases, the overall shape and position is good, but the band gap,
between HOMO and LUMO, is consistently underestimated by at least a factor of 2. In some
the minimum in its total energy curve. Typically, HF underestimates such binding energies
cases, some semiconductors have no gap in LDA, so it makes the incorrect prediction that
by about 100 kcal/mol, while LDA overbinds by about 30 kcal/mol, or 1.3 eV, or about 50
millihartrees. This meant LDA was never adopted as a general tool in quantum chemistry. For they are metals.
similar reasons, this also leads to transition state barriers that are too low, often non-existent.
Finally, qualitative errors occur for highly correlated systems, such as the solid NiO or the 10.9 Questions
molecules Cr 2 , or H 2 stretched to large distances. These systems are extremely difficult to
1. Does the reliability of LSD show that most systems are close to uniform?
study with density functional approximations, as will be discussed below.

10.5 Geometries and vibrations
However, bond lengths are extremely good in LDA, usually (but not always) being underes-
timated by about 1-2%.