JOURNAL OF CHEMICAL PHYSICS
VOLUME 110, NUMBER 16
22 APRIL 1999
Gaussian basis sets for use in correlated molecular calculations.
IX. The atoms gallium through krypton
Angela K. Wilson,a) David E. Woon,b) Kirk A. Peterson,c)
and Thom H. Dunning, Jr.d)
Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory,
Richland, Washington 99352
~Received 6 November 1998; accepted 22 January 1999!
Valence correlation consistent and augmented correlation consistent basis sets have been
determined for the third row, main group atoms gallium through krypton. The methodology,
originally developed for the first row atoms, was first applied to the selenium atom, resulting in the
expected natural groupings of correlation functions ~although higher angular momentum functions
tend to be relatively more important for the third row atoms as they were for the second row atoms!.
After testing the generality of the conclusions for the gallium atom, the procedure was used to
generate correlation consistent basis sets for all of the atoms gallium through krypton. The
correlation consistent basis sets for the third row main group atoms are as follows: cc-pVDZ:
(14s11p6d)/ @ 5s4p2d # ; cc-pVTZ: (20s13p9d1 f )/ @ 6s5 p3d1 f # ; cc-pVQZ: (21s16p12d2 f 1g)/
@ 7s6p4d2 f 1g # ; cc-pV5Z: (26s17p13d3 f 2g1h)/ @ 8s7 p5d3 f 2g1h # . Augmented sets were
obtained by adding diffuse functions to the above sets ~one for each angular momentum present in
the set!, with the exponents of the additional functions optimized in calculations on the atomic
anions. Test calculations on the atoms as well as selected molecules with the new basis sets show
good convergence to an apparent complete basis set limit. © 1999 American Institute of Physics.
@S0021-9606~99!30515-8#
I. INTRODUCTION
family of basis sets—denoted by cc-pVDZ, cc-pVTZ, ccpVQZ, cc-pV5Z, ...—that systematically expand the coverage of the atomic radial and angular spaces, approaching the
complete basis set limit as the size of the sets increase ~i.e.,
as n→` for cc-pVnZ). These sets are constructed by grouping together all of the functions that lower the atomic correlation energy by the same amount and then adding these
groups sequentially to the atomic Hartree–Fock orbitals.
Thus, for a given level of accuracy in the correlation energy,
these sets are as compact as possible. It is possible to improve on the correlation energy obtained with the correlation
consistent sets, as noted by Hashimoto et al.,5 by using contracted functions rather than primitive functions for the polarization sets. As expected, the improvements are most significant for the smaller sets and largely cancel out in
computing energy differences.
In many instances it has been found that the convergence
behavior of atomic and molecular quantities computed with
the correlation consistent sets is sufficiently regular to allow
the results to be extrapolated to the complete basis set limit
using a series of calculations with increasing n.6–9 This allows the basis set convergence and electronic structure
method errors to be clearly distinguished, often leading to
surprising results. For example, for some molecular properties, the two errors are of opposite sign so that, for a particular basis set, the convergence error is counterbalanced by the
method error and the calculation appears to be very accurate.
This is the case for the dissociation energy (D e ) of N2.
D e (N2) calculated with the MP2 method and a cc-pVTZ
basis set is very close to the experimental value: 229.2 kcal/
mol calculated versus 228.4 kcal/mol experimental.10 How-
In selecting a basis set for use in solving the electronic
Schrödinger equation for atoms and molecules, the two primary criteria to consider are size and accuracy. Because the
computational cost of atomic and molecular calculations
scale as a high power of the number of basis functions ~e.g.,
N 4 for Hartree–Fock, N 5 for second-order perturbation
theory, and even higher for more accurate methods!, it is
important to keep the set as compact as possible for a given
level of accuracy. In addition to these qualities, it is also
desirable for the basis set to be a member of a family of sets
that systematically approach the complete basis set limit.
Knowing the complete basis sets limit, it becomes theoretically possible to decompose the error in any given calculated
quantity into the sum of the basis set convergence error and
the error in the electronic structure method. This distinction
is critical if we are to understand the inherent limits of the
numerous methods that have been developed to solve the
electronic Schrödinger equation ~e.g., configuration
interaction,1 perturbation theory,2 and coupled cluster
methods.3!
The correlation consistent basis sets, developed by Dunning for the first row elements boron through neon,4 satisfy
all of the above criteria. The correlation consistent sets are a
a!
Electronic mail: [email protected]
Current address: Molecular Research Institute, 845 Page Mill Rd, Palo
Alto, California 94304. Electronic mail: [email protected]
c!
Also at the Department of Chemistry, Washington State University, Richland, Washington 99352. Electronic mail: [email protected]
d!
Electronic mail: [email protected]
b!
0021-9606/99/110(16)/7667/10/$15.00
7667
© 1999 American Institute of Physics
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7668
Wilson et al.
J. Chem. Phys., Vol. 110, No. 16, 22 April 1999
ever, further expansion of the basis set shows that, in the
complete basis set limit, the MP2 method overshoots the experimental dissociation energy by nearly 12 kcal/mol!
More recently, the approach outlined by Dunning has
been used to develop correlation consistent basis sets for the
second row atoms,11 to extend the augmented sets12 to better
describe electrical response properties,13 to develop core–
valence correlation consistent sets for the first row atoms,14
and to extend the first and second row sets to include even
larger, more accurate basis sets ~cc-pV6Z and
aug-cc-pV6Z!.15,16 Because of the unique characteristics of
the correlation consistent sets noted above, they have been
used in more than a thousand studies reported in the literature. These studies have been directed toward a number of
goals, including a better understanding of the convergence of
molecular properties with basis set, the examination of extrapolation procedures to determine complete basis set limits,
and the determination of accurate values for atomic and molecular quantities.
In this paper we extend this approach to the main group
third row atoms, gallium to krypton. We find that, as was the
case for the first and second row atoms, a family of basis sets
can be logically constructed for gallium through krypton that
become more and more accurate as the set is increased in
size. Further, we find that atomic and molecular properties
computed with these sets show convergence toward an apparent complete basis set limit. By making available correlation consistent basis sets for the third row atoms, we hope
that molecules containing third row, main group atoms can
finally be described at a level comparable to that for the first
two rows, although in many instances relativistic effects will
also need to be included to achieve this level of accuracy
~see, e.g., Ref. 17!. In Sec. II of this paper we focus on the
procedure for developing the cc-pVnZ and aug-cc-pVnZ sets
for the selenium atom. In Sec. III, we discuss the standard
and augmented basis sets for gallium through krypton, reporting Hartree–Fock energies, as well as CCSD and
CCSD~T! valence correlation energies and electron affinities.
Concluding remarks are given in Sec. IV. In subsequent papers we will report a series of benchmark calculations on
molecules containing third row atoms ~see also Refs. 18 and
19! that quantify the performance of the new basis sets.
II. METHODOLOGY
The first step in the development of the third row basis
sets was to determine Hartree–Fock primitive sets ranging
from (13s) to (26s), from (9p) to (18p), and from (4d) to
(14d) for selenium.20 As we shall see later, generating a full
complement of HF primitive sets is needed to ensure a close
match between the optimum ns-, np-, and nd-‘‘correlation’’
sets and the HF primitive sets adopted for each of the correlation consistent basis sets. These calculations were carried
out with a modified version of the atomic Hartree–Fock program of Clementi.21
To properly describe the ground state of selenium, the
orbitals in the three 3 P configurations, 4s 2 4p 2x 4 p y 4 p z ,
4s 2 4p x 4p 2y 4p z , and 4s 2 4p x 4p y 4p 2z , were forced to be symmetry equivalent by basing the HF calculations on the aver-
age of the configurations. However, the singles and doubles
configuration interaction ~CISD, or HF1112) calculations,
as well as the coupled cluster calculations, included excitations from only one of the three configurations. Only excitations from the valence 4s- and 4 p-atomic orbitals were allowed. Thus, the current basis sets are only suitable for
describing correlation of the electrons in the valence orbitals
of the third row, main group atoms.
To allow the greatest flexibility in describing the atomic
orbitals, the general contraction method of Raffenetti22 was
used. Pure spherical harmonics were used in all calculations;
this not only reduces the number of functions used in the
calculations, but also helps reduce linear dependence problems. All correlated calculations were performed using
23
MOLPRO.
Although only calculations on selenium are reported in
this section, an identical study was undertaken for the gallium atom. The general conclusions drawn from the gallium
studies were identical to those based on the selenium studies.
Thus, the results of that study are not reported here.
A. Determination of polarization sets „ f , g , h ,...… for
selenium
To begin, a large primitive set of functions was chosen
for selenium with an energy very near the HF limit ~numerical solution of the Hartree–Fock equations yields an energy
of 22399.867 61 hartree24!. A (24s16p10d) primitive set
contracted to @ 8s7 p1d # was selected as the ‘‘base’’ ~spd!
set. The @ 8s # functions are the four HF atomic orbitals plus
the four most diffuse primitive Gaussian functions in the
(24s) set. The @ 7 p # and @ 1d # sets were similarly defined.
This set yields a HF energy of 22399.867 42 hartree, just 0.2
millihartree above the HF limit.
To determine the optimum ‘‘polarization’’ functions for
use in correlated calculations, d, f, g, and h functions were
sequentially and cumulatively added to the @ 8s7p1d # base
set and the exponents of the added functions optimized. As
in previous work, the exponents were taken to be an eventempered series, given by
z li 5 a l b i21
,
l
i51, . . . ,N k ,
~1!
where N k is the number of functions in the set and ‘‘l’’
designates the angular symmetry ~i.e., d, f ,...). In Eq. ~1!, a
and b are optimized for each (l,N k ) set by minimizing the
atomic CISD/HF1112 energy. All of the added functions
were uncontracted primitive functions. Various sets of 3d
functions ~for N k ranging from 1 to 6! were added to the base
set, and the optimum ~a, b!’s determined. The (4d) set was
then added to the @ 8s7 p1d # base, to form a new base set,
@ 8s7 p5d # , and a series of 4 f functions were added to this
set, and the ~a, b!’s optimized. After that, a new base set,
@ 8s7 p5d3 f # , was constructed and ~a, b!’s optimized for 5g
functions. Finally, an even larger base set, @ 8s7p5d3 f 2g # ,
was constructed and the exponent of a single 6h function
was optimized.
The results of these calculations are summarized in
Table I. In order to interpret the results, the following definitions are useful. The incremental energy lowering,
DE k,k21 , is defined as the energy obtained by adding the
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Wilson et al.
J. Chem. Phys., Vol. 110, No. 16, 22 April 1999
7669
TABLE I. Total energies, correlation energies, and energy lowerings from CISD/HF1112 calculations on the
selenium atom. Total energies (E HF1112 ) are in hartrees; correlation energies and energy lowerings (E corr and
DE k,k21 ) are in millihartrees. The Hartree–Fock energy is 22399.867 415 hartrees.
Polarization set
(24s16p10d)/ @ 8s7p1d #
@ 8s7p1d # 1
(1d)
(2d)
(3d)
(4d)
(5d)
(6d)
@ 8s7p1d # 1(4d)1
(1 f )
(2 f )
(3 f )
@ 8s7p1d # 1(4d3 f )1
(1g)
(2g)
@ 8s7p1d # 1(4d3 f 2g)1
(1h)
~a,b!
E HF1112
E corr
¯
22399.893 032
225.617
0.3638
~0.215, 2.700!
~0.158, 2.160!
~0.1465, 2.102!
~0.1256, 1.991!
~0.1144, 1.838!
22399.960 691
22399.969 128
22399.969 851
22399.969 982
22399.970 080
22399.970 106
293.276
2101.712
2102.435
2102.566
2102.664
2102.691
267.659
28.437
20.723
20.131
20.098
20.026
0.462
~0.284, 2.499!
~0.212, 2.034!
22399.990 876
22399.993 449
22399.993 734
2123.461
2126.033
2126.318
220.894
22.573
20.285
0.570
~0.385, 2.248!
22399.998 611
22399.999 393
2131.195
2131.978
24.877
20.782
0.722
22400.000 748
2133.333
21.355
first function of a higher angular symmetry ‘‘l’’ to its base
set ~e.g., adding the first f function to the @ 8s7p5d # set!,
DE 1,05E CISD~ l ! 2E CISD~ l21 ! ,
~2!
or by extending the number N k of functions of the same
symmetry ‘‘l’’ ~e.g., by adding a second f function to the
@ 8s7p5d1 f # set!,
DE k,k21 5E CISD~ N k ! 2E CISD~ N k21 ! .
~3!
DE k,k21 provides a direct measure of the importance of each
expansion of the correlation set, providing the metric used
for the construction of the correlation consistent basis sets. In
Table I, these energy lowerings, as well as the optimized
exponents and total energies, are provided for the following
sets of polarization functions: 1d – 6d, 1 f – 3 f , 1g and 2g,
and 1h.
In Fig. 1, the energy lowerings obtained for the various
angular momentum sets are plotted. As can be seen, for the
first few members of the series, DE k,k21 , decreases nearly
exponentially. For the d sets, however, there is a pronounced
‘‘knee’’ in the plot, beginning at k54. This ‘‘knee’’ was also
found in calculations on the sulfur atom ~see Fig. 2!, although it is slightly more pronounced—if at lower
energy—in selenium than in sulfur. The presence of the knee
seems to be due to the fact that the valence orbitals of the
second and third row atoms have significant electron density
in the region occupied by the next lower shell ~the L shell for
sulfur and the M shell for Se!. This requires functions with
higher exponents than found in the standard valence set. For
both sulfur and selenium, there is a change in the exponent
pattern at the knee,11 with higher exponent functions suddenly being added to the basis set ~in an even-tempered expansion this occurs when a k11 ' a k ).
The energy lowerings for the f and g sets of sulfur and
selenium are quite similar ~see Fig. 2!. We also find that the
first function of the given angular momentum is more important in sulfur and selenium than in oxygen, but the addition
DE k,k21
of subsequent functions of that angular momentum is less
important, with the difference increasing with k.
B. Determination of sp sets for selenium
Now that optimum polarization (d, f ,g, . . . ) sets have
been determined, the s and p sets can be optimized. To optimize the s sets, we employ a (24s16p10d)/ @ 4s7p1d #
1(3d2 f 1g) base set, while for the p sets we use a
(24s16p10d)/ @ 8s3 p1d # 1(3d2 f 1g) base set. Following
the procedure outlined above, we obtain the results summarized in Table II. As for the sulfur atom,11 for all sets, save
the (2s) set, we find two distinct minima on the ~a,b! surface: one at a and one at approximately 3a. In fact, the 3a
FIG. 1. Contributions of polarization functions to the correlation energy of
selenium. The absolute values of the incremental correlation energy lowerings, u DE k,k21 u in mE h are plotted against the number of functions in the
well-tempered expansions for d, f, g, and h functions. The dashed lines
correspond to analytic fits to the first three points of the d, f, and g series.
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7670
Wilson et al.
J. Chem. Phys., Vol. 110, No. 16, 22 April 1999
FIG. 2. Contributions of d, f, and g functions to the correlation energy of oxygen, sulfur, and selenium. The absolute values of the incremental correlation
energy lowerings, u DE k,k21 u in mE h are plotted against the number of functions in the well-tempered expansions for d, f, and g functions.
sets give the lower total energies, although the differences
are small compared to the corresponding incremental energy
lowerings. As for sulfur, we have chosen to use the sets
based on the smaller a’s, following the observation that, in
constructing molecular basis sets, it is important to maintain
maximum flexibility in the outermost functions, as these
functions describe the internuclear regions of a molecule,
which are the regions that differ most from the atoms.4,11,25
In Fig. 3 we plot the incremental energy lowering in
selenium for the sp, d, f, g, and h sets, grouping the sets
together to form ‘‘correlation consistent’’ sets ~the energy
lowerings for the sp sets were approximated by summing the
energy lowerings given in Table II!. For oxygen we found
that the curves corresponding to those in Fig. 3 were well
separated and nearly flat.4 That was not the case for sulfur11
and is not the case for selenium. Here we see a very significant upward trend as the angular momentum increases. The
net result is that the energy lowerings for the low angular
momentum functions of one set can be very similar to those
for the high angular momentum functions for the next larger
set. This means that it might be possible to construct the
correlation consistent basis sets using an alternate formulation, e.g., dropping the second sp set from the cc-pVTZ set,
then including the second sp set in the cc-pVQZ set but
dropping the third sp and d set, etc. We have chosen not to
do this here. Part of the consistency observed as a result of
TABLE II. Total energies, correlation energies, and incremental correlation energies for ~s! and ~p! functions
added to selenium. Total energies (E HF1112 ) are given in hartrees; correlation energies and incremental correlation energies (E corr and DE k,k21 ) are given in millihartrees. The Hartree–Fock energy is 22399.867 421
hartrees.
(s)/(p) set
@ 4s7p1d # 1(3d2 f 1g)1
(1s)
(2s)
(3s)
(4s)
(5s)
@ 8s3p1d # 1(3d2 f 1g)1
(1p)
(2p)
(3p)
(4p)
(5p)
~a,b!
0.1467
0.409
~0.1225, 5.089!
~0.119, 2.317!
~0.323, 2.167!
~0.0956, 2.058!
~0.289, 2.081!
~0.0950, 2.000!
~0.226, 1.961!
0.1197
0.4666
~0.0963, 6.555!
~0.312, 1.993!
~0.0694, 2.361!
~0.271, 2.262!
~0.0677, 2.352!
~0.209, 2.006!
~0.0657, 2.062!
~0.1734, 1.905!
E HF1112
E corr
22399.988 398
22399.996 547
22399.996 797
22399.997 997
22399.998 153
22399.998 156
22399.998 205
22399.998 209
22399.998 230
22399.998 233
22399.971 086
22399.993 989
22399.994 662
22399.997 428
22399.997 565
22399.997 904
22399.998 085
22399.998 170
22399.998 227
22399.998 242
22399.998 258
2120.977
2129.125
2129.376
2130.575
2130.732
2130.734
2130.784
2130.787
2130.809
2130.812
2103.664
2126.567
2127.241
2130.006
2130.144
2130.483
2130.664
2130.748
2130.805
2130.821
2130.837
DE k,k21
28.148
28.399
21.450
20.157
¯
20.052
20.053
20.025
20.025
222.903
223.576
23.439
22.903
20.477
20.520
20.265
20.141
20.072
20.031
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Wilson et al.
J. Chem. Phys., Vol. 110, No. 16, 22 April 1999
FIG. 3. Incremental correlation energy lowerings u DE k,k21 u and correlation
consistent groupings of functions for selenium. The groupings correspond to
valence double zeta through valence quintuple zeta sets.
using the correlation consistent sets is due to the gradual,
ever-expanding coverage of both the atomic radial and angular spaces. The alternate formulation might break that pattern, resulting in less consistency, even though the total energies would be lower. However, it would certainly be
worthwhile to investigate such alternative formulations of
the correlation consistent sets at some point in the future.
One could use the optimum s and p sets given in Table II
in atomic and molecular calculations, combined with a suitable representation of the atomic Hartree–Fock orbitals.
However, as before, we have chosen to substitute the optimum ‘‘correlation’’ s and p exponents with the corresponding set of exponents from an appropriate Hartree–Fock set.
This is done as illustrated in Tables III and IV. The optimum
1s exponent is 0.1467. The most diffuse exponent in the HF
(13s) primitive set has an exponent of 0.138 32. Substitution
of this exponent for the optimum exponent decreases the
magnitude of the correlation energy recovered by just 0.040
mE h compared to a total energy lowering for an addition of
the 1s-correlation function of 28.148 mE h ~a 0.5% effect!.
7671
Further, as shown in Table VI, the Hartree–Fock error associated with use of the (13s) primitive set is substantially
smaller than the correlation energy error ~a desirable
attribute—it should be easier to describe the HF wave function than the correlated wave function!. Thus, the (13s)
primitive set provides a good compromise s set for the ccpVDZ set. With this choice, the @ 5s # functions in the ccpVDZ set are the 1s – 4s atomic orbitals expanded in the
(13s) set plus the single outermost primitive function in the
(13s) set. A similar process identifies the (11p) set as the
best set to use in the cc-pVDZ set, as well as the optimum s
and p sets to be used for the cc-pVTZ through cc-pV5Z sets.
We made a few minor adjustments to the above procedure. First, because of the large number of calculations reported on third row species with (14s11p5d) sets,26,27 we
selected a HF (14s) primitive set for the cc-pVDZ set instead of the (13s) primitive set. This decreases the magnitude of the correlation energy recovered by only 0.016 mE h
for selenium. Second, for the cc-pVQZ set we used the (21s)
primitive set because the (20s) set was used in the cc-pVTZ
set. Finally, for a similar reason we used the (17p) set instead of the (16p) primitive set for the cc-pV5Z set. These
choices also ensure that the errors in the correlation energy
are larger than the errors in the Hartree–Fock energy; see
Table VI.
C. Determination of combined d sets for selenium
In comparing the ‘‘correlation’’ d sets with the Hartree–
Fock d sets, it was noted that there was often a considerable
overlap between the two. To minimize the number of d functions used in the calculations, we have taken full advantage
of this overlap. In Table V we show that we can use the most
diffuse function in the HF (8d) primitive set to replace the
tighter function in the correlation 2d set. The resulting error
is just 0.047 mE h , compared to a total energy lowering of
TABLE III. Errors in the valence correlation (CISD/HF1112) energy for selenium due to substituting optimized s exponents with the exponents from
selected HF primitive sets. Total energies (E HF1112 ) are given in hartrees; correlation energies and incremental correlation energy errors (E corr and DE corr)
are given in millihartrees. DE corr is measured relative to the corresponding optimum ns-correlation set. The base set is (24s16p10d)/ @ 4s3 p1d #
1(5p3p2 f 1g) and the corresponding HF energy is 22399.867 421 hartrees.
~ns! set
Source
(1s)
Optimum
(13s)
(14s)
(15s)
(16s)
Optimum
(19s)
(20s)
(21s)
Optimum
(19s)
(20s)
(21s)
Optimum
(23s)
(24s)
(25s)
(26s)
(2s)
(3s)
(4s)
Exponents
0.1467
0.138 32
0.136 76
0.135 82
0.135 05
0.1225
0.134 12
0.106 28
0.100 35
0.1190
0.134 12
0.106 28
0.100 35
0.0956
0.097 258
0.096 214
0.093 537
0.089 328
0.623
1.4884
0.506 68
0.492 85
0.276
0.365 16
0.254 08
0.235 20
0.1967
0.226 84
0.223 85
0.215 46
0.201 43
0.639
1.4884
0.506 68
0.492 85
0.405
0.491 04
0.488 96
0.475 79
0.441 03
0.833
1.4286
1.3669
1.1981
0.940 50
E HF1112
E corr
DE corr
22399.996 619
22399.996 580
22399.996 564
22399.996 553
22399.996 544
22399.998 067
22399.997 674
22399.997 986
22399.997 943
22399.998 229
22399.997 790
22399.998 190
22399.998 171
22399.998 283
22399.998 252
22399.998 258
22399.998 272
22399.998 284
2129.198
2129.158
2129.142
2129.131
2129.122
2130.646
2130.253
2130.564
2130.521
2130.807
2130.368
2130.768
2130.749
2130.861
2130.830
2130.836
2130.850
2130.862
¯
0.040
0.056
0.066
0.075
¯
0.393
0.082
0.125
¯
0.439
0.040
0.058
¯
0.031
0.025
0.011
20.001
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7672
Wilson et al.
J. Chem. Phys., Vol. 110, No. 16, 22 April 1999
TABLE IV. Errors in the valence correlation (CISD/HF1112) energy for selenium due to substituting optimized p exponents with the exponents from
selected HF primitive sets. Total energies (E HF1112 ) are given in hartrees; correlation energies and incremental correlation energy errors (E corr and DE corr)
are given in millihartrees. DE corr is measured relative to the corresponding optimum np-correlation set. The base set is (24s16p10d)/ @ 4s3 p1d #
1(5s3d2 f 1g) and the corresponding HF energy is 22399.867 421 hartrees.
~np! set
Source
(1 p)
Optimum
(10p)
(11p)
(12p)
Optimum
(12p)
(13p)
(14p)
Optimum
(15p)
(16p)
(17p)
Optimum
(15p)
(16p)
(17p)
(2p)
(3p)
(4 p)
Exponents
0.1197
0.133 33
0.124 86
0.110 57
0.0963
0.115 07
0.888 20
0.083 064
0.0694
0.078 752
0.070 556
0.063 738
0.0677
0.078 752
0.070 556
0.063 738
0.631
1.0245
0.623 56
0.564 72
0.1639
0.206 94
0.175 92
0.151 34
0.1592
0.206 94
0.175 92
0.151 34
0.387
0.517 75
0.420 52
0.345 25
0.375
0.517 75
0.420 52
0.345 25
28.436 mE h ~Table I!. Likewise, we can replace the two
tighter functions in the correlation 3d set with the two most
diffuse functions in the HF (11d) primitive set and the three
tighter functions in the correlation 4d set with the three most
diffuse functions in the HF (12d) primitive set. This means
that the cc-pVTZ set will use a (8d)/ @ 2d # 1(1d) set, the
cc-pVQZ set will use a (11d)/ @ 3d # 1(1d) set, and the ccpV5Z set will use a (12d)/ @ 4d # 1(1d) set. For use in the
final cc-pVnZ sets, we reoptimized the exponent of the (1d)
function in each of these sets, but left the exponents involved
in the HF orbitals unchanged. This recovered some of the
correlation energy losses listed in Table V. These choices for
the cc-pVnZ sets also lead to HF errors that are smaller than
the correlation energy errors; see Table VI.
The above process does not determine which HF ~nd! set
to use for the cc-pVDZ set. For this set, we selected a HF
(5d) primitive set, again because of the widespread use of
the (14s11p5d) sets. This means that the cc-pVDZ sets determined here are very similar to the double zeta sets for the
third row atoms that have been in wide use for a number of
0.881
1.3596
0.958 99
0.749 23
E HF1112
E corr
DE corr
22399.994 017
22399.993 545
22399.993 953
22399.993 801
22399.997 463
22399.996 778
22399.997 392
22399.997 294
22399.997 954
22399.997 453
22399.997 965
22399.997 921
22399.998 223
22399.998 122
22399.998 233
22399.998 198
2126.596
2126.124
2126.532
2126.379
2130.042
2129.357
2129.971
2129.873
2130.533
2130.031
2130.543
2130.500
2130.802
2130.701
2130.812
2130.776
¯
0.472
0.064
0.216
¯
0.685
0.071
0.169
¯
0.502
20.010
0.033
¯
0.101
20.010
0.025
years.26,27 It does mean, however, that the error in the HF
energy resulting from use of the (5d) set is larger than the
error in the valence correlation energy, but, of course, the
(5d) set describes a core orbital, not a valence orbital.
D. Determination of augmenting sets for selenium
For calculations on atomic and molecular anions, as well
as to improve the description of long range interactions,28–30
potential energy surfaces for chemical reactions,31,32 proton
affinities,33–35 etc., the standard cc-pVnZ sets must be augmented with a set of diffuse functions. We follow the procedure first used by Kendall et al.12 to determine the exponents
for the functions used to augment each of the angular sets.
Namely, Hartree–Fock calculations on the negative ions are
used to optimize the additional s and p functions, while
CISD calculations are used to optimize the additional
d, f ,g,..., functions. The results of these calculations on selenium are listed in Table VII.
As was found for both oxygen and sulfur, the largest
TABLE V. Errors in the valence correlation (HF1112) energy for selenium due to substituting (n21) optimized d exponents with the exponents from
selected HF primitive sets. Total energies (E HF1112 ) are given in hartrees; correlation energies and incremental correlation energy errors (E corr and DE corr)
are given in millihartrees. DE corr is measured relative to the corresponding optimum nd-correlation set. The base set is (24s16p10d) @ 8s7p1d # and the
corresponding HF energy is 22399.867 415 hartrees.
~nd! set
Source
(2d)
Optimum
(7d)
(8d)
(9d)
Optimum
(10d)
(11d)
(12d)
Optimum
(11d)
(12d)
(13d)
(3d)
(4d)
Exponents
0.215
0.215
0.215
0.215
0.158
0.158
0.158
0.158
0.1465
0.1465
0.1465
0.1465
0.5805
0.732 59
0.607 99
0.511 80
0.3413
0.434 66
0.370 71
0.315 94
0.3079
0.370 71
0.315 94
0.269 09
0.7372
1.0367
0.8701
0.7359
0.6473
0.870 12
0.735 91
0.627 29
1.3606
1.8087
1.4988
1.2570
E HF1112
E corr
DE corr
22399.969 128
22399.967 843
22399.969 081
22399.968 764
22399.969 851
22399.969 687
22399.969 785
22399.969 764
22399.969 982
22399.969 971
22399.969 965
22399.969 925
2101.712
2100.428
2101.666
2101.349
2102.435
2102.271
2102.370
2102.349
2102.566
2102.556
2102.550
2102.510
¯
1.285
0.047
0.364
¯
0.164
0.066
0.087
¯
0.011
0.017
0.057
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Wilson et al.
J. Chem. Phys., Vol. 110, No. 16, 22 April 1999
7673
TABLE VI. Errors in the valence 4s and 4p and 3d Hartree–Fock ~HF! energies for selenium for selected HF primitive sets. Total energies (E HF) are given
in hartrees; Hartree–Fock energy errors (DE HF) are given in millihartrees. DE HF is measured relative to the (24s), (18p), or (12d) HF sets. In these
calculations, only the 4s, 4p, or 3d orbitals are expanded in the indicated sets; all other orbitals are expanded in a (24s18p12d) set. Thus, the DE HF’s
measure only the error in the HF energy resulting from the use of the indicated set to describe the selected orbital.
~ns!a
E HF
DE HF
~np!b
E HF
DE HF
~nd!c
E HF
DE HF
(13s)
(14s)
(15s)
(16s)
(17s)
(18s)
(19s)
(20s)
(21s)
(22s)
(23s)
(24s)
(25s)
22399.867 255
22399.867 283
22399.867 291
22399.867 280
22399.867 262
22399.867 250
22399.867 239
22399.867 408
22399.867 415
22399.867 416
22399.867 419
22399.867 415
22399.867 425
0.160
0.132
0.123
0.135
0.153
0.165
0.176
0.007
0.000 5
20.001
20.004
¯
20.010
(9p)
(10p)
(11p)
(12p)
(13p)
(14p)
(15p)
(16p)
(17p)
(18p)
22399.861 072
22399.862 719
22399.863 955
22399.865 888
22399.867 231
22399.867 313
22399.867 387
22399.867 415
22399.867 499
22399.867 513
6.441
4.794
3.558
1.625
0.281
0.199
0.126
0.098
0.014
¯
(4d)
(5d)
(6d)
(7d)
(8d)
(9d)
(10d)
(11d)
(12d)
22399.615 023
22399.811 057
22399.854 220
22399.864 203
22399.866 629
22399.867 249
22399.867 415
22399.867 462
22399.867 476
252.453
56.419
13.255
3.273
0.847
0.227
0.060
0.014
¯
The base set is (24s16p10d)/ @ 3s3p1d # .
The base set is (24s16p10d)/ @ 4s2p1d # .
c
The base set is (24s16p)/ @ 4s3p # .
a
b
effect on the electron affinity arises from the addition of a
diffuse p function. This is the case for all of the cc-pVnZ
selenium sets. However, the increases in the electron affinity
resulting from the addition of diffuse functions with higher
angular momentum are not negligible, e.g., for the cc-pVTZ
set, the addition of the diffuse d and f functions each increase
the electron affinity of selenium by 25% of that due to the
addition of the diffuse p function. Although this effect becomes less pronounced as the set becomes larger, in aggre-
gate the addition of diffuse higher angular momentum functions will be important for highly accurate calculations on
negative ions.
III. RESULTS AND DISCUSSION FOR THE ATOMS
GALLIUM THROUGH KRYPTON
The procedure outlined in Sec. II for selenium was used
to generate cc-pVnZ and aug-cc-pVnZ sets for the gallium
TABLE VII. Contributions from augmenting functions for the selenium anion and atom. Total energies are
given in hartrees; the electron affinity ~EA! and the change in electron affinity ~DEA! are given in eV. DEA is
the difference between the current value and the previous one. The experimental EA for selenium is 2.0206 eV.a
E HF1112
Basis set
cc-pVDZ
1(s)
1(s p)
1(s pd)
cc-pVTZ
1(s)
1(s p)
1(s pd)
1(s pd f )
cc-pVQZ
1(s)
1(s p)
1(s pd)
1(s pd f )
1(s pd f g)
cc-pV5Z
1(s)
1(s p)
1(s pd)
1(s pd f )
1(s pd f g)
1(s pd f gh)
Exponent
z s 50.048 747
z p 50.035 492
z d 50.1283
z s 50.039 201
z p 50.030 251
z d 50.0837
z f 50.188
z s 50.038 152
z p 50.026 569
z d 50.0619
z f 50.124
z g 50.263
z s 50.033 935
z p 50.024 975
z d 50.0548
z f 50.0992
z g 50.183
z h 50.402
Anion
Neutral
EA
22399.912 062
22399.913 443
22399.938 015
22399.949 982
22400.042 346
22400.042 577
22400.048 919
22400.050 764
22400.053 703
22400.062 174
22400.062 303
22400.064 486
22400.064 804
22400.065 291
22400.066 493
22400.067 813
22400.067 843
22400.069 159
22400.069 308
22400.069 431
22400.069 701
22400.070 126
22399.881 739
22399.882 034
22399.883 673
22399.888 987
22399.986 475
22399.986 506
22399.986 620
22399.986 945
22399.988 388
22399.997 697
22399.997 710
22399.997 722
22399.997 751
22399.997 892
22399.998 365
22400.000 892
22400.000 894
22400.000 900
22400.000 911
22400.000 937
22400.001 001
22400.001 214
0.825
0.855
1.479
1.660
1.520
1.526
1.695
1.737
1.777
1.755
1.758
1.817
1.825
1.834
1.854
1.821
1.822
1.857
1.861
1.864
1.869
1.875
DEA
0.030
0.624
0.181
0.005
0.169
0.041
0.041
0.003
0.059
0.008
0.009
0.020
0.001
0.036
0.004
0.003
0.006
0.006
H. Hotop and W. C. Lineberger, J. Phys. Chem. Ref. Data 14, 731 ~1985!. Note that this value has not been
corrected for spin–orbit effects. It is, therefore, not strictly comparable to the quantity computed here.
a
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7674
Wilson et al.
J. Chem. Phys., Vol. 110, No. 16, 22 April 1999
TABLE VIII. Hartree-Fock and correlation energies for gallium through
krypton from Hartree–Fock, SDCI, and coupled cluster @CCSD, CCSD~T!#
calculations with the cc-pVnZ basis sets. Only the electrons in the valence
4s- and 4p-orbitals have been correlated. Hartree–Fock energies (E HF) are
in hartrees, and correlation energies (E corr) are in millihartrees.
E corr
E corr
Atom
Basis set
E HF
SDCI
CCSD
CCSD~T!
Ga
cc-pVDZ
cc-pVTZ
cc-pVQZ
cc-pV5Z
Numericala
cc-pVDZ
cc-pVTZ
cc-pVQZ
cc-pV5Z
Numericala
cc-pVDZ
cc-pVTZ
cc-pVQZ
cc-pV5Z
Numericala
cc-pVDZ
cc-pVTZ
cc-pVQZ
cc-pV5Z
Numericala
cc-pVDZ
cc-pVTZ
cc-pVQZ
cc-pV5Z
Numericala
cc-pVDZ
cc-pVTZ
cc-pVQZ
cc-pV5Z
Numericala
21923.189 287
21923.258 599
21923.260 795
21923.260 959
21923.261 01
22075.288 178
22075.357 088
22075.359 510
22075.359 680
22075.359 73
22234.166 119
22234.236 013
22234.238 422
22234.238 601
22234.238 65
22399.793 025
22399.864 906
22399.867 363
22399.867 556
22399.867 61
22572.364 623
22572.438 556
22572.441 076
22572.441 276
22572.441 33
22751.974 872
22752.052 121
22752.054 714
22752.054 918
22752.054 98
246.169
249.831
251.196
251.583
246.229
249.849
251.206
251.593
246.807
250.842
252.336
252.764
258.944
267.818
269.923
270.759
259.291
268.216
270.333
271.173
260.083
270.206
272.623
273.570
268.978
284.921
288.412
289.841
269.630
285.848
289.382
290.822
270.360
288.781
292.888
294.516
288.714
2121.570
2130.334
2133.336
290.784
2125.099
2134.205
2137.280
291.714
2128.897
2138.932
2142.290
2106.993
2153.462
2168.070
2173.057
2110.213
2159.177
2174.531
2179.685
2111.128
2163.776
2180.460
2186.035
2124.102
2180.266
2201.466
2208.840
2127.989
2187.495
2209.884
2217.539
2128.781
2192.862
2216.918
2225.151
Ge
As
Se
Br
Kr
a
Reference 40.
through krypton atoms. In Table VIII we report the HF energies and CISD, CCSD, and CCSD~T! correlation energies
for these atoms obtained with the new cc-pVnZ basis sets
~see Fig. 4!. As can be seen, the HF energies computed with
the cc-pV5Z set are very close to the Hartree–Fock limit—
the error ranges from 0.05 to 0.06 mE h . For the cc-pVTZ
sets the errors are larger, however, the range of errors is still
relatively small—2.41 to 2.86 mE h .
In Table IX we list the exponents for the diffuse functions used to augment the standard cc-pVnZ sets as well as
the CCSD~T! energies and electron affinities ~EAs! computed with the aug-cc-pVnZ basis sets ~see Fig. 5!. Comparing the CCSD~T! energies of the neutral atoms obtained with
the aug-cc-pVnZ sets to the values of E HF1E corr reported in
Table VIII for the cc-pVnZ sets, we see that augmentation
decreases the energy obtained with the cc-pVDZ set substantially, by 1.39 to 9.85 mE h for gallium and bromine, respectively. However, the effect decreases as n increases, becoming just 0.05 to 0.64 mE h for those same atoms for the ccpV5Z set.
As can be seen in Table IX, the computed electron affinities converge to well-defined limits as n increases in the
aug-cc-pVnZ sets. The difference between the aug-cc-pVQZ
FIG. 4. Absolute values of the valence correlation energies of the third row
atoms, Ga–Kr, from CCSD~T!/RCCSD~T! calculations with the new correlation consistent basis sets (cc-pVnZ, n5D25). In millihartress (mE h ).
and aug-cc-pV5Z results are all less than 0.03 eV, while that
between the aug-cc-pVDZ and aug-cc-pV5Z results can exceed 0.3 eV. In Table IX, it was only possible to correct for
spin–orbit effects for Ge and Br. In both cases the calculations slightly overestimate the measured electron
affinity—by 0.028 eV for Ge and 0.008 eV for Br. These
errors could be due to a number of effects neglected in the
current calculations, including scalar relativistic effects and
core–valence correlation effects, as well as errors in the
CCSD~T! method itself. For the other atoms, Ga, As, and Se,
there is insufficient information on the negative ions to correct for spin–orbit effects. Thus, the differences between the
CCSD~T! and measured results listed in Table IX for these
three atoms include this source of error as well.
IV. BENCHMARK CALCULATIONS WITH THE NEW
CORRELATION CONSISTENT BASIS SETS
The cc-pVnZ and aug-cc-pVnZ sets have already been
used in a wide ranging series of benchmark calculations.
Molecules that have been studied include a full range of
diatomics (A 2 and AB, where A represents a third row atom
and B represents a hydrogen, halogen, oxygen, sulfur, or
other third row atoms! as well as several larger systems.36–38
These calculations, as well as those reported by Peterson,18,19
indicate that the new basis sets provide an unparalleled level
of accuracy for molecular calculations. In these studies, we
observed the same general convergence behavior seen in the
molecular properties computed with the first and second row
correlation consistent basis sets. However, we observed an
interesting behavior in a number of benchmark calculations
involving molecules formed from atoms at the left side of the
third row, main group combined with oxygen and fluoride. In
such molecules, the 3 d s gallium/germanium and 2s s
oxygen/fluorine orbitals in the HF calculations are nearly
degenerate. This behavior confounds the separation between
core and valence orbitals, resulting in inaccurate descriptions
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Wilson et al.
J. Chem. Phys., Vol. 110, No. 16, 22 April 1999
7675
TABLE IX. Optimized exponents for the augmenting functions for the cc-pVnZ sets, anion and neutral energies, and calculated and experimental electron
affinities for the third row atoms gallium through bromine. Total energies @ E CCSD~T!# are given in hartrees; electron affinities ~EA! are in electron-volts. Only
the electrons in the valence 4s and 4 p orbitals have been correlated. Exponents for the augmenting functions for krypton were obtained by extrapolation from
the exponents for the preceding atoms.
Exponents of augmenting functions
Atom
Basis set
zs
zp
zd
zf
Ga
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
aug-cc-pV5Z
0.024 348
0.014 398
0.018 475
0.017 301
0.015 164
0.019 300
0.011 406
0.011 050
0.0537
0.0387
0.0279
0.0260
0.0980
0.0655
0.0511
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
aug-cc-pV5Z
0.033 961
0.027 370
0.026 390
0.024 274
0.023 945
0.021 368
0.018 550
0.017 593
0.0771
0.0528
0.0397
0.0364
0.1323
0.0884
0.0705
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
aug-cc-pV5Z
0.041 152
0.033 407
0.032 499
0.029 418
0.031 268
0.026 799
0.023 698
0.022 043
0.1078
0.0700
0.0531
0.0488
0.169
0.1132
0.0899
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
aug-cc-pV5Z
0.048 747
0.039 201
0.038 152
0.033 935
0.035 492
0.030 251
0.026 569
0.024 975
0.1283
0.0837
0.0619
0.0548
0.188
0.124
0.0992
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
aug-cc-pV5Z
0.056 946
0.045 593
0.044 27
0.039 106
0.041 049
0.035 142
0.030 513
0.029 052
0.1719
0.1047
0.0829
0.0781
0.258
0.1748
0.1388
aug-cc-pVDZ
aug-cc-pVTZ
aug-cc-pVQZ
aug-cc-pV5Z
0.065 145
0.051 985
0.050 388
0.044 277
0.046 606
0.040 033
0.034 457
0.033 129
0.2155
0.1257
0.1039
0.1014
0.328
0.2256
0.1784
Ge
As
Se
Br
Kr
a
E CCSD~T!
zg
zh
Anion
Neutral
EA
0.254
21923.248 041
21923.321 939
21923.326 259
21923.326 925
0.320
22075.397 657
22075.477 920
22075.482 903
22075.484 112
0.367
22234.255 447
22234.349 934
22234.358 436
22234.360 848
0.402
22399.958 658
22400.067 882
22400.082 088
22400.086 142
0.311
0.219
0.491
22572.607 169
22572.730 376
22572.751 244
22572.757 490
21923.237 484
21923.309 823
21923.313 227
21923.313 771
Experimentala
22075.351 141
22075.427 975
22075.432 350
22075.433 352
Experimentala
22234.241 188
22234.325 877
22234.331 667
22234.333 285
Experimentala
22399.893 228
22399.996 228
22400.007 105
22400.010 226
Experimentala
22572.485 605
22572.605 994
22572.622 846
22572.627 949
Experimentala
0.287
0.330
0.355
0.358
0.3
1.266
1.359
1.376
1.381
1.353
0.388
0.655
0.728
0.750
0.81
1.780
1.950
2.040
2.066
2.0206
3.308
3.385
3.494
3.525
3.517
0.359
0.255
0.580
0.168
0.114
0.2143
0.146
0.239
0.1655
0.263
0.183
H. Hotop and W. C. Lineberger, J. Phys. Chem. Ref. Data 14, 731 ~1985!. For comparison to calculated results, the experimental EAs must be corrected for
spin–orbit effects by averaging the atomic multiplets. The needed atomic data for averaging over the multiplets was only available for Ge and Br.
of the systems if frozen core calculations are performed. The
problem can be addressed either by unscrambling the ‘‘arbitrarily’’ mixed (3 d s 1l2s s ) and (2s s 2l3 d s ) orbitals
or by including the 3d orbitals in the correlated calculations.
The latter approach is, of course, far more expensive than the
former. All of these results will be reported shortly.36–38
V. CONCLUSIONS
In this work, we extended the correlation consistent basis sets to the third row, main group atoms gallium through
krypton. Both standard, cc-pVnZ, and augmented,
aug-cc-pVnZ, sets are reported for n5D, T, Q, and 5. The
standard sets are
cc-pVDZ: ~ 14s11p6d ! / @ 5s4 p2d # ,
cc-pVTZ: ~ 20s13p9d1 f ! / @ 6s5 p3d1 f # ,
cc-pVQZ: ~ 21s16p12d2 f 1g ! / @ 7s6 p4d2 f 1g # ,
cc-pV5Z: ~ 26s17p13d3 f 2g1h ! / @ 8s7 p5d3 f 2g1h # .
FIG. 5. Electron affinities ~EA! for gallium through bromine from
CCSD~T!/RCCSD~T! calculations with the new augmented correlation consistent basis sets (aug-cc-pVnZ, n5D25). In eV.
The augmented sets include an additional function for each
angular momentum present in the standard set. The new basis sets can be retrieved from the EMSL Gaussian Basis Set
Library at http://www.emsl.pnl.gov:2080/forms/basisform.
html. The availability of these sets, which appear to systematically converge toward the complete basis set limit, provide
a means for calculating accurate, nonrelativistic values for a
wide range of molecular properties.
The current correlation consistent basis sets are appropriate for correlating the electrons in the valence 4s and 4p
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7676
J. Chem. Phys., Vol. 110, No. 16, 22 April 1999
orbitals of the third row atoms, gallium through krypton. As
we will show in subsequent molecular benchmark
studies,36–38 inclusion of only valence correlation effects
does yield high accuracy results for many molecular properties in molecules containing third row atoms. In making this
statement, however, one must be careful to avoid
pseudocore/valence effects ~see Sec. III and Ref. 37!. It
should also be noted that, for accurate predictions of some
molecular properties, it will be necessary to include relativistic effects ~both spin–orbit and scalar effects!. Work is currently underway on core–valence basis sets for Ga–Kr.39
ACKNOWLEDGMENTS
The authors wish to acknowledge the support of the
Chemical Sciences Division in the Office of Basic Energy
Sciences of the U.S. Department of Energy. The work was
carried out at Pacific Northwest National Laboratory, a multiprogram national laboratory operated for the U.S. Department of Energy by Battelle Memorial Institute under Contract No. DE-AC06-76RLO 1830. This research was also
supported by Associated Western Universities, Inc., Northwest Division, under Grant No. DE-FG06-89ER-75522, with
the U.S. Department of Energy. All calculations were performed on a SGI Power Challenge at PNNL purchased with
funding provided by the Chemical Sciences Division. We
thank Dr. David F. Feller and Dr. Andreas Nicklass for their
comments on the manuscript.
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