Page 33 - 71 the abc of dft_opt

66 CHAPTER 7. KOHN-SHAM
is the kinetic energy of non-interacting electrons. We have implicitly assumed that the Kohn-
Sham wavefunction is a single Slater determinant, which is true most of the time. We denote
the minimizing wavefunction in Eq. (7.4) by Φ[n]. This gives us a verbal definition of the
Kohn-Sham wavefunction. The Kohn-Sham wavefunction of density n(r) is that wave-
@ @
Chapter 7 function that yields n(r) and has least kinetic energy. Obviously T S [n] = %Φ[n] @ T @ Φ[n]&,
@
@
@ ˆ@
which differs from T[n].
Kohn-Sham Exercise 29 Kinetic energies
Show that T[n] ≥ T S [n]. What is the relation between T HF [n] and T S [n]? Is there a simple
relation between T[n] and T HF [n]?
The most important step, establishing modern density functional theory as a useful tool,
Now, write the ground-state functional of an interacting system in terms of the non-
comes with the introduction of the Kohn-Sham equations, which allow almost all the interacting kinetic energy:
kinetic energy to be calculated exactly.
F[n] = T S [n] + U[n] + E XC [n]. (7.5)
7.1 Kohn-Sham equations We have explicitly included the Hartree energy, as we know this will be a large part of the
remainder, and we know it explictly as a density functional. The rest is called the exchange-
The major error in the Thomas-Fermi approach comes from approximating the kinetic energy correlation energy. Inserting F[n] into Eq. (6.13) and comparing with Eq. (7.3), we find
as a density functional. We saw in chapter one that local approximations to the kinetic
energy are typically good to within 10-20%. However, since for Coulombic systems, the 1 3 n(r) δE XC
v S (r) = v ext (r) + d r + v XC [n](r) v XC (r) = (7.6)
kinetic energy equals the absolute value of the total energy, errors of 10% are huge. Even |r − r | δn(r)
#
if we could reduce errors to 1%, they would still be too large. A major breakthrough in this
This is the first and most important relationship of exact density functional theory: from the
area is provided by the Kohn-Sham construction of non-interacting electrons with the same functional dependence of F[n], we can extract the potential felt by non-interacting electrons
density as the physical system, because solution of the Kohn-Sham equations produces the of the same density.
exact non-interacting kinetic energy, which includes almost all the true kinetic energy.
We note several important points:
We now have the theoretical tools to immediately write down these KS equations. Recall
from the introduction, that the KS system is simply a fictitious system of non-interacting • The Kohn-Sham equations are exact, and yield the exact density. For every physical
electrons, chosen to have the same density as the physical system. Then its orbitals are given system, the Kohn-Sham alter ego is well-defined and unique (recall Fig. 1.2). There is
by Eq. (1.5), i.e., nothing approximate about this.
2 1 3
2
− ∇ + v S (r) φ i (r) = & i φ i (r), (7.1) • The Kohn-Sham equations are a set of single-particle equations, and so are much easier
2
to solve than the coupled Schr¨odinger equation, especially for large numbers of electrons.
and yield
N However, in return, the unknown exchange-correlation energy must be approximated.
- 2
n(r) = |φ i (r)| . (7.2) (We do not know this functional exactly, or else we would have solved all Coulomb-
i=1
interacting electronic problems exactly.)
The subscript s denotes single-electron equations. But the Euler equation that is equivalent
to these equations is • While the KS potential is unique by the Hohenberg-Kohn theorem (applied to non-
interacting electrons), there are known examples where such a potential cannot be found.
δT S
+ v S (r) = µ, (7.3)
δn(r) In common practice, this has never been a problem.
where
@ @ • The great advantage of the KS equations over Thomas-Fermi theory is that almost all
@
T S [n] = min%Φ @ T @ Φ&, (7.4)
@
@ ˆ@
Φ→n the kinetic energy (T S ) is treated exactly.
65