Page 77 - 71 the abc of dft_opt

18.3. EFFECTS OF GRADIENT CORRECTIONS 153 154 CHAPTER 18. GENERALIZED GRADIENT APPROXIMATION
We can repeat this analysis to understand the effects of spin polarization. Must fill this in. will be too low in an LDA calculation, and be raised by a GGA calculation. Notice though
that this depends on the coordination argument. If the transition state is less coordinated,
Exercise 85 Deduce the maximum value for F XC allowed by the Lieb-Oxford bound.
the result is the reverse: LDA overestimates the barrier, and GGA will weaken it. This is
Exercise 86 The lines of different r s values never cross in Fig. Y. What does this imply often the case for, eg. surface diffusion, where the transition state (e.g. bridge-bonded)
about the functional? has lower symmetry than the initial state (e.g. atop bonded).
• By the same reasoning as for atomization energies, consider the process of stretching a
18.3 Effects of gradient corrections bond. As a bond is stretched, LDA’s overbinding tendancy will reduce. Thus equilibrium
bond lengths are usually too small in LDA, and GGA stretches them. The only exception
In this section, we survey some of the many properties that have been calculated in DFT to this is the case of bonds including H atoms. One can show that typically GGA favors
electronic structure calculations, and try to understand the errors made LSD, and why GGA a process in which
corrects them the way it does. ∆%s& ∆%r s &
≥ P + Q∆|ζ|&, (18.9)
• Both the large underestimate in the magnitude of total exchange energies and the over- %s& 2%r s &
estimate in the magnitude of correlation energies can be seen immediately from Fig. where %s&, etc., are precisely chosen averages, and where P is typically close to 1, and
18.6. Ignoring gradients ignores the enhancement of exchange and the truncation of Q ≈ 0. Usually fractional changes in the density are very small, due to the presence of
core electrons, and the right hand side is thought of as zero. This leads to the generic
correlation.
claim that GGA prefers inhomogeneity, i.e., LDA overfavors homogeneity. For example, in
• In studying ionization energies of atoms, one finds LSD overbinds s electrons relative to our cases above, when a molecule is atomized, its mean gradient increases. The inequality
p, p relative to d, and so forth. Since core electrons have higher density than outer ones, means that GGA will like this more than LDA, and so have a smaller atomization energy.
the local approximation is less accurate for them.
Neglecting the right-hand-side works for all atomization energies, in which the changes
• When a molecule is formed from atoms, the density becomes more homogeneous than are large. But for a bond-stretch, the fractional gradient changes are much smaller. In
before. The region of the chemical bond is flatter than the corresponding regions of the case of H atoms, the density is sufficiently low as to make its fractional change larger
exponential decay in atoms. Thus LDA makes less of an underestimate of the energy of than that of the gradient, when stretching a bond with an H atom. Thus in that case,
a molecule than it does of the constituent atoms. Write the gradient corrections have the reverse effect, and those bonds are shortened.
E XC (system) = E LDA (system) + ∆E LDA (system) (18.5)
XC XC
18.4 Satisfaction of exact conditions
Then ∆E XC (atoms) < ∆E XC (molecules), remebering that both are negative. The
atomization energy of a molecule is We can now look back on the previous chapters, and check that the real-space cutoff GGA
satisfies conditions that LDA satisfies, and maybe a few more.
atmiz
E = E XC (molecule) − E XC (atoms) (18.6)
XC
• Size-consistency Obviously, GGA remains size consistent.
so that
• The Lieb-Oxford bound.
E XC (atom) = E LDA (atom) + ∆E LDA (molecule) − E LDA (atom) (18.7)
XC XC XC The LO bound gives us a maximum value for the enhancement factor of 1.804. The
Thus molecules are overbound in LDA, typically by as much as 30 kcal/mol. real-space cutoff will eventually violate this bound, but only at large s, where we do not
trust it anyhow.
• Transition state barriers can be understood in much the same way. A typical transition
state in quantum chemistry is one of higher symmetry and coordination than the reac- • Coordinate scaling
tants. Thus comparing the transition state to the reactants, in just the way done above For exchange, clear cos only s dependence.
for a bond and its constituent atoms, we find that the transition-state barrier, defined as Show F XC lines never cross.
E trans = −E XC (reactants) + E XC (transitionstate) (18.8) • Virial theorem Trivially true.
XC