Chemistry Reference
In-Depth Information
an initial guess at the wave functions or charge density, the Poisson equation
is solved to yield the electrostatic potential, and the exchange-correlation
potential is computed. This yields the effective potential in the Kohn-
Sham approach, and then we solve the eigenvalue problem approximately
to get new approximations to the eigenfunctions. In an MG approach, we
can then make a choice as to how to proceed. The first way, which follows
closely standard electronic structure methods, is to perform an MG V-cycle
for the eigenvalue problem, holding the effective potential fixed during
the iterations. Then, at the end of the V-cycle, the charge density and effec-
tive potential can be updated on the finest level. The solver goes back and
forth between solution of the eigenvalue problem and generation of the
effective potential.
A second strategy would be to update the effective potential on coarse
levels
during
the V-cycle so that the eigenfunctions and effective potential
evolve together.
116
This approach fits nicely into the overall MG philosophy
and has been implemented successfully in DFT calculations.
119,120
However, it
was found in Ref. 119 that simultaneously updating the potential on coarse
levels does not lead to a significant improvement in overall efficiency. The rea-
sons for this are not entirely clear.
LINEAR SCALING FOR ELECTRONIC STRUCTURE?
It was argued above that many chemical systems exhibit some degree of
localization. What this means is that if one atom moves by a small amount, the
electrons and nuclei relax in a way that screens out, over some length scale,
the effects of that movement. This concept can be quantified by looking at
the one-electron density matrix.
110
In a Kohn-Sham calculation, the electron
density at a point
r
is given by
2
X
N
e
=
2
2
r
ð
r
Þ¼
1
j
c
i
ð
r
Þj
½
44
i
¼
A generalization of Eq. [44] yields the one-electron density matrix:
2
X
N
e
=2
g
ð
r
;
r
0
Þ¼
1
c
i
ð
r
Þ
c
i
ð
r
0
Þ
½
45
i
¼
The total energy in Hartree-Fock theory or DFT can be expressed entirely in
terms of this density matrix. It can be shown that, for systems with a band gap
(that is, a separation between the highest occupied molecular orbital (HOMO)
and the lowest unoccupied molecular orbital (LUMO) energies)
121
or for
