Theor Chem Acc (2015) 134:74
1 3
Gaussian 09 suite [ 53 ]. The computed total energies are
displayed in Table 2 , while the number of basis functions is
collected in Table 3 .
Our results show that increasing the size of the system, two s -type BFs are not suffi cient any more at the
B3LYP level to obtain energies of quality similar to that
of r6-31G**. However, with three BFs, the calculated total
energies are lower by 1.2–4.6 m E h than the corresponding
values obtained with r6-31G**. If four BFs are employed,
even the smallest difference is 4.7 m E h , , but in this case,
the number of functions in the BF basis is somewhat larger.
Taking into account these observations in the further calculations, the basis sets optimized with three s -type BFs were
used for the C–H and C–C bonds.
2.2.2 Unsaturated hydrocarbons
In the next step, we constructed BF basis sets for the double and triple bonds present in unsaturated hydrocarbons.
The AO basis sets optimized for the C and H atoms as well
as the three s -type BFs selected for the saturated molecules
were placed on the ethene [ 44 ] and ethine [ 44 ] molecules,
and the parameters for the BFs of C=C and C≡C bonds
were optimized. To be consistent with the chemical picture,
p -type BFs were used for the description of π bonds; thus,
at least one and two p functions were used for the double
and triple bonds, respectively. Inspecting the results in
Table 1 , we can conclude that the 6-31G** AO basis sets
reoptimized for methane perform less well for ethene and
ethine than the original Pople basis sets; therefore, the performance of the basis sets including BFs was compared to
the latter. For ethene, an additional s function is required
beside the 1 s 2 p minimal BF set for an accurate energy,
while for ethine, one s and two p functions are suffi cient.
To assess the performance of the optimized basis sets
for unsaturated hydrocarbons, test calculations were carried
out at the DFT level. For reference, the energies obtained
with Dunning’s cc-pVTZ basis set were used. The energies
are collected in Table 4 , while the number of functions in
the various basis sets is presented in Table 5 .
As can be seen, the mean absolute error with respect to
the reference value is smaller by 33 % with the basis set
augmented with the selected BFs (6-31G–BF) than with
the reoptimized Pople basis set containing polarization
functions, while the total number of functions is lower by
6 %. It means that the decrease in the size of the basis set is
not dramatic; however, in the BF basis sets, there are no d
functions at all, while there are, in average, 24 ones in the
6-31G** basis sets for the considered molecules, as well as
the number of p functions also decreases by about 32 %. In
turn, the number of s functions is signifi cantly higher; however, the evaluation of two-electron integrals over s functions is substantially faster than that for basis functions of
higher angular momenta.
2.2.3 Heteroatom-containing hydrocarbons
Of the heteroatom-containing organic compounds, the
nitrogen- and oxygen-substituted ones were considered. As
for the BFs of hydrocarbons, the BFs for a particular bond
type were optimized for the smallest molecule in which the
bond of that type can be found. In this case, only the AO
basis functions centered on the N and O atoms were reoptimized, and for the H and C atoms, just as for the C–H
and C–C bonds, the basis functions were taken over from
the 6-31G–BF basis optimized for the hydrocarbons and
were not varied. First, for the description of the O–H and
N–H bonds, we considered the water [ 43 ] and ammonia
Table 6 B3LYP total energies
in E h for heteroorganic
molecules with various basis
sets
Molecule
Basis set
6-31G**
r6-31G**
6-31G–BF
cc-pVTZ
2-Aminopropan-1-ol
−249.55731
−249.57884
−249.60150
−364.17802
But-3-en-2-imine
−211.22286
−211.23126
−211.25137
−211.29976
But-3-en-2-one
−231.09874
−231.10993
−231.13414
−231.18563
Dimethylacetamide
−287.65991
−287.68500
−287.70126
−287.76399
Dimethylamine
−135.07599
−135.08945
−135.09761
−135.12468
Dimethyl ether
−154.93346
−154.94772
−154.96355
−154.99341
Ethanol
−154.94764
−154.96156
−154.97903
−155.01009
Ethyl cyanate
−247.12216
−247.14223
−247.16181
−247.21739
Ethyl methyl carbonate
−381.49093
−381.50859
−381.55443
−381.64087
Methyl propanoate
−307.53089
−307.55567
−307.58037
−307.64626
Propan-1-amine
−174.37198
−174.38821
−174.39859
−174.43428
Propanoic acid
−267.02444
−267.03636
−267.07153
−267.13211
Mean absolute error
0.089
0.072
0.050
211
Reprinted from the journal
1 3
Gaussian 09 suite [ 53 ]. The computed total energies are
displayed in Table 2 , while the number of basis functions is
collected in Table 3 .
Our results show that increasing the size of the system, two s -type BFs are not suffi cient any more at the
B3LYP level to obtain energies of quality similar to that
of r6-31G**. However, with three BFs, the calculated total
energies are lower by 1.2–4.6 m E h than the corresponding
values obtained with r6-31G**. If four BFs are employed,
even the smallest difference is 4.7 m E h , , but in this case,
the number of functions in the BF basis is somewhat larger.
Taking into account these observations in the further calculations, the basis sets optimized with three s -type BFs were
used for the C–H and C–C bonds.
2.2.2 Unsaturated hydrocarbons
In the next step, we constructed BF basis sets for the double and triple bonds present in unsaturated hydrocarbons.
The AO basis sets optimized for the C and H atoms as well
as the three s -type BFs selected for the saturated molecules
were placed on the ethene [ 44 ] and ethine [ 44 ] molecules,
and the parameters for the BFs of C=C and C≡C bonds
were optimized. To be consistent with the chemical picture,
p -type BFs were used for the description of π bonds; thus,
at least one and two p functions were used for the double
and triple bonds, respectively. Inspecting the results in
Table 1 , we can conclude that the 6-31G** AO basis sets
reoptimized for methane perform less well for ethene and
ethine than the original Pople basis sets; therefore, the performance of the basis sets including BFs was compared to
the latter. For ethene, an additional s function is required
beside the 1 s 2 p minimal BF set for an accurate energy,
while for ethine, one s and two p functions are suffi cient.
To assess the performance of the optimized basis sets
for unsaturated hydrocarbons, test calculations were carried
out at the DFT level. For reference, the energies obtained
with Dunning’s cc-pVTZ basis set were used. The energies
are collected in Table 4 , while the number of functions in
the various basis sets is presented in Table 5 .
As can be seen, the mean absolute error with respect to
the reference value is smaller by 33 % with the basis set
augmented with the selected BFs (6-31G–BF) than with
the reoptimized Pople basis set containing polarization
functions, while the total number of functions is lower by
6 %. It means that the decrease in the size of the basis set is
not dramatic; however, in the BF basis sets, there are no d
functions at all, while there are, in average, 24 ones in the
6-31G** basis sets for the considered molecules, as well as
the number of p functions also decreases by about 32 %. In
turn, the number of s functions is signifi cantly higher; however, the evaluation of two-electron integrals over s functions is substantially faster than that for basis functions of
higher angular momenta.
2.2.3 Heteroatom-containing hydrocarbons
Of the heteroatom-containing organic compounds, the
nitrogen- and oxygen-substituted ones were considered. As
for the BFs of hydrocarbons, the BFs for a particular bond
type were optimized for the smallest molecule in which the
bond of that type can be found. In this case, only the AO
basis functions centered on the N and O atoms were reoptimized, and for the H and C atoms, just as for the C–H
and C–C bonds, the basis functions were taken over from
the 6-31G–BF basis optimized for the hydrocarbons and
were not varied. First, for the description of the O–H and
N–H bonds, we considered the water [ 43 ] and ammonia
Table 6 B3LYP total energies
in E h for heteroorganic
molecules with various basis
sets
Molecule
Basis set
6-31G**
r6-31G**
6-31G–BF
cc-pVTZ
2-Aminopropan-1-ol
−249.55731
−249.57884
−249.60150
−364.17802
But-3-en-2-imine
−211.22286
−211.23126
−211.25137
−211.29976
But-3-en-2-one
−231.09874
−231.10993
−231.13414
−231.18563
Dimethylacetamide
−287.65991
−287.68500
−287.70126
−287.76399
Dimethylamine
−135.07599
−135.08945
−135.09761
−135.12468
Dimethyl ether
−154.93346
−154.94772
−154.96355
−154.99341
Ethanol
−154.94764
−154.96156
−154.97903
−155.01009
Ethyl cyanate
−247.12216
−247.14223
−247.16181
−247.21739
Ethyl methyl carbonate
−381.49093
−381.50859
−381.55443
−381.64087
Methyl propanoate
−307.53089
−307.55567
−307.58037
−307.64626
Propan-1-amine
−174.37198
−174.38821
−174.39859
−174.43428
Propanoic acid
−267.02444
−267.03636
−267.07153
−267.13211
Mean absolute error
0.089
0.072
0.050
211
Reprinted from the journal