Theor Chem Acc (2015) 134:74
1 3
sought to answer two questions: (1) To what extent does
the energy decrease if the conventional BFs are replaced by
EBFs, and (2) can conventional p -type BFs be replaced by
s -type EBFs, e.g., for heteroatom–carbon bonds, in order
to further reduce the average angular momentum quantum
number of the basis functions? To that end, for conventional BF basis sets containing only s functions, EBF sets
with the same number of s -type EBFs were optimized. For
the BF bases including also p functions, in the corresponding EBF basis set, each p shell was replaced by three s -type
EBFs. For comparison, in the latter case, we also constructed a GTO BF basis with the same number of s -type
BFs. The parameters of the functions were optimized as
described in Sect. 2 .
The results of the optimizations are summarized in
Table 8 , where 6-31G–BF( p ) refers to the original BF basis
sets containing p functions, if any, under 6-31G–BF( s ), the
results obtained with BF basis sets including exclusively
s -type GTO BFs (the original BF basis if it does not contain p functions, and the new GTO BF basis optimized for
this purpose otherwise) are presented, whereas 6-31G–
EBF( s ) stands for the EBF basis sets. As can be seen from
the table, due to the variation of the additional parameter of
the EGTOs, the stretch factor, for molecules where all the
parameters of the basis sets are optimized at a time (methane, ammonia, and water), the energy computed with the
EBF sets is lower by less than a m E h per bond with respect
to that obtained with the corresponding BF basis set including only s functions. For the remaining molecules, such as
ethane, methylamine, and methanol, where only the EBFs
of the C–C, N–C, and O–C bonds are optimized, respectively, and other EBFs, just as the AO basis, are taken from
methane, ammonia, and water, the energy either decreases
(ethane and methanol) or practically does not change
(methylamine). Concerning the difference with respect
to the basis sets including p functions, the results suggest
that the accuracy of the latter cannot be approached if the p
functions are replaced by the EGTOs.
To further study the performance of the various BF basis
sets, we compiled a test set of ten molecules including
hydrocarbons, alcohols, ethers, and various amines. The
results are presented in Table 9 , where the cc-pVTZ energies are used again as reference values, and the same notations are employed as in Table 8 .
The trends are similar to those we observed for the molecules used in the basis set optimizations. In average, EBFs
decrease the energy by 10 m E h , and thus, the error by 15 %
with respect to purely s -type GTO BF basis sets, which
correspond to sub-m E h decrease per bond. The energy
decrease is consistent, and the only exception is methoxyethane, where a slight increase can be noticed. Compared
to the BF sets including p functions, the EBF bases are still
not fl exible enough to defeat them, but the error is considerably smaller than with the simple GTO BF bases. Nevertheless, these results prove the potential of EGTOs and
also suggest that it is worthwhile testing the performance of
p -type EBFs in future.
4 Conclusions and outlook
In this paper, new types of BF basis sets have been tested.
First, we considered conventional GTO BFs, but optimized
the center of each BF shell separately. Our results show
that using these BF sets, HF and DFT energies of similar
quality can be achieved as with conventional AO basis sets
including polarization functions, while the number of basis
functions is somewhat smaller and the angular momentum
quantum numbers of the basis functions are in average signifi cantly lower resulting in less expensive molecular integrals. Second, we tested ellipsoidal GTO BFs. The results
show that s -type EBFs decrease the error by about 15 %
with respect to basis sets with simple GTO BFs. All in all,
for total energies, the performance of the new types of BF
basis sets proposed here is encouraging, and further studies will be required to test them for other types of chemical
bonds and with larger AO basis sets.
Further investigations are also desirable to test the performance of our approach for energy differences, such as
reaction energies or barrier heights. However, here we must
also tackle with the general problems of the BF methodology. Though the proposed scheme for the positioning
of the BFs guarantees smooth potential energy surfaces
(PESs) and unbiased results for the energies of most reactions, for particular reactions and transition states, the situation is ambiguous. The simplest example is probably the
ring opening of cyclopropane to form propane, where we
have three C–C bonds in the educt, while only two ones
in the product. If the entire PES along the reaction coordinate is computed, the BFs originally located on the splitting
bond will migrate close to the middle carbon atom of propane. This is not elegant, but more importantly, the reaction energy will be different as if it was directly calculated
from the total energies of the reactant and the product. A
possible alternative approach, which resolves this problem
but partly retains the advantages of BFs, is to use atom-centered ellipsoidal Gaussian functions stretched in the direction of the bonds with geometry-dependent stretch factors.
If the latter are determined by an algorithm which guarantees that they converge to unity upon bond dissociation, the
ellipsoidal Gaussians will reduce to simple atom-centered
spherical Gaussians. The calculation of energy differences
and the development of atom-centered ellipsoidal Gaussian
basis sets will be the subject of subsequent papers.
216
Reprinted from the journal
1 3
sought to answer two questions: (1) To what extent does
the energy decrease if the conventional BFs are replaced by
EBFs, and (2) can conventional p -type BFs be replaced by
s -type EBFs, e.g., for heteroatom–carbon bonds, in order
to further reduce the average angular momentum quantum
number of the basis functions? To that end, for conventional BF basis sets containing only s functions, EBF sets
with the same number of s -type EBFs were optimized. For
the BF bases including also p functions, in the corresponding EBF basis set, each p shell was replaced by three s -type
EBFs. For comparison, in the latter case, we also constructed a GTO BF basis with the same number of s -type
BFs. The parameters of the functions were optimized as
described in Sect. 2 .
The results of the optimizations are summarized in
Table 8 , where 6-31G–BF( p ) refers to the original BF basis
sets containing p functions, if any, under 6-31G–BF( s ), the
results obtained with BF basis sets including exclusively
s -type GTO BFs (the original BF basis if it does not contain p functions, and the new GTO BF basis optimized for
this purpose otherwise) are presented, whereas 6-31G–
EBF( s ) stands for the EBF basis sets. As can be seen from
the table, due to the variation of the additional parameter of
the EGTOs, the stretch factor, for molecules where all the
parameters of the basis sets are optimized at a time (methane, ammonia, and water), the energy computed with the
EBF sets is lower by less than a m E h per bond with respect
to that obtained with the corresponding BF basis set including only s functions. For the remaining molecules, such as
ethane, methylamine, and methanol, where only the EBFs
of the C–C, N–C, and O–C bonds are optimized, respectively, and other EBFs, just as the AO basis, are taken from
methane, ammonia, and water, the energy either decreases
(ethane and methanol) or practically does not change
(methylamine). Concerning the difference with respect
to the basis sets including p functions, the results suggest
that the accuracy of the latter cannot be approached if the p
functions are replaced by the EGTOs.
To further study the performance of the various BF basis
sets, we compiled a test set of ten molecules including
hydrocarbons, alcohols, ethers, and various amines. The
results are presented in Table 9 , where the cc-pVTZ energies are used again as reference values, and the same notations are employed as in Table 8 .
The trends are similar to those we observed for the molecules used in the basis set optimizations. In average, EBFs
decrease the energy by 10 m E h , and thus, the error by 15 %
with respect to purely s -type GTO BF basis sets, which
correspond to sub-m E h decrease per bond. The energy
decrease is consistent, and the only exception is methoxyethane, where a slight increase can be noticed. Compared
to the BF sets including p functions, the EBF bases are still
not fl exible enough to defeat them, but the error is considerably smaller than with the simple GTO BF bases. Nevertheless, these results prove the potential of EGTOs and
also suggest that it is worthwhile testing the performance of
p -type EBFs in future.
4 Conclusions and outlook
In this paper, new types of BF basis sets have been tested.
First, we considered conventional GTO BFs, but optimized
the center of each BF shell separately. Our results show
that using these BF sets, HF and DFT energies of similar
quality can be achieved as with conventional AO basis sets
including polarization functions, while the number of basis
functions is somewhat smaller and the angular momentum
quantum numbers of the basis functions are in average signifi cantly lower resulting in less expensive molecular integrals. Second, we tested ellipsoidal GTO BFs. The results
show that s -type EBFs decrease the error by about 15 %
with respect to basis sets with simple GTO BFs. All in all,
for total energies, the performance of the new types of BF
basis sets proposed here is encouraging, and further studies will be required to test them for other types of chemical
bonds and with larger AO basis sets.
Further investigations are also desirable to test the performance of our approach for energy differences, such as
reaction energies or barrier heights. However, here we must
also tackle with the general problems of the BF methodology. Though the proposed scheme for the positioning
of the BFs guarantees smooth potential energy surfaces
(PESs) and unbiased results for the energies of most reactions, for particular reactions and transition states, the situation is ambiguous. The simplest example is probably the
ring opening of cyclopropane to form propane, where we
have three C–C bonds in the educt, while only two ones
in the product. If the entire PES along the reaction coordinate is computed, the BFs originally located on the splitting
bond will migrate close to the middle carbon atom of propane. This is not elegant, but more importantly, the reaction energy will be different as if it was directly calculated
from the total energies of the reactant and the product. A
possible alternative approach, which resolves this problem
but partly retains the advantages of BFs, is to use atom-centered ellipsoidal Gaussian functions stretched in the direction of the bonds with geometry-dependent stretch factors.
If the latter are determined by an algorithm which guarantees that they converge to unity upon bond dissociation, the
ellipsoidal Gaussians will reduce to simple atom-centered
spherical Gaussians. The calculation of energy differences
and the development of atom-centered ellipsoidal Gaussian
basis sets will be the subject of subsequent papers.
216
Reprinted from the journal