Chemistry Reference
In-Depth Information
number of basis functions), but the most expensive step under the RI approx-
imation scales as
N
aux
N
bf
Þ
(with
N
aux
the number of auxiliary functions).
Another way to reduce the cost of a given computation is to apply local
approximations. Because
Oð
the
electron repulsion between the product
m
ð
r
1
Þ
n
ð
r
1
Þ
depends inversely on the dis-
tance
r
12
between electrons 1 and 2, at sufficiently large distances this interaction
can be approximated by multipole expansions or even neglected entirely. Such
local approximations have made Hartree-Fock and density functional theory
computations feasible for systems with hundreds of atoms or more.
75,76
Simi-
larly, one can employ local approximations to correlated electronic structure the-
ories.
70,77-85
These approaches, pioneered by Pulay and Saebø,
77-79
localize the
molecular orbitals and then exploit the fact that the motion of two electrons will
only be correlated if those electrons are in nearby orbitals. Moreover, one may
safely neglect virtual (unoccupied) orbitals that are not spatially close to the occu-
pied orbitals in a given excitation. One advantage of local correlation methods is
that they neglect some of the terms that lead to basis set superposition error.
86
To date, very little work has been done to investigate either RI or local
correlation approximations
87,88
and
r
ð
r
2
Þ
s
ð
r
2
Þ
in the integral
ð
mn
j
rs
Þ
interactions. However, it is likely that
both will be very beneficial for speeding up computations significantly while
introducing only small errors. The RI approximation will be particularly use-
ful for computations involving large basis sets. Local approximations, on the
other hand, only become helpful for computations of large molecules because
they involve a certain ''overhead'' cost that can only be recovered once the
molecule reaches a certain size (called the crossover point). It is possible to
employ both approximations simultaneously, and in 2003 Werner, Manby,
and Knowles reported a linear scaling resolution of the identity (RI) local
MP2 method (RI-LMP2 or DF-LMP2).
70
This method has since been made
available in the MOLPRO program.
89
To demonstrate how these approximations may be helpful in computa-
tions of
for
p
interactions, Table 4 provides interaction energies for three prototype
configurations of the benzene dimer using canonical MP2, the resolution of the
p
Table 4
MP2 Interaction Energies and Errors for Approximations to MP2 (kcal mol
1
)
for Various Configurations of the Benzene Dimer
a
MP2
RI-MP2
LMP2
RI-LMP2
Sandwich
3.195
(14.2)
0.005
(3.3)
0.015
(29.4)
0.013
(3.9)
T-Shaped
3.355
(13.7)
0.005
(3.8)
0.002
(28.3)
0.004
(4.3)
Parallel Displaced
4.659 (17.1) 0.008 (2.7) 0.155 (35.4) 0.155 (4.5)
a
Evaluated using the aug-cc-pVTZ basis set and the rigid monomer geometries of Gauss and
Stanton (Ref. 39) with the counterpoise-corrected MP2/aug-cc-pVDZ optimized intermonomer
distances (Ref. 38). RI energies and errors for sandwich and T-shaped configurations from
Ref. 88. Computations performed on a 3.2-GHz Intel EM64T Nocona workstation. RI and local
computations were performed without the use of point-group symmetry. Local methods not
counterpoise corrected. CPU times in parentheses (h).
