Chemistry Reference
In-Depth Information
data points are used in the extrapolation.
129,130
For example, if MP2 correla-
tion energies are available for cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pV5Z, and
cc-pV6Z basis sets, then the most reliable extrapolation schemes can be
obtained by fitting the Q/5/6 data points or just the 5/6 data points. Some
examples of these extrapolation procedures are presented in the tutorial
below.
Explicitly Correlated Methods
A serious implication of the equations presented in the previous section is
that the correlation energy converges to the CBS limit slowly with respect to
the cardinal number (or angular momentum) of the basis set. In response, dra-
matic progress has been made in the development of explicitly correlated R12
methods that '' bypass the slow convergence of conventional methods, by aug-
menting the traditional orbital expansions with a small number of terms that
depend explicitly on the interelectronic distance
r
12.''
131
Through various
approximations (e.g., the resolution of the identity) and by changing the linear
r
12 dependence to a different functional form (
f
12), these R12 and F12 meth-
ods can provide correlation energies (typically at the MP2 level) that are con-
verged to the CBS limit with only TZ or possibly even DZ quality basis sets.
Readers interested in more details are strongly encouraged to consult the out-
standing review by Klopper and co-workers.
Scaling Problem
Thanks to the concurrent development of more efficient computer algo-
rithms and affordable high-performance computing hardware, the sophisti-
cated electronic structure techniques described in the primer on electron
correlation above can be brought to bear on weakly bound clusters of ever-
increasing size (and with larger/better basis sets such as the correlation-
consistent basis sets described in the preceding section). The drawback of these
correlated electronic structure techniques is that their computational demands
(memory, CPU time, disk space) increase sharply with the size of the system.
For example, the ''gold standard'' of single-reference, ground-state quantum
chemistry [i.e., the CCSD(T) method] scales as the 7th power of the size of
the system,
N
7
. The practical consequences of this are devastating. Sup-
pose you have the facilities to perform a CCSD(T) computation on the water
hexamer with the aug-cc-pVTZ basis set (552 basis functions). By the time this
chapter is printed, high-performance personal computers
might
be fast enough
to perform a serial (nonparallel) computation of this magnitude in approxi-
mately 1 week. If one wishes to perform the same calculation on (H
2
O)
18
,
the computational requirements will increase by 3
7
Oð
Þ
2187 since the system
has tripled in size. That means the time required to perform the computation
would increase from 1 week to more than 42 years on the same computer. (A
nice overview of the scaling requirements of some popular methods can be
¼
