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