Theor Chem Acc (2015) 134:74
1 3
Gaussian-type orbitals (EGTOs) was laid down in a report
by Browne and Poshusta [ 32 ]. They introduced the following class of functions:
and the corresponding molecular integrals were also
derived and presented. It is easy to recognize that when
α 1 = α 2 = α 3 , the ellipsoidal case simplifi es to a conventional GTO. They expected that the allowance for asymmetry in the exponents will improve the description of electron distribution.
The analytical formulas for the integrals of generalized
Hermite Gaussian functions were presented by Katriel [ 33 ]
and Katriel and Adam [ 34 ]. They also investigated the
effects of these basis functions for H 2 and He
2+
2 test systems. A serious drawback of their approach is its coordinate dependence.
Bishop and Leclerc [ 35 ] investigated several unconventional basis functions including EGTOs, generalized
Slater–Gauss-type, non-integer n Slater, rational, Hulthén,
and Bessel functions and found that the non-integer n
Slater basis performed best. However, the scope of their
study was fairly limited; only one system, the H 2 molecule,
was probed.
More detailed studies dealing with EGTOs were conducted by Cohen and Basch [ 36 , 37 ]. In their fi rst paper
[ 36 ], new, effi cient analytic integral evaluation formulas
were developed and tested for the calculation of the total
energies of HF and CO 2 . In their work, two single- ζ basis
sets (3 s 1 p /1 s ), a spherical and an ellipsoidal one, were generated and compared with a double- ζ basis set (4 s 2 p /2 s ) for
HF calculations. In the single- ζ sets, both the spherical and
ellipsoidal exponents were optimized for the valence orbitals of the atoms. They found that the optimized ellipsoidal
single- ζ basis set results approached the double- ζ values
and also performed better than the optimized spherical
single- ζ basis set. In the second report [ 37 ], further molecular systems were investigated using the methodology developed in their fi rst study [ 36 ]. However, beside the total
energy, several one-electron properties were also examined
and compared to the double- ζ basis set results. Based on
the one-electron properties, they concluded that the ellipsoidal valence orbitals in the single- ζ basis set were too
rigid to represent simultaneously both the spherical atomic
and the ellipsoidal bonding regions of the molecules and
exaggerate the transfer of charge density from the atomic
regions to the bonding regions of the molecules. They suggested that a more selective use of ellipsoidal basis functions will yield better results.
To describe the polarization caused by the molecular environment, Szalay and Surján [ 38 ] considered AC
Slater-type orbitals whose exponents depended on the polar
(1)
χ = x
i y
j z
k exp
−
α 1 x
2
+ α 2 y
2
+ α 3 z
2
,
angles. For the two-center one-electron problem, H
+
2 , they
presented the necessary molecular integrals and demonstrated that the potential curve of H
+
2 obtained with these
distorted s functions in the minimal basis agrees well with
the exact curve.
Here, we also refer to some other studies that are less
directly related to our present work. Gaussian Lobe Functions were introduced by Preuss [ 39 ]. In this representation,
the higher angular momentum functions are constructed
as a linear combination of usually two or more s -type
GTOs located on different centers. The study of Tasi and
Császár [ 40 ] is also notable. In their work, in addition to
the molecular orbital coeffi cients, both the position and the
exponents of the GTOs were treated as variational parameters during the solution of the HF equations. They were able
to obtain HF-limit energies for small atomic and molecular
systems with the help of a few dozen s -type GTOs.
The purpose of this study is twofold. First, we develop
new types of BF basis sets constituted of conventional
GTOs. We optimize both the exponents and the positions of
the BFs, but, in contrast to previous studies, the position of
each BF shell is optimized separately. Second, we propose
new types of basis functions for quantum chemical calculations, the general EGTOs. To ensure the coordinate-system invariance of the results, the EGTOs are also used as
BFs, and BF basis sets including EGTOs are optimized and
compared to those containing only conventional GTO BFs.
In both cases, we assess the performance of the developed
BF basis sets for HF and DFT calculations against conventional AC GTO basis sets including polarization functions.
2 Bond-centered spherical Gaussian functions
Our fi rst aim was to study and compare the effects of spherical BC GTOs with those of AC higher angular momentum
functions. In the following, the optimization of the GTO
parameters is described and the results obtained with the
application of BC GTOs are discussed in light of the data
calculated with conventional polarization basis sets.
2.1 Optimization of function parameters
Basis sets most frequently used in quantum chemistry
made up of contracted GTOs which are by defi nition linear
combinations of the
Cartesian GTOs. In the above expression, r = (x, y, z) designates the electron’s coordinates, A = (A x , A y , A z ) contains
the coordinates of the function’s center, and l = (i, j, k)
determines the function’s type, for example, in the case of
(2)
χ(r, A, l, α) = (x − A x )
i
y − A y
j (z − A z )
k e
−α(r−A) 2
207
Reprinted from the journal
1 3
Gaussian-type orbitals (EGTOs) was laid down in a report
by Browne and Poshusta [ 32 ]. They introduced the following class of functions:
and the corresponding molecular integrals were also
derived and presented. It is easy to recognize that when
α 1 = α 2 = α 3 , the ellipsoidal case simplifi es to a conventional GTO. They expected that the allowance for asymmetry in the exponents will improve the description of electron distribution.
The analytical formulas for the integrals of generalized
Hermite Gaussian functions were presented by Katriel [ 33 ]
and Katriel and Adam [ 34 ]. They also investigated the
effects of these basis functions for H 2 and He
2+
2 test systems. A serious drawback of their approach is its coordinate dependence.
Bishop and Leclerc [ 35 ] investigated several unconventional basis functions including EGTOs, generalized
Slater–Gauss-type, non-integer n Slater, rational, Hulthén,
and Bessel functions and found that the non-integer n
Slater basis performed best. However, the scope of their
study was fairly limited; only one system, the H 2 molecule,
was probed.
More detailed studies dealing with EGTOs were conducted by Cohen and Basch [ 36 , 37 ]. In their fi rst paper
[ 36 ], new, effi cient analytic integral evaluation formulas
were developed and tested for the calculation of the total
energies of HF and CO 2 . In their work, two single- ζ basis
sets (3 s 1 p /1 s ), a spherical and an ellipsoidal one, were generated and compared with a double- ζ basis set (4 s 2 p /2 s ) for
HF calculations. In the single- ζ sets, both the spherical and
ellipsoidal exponents were optimized for the valence orbitals of the atoms. They found that the optimized ellipsoidal
single- ζ basis set results approached the double- ζ values
and also performed better than the optimized spherical
single- ζ basis set. In the second report [ 37 ], further molecular systems were investigated using the methodology developed in their fi rst study [ 36 ]. However, beside the total
energy, several one-electron properties were also examined
and compared to the double- ζ basis set results. Based on
the one-electron properties, they concluded that the ellipsoidal valence orbitals in the single- ζ basis set were too
rigid to represent simultaneously both the spherical atomic
and the ellipsoidal bonding regions of the molecules and
exaggerate the transfer of charge density from the atomic
regions to the bonding regions of the molecules. They suggested that a more selective use of ellipsoidal basis functions will yield better results.
To describe the polarization caused by the molecular environment, Szalay and Surján [ 38 ] considered AC
Slater-type orbitals whose exponents depended on the polar
(1)
χ = x
i y
j z
k exp
−
α 1 x
2
+ α 2 y
2
+ α 3 z
2
,
angles. For the two-center one-electron problem, H
+
2 , they
presented the necessary molecular integrals and demonstrated that the potential curve of H
+
2 obtained with these
distorted s functions in the minimal basis agrees well with
the exact curve.
Here, we also refer to some other studies that are less
directly related to our present work. Gaussian Lobe Functions were introduced by Preuss [ 39 ]. In this representation,
the higher angular momentum functions are constructed
as a linear combination of usually two or more s -type
GTOs located on different centers. The study of Tasi and
Császár [ 40 ] is also notable. In their work, in addition to
the molecular orbital coeffi cients, both the position and the
exponents of the GTOs were treated as variational parameters during the solution of the HF equations. They were able
to obtain HF-limit energies for small atomic and molecular
systems with the help of a few dozen s -type GTOs.
The purpose of this study is twofold. First, we develop
new types of BF basis sets constituted of conventional
GTOs. We optimize both the exponents and the positions of
the BFs, but, in contrast to previous studies, the position of
each BF shell is optimized separately. Second, we propose
new types of basis functions for quantum chemical calculations, the general EGTOs. To ensure the coordinate-system invariance of the results, the EGTOs are also used as
BFs, and BF basis sets including EGTOs are optimized and
compared to those containing only conventional GTO BFs.
In both cases, we assess the performance of the developed
BF basis sets for HF and DFT calculations against conventional AC GTO basis sets including polarization functions.
2 Bond-centered spherical Gaussian functions
Our fi rst aim was to study and compare the effects of spherical BC GTOs with those of AC higher angular momentum
functions. In the following, the optimization of the GTO
parameters is described and the results obtained with the
application of BC GTOs are discussed in light of the data
calculated with conventional polarization basis sets.
2.1 Optimization of function parameters
Basis sets most frequently used in quantum chemistry
made up of contracted GTOs which are by defi nition linear
combinations of the
Cartesian GTOs. In the above expression, r = (x, y, z) designates the electron’s coordinates, A = (A x , A y , A z ) contains
the coordinates of the function’s center, and l = (i, j, k)
determines the function’s type, for example, in the case of
(2)
χ(r, A, l, α) = (x − A x )
i
y − A y
j (z − A z )
k e
−α(r−A) 2
207
Reprinted from the journal