Chemistry Reference
In-Depth Information
metals at finite temperature,
122
the density matrix defined above decays expo-
nentially with distance:
g
ð
r
;
r
0
Þ/
j
r
r
0
jÞ
exp
ð
½
46
c
where
c
is a constant that depends on the magnitude of the band gap and/or
the temperature. (Metals at zero temperature are more difficult to handle since
the decay with distance is then only algebraic.
55
) Thus, in principle, we should
be able to represent the density matrix in some basis of localized functions and
discard information beyond a specified cutoff. This truncation should have lit-
tle impact on computed energies if it is chosen to be large enough to include
most of the relevant physical information (oscillations in the density matrix).
The MG eigenvalue method discussed above possesses computational
scaling that is more than linear in several parts of the algorithm. A goal of
modern computational chemistry is to drive that scaling down to near linear,
and a great deal of effort has been directed at achieving this optimal scaling.
92
Attempts at linear scaling have been made in traditional basis set algo-
rithms,
123-126
and in real-space solvers;
90,112,113,127
here we discuss the real-
space approaches. If the wave functions span the whole physical domain,
the basic relaxation step scales as
qN
g
,or
N
e
. The only way to reduce this scal-
ing is to enforce some form of localization on the orbitals being updated. If the
orbitals are restricted to have nonzero values only within some specified dis-
tance from a chosen center, the relaxation step will then scale as the number of
orbitals times the constant number of grid points within the local domain.
Even more challenging are the
N
e
-scaling orthogonalization and Ritz pro-
jection steps. If the problem is reformulated in a localized-orbital representation,
those costly operations can be reduced or eliminated, at the expense of other
processes that crop up. One MG approach to solving this problem was devel-
oped by Fattebert and Bernholc,
90
who utilized localized, nonorthogonal orbi-
tals to represent the electronic states. In this method the total electronic energy is
minimized while enforcing the localization constraint. The formulation requires
computation of the overlap matrix and its inverse and leads to a generalized
eigenvalue problem in the basis of the localized orbitals. Thus, there still remain
N
e
operations in the solver, but their prefactor tends to be small. Fattebert and
Bernholc
90
observed that the cubic-scaling operations comprise only a small
amount of the total work for large systems. This localized-orbital method has
been generalized to an FAS-MG form by Feng and Beck.
127
Two apparent drawbacks of the localized-orbital approach exist. First, the
convergence of the total energy stalls at a value above the exact numerical solu-
tion depending on the size of the cutoff radius for the orbitals.
128
This makes
sense because physical information is lost in the truncation. Second, the conver-
gence rate appears to slow somewhat with increasing system size.
127
The
observed convergence rates are still good, and competitive with other numerical
methods, but this slowdown does not fit with standard MG orthodoxy—the
