Basis sets for correlated methods
All of the basis sets discussed up
to now have been developed for use at the Hartree-Fock level of theory, even
though corrections have been included in some cases to better reproduce
experimental data. Basis sets specifically designed for use with correlated methods
have been developed more recently. These basis sets are usually at least of double zeta
quality and also include polarization functions. One typical example
is the 6-311G(d,p) basis set.^{14} This basis set consists
of a core region of six contracted Gaussian primitives and a triply split
valence basis set using, respectively, three, one, and one Gaussian primitives.
Polarization functions on first-row elements consist of one set of five pure d-functions, while a set of
three p-type orbitals are added on hydrogen. On total this yields 18 basis
functions for each non-hydrogen atom of the first row and 6 basis functions for
hydrogen. The basis sets for carbon and hydrogen are:
C 0
S 6 1.00
.4563240000D+04 .1966650000D-02
.6820240000D+03 .1523060000D-01
.1549730000D+03 .7612690000D-01
.4445530000D+02 .2608010000D+00
.1302900000D+02 .6164620000D+00
.1827730000D+01 .2210060000D+00
SP 3 1.00
.2096420000D+02 .1146600000D+00 .4024870000D-01
.4803310000D+01
.9199990000D+00 .2375940000D+00
.1459330000D+01
-.3030680000D-02 .8158540000D+00
SP 1 1.00
.4834560000D+00
.1000000000D+01 .1000000000D+01
SP 1 1.00
.1455850000D+00
.1000000000D+01 .1000000000D+01
D 1 1.00
.6260000000D+00 .1000000000D+01
H 0
S 3 1.00
.3386500000D+02 .2549380000D-01
.5094790000D+01 .1903730000D+00
.1158790000D+01 .8521610000D+00
S 1 1.00
.3258400000D+00 .1000000000D+01
S 1 1.00
.1027410000D+00 .1000000000D+01
P 1 1.00
.7500000000D+00 .1000000000D+01
Orbital exponents
and expansion coefficients of the 6-311G(d,p) basis set have been chosen such
that the ground state UMP2(FC) atomic energies were minimized. The
frozen core
(FC) calculation eliminates the non-valence electrons in the inner core from
the electron correlation calculations. Unlike the small 6-31G(d) basis set, the
6-311G(d,p) basis set uses different exponents for the polarization functions
on different nuclei.
For methanol the 6-311G(d,p) basis
set consists of 60 basis functions which are in turn composed of 94 primitive
Gaussians.
%Kjob L301 #P HF/6-311G(d,p) GFInput GFPrint methanol basis set 0,1 C1 O2 1 r2 H3 1 r3 2 a3 H4 1 r4 2 a4 3 d4 r2=1.20 r3=1.0 r4=1.0 a3=120. a4=120. d4=180. |
Kjob command kills the job after checking the input The GFInput (“Gaussian Function Input”) output generation keyword causes the current basis set to be printed in a form suitable for use as general basis set input, and can thus be used in adding to or modifying standard basis sets. GFPrint command: This output generation keyword prints the current basis set in tabular form. |
1.7
correlation consistent basis sets
A series of basis sets for
correlated calculations has also been developed by Dunning et al.^{15} These basis sets
are referred to as correlation consistent (or cc) and are designed such that a base
set of sp functions is combined with correlation functions. These latter
functions are chosen such that all functions in a given set lead to a similar
lowering of the atomic correlation energy (calculated by CISD). The smallest
member of this series and thus often the starting point for correlated
calculations is the correlation consistent polarized double zeta basis set
designated "cc-pVDZ".^{15} The cc-pVDZ basis
for oxygen is:
O 0
S 9 1.00
.1172000000D+05 .7100000000D-03
.1759000000D+04 .5470000000D-02
.4008000000D+03 .2783700000D-01
.1137000000D+03 .1048000000D+00
.3703000000D+02 .2830620000D+00
.1327000000D+02 .4487190000D+00
.5025000000D+01 .2709520000D+00
.1013000000D+01 .1545800000D-01
.3023000000D+00 -.2585000000D-02
S 9 1.00
.1172000000D+05 -.1600000000D-03
.1759000000D+04 -.1263000000D-02
.4008000000D+03 -.6267000000D-02
.1137000000D+03 -.2571600000D-01
.3703000000D+02 -.7092400000D-01
.1327000000D+02 -.1654110000D+00
.5025000000D+01 -.1169550000D+00
.1013000000D+01 .5573680000D+00
.3023000000D+00 .5727590000D+00
S 1 1.00
.3023000000D+00 .1000000000D+01 „generic contractions“
P 4 1.00
.1770000000D+02 .4301800000D-01
.3854000000D+01 .2289130000D+00
.1046000000D+01 .5087280000D+00
.2753000000D+00 .4605310000D+00
P 1 1.00
.2753000000D+00 .1000000000D+01
D 1 1.00
.1185000000D+01 .1000000000D+01
This smallest
member of the cc-family is a
[3s,2p,1d] contraction of a (9s,4p,1d) basis set. A (1s,1p,1d) subset of this
basis set has been optimized such that the energy for the oxygen atom is
minimized in a CISD calculation.
The next member of the family is
obtained by starting from a somewhat larger (10s,5p) atomic basis set obtained
at the Hartree-Fock level and adding 2d and 1f polarization functions to obtain
the final [4s,3p,2d,1f] contraction. Addition of the two d-type polarization
function yield in this case the same amount of correlation energy as addition
of a single f function. The higher members of the cc-family of basis sets are
obtained in a similar manner:
basis set contraction functions for O
cc-pVDZ [3s2p1d] 14
cc-pVTZ [4s3p2d1f] 30
cc-pVQZ [5s4p3d2f1g] 55
cc-pV5Z [6s5p4d3f2g1h] 91
cc-pV6Z [7s,6p,5d,4f,3g,2h,1i] 140
An extension of the cc-basis sets by more diffuse functions
is mandatory for charged systems. A systematic extension is represented by the
aug-cc-pVxZ series which derives from the cc-pVxZ series through addition of
one set of diffuse functions for each angular momentum type already present in
the respective cc-basis. The cc-pVDZ thus addes a diffuse (1s1p1d) set to build the final [4s3p2d] aug-cc-pVDZ set. The
complete family of aug-cc-pVXZ basis set is then:
basis set contraction functions for O
aug-cc-pVDZ [4s3p2d] 23
aug-cc-pVTZ [5s4p3d2f] 46
aug-cc-pVQZ [6s5p4d3f2g] 80
aug-cc-pV5Z [7s6p5d4f3g2h] 127
The exponents of
the most diffuse basis functions in the aug-cc-pVXZ basis sets have been
optimized through HF and CISD calculations on the negatively charged atoms. These basis sets are generally recommend for
use with positively and negatively charged systems (Table 2).
Table 2. Proton
Affinities for Hydrogen Fluoride [CCSD(T) calculations]
Basis set PA
[kcal/mol]
cc-pVDZ 129.14
cc-pVTZ 125.31
cc-pVQZ 122.97
cc-pV5Z 121.94
aug-cc-pVDZ 118.97
aug-cc-pVTZ 121.63
aug-cc-pVQZ 121.62
aug-cc.pV5Z 121.54
exp. 122±1
Despite the fact
that these basis sets have been developed specifically for charged systems, the
perfomance of the aug-cc-basis sets is often superior to the cc-basis sets even
for neutral molecules. The complexation energies calculated for the water dimer
(Table 3) might serve as an example.
Table 3. Interaction Energies for the Water Dimer
[RI-MP2(FC) calculations]
Basis set DE [kcal/mol]
cc-pVDZ -7.54
cc-pVTZ -6.08
cc-pVQZ -5.47
cc-pV5Z -5.09
aug-cc-pVDZ -5.31
aug-cc-pVTZ -5.18
aug-cc-pVQZ -5.09
aug-cc.pV5Z -4.98
DH(375)[theo.] -3.2±0.1
DH(375)[exp.] -3.6±0.5
%Kjob L301 #P HF/aug-cc-pVDZ GFInput GFPrint methanol basis set 0,1 C1 O2 1 r2 H3 1 r3 2 a3 H4 1 r4 2 a4 3 d4 r2=1.20 r3=1.0 r4=1.0 a3=120. a4=120. d4=180. |
Kjob command kills the job after checking the input The GFInput (“Gaussian Function Input”) output generation keyword causes the current basis set to be printed in a form suitable for use as general basis set input, and can thus be used in adding to or modifying standard basis sets. GFPrint command: This output generation keyword prints the current basis set in tabular form. |
last changes: 01.04.2008, AS questions & comments to: axel.schulz@uni-rostock.de