Extension of Gaussian2 (G2) theory to bromine and iodinecontaining
molecules: Use of effective core potentials
Mikhail N. Glukhovtsev, Addy Pross, Mark P. McGrath, and Leo Radom
Citation: J. Chem. Phys. 103, 1878 (1995); doi: 10.1063/1.469712
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Extension of Gaussian-2 (G2) theory to bromine- and iodine-containing
molecules: Use of effective core potentials
Mikhail N. Glukhovtsev and Addy Prossa)
School of Chemistry, University of Sydney, Sydney, NSW 2006, Australia
Mark P. McGrath
Department of Chemistry, University of California, Irvine, California 92717
Leo Radom
Research School of Chemistry, Australian National University, Canberra, ACT 0200, Australia
~Received 28 November 1994; accepted 20 April 1995!
Basis sets have been developed for carrying out G2 calculations on bromine- and iodine-containing
molecules using all-electron ~AE! calculations and quasirelativistic energy-adjusted
spin–orbit-averaged seven-valence–electron effective core potentials ~ECPs!. Our recommended
procedure for calculating G2@ECP# energies for such systems involves the standard G2 steps
introduced by Pople and co-workers, together with the following modifications: ~i! second-order
Mo” ller–Plesset ~MP2! geometry optimizations use polarized split-valence @31,31,1# basis sets for
bromine and iodine together with 6-31G(d) for first- and second-row atoms; ~ii! single-point
higher-level energies are calculated for these geometries using our new supplemented bromine and
iodine valence basis sets along with supplemented 6-311G and McLean–Chandler 6-311G bases for
first- and second-row atoms, respectively; and ~iii! first-order spin–orbit corrections are explicitly
taken into account. An assessment of the results obtained using such a procedure is presented. The
results are also compared with corresponding all-electron calculations. We find that the G2@ECP#
calculations give results which are generally comparable in accuracy to those of the G2@AE#
calculations but which involve considerably lower computational cost. They are therefore
potentially useful for larger bromine- and iodine-containing molecules for which G2@AE#
calculations would not be feasible. © 1995 American Institute of Physics.
I. INTRODUCTION
The GAUSSIAN-2 ~G2! theoretical procedure, introduced
by Pople and co-workers for the purpose of making reliable
theoretical thermochemical predictions,1 has been found to
have widespread utility, consistently reproducing atomization
energies, ionization energies, electron affinities, gas-phase
acidities, and proton affinities to within 10 kJ mol1.1– 4 The
basis sets required to carry out G2 calculations have been
defined to date for first- and second-row atoms,1 and for
bromine.5 In a number of current studies,6 we have been
making comparisons of series of systems and reactions involving the halogens F, Cl, Br, and I. Although we were able
to use G2 theory to obtain high quality thermochemical data
for the F, Cl, and Br systems, this was not immediately possible for the iodine-containing molecules since G2 basis sets
for iodine were not yet defined. In order to redress this problem, we have developed basis sets suitable for carrying out
G2 calculations on iodine-containing molecules, and these
are presented and tested in the current paper.
G2 calculations have previously been based exclusively
on all-electron ~AE! calculations. An attractive alternative
for atoms ~such as bromine and iodine! of the lower periods
of the periodic table is to use effective core potentials
~ECPs!, which have received increasing attention in recent
years.7–11 Such calculations are significantly less expensive
a!
Permanent address: Department of Chemistry, Ben Gurion University, Beer
Sheva, Israel.
1878
J. Chem. Phys. 103 (5), 1 August 1995
computationally than the corresponding AE calculations. It
would be desirable to develop ECP-based basis sets suitable
for what might be called G2@ECP# calculations. A G2@ECP#
procedure would be potentially applicable to considerably
larger bromine- and iodine-containing molecules than those
for which all-electron G2 calculations would be feasible. We
have taken such an approach here, constructing bromine and
iodine basis sets utilizing the so-called energy-adjusted ECPs
of the Stuttgart group (S) 10,11 and the shape-consistent
~orbital-adjusted! ECPs of Hay and Wadt ~HW!.9 However,
we may of course continue to treat the heavier atoms by the
conventional but computationally demanding all-electron
calculations,5,12 and so we also construct here all-electron
iodine basis sets for G2@AE# calculations.
Thus, in this paper, we present basis sets for iodine suitable for G2@AE# calculations to complement those previously derived for bromine,5 together with basis sets for bromine and iodine suitable for G2@ECP# calculations. We
compare the G2@AE# results for bromine- and iodinecontaining systems with those obtained with the G2@ECP#
procedure, and assess all of the theoretical results through
comparisons with experimental data.
II. THEORETICAL PROCEDURES
Standard ab initio molecular orbital calculations13 were
performed at the G2 level with the GAUSSIAN 92 series of
programs.14 G2 theory corresponds effectively to calcula-
0021-9606/95/103(5)/1878/8/$6.00
© 1995 American Institute of Physics
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Glukhovtsev et al.: Extension of G2 to Br- and I-containing molecules
1879
TABLE I. Energy-optimized exponents of polarization d and f functions and diffuse s and p functions for
augmentation of the valence basis sets in AE and ECP calculations of bromine- and iodine-containing species.
Br
Br
Br
I
I
I
c
AE
ECP~HW!
ECP~S!
AE
ECP~HW!
ECP~S!
da
db
f
s ~diffuse!
p ~diffuse!
0.389
0.391d
0.381e
0.266
0.279d
0.267e
0.451
0.427
0.410
0.302
0.292
0.276
0.56
0.574
0.590
0.38
0.441
0.434
0.0462
0.0640
0.0493
0.0468
0.0569
0.0405f
0.0347
0.0402
0.0363
0.0286
0.0330
0.0328f
a
Values to be used with valence basis sets in geometry optimizations.
Values to be used with valence basis sets in single-point higher-level calculations.
c
From Ref. 5.
d
Values to be used with valence @21,21# basis sets ~Ref. 9!.
e
Values to be used with valence @31,31# basis sets ~Ref. 11!.
f
Diffuse function exponents optimized at the QCISD~T! level are 0.0591 (s) and 0.0333 (p).
b
tions at the QCISD~T!/6-3111G(3d f ,2p)//MP2/6-31G(d)
level with zero-point vibrational energy and higher-level
corrections.1
In our all-electron calculations, equilibrium structures
were optimized at the MP2 level, allowing the full set of
molecular orbitals to be used in the electron correlation procedure. The SV4P basis set15 was used for bromine and iodine in conjunction with 6-31G(d) for first- and second-row
elements. The bromine basis sets for the higher-level singlepoint calculations required for G2@AE# calculations have
been described previously.5 Corresponding iodine sets were
constructed in an analogous manner. The (15s,11 p,6d) basis of Stromberg et al.16 was augmented with another p shell
and the five valence s p exponents optimized, resulting in a
@5211111111,411111111,3111# contraction scheme. Appropriate numbers of optimized d and f polarization functions and
s and p diffuse functions ~Table I! were added to the iodine
G2@AE# bases. The single-point energies were computed using the frozen-core approximation,13 with the inclusion of
the 3d electrons of bromine and the 4d electrons of iodine in
the frozen core. The very high-lying virtual molecular orbitals, corresponding to 1s-like antibonding bromine and iodine
orbitals, were also excluded from the correlation treatments.
The iodine basis functions are summarized in the Appendix.
Two types of effective core potentials were examined in
detail. Quasirelativistic energy-adjusted spin–orbit-averaged
seven-valence–electron ECPs for bromine and iodine atoms
were taken from work of the Stuttgart group.10,11 These
pseudopotentials were derived11 in a basis-independent way
from quasirelativistic all-electron calculations of the neutral
atoms and anions. Atomic excitation and ionization energies
served as reference data for the parameter adjustment. These
ECPs include relativistic contributions11,17,18 associated with
the inner shells of the halogen atoms, and have been previously used for calculations of various halogen-containing
species.10,19
The shape-consistent ~orbital-adjusted! seven-valence–
electron ECPs were generated by Hay and Wadt from nonrelativistic numerical Hartree–Fock ~HF! atomic wave functions for Br and from relativistic HF wave functions for I.9
These ECPs have also been widely employed in calculations
of halogen-containing molecules.20
In order to obtain gas-phase proton affinities and methyl
cation affinities at 298 K, vibrational contributions to temperature corrections13 were calculated using harmonic frequencies computed at HF/6-31G(d) and scaled by 0.8929.1
The most naturally abundant isotopes were used ~e.g., 79Br!.
For simplicity in nomenclature, the basis sets for bromine and iodine used in conjunction with 6-31G(d) for firstand second-row elements are referred to also as 6-31G(d),
while those used in conjunction with supplemented 6-311G
or McLean–Chandler 6-311G basis sets for first- and secondrow elements, respectively, are referred to also as supplemented 6-311G sets. The three types of G2 calculations described in the remainder of this paper are distinguished by
the nomenclature G2@AE#, G2@ECP~HW!# and G2@ECP~S!#
for the all-electron, Hay–Wadt ECP and Stuttgart ECP calculations, respectively.
III. CONSTRUCTION OF VALENCE BASIS SETS FOR
ECP PROCEDURES
The sp parts of the valence basis sets to be used in
single-point calculations with the Stuttgart energy-adjusted
ECPs10,11 for bromine and iodine were taken as uncontracted
(4s,4 p)→@1111,1111# sets.11 These were augmented by appropriate numbers of d and f polarization as well as by s and
p diffuse functions. The d function exponents for bromine
and iodine were optimized through calculations with a
@1111,1111,1# valence basis at the QCISD~T! level on HBr
and HI in their experimental geometries.21 The f function
exponents were then obtained by adding a single uncontracted f function to a @1111,1111,11# valence basis set and
again optimizing at the QCISD~T! level for the HBr and HI
molecules. In creating multiple sets of d functions from a
single optimized function, the normal procedure is to use
exponents that are multiples n a d and fractions a d /n of the
single optimized exponent ad . For example, for first- and
second-row elements, n52 is used for splitting into two
functions and n54 is used for splitting into three functions.22
However, it has been found that such a geometric progression is unsuitable for third-row elements, often leading to an
increase in energy compared with that calculated with a
single d function.12,23 Values of n53 for the (3d) splitting in
third-row elements,12 and n51.5 and n52 for the (2d) and
(3d) splittings, respectively, for fourth-row elements have
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Glukhovtsev et al.: Extension of G2 to Br- and I-containing molecules
1880
TABLE II. G2@AE# and G2@ECP# energies ~in hartrees! of brominecontaining species ~0 K!.
Species
G2@AE#
2
22572.536 42
22575.658 49
22572.105 89a
25145.141 48
25145.232 10
25144.756 18a
22573.173 25
22573.394 12
22572.747 77a
22612.390 46
22612.641 42
Br~ P!
Br2
Br1~3P!
Br2
2 1
Br2
2 ~ Su !
1 2
Br2 ~ Pg !
HBr
HBrH1
HBr1~2P!
CH3Br
CH3BrH1
a
G2@ECP~HW!#
G2@ECP~S!#
Species
G2@AE#
G2@ECP~HW!#
G2@ECP~S!#
213.108 25
213.229 29
212.680 42a
226.282 52
226.374 17
225.898 91a
213.743 98
213.965 64
213.320 61a
252.961 92
253.213 69
213.308 71
213.429 88
212.879 58a
226.686 11
226.776 71
226.302 34a
213.941 43
14.162 67
213.516 89a
253.159 21
253.410 61
HBr
HBrH1
CH3Br
CH3BrH1
22573.169 95
22573.390 32
22612.386 39
22612.636 47
213.740 68
213.961 84
252.957 85
253.208 72
213.938 13
214.158 87
253.155 14
253.405 65
HI
HIH1
CH3I
CH3IH1
26917.605 10
26917.842 32
26956.825 85
26957.088 16
211.947 64
212.186 01
251.168 64
251.432 44
211.957 86
212.193 94
251.178 69
251.439 57
a
a
Including first-order spin–orbit corrections for Br ~25.78 mhartree!, Br1
1
~26.77 mhartree!, Br1
~26.26 mhartree!,
2 ~26.55 mhartree!, and HBr
obtained from spin–orbit configuration interaction energies @Ref. 27~a!#.
The experimental values, calculated as the difference between the lowest
spin–orbit coupled state and the weighted J- or V-averaged state ~Refs. 5
and 12! are as follows: Br ~25.60 mhartree! ~Ref. 33!, Br1 ~26.71 mhar1
tree! ~Ref. 33!, Br1
2 ~26.42 mhartree! ~Ref. 21!, and HBr ~26.06 mhartree! ~Ref. 34!.
a
been found to be more suitable. We find, through appropriate
optimizations, that these AE n values are also suitable for our
G2@ECP# basis sets. The diffuse s and p exponents were
optimized independently for the ground states of the Br2 and
I2 anions at the HF level with the @1111,1111,1# valence
basis sets. Optimization of these exponents at the QCISD~T!
level leads to results for iodine-containing species almost the
same as those obtained with the HF-derived s and p diffuse
exponents ~see Table IX below!. For geometry optimizations
at the HF and MP2 levels, the @31,31# basis sets11 were augmented by d functions with exponents optimized at the MP2
level for the HBr and HI molecules.
For the orbital-adjusted Hay–Wadt ECPs,9 an uncontracted (3s,3p)→@111,111# basis set9 was augmented by d
and f polarization as well as by s and p diffuse functions on
TABLE III. G2@AE# and G2@ECP# energies ~in hartrees! of iodinecontaining species ~0 K!.
Species
G2@AE#
2
26 916.997 23
26 917.107 59
26 916.614 26a
213 834.046 64
213 834.135 31
213 833.704 88a
26 917.608 40
26 917.846 15
26 917.231 10a
26 956.829 99
26 957.093 25
I~ P!
I2
I1~3P!
I2
2 1
I2
2 ~ Su !
2
I1
2 ~ Pg !
HI
HIH1
HI1~2P!
CH3I
CH3IH1
a
TABLE IV. G2@AE# and G2@ECP# energies ~in hartrees! of bromine- and
iodine-containing species ~298 K!.
G2@ECP~HW!#
G2@ECP~S!#
211.337 54
211.446 98
210.954 26a
222.727 56
222.814 57
222.386 96a
211.950 94
212.189 84
211.572 85a
251.172 78
251.437 50
211.351 84a
211.460 56
210.969 33a
222.751 23
222.842 16
222.410 09a
211.961 64
212.197 77
211.584 52a
251 182 84
251.444 67
a
Including first-order spin–orbit corrections for I ~211.49 mhartree!, I1
1
~213.52 mhartree!, I1
~212.26 mhartree!,
2 ~211.63 mhartree!, and HI
obtained from spin–orbit configuration interaction energies @Ref. 27~b!#.
The experimental values, calculated as the difference between the lowest
spin–orbit coupled state and the weighted J- or V-averaged state ~Refs. 5
and 12!, are as follows: I ~211.55 mhartree! ~Ref. 33!, I1 ~214.03 mhar1
tree! ~Ref. 33!, I1
2 ~211.80 mhartree! ~Ref. 21!, and HI ~212.20 mhartree!
~Ref. 35!.
bromine and iodine in a manner similar to that described
above for the valence basis sets for the energy-adjusted
ECPs. However, the diffuse s and p exponents were optimized at the QCISD~T! level and geometrical (2d) and (3d)
splittings were used. We find ~see Table VIII below! that
calculations on iodine species with reduced (2d) and (3d)
splittings lead to little or no change to the results. The
@21,21# basis sets,9 augmented by d exponents optimized at
the MP2 level for the HBr and HI molecules, were used for
geometry optimizations at the HF and MP2 levels.
The exponents used for supplementary functions for bromine and iodine for all of the basis sets are summarized in
Table I.
IV. EVALUATION
In attempting to obtain accurate results for molecules
and ions containing heavy atoms, we are faced with the problem of how to treat spin–orbit coupling and other relativistic
effects which increase with atomic number.7,18,24 –26 The quasirelativistic ECPs9–11 that we have used take into account
the contributions of two of the most important relativistic
effects, namely, the mass–velocity and Darwin
contributions.18,25 But to favorably compare our reaction energies with experiment, we must explicitly take account of
spin–orbit coupling effects.24 –26
When spin–orbit effects are introduced in a perturbational manner, it is found that first-order corrections result
when the ground state of the atom or molecule of interest is
TABLE V. Comparison of MP2@AE# and MP2@ECP# bond lengths ~in Å! for
bromine-containing species with experimental values.a
a
Bond length
AE
ECP~HW!
ECP~S!
Expt.b
Br—Br
Br—Br2
Br—Br1
H—Br
H—BrH1
H—Br1
H3C—Br
H3C—BrH1
2.3088
2.8676
2.2315
1.4348
1.4539
1.4627
1.9485
1.9918
2.3346
2.8933
2.2520
1.4352
1.4548
1.4623
1.9532
2.0028
2.3124
2.8634
2.2299
1.4297
1.4491
1.4571
1.9498
2.0008
2.278
a
2.18c
1.414
1.448
1.934d
See the text for the description of basis sets used in the geometry optimizations.
b
Experimental values taken from Ref. 21 unless otherwise indicated.
c
Reference 28.
d
Reference 29.
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Glukhovtsev et al.: Extension of G2 to Br- and I-containing molecules
TABLE VI. Comparison of MP2@AE# and MP2@ECP# bond lengths ~in Å!
for iodine-containing species with experimental values.a
Bond length
AE
ECP~HW!
ECP~S!
Expt.b
I—I
I—I2
I—I1
H—I
H—IH1
H—I1
H2C—I
H3C—IH1
2.7159
3.2718
2.6448
1.6413
1.6538
1.6606
2.1624
2.1774
2.6900
3.2606
2.6080
1.6150
1.6262
1.6330
2.1417
2.1613
2.7113
3.2841
2.6433
1.6262
1.6394
1.6460
2.1607
2.1893
2.666
1.609
1.62
2.132c
a
See the text for the description of basis sets used in the geometry optimizations.
b
Experimental values taken from Ref. 21 unless otherwise indicated.
c
From Ref. 30.
spatially degenerate ~e.g., 2P, 3P for atoms and 2P for molecules in this work!.24 –26 Our previous calculations have
shown that it is important to take into account these firstorder spin–orbit corrections, as the resulting energy lowerings are already essential in obtaining accurate reaction energies for the bromine-containing systems.5,12 We have
therefore included first-order spin–orbit corrections, obtained
from
spin–orbit
configuration
interaction
calculations,27 in our G2 total energies. The incorporation of
calculated first-order spin–orbit corrections into the G2 total
energies is consistent with our recent all-electron extension
of G2 theory to molecules containing third-row elements.12
The G2@AE# and G2@ECP# energies for X, X2, X1, X2 ,
2
1
1
1
X2 , X1
2 , HX, HXH , HX , CH3X, and CH3XH are given
1881
TABLE VIII. Comparison of G2@AE# and G2@ECP# predictions of ionization energies, electron affinities, and atomization energies ~in eV! for
bromine-containing species with experimental values.
Br→Br1
Br2→Br1
2
HBr→HBr1
2
Br →Br
Br2
2 →Br2
Br2→Br1Br
HBr→H1Br
CH3Br→C1Br13H
AEa
ECP~HW!
ECP~S!
Expt.b
11.72~11.71!
10.48~10.47!
11.58~11.57!
3.32~3.31!
2.47~2.46!
1.87~1.85!
3.72~3.72!
15.50~15.50!
11.64
10.44
11.52
3.29
2.49
1.80
3.69
15.49
11.68
10.44
11.55
3.30
2.47
1.87
3.61
15.41
11.81
10.52
11.66
3.37
2.5c
1.97
3.77
15.55
Values calculated with geometrical ~i.e., the same as for first- and secondrow elements! splitting factors for the d functions required for the (3d)
parts of the basis sets are shown in parentheses.
b
Experimental values ~at 0 K! are taken from Ref. 32.
c
Other experimental values are 2.4, 2.6, and 2.9 eV, see Ref. 32.
a
in Tables II ~X5Br! and III ~X5I!. Corresponding energies
at 298 K for HX, HXH1, CH3X, and CH3XH1 ~X5Br and I!
are presented in Table IV. The MP2 optimized geometries of
these systems are compared with available experimental
data21,28 –31 in Tables V ~X5Br! and VI ~X5I!. Additional
comparisons of geometries and vibrational frequencies for a
series of A–X systems ~X5Br or I! are presented in Table
VII. Ionization energies ~IEs!, electron affinities ~EAs!, and
atomization energies, calculated using both the G2@AE# and
G2@ECP# procedures, are compared with experiment32 in
Tables VIII and IX, while the data collected in Table X illustrate the role of spin–orbit corrections21,27,33–35 in calcula-
TABLE VII. Deviations of AE and ECP bond lengths and harmonic vibrational frequencies from experimental
values.a
vMP2c
r MP2b
vHFd
AE
HW
S
AE
HW
S
AE
HW
S
H—Br
Li—Br
Na—Br
B—Br
Al—Br
F—Br
Cl—Br
umeanue
0.021
0.036
0.030
0.015
0.016
0.031
0.030
0.026
0.021
0.057
0.034
0.015
0.026
0.033
0.029
0.031
0.016
0.052
0.040
0.016
0.021
0.033
0.029
0.030
10.9
20.2
21.7
11.8
11.9
13.3
10.7
1.5
10.9
10.2
11.0
12.8
13.4
11.3
20.2
1.4
10.5
22.4
22.3
11.0
11.9
10.7
10.5
1.3
25.5
212.1
213.6
211.5
212.4
13.3
22.5
8.7
27.0
211.2
212.5
212.1
211.9
11.5
24.5
8.7
26.2
212.9
213.7
211.9
212.1
11.8
23.3
8.8
H—I
Li—I
Na—I
B—I
Al—I
F—I
Cl—I
umeanue
0.032
0.042
0.037
0.03
0.031
0.032
0.048
0.036
0.006
0.052
0.041
0.01
0.015
0.006
0.023
0.22
0.017
0.052
0.041
0.02
0.020
0.034
0.047
0.033
10.2
11.7
12.2
20.1
11.6
15.0
10.3
1.6
11.1
21.0
10.8
11.2
12.5
11.8
20.5
1.3
20.0
21.2
11.2
10.5
12.8
12.3
21.5
1.4
26.1
211.5
211.4
214.1
213.4
12.8
24.2
9.1
25.9
212.5
211.7
213.3
212.4
10.4
25.4
8.8
26.8
212.5
211.1
214.0
212.4
11.9
26.1
9.3
Experimental bond lengths ~in Å! and harmonic vibrational frequencies ~in cm21! are 1.414, 2649 ~HBr!,
2.170, 563 ~LiBr!, 2.502, 302 ~NaBr!, 1.888, 684 ~BBr!, 2.295, 378 ~AlBr!, 1.759, 671 ~FBr!, 2.136, 444
~ClBr!, 1.609, 2309 ~HI!, 2.392, 498 ~LiI!, 2.711, 258 ~NaI!, 2.13, 575 ~BI!, 2.537, 316 ~AlI!, 1.910, 610 ~FI!,
and 2.321, 384 ~CII!. All experimental values are from Ref. 21 except those for BI which come from Ref. 31.
b
Values tabulated are r MP22r expt in Å.
c
Values tabulated are 100~vMP22vexpt!/vexpt~%!.
d
Values tabulated are 100~vHF2vexpt!/vexpt~%!, where the vHF values are the directly calculated harmonic
frequencies, scaled by 0.8929.
e
Mean absolute deviations.
a
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Glukhovtsev et al.: Extension of G2 to Br- and I-containing molecules
1882
TABLE IX. Comparison of G2@AE# and G2@ECP# predictions of ionization
energies, electron affinities, and atomization energies ~in eV! for iodinecontaining species with experimental values.
AE
1
I→I
I2→I1
2
HI→HI1
2
I →I
I2
2 →I2
I2→I1I
HI→H1I
CH3I→C1I13H
ECP~HW!
10.42
9.30
10.27
3.00
2.41
1.42
3.02
14.92
10.43
9.27
10.29
2.98
2.37
1.43
3.09
14.99
ECP~S!
10.41
9.28
10.26
2.96
2.47
1.29
2.99
14.88
b
TABLE XI. Comparison of G2 proton affinities ~PA! for HBr, CH3Br, HI,
and CH3I with experimental PA values.a
Expt.a
10.45
9.40~9.31!c
10.39~10.38!c
3.06
2.5d~2.56!c
1.54
3.05
14.94
Experimental values ~at 0 K! taken from Ref. 32 unless otherwise indicated.
G2@ECP~S!# calculations in which the diffuse s and p exponents on iodine
have been optimized at the QCISD~T! level ~instead of the HF level! give
the same values for all the energies except the ionization energy of I, which
is 10.40 eV.
c
Reference 21.
d
Other experimental values are 2.4 and 2.6 eV; see Ref. 32.
a
Species
AE
ECP~HW!
ECP~S!
Expt.
HBr
CH3Br
HI
CH3I
584.8
662.8
629.0
694.9
586.9
664.9
632.0
698.8
585.7
663.9
626.0
691.1
582b
662.3c
618d
683.1c
All PA values are given in kJ mol21 at 298 K.
Reference 32. A more recent PA298~HBr! value is 580.7 kJ mol21, see Ref.
36.
c
Experimental PA~CH3Br! and PA~CH3I! values are taken from Ref. 6~b!.
d
T. B. McMahon, private communication. The value of PA298~HI! given in
Ref. 32 is 615 kJ mol21.
a
b
b
tions of these quantities. Finally, the G2@ECP# and G2@AE#
calculated gas-phase proton affinities ~PA! of HX and CH3X
as well as the gas-phase methyl cation affinities ~MCA! of
HX ~X5Br and I! are compared with experiment6~b!,32,36 in
Tables XI and XII, respectively.
A. Bond lengths and harmonic vibrational
frequencies
The results in Tables V and VI show that all three procedures, MP2@AE#, MP2@ECP~HW!# and MP2@ECP~S!#,
when used in conjunction with the singly polarized splitvalence bases defined above, consistently overestimate the
length of bonds to bromine and iodine. For the brominecontaining systems ~Table V!, there is little difference beTABLE X. Effect of spin–orbit corrections on the calculated G2@AE# ionization energies, electron affinities, and atomization energies ~in eV! of
bromine- and iodine-containing species.
Without spin–orbit
corrections
With first-order spin–
orbit correctionsa
Expt.b
Br→Br1
Br2→Br1
2
HBr→HBr1
Br2→Br
Br2
2 →Br2
Br2→Br1Br
HBr→H1Br
CH3Br→C1Br13H
11.74
10.66
11.75
3.48
2.47
2.18
3.88
15.66
11.72
10.48
11.58
3.32
2.47
1.87
3.72
15.50
11.81
10.52
11.66
3.37
2.5
1.97
3.77
15.55
I→I1
I2→I1
2
HI→HI1
2
I →I
I2
2 →I2
I2→I1I
HI→H1I
CH3I→C1I13H
10.48
9.62
10.60
3.31
2.41
2.05
3.33
15.23
10.42
9.30
10.27
3.00
2.41
1.42
3.02
14.92
10.45
9.40
10.39
3.06
2.5
1.54
3.05
14.94
The spin–orbit corrections in eV are: 20.1573 ~Br!, 20.1842 ~Br1!,
1
1
20.1782 ~Br1
2 !, 20.1703 ~HBr !, 20.3128 ~I!, 20.3679 ~I !, 20.3165
1
1
~I2 !, and 20.3337 ~HI ! ~Ref. 27!. See also the footnotes to Tables II and
III.
b
Experimental data were taken from Ref. 32. See also Tables VIII and IX.
tween the three procedures, the mean absolute deviations
from experimental bond lengths being 0.027 Å ~AE!, 0.029
Å @ECP~HW!#, and 0.025 Å @ECP~S!#. For the iodinecontaining systems ~Table VI! there are larger differences,
the mean absolute deviations being 0.038 Å ~AE!, 0.013 Å
@ECP~HW!#, and 0.031 Å @ECP~S!#. However, the small set
of comparisons makes the significance of these differences
uncertain. Interestingly, similar trends are observed in the
different sample of systems shown in Table VII. Thus, for the
bromine-containing systems, the mean absolute deviations
from experimental bond lengths are 0.026 Å ~AE!, 0.031 Å
@ECP~HW!#, and 0.030 Å @ECP~S!#, whereas for the iodinecontaining systems, the mean absolute deviations are 0.036
Å ~AE!, 0.022 Å @ECP~HW!#, and 0.033 Å @ECP~S!#.
All three MP2-level procedures fare very well at predicting harmonic vibrational frequencies ~Table VII!. The mean
absolute percentage deviations in the vibrational frequencies,
given as 100~vMP22ve !/ve ~%!, come out as 1.5% ~AE!,
1.4% @ECP~HW!#, and 1.3% @ECP~S!#, while for the iodinecontaining systems, the percentage deviations are 1.6%
~AE!, 1.3% @ECP~HW!#, and 1.4% @ECP~S!#. At the scaled
HF level, while the deviations from experimental frequencies
are uniformly larger, the three procedures again perform
similarly, with mean absolute deviations for the three methods lying between 8.7% and 8.8% for the brominecontaining systems and between 8.8% and 9.3% for the
iodine-containing systems. Because the efficient calculation
of MP2 force constants is now routine,14 it is unfortunate that
scaled HF zero-point vibrational energies are used in standard G2 theory, especially when the scaling is found not to
improve the agreement with experiment compared with the
unscaled MP2 results.
TABLE XII. Comparison of G2 methyl cation affinities ~MCA! for HBr and
HI with experimental values.a
a
Species
AE
ECP~HW!
ECP~S!
Expt.b
HBr
HI
222.5
265.9
226.4
270.5
225.1
262.3
227.7
257.4
All MCA values are given in kJ mol21 at 298 K. The G2 energy of CH1
3 is
239.381 79 hartrees at 298 K.
b
Reference 6~b!.
a
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Glukhovtsev et al.: Extension of G2 to Br- and I-containing molecules
B. Ionization energies, electron affinities, and
atomization energies
For bromine-containing species, the G2@AE# results for
the energetic quantities shown in Table VIII are somewhat
better than those obtained at G2@ECP#. Thus, mean ~and
maximum! absolute deviations from experimental values for
G2@AE# are 0.06 ~0.10! eV, while the corresponding deviations for G2@ECP~HW!# are 0.10 ~0.17! eV, and for
G2@ECP~S!# they are 0.10 ~0.16! eV. The G2@AE# results
obtained with the geometrical (3d) splitting parameter n54
differ only slightly ~by 0.00–0.02 eV! from those obtained
with the standard third-row (3d) splitting parameter n53
~Table VIII!. For iodine-containing species, the performance
comparison ~Table IX! is similar, with one exception. Thus,
the mean ~and maximum! absolute deviations are 0.07 ~0.12!
eV for G2@AE#, 0.08 ~0.13! eV for G2@ECP~HW!#, and 0.10
~0.25! eV for G2@ECP~S!#. The reason for the significantly
larger error in the G2@ECP~S!# atomization energy of I2 is not
apparent.
It is perhaps surprising that overall the G2@AE# results
are at least as good as those obtained from the G2@ECP#
procedures. Apparently, a relativistic treatment of the halogen core is not necessary to achieve comparably accurate
reaction energies, at least for the reactions shown in Tables
VIII and IX. As a further test of this point, we carried out
MP4 calculations of the IEs and EAs of Br and I using both
the relativistic and nonrelativistic ~V R and V NR! ECPs defined by Schwerdtfeger et al.,37 coupled with their uncontracted (9s,6p,2d) basis,37 which we supplemented with an
f function ~a f 50.5!. The relativistic ~V R! ECPs yield MP4
reaction energies for the four comparisons that are 0.03–0.06
eV lower than the nonrelativistic ~V NR! values. The same
trend is shown in Tables VIII and IX, where in seven out of
eight comparisons the G2@ECP# IEs and EAs are lower ~further away from the experimental values! than the corresponding G2@AE# values by 0.01–0.07 eV.
As expected, the first-order spin–orbit corrections are
quite significant for attaining accurate reaction energies
~Table X!. When spin–orbit coupling is not taken into account, the mean absolute G2@AE# deviation from experiment
nearly doubles for the bromine-containing systems, and more
than triples for the iodine-containing systems. It is unlikely
that application of the present G2 procedures to the analogous astatine-containing species would result in reaction energies of quality comparable to that demonstrated in Tables
VIII and IX. Second-order spin–orbit corrections, which
arise from the interaction of the spin–orbit operator between
the unperturbed ground state and its excited states, will not
be negligible for the astatine-containing species, since their
magnitudes are expected to be comparable to the nonzero
first-order corrections of the iodine-containing species. In addition, there are other corrections that we have not considered ~e.g., core penetration!17 that are likely be important for
some astatine-containing systems.
C. Proton and methyl cation affinities
Calculations of gas-phase proton affinities and methyl
cation affinities are very sensitive to the computational level
1883
employed.3f,3i,38 Therefore, it is instructive to test both the
G2@AE# and G2@ECP# schemes in calculations of PA~HX!
and PA~CH3X! as well as of MCA~HX!, X5Br and I. The
best performance is provided by G2@ECP~S!#, which shows a
mean absolute deviation from the experimental results in
Tables XI and XII of 0.05 eV, with a maximum deviation of
0.08 eV. For G2@AE#, the mean and maximum deviations are
0.07 and 0.12 eV, respectively, while for G2@ECP~HW!#, the
corresponding values are 0.09 and 0.16 eV.
D. The choice of standard G2[AE] and G2[ECP]
procedures
The data presented in Tables VIII and IX show that
G2@ECP~S!# and G2@ECP~HW!# generally demonstrate similar accuracy, the sole exception being the relatively poor
G2@ECP~S!# atomization energy of I2 . However, ECP~S! has
the better performance in calculations of gas-phase proton
affinities and methyl cation affinities ~Tables XI and XII!.
On balance, we have selected the G2@ECP~S!# scheme,
based on the quasirelativistic energy-adjusted spin–orbitaveraged seven-valence–electron ECPs for bromine and iodine atoms derived by the Stuttgart group, as the standard
G2@ECP# scheme, although it must be admitted that our
choice is somewhat arbitrary.39 The G2@AE# scheme for bromine has been presented previously,5 while for iodine we
recommend the basis sets and procedures described in the
present paper.
G2@ECP# calculations with quasirelativistic ECPs are
generally comparable in accuracy to the G2@AE# scheme
~Tables V–XII!. At the same time, particularly for iodinecontaining species, the ECP scheme gives a substantial reduction in computational time in comparison with the AE
calculations.40 Although, as a result of the frozen core approximation, the G2@AE# single-point energies also do not
correlate 46 electrons per iodine atom, a price has to be paid
for the greater number of virtual spin orbitals in the AE calculations. For example, with the 6-311G(2d f ) basis set,
while there are 59 virtual spin orbitals ~29 alpha and 30 beta!
per iodine atom in the ECP calculations, there are 93 ~46
alpha and 47 beta! in the AE calculations. For MP4 calculations on molecules of the size considered in this work ~two
heavy atoms!, the computational cost increases roughly as
the number of active alpha ~or beta! spin orbitals to the fifth
power, while for increasingly larger molecules, the cost will
approach a seventh power dependence. Thus the smaller
number of virtual spin orbitals in the ECP scheme can lead to
significant computational savings.
V. CONCLUDING REMARKS
This paper introduces the use for the first time of quasirelativistic effective core potentials ~ECPs! in G2 calculations, enabling the efficient application of this high quality
procedure to heavier atoms, in the first instance bromine and
iodine. The computational results for ionization energies,
electron affinities, atomization energies, and proton and methyl cation affinities of simple but representative sets of
bromine- and iodine-containing species are encouraging. An
accuracy comparable to that of G2 calculations on molecules
J. Chem. Phys., Vol. 103, No. 5, 1 August 1995
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Glukhovtsev et al.: Extension of G2 to Br- and I-containing molecules
1884
TABLE XIII. Iodine 6-311G ~and supplementary function! Gaussian exponents ~ai !, and contraction coefficients (c i ).
1s 8
1s 9
2s 8
2s 9
3s 8
3s 9
4s 8
4s 9
5s 8
5s 9
2p 8
2p 9
3p 8
3p 9
4p 8
4p 9
5p 8
5p 9
5p 3d 8
3d 9
4d 8
4d 9
ai
ci
444 750
66 127
14 815
4 144.9
1 361.2
508.44
209.59
81.959
36.805
13.495
6.885 9
2.552 0
1.208 8
0.273 4
0.100 9
2 953.6
712.61
236.71
92.631
39.732
17.273
7.957 0
3.152 9
1.332 8
0.494 7
0.216 0
0.082 93
261.95
76.734
27.551
10.606
3.421 7
1.137 0
0.000 89
0.006 94
0.036 09
0.135 68
0.338 78
0.436 59
0.183 75
1
1
1
1
1
1
1
1
0.012 21
0.085 87
0.294 93
0.478 49
1
1
1
1
1
1
1
1
0.031 44
0.190 28
0.472 47
1
1
1
Supplementary functions
s diff
p diff
d
f
0.046 8
0.028 6
0.302
0.38
1
1
1
1
and ions containing only first- and second-row atoms is often
achieved. This extension of G2 theory opens horizons for
analogous extensions to other fourth-row main group elements.
ACKNOWLEDGMENTS
We gratefully acknowledge a generous allocation of time
on the Fujitsu VP-2200 supercomputer of the Australian National University Supercomputing Facility and the award ~to
A.P.! of an ARC Senior Research Fellowship. We thank Dr.
L. A. Curtiss, Professor H. Stoll, Professor P. Pyykkö, and
Professor G. Frenking for helpful discussions, and Dr. J. P.
Blaudeau and Dr. L. A. Curtiss for providing the first-order
spin–orbit corrections prior to publication.
APPENDIX
The iodine basis functions are summarized in Table XIII.
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39
The performance of G2@ECP~HW!# is not substantially different from
G2@ECP~S!# and we have, in fact, used ECP~HW! in a so-called G2~1!
scheme in some applications @Refs. 6~a! and 6~c!#.
40
For example, an all-electron MP4/6-311G(2d f ) calculation on I2 requires
89 min of CPU time on an IBM RISC/6000 series 355 workstation, in
contrast to 10 min of CPU time for the analogous ECP~S! calculation.
J. Chem. Phys., Vol. 103, No. 5, 1 August 1995
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