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since there are no Hartree and xc terms in the absence of interactions. The
density solving this Euler equation is ns(r). Comparing this with Eq. (68)
we find that both minimizations have the same solution ns(r) a" n(r), if vs
is chosen to be
vs(r) = v(r) + vH(r) + vxc(r). (70)
Consequently, one can calculate the density of the interacting (many-body)
system in potential v(r), described by a many-body Schrödinger equation
of the form (2), by solving the equations of a noninteracting (single-body)
system in potential vs(r).34
In particular, the Schrödinger equation of this auxiliary system,
¯2"2
-h + vs(r) Æi(r) = %EÅ‚iÆi(r), (71)
2m
yields orbitals that reproduce the density n(r) of the original system (these
are the same orbitals employed in Eq. (54)),
N
n(r) a" ns(r) = fi |Æi(r)|2, (72)
i
where fi is the occupation of the i th orbital.35 Eqs. (70) to (72) are the
celebrated Kohn-Sham (KS) equations. They replace the problem of mini-
mizing E[n] by that of solving a noninteracting Schrödinger equation. (Recall
34
The question whether such a potential vs(r) always exists in the mathematical sense
is called the noninteracting v-representability problem. It is known that every interacting
ensemble v-representable density is also noninteracting ensemble v-representable, but, as
mentioned in Sec. 3.2, only in discretized systems has it been proven that all densities are
interacting ensemble v-representable. It is not known if interacting ensemble-representable
densities may be noninteracting pure-state representable (i.e, by a single determinant),
which would be convenient (but is not necessary) for Kohn-Sham calculations.
35
Normally, the occupation numbers fi follows an Aufbau principle (Fermi statistics)
with fi = 1 for i N, and 0 d" fi d" 1 for i = N (i.e., at most
33
that the minimization of E[n] originally replaced the problem of solving the
many-body Schrödinger equation!)
Since both vH and vxc depend on n, which depends on the Æi, which in
turn depend on vs, the problem of solving the KS equations is a nonlinear
one, just as is the one of solving the (much more complicated) Dyson equation
(38). The usual way of solving such problems is to start with an initial guess
for n(r), calculate the corresponding vs(r), and then solve the differential
equation (71) for the Æi. From these one calculates a new density, using (72),
and starts again. The process is repeated until it converges. The technical
name for this procedure is  self-consistency cycle . Different convergence
criteria (such as convergence in the energy, the density, or some observable
calculated from these) and various convergence-accelerating algorithms (such
as mixing of old and new effective potentials) are in common use. Only rarely
it requires more than a few dozen iterations to achieve convergence, and even
rarer are cases where convergence seems unattainable, i.e., a self-consistent
solution of the KS equation cannot be found.
Once one has a converged solution n0, one can calculate the total energy
from Eq. (55) or, equivalently and more conveniently, from36
N q2
E0 = %EÅ‚i - d3r d3r2 n0(r)n0(r2 ) - d3r vxc(r)n0(r) + Exc[n0]. (73)
i
2 |r-r2 |
Equation (73) follows from writing V [n] in (55) by means of (70) as
V [n] = d3r v(r)n(r) = d3r [vs(r) - vH(r) - vxc(r)]n(r) (74)
= Vs[n] - d3r [vH(r) + vxc(r)]n(r), (75)
the highest occupied orbital can have fractional occupation). Some densities that are not
noninteracting v-representable by a single ground-state Slater determinant, may still be
obtained from a single determinant if one uses occupation numbers fi that leave holes
below the HOMO (the Fermi energy in a metal), so that fi = 1 even for some i
[31], but this is not guaranteed to describe all possible densities. Alternatively (see Sec.
3.2 and footnote 34) a Kohn-Sham equation may be set up in terms of ensembles of
determinants. This guarantees noninteracting v-representability for all densities that are
interacting ensemble v-representable. For practical KS calculations, the most important
consequence of the fact that not every arbitrary density is guaranteed to be noninteracting
v-representable is that the Kohn-Sham selfconsistency cycle is not guaranteed to converge.
36
All terms on the right-hand side of (73) except for the first, involving the sum of the
single-particle energies, are sometimes known as double-counting corrections, in analogy
to a similar equation valid within Hartree-Fock theory.
34
and identifying the energy of the noninteracting (Kohn-Sham) system as
N
Es = %EÅ‚i = Ts + Vs.
i
4.2.2 The eigenvalues of the Kohn-Sham equation
Equation (73) shows that E0 is not simply the sum37 of all %EÅ‚i. In fact, it
should be clear from our derivation of Eq. (71) that the %EÅ‚i are introduced as
completely artificial objects: they are the eigenvalues of an auxiliary single-
body equation whose eigenfunctions (orbitals) yield the correct density. It is
only this density that has strict physical meaning in the KS equations. The
KS eigenvalues, on the other hand, in general bear only a semiquantitative
resemblance with the true energy spectrum [61], but are not to be trusted
quantitatively.
The main exception to this rule is the highest occupied KS eigenvalue.
Denoting by %EÅ‚N(M) the N th eigenvalue of a system with M electrons, one
can show rigorously that %EÅ‚N(N) = -I, the negative of the first ionization
energy of the N-body system, and %EÅ‚N+1(N + 1) = -A, the negative of the
electron affinity of the same N-body system [58, 62, 63]. These relations
hold for the exact functional only. When calculated with an approximate
functional of the LDA or GGA type, the highest eigenvalues usually do not
provide good approximations to the experimental I and A. Better results
for these observables are obtained by calculating them as total-energy differ-
ences, according to I = E0(N - 1) - E0(N) and A = E0(N) - E0(N + 1),
where E0(N) is the ground-state energy of the N-body system. Alterna-
tively, self-interaction corrections can be used to obtain improved ionization
energies and electron affinities from Kohn-Sham eigenvalues [64].
Figure 2 illustrates the role played by the highest occupied and lowest
unoccupied KS eigenvalues, and their relation to observables. For molecules,
HOMO(N) is the highest-occupied molecular orbital of the N-electron sys-
tem, HOMO(N+1) that of the N + 1-electron system, and LUMO(N) the [ Pobierz całość w formacie PDF ]