Theor Chem Acc (2015) 134:74
1 3
functions. Frost and Preuss were among the fi rst in the
1960s who pioneered the use of BFs [ 18 , 19 ]. Preuss [ 18 ]
studied the electron correlation problem in the H
+
3 ion with
s -type GTOs positioned at the atomic nuclei and the three
bond centers as well as the centroid of the equilateral triangle structure. Frost described a simple fl oating spherical
Gaussian orbital model (FSGO) and applied it to the LiH
molecule [ 19 ]. His model can be used for the qualitative
description of singlet ground states; each pair of electrons
is modeled by an FSGO, the centers and exponents of
which are determined to satisfy the variational principle.
The fl oating orbital geometry optimization (FOGO) procedure was developed by Huber [ 20 ], and its feasibility was
demonstrated in subsequent papers [ 21 – 23 ]. In the FOGO
procedure, the energy gradient and the Hellmann–Feynman
force are simultaneously minimized letting the nuclei and
orbitals move independently. It is worth noting that the additional calculation of the Hellmann–Feynman force does not
cause any serious computational overhead because it does
not require the calculation of the two-electron derivatives.
The agreement between the experimental and FOGO geometries and dipole moments was found to be excellent [ 22 ].
Furthermore, rotational barriers obtained in FOGO using
a double- ζ basis set without polarization functions were in
line with those calculated using energy gradient optimization and basis sets including polarization functions [ 23 ].
Helgaker and Almlöf [ 24 ] studied the fi rst- and second-order molecular properties of small molecules
at the Hartree–Fock (HF) level using double- ζ quality basis sets including fl oating GTOs. They found that
although the electronic energies were only slightly better in comparison with the results obtained with the large
6-311++G(3df,3pd) basis set, the accuracy of calculated
properties greatly improved in many cases.
Rothenberg and Schaefer [ 25 ] investigated the effectiveness of various polarization schemes including BFs
and concluded that “bond functions provide a reasonable
alternative to the more conventional type of polarization
function.”
The fi rst thorough studies about BFs were conducted
by Neisius and Verhaegen [ 26 , 27 ]. First, BF parameters,
positions and exponents, were determined for the C–C and
C–H bonds as well as carbon lone pairs [ 26 ]. Later, they
extended their basis set work for molecules containing C,
N, O, and H atoms [ 27 ]. In both studies, the s - and p -type
functions were restricted to be centered at the same position, had identical exponents, and were optimized to be
used with the 6-31G basis set. Their results showed that
for less computational cost, the 6-31G basis set with additional BFs provided on average as accurate energies as the
6-31G* basis set.
The fi rst extensive BF study with correlation methods was performed by Martin et al. [ 28 ]. The total and
dissociation energies of fi rst-row hydrides were calculated using fourth-order Møller–Plesset perturbation theory with hybrid bond/polarization basis sets. Their hybrid
basis sets based on the 6-31G basis set and contained both
higher angular momentum functions and BFs. The results
obtained with the hybrid basis sets were competitive with
those calculated using considerably larger polarizationonly basis sets. Furthermore, the computational savings
gained were also substantial, and the cost of the calculations was reduced by a factor of 4–20.
Bauschlicher and Partridge [ 29 ] investigated the bond
energies of eight diatomics using a series of correlation
consistent basis sets with and without BFs. They concluded that although the addition of BFs to a given basis set
always improved the bond energies, the use of BFs does not
offer any advantages over AC basis sets when the results
were extrapolated to the basis set limit. Nevertheless, for
the double- and triple- ζ basis sets, they reported a sizable
improvement in the bond energetics due to the incorporation of BFs.
Jensen et al. [ 30 ] examined the basis set convergence
properties of BFs in density functional theory (DFT) calculations using the H 2 , N 2 , and N 4 species as test systems. They compared the convergence patterns of basis
sets consisting of BFs only, AC polarization functions
only, and a mixture of BFs and AC polarization functions.
They showed that the BFs only basis sets yield the slowest convergence toward the basis set limit. The convergence behavior of the other two basis set types was faster
and similar. Some of their results also indicated that optimal BFs might have fairly high angular momentum. It was
concluded that although the use of low angular momentum
BFs can reduce the number of high angular momentum AC
polarization functions, BFs do not provide any computational advantage over pure AC basis functions.
Styszy n ´ ski et al. [ 31 ] studied the performance of BFs in
relativistic HF and non-relativistic and relativistic electron
correlation calculations for the BF, AlF, and GaF molecules. It was found that at both the non-relativistic and relativistic HF levels, the diffuse and polarization functions of
an AC basis set can be effectively substituted by BFs unless
the calculations with the AC basis itself yield near-HF-limit
energies. However, in correlation calculations, the benefi ts
of BFs were not obvious because when many BFs were
used, the rate of convergence was usually ruined. Nevertheless, when the number of BFs was relatively small, a considerable improvement could be observed in the correlation
energy at practically no additional computational cost.
Besides BFs, there have been numerous attempts made
to introduce unconventional AC basis functions which
improve upon conventional GTOs. Nearly a decade
after Boys’ [ 14 ] paper on GTOs, the mathematical background of the application of special ellipsoidal (elliptical)
206
Reprinted from the journal
1 3
functions. Frost and Preuss were among the fi rst in the
1960s who pioneered the use of BFs [ 18 , 19 ]. Preuss [ 18 ]
studied the electron correlation problem in the H
+
3 ion with
s -type GTOs positioned at the atomic nuclei and the three
bond centers as well as the centroid of the equilateral triangle structure. Frost described a simple fl oating spherical
Gaussian orbital model (FSGO) and applied it to the LiH
molecule [ 19 ]. His model can be used for the qualitative
description of singlet ground states; each pair of electrons
is modeled by an FSGO, the centers and exponents of
which are determined to satisfy the variational principle.
The fl oating orbital geometry optimization (FOGO) procedure was developed by Huber [ 20 ], and its feasibility was
demonstrated in subsequent papers [ 21 – 23 ]. In the FOGO
procedure, the energy gradient and the Hellmann–Feynman
force are simultaneously minimized letting the nuclei and
orbitals move independently. It is worth noting that the additional calculation of the Hellmann–Feynman force does not
cause any serious computational overhead because it does
not require the calculation of the two-electron derivatives.
The agreement between the experimental and FOGO geometries and dipole moments was found to be excellent [ 22 ].
Furthermore, rotational barriers obtained in FOGO using
a double- ζ basis set without polarization functions were in
line with those calculated using energy gradient optimization and basis sets including polarization functions [ 23 ].
Helgaker and Almlöf [ 24 ] studied the fi rst- and second-order molecular properties of small molecules
at the Hartree–Fock (HF) level using double- ζ quality basis sets including fl oating GTOs. They found that
although the electronic energies were only slightly better in comparison with the results obtained with the large
6-311++G(3df,3pd) basis set, the accuracy of calculated
properties greatly improved in many cases.
Rothenberg and Schaefer [ 25 ] investigated the effectiveness of various polarization schemes including BFs
and concluded that “bond functions provide a reasonable
alternative to the more conventional type of polarization
function.”
The fi rst thorough studies about BFs were conducted
by Neisius and Verhaegen [ 26 , 27 ]. First, BF parameters,
positions and exponents, were determined for the C–C and
C–H bonds as well as carbon lone pairs [ 26 ]. Later, they
extended their basis set work for molecules containing C,
N, O, and H atoms [ 27 ]. In both studies, the s - and p -type
functions were restricted to be centered at the same position, had identical exponents, and were optimized to be
used with the 6-31G basis set. Their results showed that
for less computational cost, the 6-31G basis set with additional BFs provided on average as accurate energies as the
6-31G* basis set.
The fi rst extensive BF study with correlation methods was performed by Martin et al. [ 28 ]. The total and
dissociation energies of fi rst-row hydrides were calculated using fourth-order Møller–Plesset perturbation theory with hybrid bond/polarization basis sets. Their hybrid
basis sets based on the 6-31G basis set and contained both
higher angular momentum functions and BFs. The results
obtained with the hybrid basis sets were competitive with
those calculated using considerably larger polarizationonly basis sets. Furthermore, the computational savings
gained were also substantial, and the cost of the calculations was reduced by a factor of 4–20.
Bauschlicher and Partridge [ 29 ] investigated the bond
energies of eight diatomics using a series of correlation
consistent basis sets with and without BFs. They concluded that although the addition of BFs to a given basis set
always improved the bond energies, the use of BFs does not
offer any advantages over AC basis sets when the results
were extrapolated to the basis set limit. Nevertheless, for
the double- and triple- ζ basis sets, they reported a sizable
improvement in the bond energetics due to the incorporation of BFs.
Jensen et al. [ 30 ] examined the basis set convergence
properties of BFs in density functional theory (DFT) calculations using the H 2 , N 2 , and N 4 species as test systems. They compared the convergence patterns of basis
sets consisting of BFs only, AC polarization functions
only, and a mixture of BFs and AC polarization functions.
They showed that the BFs only basis sets yield the slowest convergence toward the basis set limit. The convergence behavior of the other two basis set types was faster
and similar. Some of their results also indicated that optimal BFs might have fairly high angular momentum. It was
concluded that although the use of low angular momentum
BFs can reduce the number of high angular momentum AC
polarization functions, BFs do not provide any computational advantage over pure AC basis functions.
Styszy n ´ ski et al. [ 31 ] studied the performance of BFs in
relativistic HF and non-relativistic and relativistic electron
correlation calculations for the BF, AlF, and GaF molecules. It was found that at both the non-relativistic and relativistic HF levels, the diffuse and polarization functions of
an AC basis set can be effectively substituted by BFs unless
the calculations with the AC basis itself yield near-HF-limit
energies. However, in correlation calculations, the benefi ts
of BFs were not obvious because when many BFs were
used, the rate of convergence was usually ruined. Nevertheless, when the number of BFs was relatively small, a considerable improvement could be observed in the correlation
energy at practically no additional computational cost.
Besides BFs, there have been numerous attempts made
to introduce unconventional AC basis functions which
improve upon conventional GTOs. Nearly a decade
after Boys’ [ 14 ] paper on GTOs, the mathematical background of the application of special ellipsoidal (elliptical)
206
Reprinted from the journal