Page 34 - 71 the abc of dft_opt

7.2. EXCHANGE 67 68 CHAPTER 7. KOHN-SHAM
• The KS orbitals supercede the HF orbitals, in providing an exact molecular orbital theory. atom E T V ext V ee T S U E X T C U C E C
With exact KS theory, we see now how an orbital calculation can provide exact energetics. He -2.904 2.904 -6.753 0.946 2.867 2.049 -1.025 0.037 -0.079 -0.042
The HF orbitals are better thought of as approximations to exact KS orbitals. So the Be -14.667 14.667 -33.710 4.375 14.594 7.218 -2.674 0.073 -0.169 -0.096
standard figure, Fig. 1.3, and other orbital pictures, are much more usefully interpreted Ne -128.94 128.94 -311.12 53.24 128.61 66.05 -12.09 0.33 -0.72 -0.39
as pictures of KS atomic and molecular orbitals.
Table 7.1: Energy components for first three noble gas atoms, found from the exact densities.
• Note that pictures like that of Fig. 1.2 tell us nothing about the functional dependence of
v S (r). That KS potential was found simply by inverting the non-interacting Schr¨odinger 7.3 Correlation
equation for a single orbital. This gives us the exact KS potential for this system, but
In density functional theory, we then define the correlation energy as the remaining unknown
tells us nothing about how that potential would change with the density.
piece of the energy:
• By subtracting T S and U from F, what’s left (exchange and correlation) will turn out to E C [n] = F[n] − T S [n] − U[n] − E X [n]. (7.8)
be very amenable to local-type approximations.
We will show that it usually better to approximate exchange-correlation together as a single
entity, rather than exchange and correlation separately. Inserting the definition of F above,
7.2 Exchange
we find that the correlation energy consists of two separate contributions:
In density functional theory, we define the exchange energy as E C [n] = T C [n] + U C [n] (7.9)
@ @
@
@
E X [n] = %Φ[n] @ V ee@ Φ[n]& − U[n], (7.7) where T C is the kinetic contribution to the correlation energy,
@ˆ @
i.e., the electron-electron repulsion, evaluated on the Kohn-Sham wavefunction, yields a direct T C [n] = T[n] − T S [n] (7.10)
contribution (the Hartree piece) and an exchange contribution. In most cases, the Kohn-Sham (or the correlation contribution to kinetic energy), and U C is the potential contribution to the
wavefunction is simply a single Slater determinant of orbitals, so that E X is given by the Fock correlation energy,
integral, Eq. (5.13) of the KS orbitals. These differ from the Hartree-Fock orbitals, in that U C [n] = V ee [n] − U[n] − E X [n]. (7.11)
they are the orbitals that yield a given density, but that are eigenstates of a single potential For many systems, T C ∼ −E C ∼ −U C /2.
(not orbital-dependent). Note that Eq. (5.13) does not give us the exchange energy as an
Exercise 32 Correlation energy from Hamiltonian
explicit functional of the density, but only as a functional of the orbitals. The total energy in
a Hartree-Fock calculation is extremely close to T S + U + V ext + E X . Define the DFT correlation energy in terms of the Hamiltonian evaluated on wavefunctions.
The differences between HF exchange and KS-DFT exchange are subtle. They can be Exercise 33 Correlation energy
thought of as having two different sources. Prove E C ≤ 0, and say which is bigger: E C or E trad .
C
1. The KS exchange is defined for a given density, and so the exact exchange of a system Finally, to get an idea of how large these energies are for real systems, in Table 7.3 we
is the exchange of the KS orbitals evaluated on the exact density. The HF exchange is give accurate values for three noble gas atoms. These numbers were found as follows. First,
evaluated on the HF orbitals for the system. a highly accurate solution was found of the full Schr¨odinger equation for each atom. From
2. To eliminate the density difference, we can compare KS E X [n HF ] with that from HF. The it, the total ground-state energy and its various components could be calculated. Also, the
remaining difference is due to the local potential for the KS orbitals. ground-state density was extracted from the wavefunction. Next, a search was made for the
unique KS potential that corresponded to that density. This can be thought of as guessing
Exercise 30 HF versus DFT exchange the potential, solving for its density, and comparing with the exact density. Then changing the
Prove E HF ≤ T S [n HF ] + U[n HF ] + V ext [n HF ] + E X [n HF ] for any problem.
potential to shift the computed density toward the exact one, and repeating until converged.
Exercise 31 Exchange energies Once the KS potential and its orbitals are found, it is straightforward to evaluate its kinetic,
Calculate the exchange energy for the hydrogen atom and for the three-dimensional harmonic Hartree, and exchange energies. The last set of columns were found by subtracting KS
oscillator. If I construct a spin singlet, with both electrons in an exponential orbital, what is quantities from their exact counterparts.
the exchange energy then? There are many points to note in this table: