JOURNAL OF CHEMICAL PHYSICS
VOLUME 118, NUMBER 8
22 FEBRUARY 2003
Performance of coupled cluster theory in thermochemical calculations
of small halogenated compounds
David Feller,a) Kirk A. Peterson,b) Wibe A. de Jong,c) and David A. Dixond)
Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, MS K8-91,
Richland, Washington 99352
共Received 25 September 2002; accepted 1 November 2002兲
Atomization energies at 0 K and heats of formation at 298 K were obtained for a collection of small
halogenated molecules from coupled cluster theory including noniterative, quasiperturbative triple
excitations calculations with large basis sets 共up through augmented septuple zeta quality in some
cases兲. In order to achieve near chemical accuracy 共⫾1 kcal/mol兲 in the thermodynamic properties,
we adopted a composite theoretical approach which incorporated estimated complete basis set
binding energies based on frozen core coupled cluster theory energies and 共up to兲 five corrections:
共1兲 a core/valence correction; 共2兲 a Douglas–Kroll–Hess scalar relativistic correction; 共3兲 a
first-order atomic spin–orbit correction; 共4兲 a second-order spin–orbit correction for heavy
elements; and 共5兲 an approximate correction to account for the remaining correlation energy. The
last of these corrections is based on a recently proposed approximation to full configuration
interaction via a continued fraction approximant for coupled cluster theory 关CCSD共T兲-cf兴. Failure to
consider corrections 共1兲 to 共4兲 can introduce errors significantly in excess of the target accuracy of
⫾1 kcal/mol. Although some cancellation of error may occur if one or more of these corrections is
omitted, such a situation is by no means universal and cannot be relied upon for high accuracy. The
accuracy of the Douglas–Kroll–Hess approach was calibrated against both new and previously
published four-component Dirac Coulomb results at the coupled cluster level of theory. In addition,
vibrational zero-point energies were computed at the coupled cluster level of theory for those
polyatomic systems lacking an experimental anharmonic value. © 2003 American Institute of
Physics. 关DOI: 10.1063/1.1532314兴
I. INTRODUCTION
The ability to theoretically model the thermochemistry
of halogenated compounds is of importance due to the scarcity of experimental information, especially for Br- and
I-containing molecules. In a recent large-scale benchmark
study of 17 small halogenated molecules, as well as H2 and
CH4 , Lazarou et al.2,3 found the B3P86 density functional
theory 共DFT兲 method4,5 共averaged over 37 basis sets兲 to be
the most accurate procedure for computing bond dissociation
energies. However, B3P86 displayed a systematic tendency
to overbind open-shell molecules, a tendency which increased with the size of the system. By applying an empirical
correction to increase the total electronic energy of openshell systems, the root-mean-squared 共rms兲 deviation with
respect to experiment, ␧ rms , was reduced from 3.0 to 1.9
kcal/mol. Isolated atoms were excluded from this correction.
Note that this is a second level of empirical correction, as
Becke’s three-parameter exchange functional is already empirically fit to experimental thermodynamic data for small
molecules. While performing very well for bond dissociation
energies, B3P86 did rather poorly for heats of formation,
underestimating experiment by ⬃7 kcal/mol. The parametrized Gaussian-2 model chemistry,6 B3PW91,7 and
B3LYP8 proved to be the most accurate methods for heats of
formation. In all, Lazarou et al.2 examined over 800 different
combinations of basis sets and levels of theory.
Somewhat surprisingly, Lazarou et al. found that
coupled cluster theory with quasiperturbative connected
Halogenated compounds constitute a diverse class of
molecules that has received much attention in recent years as
a result of its importance in the area of atmospheric chemistry. Photolysis by sunlight, which gives rise to free halogen
atoms, is the first step in the stratospheric degradation of
halogenated compounds. It is these free halogens that catalytically contribute to the destruction of atmospheric ozone.
Of key interest are halocarbons that contain bromine, as bromine is several orders of magnitude more catalytically active
in ozone destruction than chlorine. Sources for brominated
hydrocarbons in the atmosphere include those arising from
natural as well as anthropogenic processes. Examples of the
former include bromoform (CHBr3 ), which is synthesized
and released by various forms of marine algae and phytoplankton. Examples of the latter include small brominated
compounds that are released from fire-fighting systems 共halons兲 and from fumigants 共methyl bromide, CH3 Br). To
place these sources in perspective, bromoform from natural
sources contributes more stratospheric bromine than all of
the combined anthropogenic sources of halons and methyl
bromide.1
a兲
Electronic mail: [email protected]
Also at the Department of Chemistry, Washington State University. Electronic mail: [email protected]
c兲
Electronic mail: [email protected]
d兲
Electronic mail: [email protected]
b兲
0021-9606/2003/118(8)/3510/13/$20.00
3510
© 2003 American Institute of Physics
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J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
triples 关CCSD共T兲兴9–11 was much less accurate than the best
performing DFT methods. For example, the uncorrected
B3P86 rms deviation with respect to experiment was 3.0
kcal/mol for bond dissociation energies, whereas the
CCSD共T兲 error was reported to be 7.0 kcal/mol. Three basis
sets were used with CCSD共T兲: 6-311⫹⫹G(2d f ,pd),
LANL2DZ⫹⫹(2d f , p), and LANL2DZ⫹⫹(3d2 f ,2pd).
For heats of formation, DFT with the B3LYP functional4,8
and a 3-21⫹⫹G(3d f ,2p) basis set gave a value of ␧ rms
⫽2.2 kcal/mol, whereas the errors at the CCSD共T兲 level
were in the range of 4.5 to 10.6 kcal/mol, depending upon
the choice of basis set. The authors noted that the poor showing of CCSD共T兲 and the related QCISD共T兲 method as compared to the DFT approaches could ‘‘be attributed to deficiencies of the basis sets employed.’’ 2 It should also be
noted that increasing the size of the basis set used with
B3LYP increased the size of ␧ rms to 3.5 kcal/mol.
Previously, we have examined a small number of halogenated compounds in conjunction with several CCSD共T兲
studies of the G2/97 collection of molecules12,13 and have
generally found this method to be capable of high
accuracy.14,15 The approach adopted in these studies begins
with a systematic sequence of extended basis set, frozen core
CCSD共T兲共FC兲 calculations that approach the complete basis
set 共CBS兲 limit. The resulting total energies are extrapolated
to the CBS limit in an attempt to eliminate basis set truncation error. They are further adjusted to include core–valence
correlation, molecular scalar relativistic corrections, and
atomic spin–orbit corrections.16 –23 Our composite, nonparametrized approach implicitly assumes that the effects of
the smaller corrections are additive to the extrapolated CBS
energies which only account for valence correlation effects.
Whenever possible, a further correction is applied to the
total dissociation energy in order to account for the intrinsic
error in CCSD共T兲. Ideally, this correction should be based on
a full configuration interaction 共FCI兲 wave function using a
basis set that is capable of at least semiquantitatively reflecting the true magnitude of the effect. For small systems such
as H2 O and OH, the use of FCI and valence triple zeta basis
sets in combination with the composite approach described
above can be used to differentiate between experimental
heats of formation that differ by as little as 0.5 kcal/mol.24,25
Unfortunately, the use of FCI is currently limited to
small basis sets and small chemical systems. Alternatively,
we have sometimes elected to use coupled cluster theory
including fully iterative triple excitations 共CCSDT兲, which
involves an iterative ⬃n 8 step, where n is the number of
basis functions. In previous work, we have carried out
CCSDT calculations on systems as large as benzene with a
triple zeta basis set.21 Earlier studies of the effects of iterative
triple excitations via the CCSDT method produced mixed
results in terms of the level of agreement with FCI or experimental dissociation energies. On average, CCSDT produced
a small improvement. This finding is consistent with a comparison of CCSD共T兲, CCSDT, and FCI total energies by He
et al.,26 who reported a decrease in the error with respect to
FCI, ␧ MAD⫽0.49E h 关CCSD共T兲兴 vs 0.30 E h 共CCSDT兲, for
molecules without significant multiconfiguration character.
However, the work of He et al. was limited to small double
Thermochemical calculations
3511
zeta quality basis sets, which are likely to be too small to
reflect the true effect of including iterative triples. In a study
of diatomic molecules, we found that for systems with significant multiconfiguration character 共e.g., C2 and CN兲, the
CCSDT method predicted changes with respect to CCSD共T兲
results which were of the opposite sign to the change predicted by FCI calculations.19
Finally, for light systems we have also computed diagonal corrections due to the Born–Oppenheimer
approximation.24,25 These calculations used the formulas as
implemented in MOLPRO27 by Schwenke28 at the complete
active space self-consistent field level 共CASSCF兲 level of
theory.
Zero-point vibrational energies, ZPEs, are needed for the
calculation of zero-point inclusive atomization energies,
⌺D 0 . Because the magnitude of the ZPE correction can easily be tens of kcals/mol for molecules such as those examined in this study, care must be taken so as not to introduce
significant errors into the dissociation energies. ZPEs can be
taken from experiment, from quartic theoretical force fields,
from a combination of experimental frequencies including
anharmonic corrections and calculated CCSD共T兲 harmonic
frequencies, or, as a last resort, from CCSD共T兲 harmonic
frequencies when no experimental data exist.
In general, the composite CCSD共T兲 approach described
above appears capable of achieving near chemical accuracy,
i.e., ⫾1 kcal/mol with respect to experiment, in thermochemical calculations for chemical systems composed of
first- and second row elements, as documented for nearly
300 compounds in the Environmental and Molecular Sciences Laboratory Computational Results Database.29
Very recent work by Dixon et al.30 on the three brominated compounds, CBr, CHBr, and CBr2 , failed to uncover
any systematic error with coupled cluster theory, although
the large size of the experimental error bars made it difficult
to draw any definitive conclusion regarding the accuracy of
CCSD共T兲. In order to determine if the CCSD共T兲 calculations
of Lazarou et al.,2 which led to anomalously large errors for
the heats of formation of halogenated compounds, are due, as
suggested, to the relatively small size of their basis sets, it is
necessary to perform calculations with basis sets that converge properly to the complete basis set limit. The purpose of
the present study is to determine the intrinsic accuracy of
coupled cluster theory through triple excitations for predicting atomization energies of small halogenated compounds,
such as those examined by Lazarou et al.2
II. METHODS
Experience has shown that, in most CCSD共T兲 calculations of atomization energies 共or heats of formation兲, the
largest source of error typically arises from the finite basis
set approximation. Consequently, our composite approach
makes use of the systematic convergence properties of the
valence correlation consistent family of basis sets containing
additional diffuse functions. These basis sets are conventionally denoted aug-cc-pVnZ, n⫽D⫺7.15,31,32 Only the spherical component subset 共e.g., five-term d functions, seven-term
f functions, etc.兲 of the Cartesian polarization functions
were used. All CCSD共T兲 calculations were performed with
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3512
Feller et al.
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
33
and DALTON 1.234 on a single
processor of an SGI Origin 2000. The largest CCSD共T兲 calculation undertaken in the present study, corresponding to
C2 Cl4 in D 2h symmetry, involved 798 total basis functions
and required 5.2 days of computer time. Some preliminary
second-order Møller–Plesset perturbation theory geometry
optimizations were run with NWCHEM35 on the 512 node
IBM SP massively parallel computer in the Molecular Science Computing Facility.
Optimized bond lengths and harmonic frequencies for
the diatomic molecules examined in this study were obtained
from a seventh degree Dunham fit of the potential energy
surface.36 For polyatomic molecules, geometry optimizations
were performed with a convergence threshold on the gradient
of approximately 10⫺4 E h /bohr or smaller. Those optimizations, performed with GAUSSIAN 98, used the ‘‘Opt⫽tight’’
option. Explicit geometry optimization of polyatomic molecules with the large aug-cc-pV5Z basis set proved to be
prohibitively expensive. Therefore, these geometries were
estimated by performing an exponential extrapolation of the
aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ bond
lengths. Bond angles were taken from the aug-cc-pVQZ values, which experience suggests are typically within 0.1° of
the basis set limit.
Of the three reported coupled cluster approaches to handling open-shell systems, we have chosen to use the completely restricted method. This method, which is conventionally labeled RCCSD共T兲,37–39 is based on restricted openshell Hartree–Fock 共ROHF兲 orbitals and retains the spin
restriction throughout the coupled cluster portion of the calculation. The completely unrestricted approach is labeled
UCCSD共T兲. For small systems, such as those in the present
study, the difference in atomization energies between the use
of RCCSD共T兲 and UCCSD共T兲 atomic asymptotes is ⬍0.8
kcal/mol. However, this difference grows with the size of the
molecule, and can meet or exceed the target⫾1 kcal/mol for
some of the G2/97 molecules.
The largest basis set used in the present study was of
valence septuple zeta quality. In keeping with the correlation
consistent convention, wherein both the number of functions
in each angular momentum shell and ᐉ max 共the highest angular momentum functions present in a given basis set兲 simultaneously increase as the basis set approaches completeness,
the aug-cc-pV7Z basis set would be expected to contain two
sets of k functions (ᐉ max⫽7). However, software limitations
prevented us from explicitly including k functions in our
molecular calculations. Instead, their contribution to the total
energy was estimated by performing an exponential extrapolation of the incremental correlation energy contributions due
to h (ᐉ⫽5) and i (ᐉ⫽6) functions. Checks on the accuracy
of this method of approximating the small k function contribution were obtained from atomic calculations at the configuration interaction singles and doubles 共CISD兲 and
CCSD共T兲 levels of theory using computer codes capable of
explicitly handling k functions. These tests suggest that the
exponential extrapolation of the correlation energy contributed by the missing k functions should be accurate to better
than 10⫺4 hartree.
For second row elements it is known that the presence of
GAUSSIAN 98,
27
MOLPRO-2000,
TABLE I. Valence basis set composition.
Name
Elements
aug-cc-pVTZ
H
C, F
Br
H
C, F
Br
H
C, F
Br
H
C, F
H
C, F
Cl
Cl
Cl
Cl
I
I
I
aug-cc-pVQZ
aug-cc-pV5Z
aug-cc-pV6Z
aug-cc-pV7Z
aug-cc-pV(T⫹d)Z
aug-cc-pV(Q⫹d)Z
aug-cc-pV(5⫹d)Z
aug-cc-pV(6⫹d)Z
aug-cc-pRVTZ/RECPa
aug-cc-pRVQZ/RECPa
aug-cc-pRV5Z/RECPa
Contraction
关 4s,3p,2d 兴
关 5s,4p,3d,2 f 兴
关 7s,6p,4d,2 f 兴
关 5s,4p,3d,2 f 兴
关 6s,5p,4d,3f ,2g 兴
关 8s,7p,5d,3f ,2g 兴
关 6s,5p,4d,3f ,2g 兴
关 7s,6p,5d,4f ,3g,2h 兴
关 9s,8p,6d,4f ,3g,2h 兴
关 7s,6p,5d,4f ,3g,2h 兴
关 8s,7p,6d,5f ,4g,3h,2i 兴
关 8s,7p,6d,5f ,4g,3h,2i 兴
关 9s,8p,7d,6f ,5g,4h,3i,(2k) 兴
关 6s,5p,4d,2 f 兴
关 7s,6p,5d,3f ,2g 兴
关 8s,7p,6d,4f ,3g,2h 兴
关 9s,8p,7d,5f ,4g,3h,2i 兴
关 6s,5p,4d,2 f 兴
关 7s,7p,5d,3f ,2g 兴
关 8s,8p,6d,4f ,3g,2h 兴
a
The I RECP has a 28 electron core, leaving 25 electrons to be explicitly
treated.
tight polarization functions are important at the Hartree–
Fock 共HF兲 level of theory. In recognition of this fact, the
original correlation consistent basis sets40 for second row
elements have been superceded by the aug-cc-pV(n⫹d)Z
sequence of basis sets,41 which contain an additional tight d
function. Compared with the original second row basis sets,
the aug-cc-pV(n⫹d)Z sets converge to the complete basis
set limit significantly more rapidly and reduce the error accompanying basis set extrapolation 共discussed below兲. Consequently, we have used the aug-cc-pV(n⫹d)Z basis sets for
chlorine.
The bromine valence basis sets are taken from the
aug-cc-pVnZ sets of Wilson et al.42 It should be noted that
these sets are intended for use with a 14-orbital frozen core,
i.e. (1s,2s,2p x ,2p y ,2p z ,3s,3p x ,3p y ,3p z ,3d z2 ,3d x2⫺y2 ,3d xy ,
3d xz ,3d yz ) atomic orbitals. This differs from the default in
GAUSSIAN 98, which retains the 3d space as active.
For iodine, we adopted preliminary versions of newly
created valence basis sets designed for use with a small core
relativistic effective core potential 共RECP兲.43 The RECP subsumes the (1s 2 , 2s 2 , 2 p 6 , 3s 2 , 3 p 6 , and 3d 10) orbital space
into the 28-electron core set, leaving the (4s 2 , 4p 6 , 5s 2 ,
4d 10, and 5p 5 ) space with 25 electrons to be handled explicitly. Of the latter, only the (5s 2 ,5p 5 ) are active in our valence correlation treatment. These polarized relativistic basis
sets are denoted aug-cc-pRVnZ. 43 The contracted function
composition of the valence basis sets is shown in Table I.
The present basis sets are among the largest currently
being used in ab initio electronic structure calculations on
polyatomic molecules. Nonetheless, due to the well-known
slow convergence of one-electron functions to the CBS limit,
the remaining basis set truncation error remains unacceptably
large if accuracy on the order of ⫾1 kcal/mol is desired. For
example, raw aug-cc-pV5Z basis set atomization energies for
systems the size of benzene can differ by as much as 3– 4
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J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
Thermochemical calculations
FIG. 1. The Ne atom incremental valence CCSD共T兲 correlation energy as a
function of the basis set size.
kcal/mol from the CBS limit. For small systems, such as
isolated atoms, basis sets of 8-zeta quality or better are required in order to achieve convergence in the frozen core
total energy to ⬃0.001 E h . This can be seen in Fig. 1 for the
neon atom, where the incremental valence correlation energy
is plotted as a function of the basis set size. The cc-pV8Z
basis set corresponds to a 关 9s,8p,7d,6f ,5g,4h,3i,2k,1l 兴 contraction, and the k- and l functions were treated explicitly in
the calculations.
Fortunately, by exploiting the systematic convergence
properties of the correlation consistent basis sets, it is possible to obtain reasonably accurate estimates of the CBS
limit without having to resort to such extremely large basis
sets that would unavoidably limit the use of our composite
approach to small diatomic molecules. In previous work, we
based our CBS estimates on one or more of the following
formulas: a mixed exponential/Gaussian function of the
form44
E 共 n 兲 ⫽E CBS⫹b * exp关 ⫺ 共 n⫺1 兲兴 ⫹c * exp关 ⫺ 共 n⫺1 兲 2 兴 ,
共1兲
where n⫽2(aVDZ), 3共aVTZ兲, 4共aVQZ兲;
a simple exponential function45– 47
E 共 n 兲 ⫽E CBS⫹b * exp共 ⫺cx 兲 ,
共2兲
or one of three formulas that involves the reciprocal of
ᐉ max48 –51
E 共 ᐉ max兲 ⫽E CBS⫹B/ 共 ᐉ max⫹0.5兲 4 ,
共3a兲
3
4
E 共 ᐉ max兲 ⫽E CBS⫹B/ᐉ max
⫹C/ᐉ max
,
共3b兲
3
.
E 共 ᐉ max兲 ⫽E CBS⫹B/ᐉ max
共3c兲
The latter three formulas are formally to be applied to the
correlation component of the total energy only, with the HF
3513
FIG. 2. CBS estimates of the frozen core total energy of the neon atom as a
function of the basis sets used in the extrapolations. The largest value of
ᐉ max used in the extrapolation is given on the x axis. For example, n⫽4
implies that the double through quadruple zeta basis sets were used in the
exponential and mixed extrapolations, and the triple and quadruple zeta sets
were used for the 1/(ᐉ max⫹0.5) 4 and 1/ᐉ max3 extrapolations.
component extrapolated separately or taken from the largest
basis set value. In practice, the effect on energy differences
of treating the HF component separately or extrapolating the
total energy is small. Other extrapolation approaches have
also been proposed.52–57
Experience has shown that the ‘‘best’’ extrapolation formula varies with the level of basis set and the molecular
system and there is no universally agreed upon definition of
best. Equations 共1兲 and 共2兲 are based on the observed convergence pattern displayed by the double through quadruple
zeta correlation consistent basis sets. In a large number of
comparisons of computed and experimental atomization energies, Eq. 共1兲 was statistically slightly superior to Eq. 共3a兲
when the largest affordable basis sets were of quadruple zeta
quality.14,15 Both of these expressions, in turn, were better
than Eq. 共2兲, the simple exponential fit. Equations 共3a兲 to
共3c兲 and similar expressions involving 1/ᐉ max are best suited
for basis sets beyond quadruple zeta, since they are motivated by the 1/Z perturbation theory work of Schwartz, who
dealt with two-electron systems in the case where each angular momentum space was saturated.58
In Fig. 2, a variety of Ne ( 1 S) CBS estimates obtained
from Eqs. 共1兲, 共2兲, 共3a兲, and 共3c兲 is plotted as a function of
the basis sets used in the extrapolation. Although not apparent in the figure, all of the extrapolations represent a considerable improvement over the raw total energies. Equation
共3a兲 provides the best overall agreement with the CCSD共T兲R12B result of Klopper,59 especially for sextuple zeta and
larger basis sets. Interestingly, the simple exponential fit
which significantly underestimates the CCSD共T兲-R12B energy for small ᐉ max , gradually converges to the energies predicted by the 1/(ᐉ max⫹0.5) 4 formula 关Eq. 共3a兲兴 for ᐉ max⫽7
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3514
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
and 8. The spread in CBS estimates can serve as a crude
measure of the uncertainty in the CBS extrapolation.
Throughout the remainder of this paper we will use Eq. 共3a兲
to obtain CBS estimates of the total energy and Eq. 共2兲 for
CBS estimates of bond lengths and harmonic frequencies.
Additional insight into the absolute accuracy of the CBS
extrapolations can be obtained from examining the results of
all-electron calculations. Equation 共3a兲 yields a neon atom
energy of ⫺128.9372 E h when extrapolating from cc-pV6Z
and cc-pV7Z energies, compared to a CCSD共T兲-R12B value
of ⫺128.9376 E h . The latter value is somewhat fortuitously
in exact agreement with the latest exact nonrelativistic energy reported by Chakravorty et al.60 Because thermochemical properties such as atomization energies involve energy
differences, high absolute accuracy in the total energy is not
a prerequisite for obtaining good agreement with experiment.
Some cancellation of error between the CBS energy estimates for a molecule and its constituent atoms can be expected.
Normally, even with the use of large basis sets, it would
be necessary to correct our theoretical binding energies for
the undesirable effects of basis set superposition error
共BSSE兲 when attempting to achieve accurate results. To the
extent that simple formulas such as Eqs. 共1兲–共3兲 provide an
effective means of estimating the CBS limit, they allow us to
circumvent the controversy surrounding how best to correct
for BSSE.
To convert vibrationless atomization energies; ⌺D e , to
⌺D 00 , and ultimately ⌬H 298
f , we require accurate molecular
zero-point vibrational energy corrections, ⌬E ZPE . For this
purpose, we rely on anharmonic zero-point energies obtained
from experimental or theoretical sources, whenever possible.
In the case of the 13 diatomics examined in this study, we
used the values given by Huber and Herzberg.61 For larger
systems we estimated the anharmonic zero-point energy by
following the suggestion of Grev et al.62 They observed that
by averaging the zero-point energies based on calculated harmonic frequencies, 0.5⌺ ␻ i , and experimental fundamentals,
0.5⌺ ␯ i , one can obtain a better approximation to the true
zero-point energy than with either set of frequencies alone.
In a previous study, we compared the accuracy of averaging
the harmonic and fundamental frequencies for 29 molecules
for which anharmonic zero-point energies were available.21
The root-mean-square errors, ␧ rms , in the ⌬E ZPE’s based on
aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-VQZ harmonic frequencies are 0.23, 0.11, and 0.09 kcal/mol, respectively.
Since experimental fundamentals are available for most of
the polyatomic molecules in this study, we made use of this
procedure in the majority of cases. Finally, in cases where
experimental fundamentals have not been reported, or if
there is a predominantly incomplete set of fundamentals, we
rely on the best available theoretical harmonic frequencies.
All theoretical normal mode frequencies in the present work
were obtained at the CCSD共T兲/aug-cc-pVDZ level of theory.
Most electronic structure calculations invoke the frozen
core approximation, in which the energetically lower lying
orbitals, e.g. (1s) in carbon or (1s,2s,2p) in chlorine, are
excluded from the correlation treatment. However, in order
to achieve thermochemical properties within ⫾1 kcal/mol of
Feller et al.
experiment it is necessary to account for both core–core 共intrashell 1s 2 in C兲 and core–valence 共intershell 1s 2 – 2s 2 2p n
in C兲 correlation energy effects. Core–valence 共CV兲 calculations were carried out with new weighted core–valence basis
sets, i.e., cc-pwCVnZ, or their diffuse function augmented
counterpart, aug-cc-pwCVnZ. 63 Compared to the older
cc-pCVnZ basis sets,64 which were optimized with respect to
the difference between all-electron and valence-electron calculations, the new sets emphasize the intershell component
of the CV correlation energy. This component tends to dominate molecular properties obtained from calculations where
CV correlation energy contributions are included. Thus, the
weighted sets show improved convergence characteristics for
most spectroscopic properties. All CV calculations utilized a
quadruple zeta level basis set. For Br and I, the cc-pwCVnZ
basis set contains up through h functions in order to provide
a consistent degree of angular correlation for the now active
3d electrons. The cc-pwCVnZ basis set for I is based on the
cc-pRVQZ basis set and accompanying small core RECP.
Core/valence calculations for I involve all 25 electrons outside the RECP core, i.e. (4s 2 , 4p 6 , 5s 2 , 4d 10, and 5p 5 ).
Up to three adjustments to ⌺D 0 are necessary in order to
account for relativistic effects in atoms and molecules. The
first correction lowers the sum of the atomic energies 共decreasing ⌺D 0 ) by replacing energies that correspond to an
average over the available spin multiplets with energies for
the lowest multiplets. Most electronic structure codes are
only capable of producing spin multiplet averaged wave
functions. The atomic spin–orbit corrections, ⌬E SO , were
based on the tables of Moore,65 and are as follows 共in kcal/
mol兲: 0.08 共C兲, 0.39 共F兲, 0.84 共Cl兲, 3.51 共Br兲, and 7.24 共I兲.
A second relativistic correction to the atomization energy
was designed to account for molecular scalar relativistic effects, ⌬E SR . In previous work, we evaluated ⌬E SR by using
expectation values for the two dominant terms in the Breit–
Pauli Hamiltonian, the so-called mass-velocity and oneelectron Darwin 共MVD兲 corrections from configuration interaction singles and doubles 共CISD兲 calculations. Explicitly,
⌬E SR was obtained from CISD wave functions with a VTZ
basis set at the optimal CCSD共T兲/aVTZ geometry. Although
the CISD共MVD兲 approach yields ⌬E SR values in good
agreement 共⫾0.3 kcal/mol兲 with more accurate values for
most G2/97 molecules, Bauschlicher66 has suggested that
this approach can sometimes be in error by as much as 0.6
kcal/mol, an unacceptable amount in light of our target accuracy. Consequently, in the present work, we used the
spin-free
one-electron
Douglas–Kroll–Hess
共DKH兲
Hamiltonian.67– 69 For molecules treated with the DKH approach, ⌬E SR was defined as the difference in atomization
energy between the result obtained from basis sets recontracted for DKH calculations70 and the atomization energy
obtained with the normal valence basis set of the same quality. DKH calculations were carried out at the CCSD共T兲/ccpVQZ and the CCSD共T兲/cc-pVQZ – DK levels of theory.
A potential problem arises in computing the scalar relativistic correction for molecules in this study which contain a
combination of iodine and one or more lighter elements. Attempts to combine the DKH approach with an RECP proved
unsuccessful with our software. Furthermore, there is the
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J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
possibility of ‘‘double counting’’ the relativistic effect on iodine when applying an MVD correction to an energy which
already includes some relativistic effects via the RECP.
However, because the MVD and DKH operators mainly
sample the core region where the pseudo-orbitals are small, it
is tempting to assume that any double counting would be
small.
In order to determine the size of any double counting, a
series of RECP and all-electron calculations was run on Br2
with configuration interaction at the singles and doubles
共CISD兲 level. The small-core RECP and all-electron DKH
dissociation energies agreed to within 0.06 kcal/mol, in contrast to the nonrelativistic value, which was 0.5 kcal/mol
larger. The RECP and RECP⫹MVD results differed by only
0.01 kcal/mol, indicating that any double counting was minimal. The RECP⫹MVD total energy was ⫺0.748 E h lower
than the straight RECP energy, suggesting that there is a
significant residual relativistic contribution from the valence
electrons. Fortunately, this component seems to cancel completely between Br2 and two Br atoms. RECP and RECP
⫹MVD calculations on I2 also yielded dissociation energies
within 0.01 kcal/mol of each other. A final test on the heteronuclear diatomic BrCl showed that the RECP⫹MVD value
of D e was within 0.02 kcal/mol of the all-electron DKH
value. Since the sign of ⌬E SR is negative, and most of the
iodine-containing molecules overestimate the experimental
atomization energy 共with the exception of IF兲, the inclusion
of MVD contributions from other elements in the molecule
would tend to improve agreement with experiment, although
the effect might be small.
A third relativistic correction was applied to molecules
containing the heavy bromine and iodine atoms. For these
molecules, second-order spin–orbit corrections were carried
out by using new relativistic, small-core pseudopotentials
with correlation consistent-like basis sets, denoted
cc-pRVnZ, that are currently under development.43 The
pseudopotentials are similar to those previously constructed
by Metz et al.,71 where the accompanying basis sets have
been constructed in a manner similar to the usual all-electron
cc-pVnZ sets but with the pseudopotentials included at all
stages. The lowest spin–orbit coupled eigenstates were obtained by diagonalizing relatively small spin–orbit matrices
in a basis of pure spin 共⌳–S兲 eigenstates. In each case, the
identity of the electronic states used as an expansion basis
was restricted to all states 共singlets and triplets兲 that correlated in the dissociation limit to ground-state atomic products. For the dihalogens, this corresponded to 12 states 共6
singlets and 6 triplets兲, whereas for the hydrogen halides 6
states 共3 singlets and 3 triplets兲 correlating to ground-state
products were used.72 Basis sets of triple-zeta quality augmented with diffuse functions were used, and the electronic
states and SO matrix elements were obtained in singles-only
multireference configuration interaction calculations with a
full valence complete active space 共CAS兲 reference function.
III. RESULTS
Values of selected bond distances obtained from frozen
core CCSD共T兲 calculations carried out with the largest available valence basis sets are compared with experimental
Thermochemical calculations
3515
values61,73,74 in Table II. The mean absolute deviation, ␧ MAD ,
for the 13 diatomics, with r e values taken from the compilation of Huber and Herzberg,61 is 0.003 Å, with a maximum
deviation of 0.014 Å 共for Br2 ) and a signed average error of
0.003 Å. Deviations for the polyatomic molecules are somewhat worse, although the experimental uncertainty is larger.
Extrapolating to the CBS limit leads to a slight shortening of the bond lengths. For example, the raw
CCSD(T)/aug-cc-pV(6⫹d)Z optimized bond length of Cl2
is 1.9907 Å compared to the estimated CBS limit of 1.9891
Å, obtained from extrapolating the aug-cc-pV(n⫹d)Z, n
⫽Q, 5 and 6 basis sets with an exponential fit. The relatively
small magnitude of the CBS correction is a consequence of
the large size of the underlying basis sets and the relatively
rapid rate with which r e converges as a function of basis set
size.
Introducing core/valence correlation at the aug-ccpCVQZ 共or aug-cc-pwCVQZ兲 level produces a further shortening of the bond lengths which is typically 3 to 4 times
larger than the shortening accompanying the CBS correction
as compared to the largest basis set used. For example, in Cl2
the CCSD共T兲/aug-cc-pwCVQZ CV correction is ⫺0.0042 Å
and the basis set extrapolation correction is ⫺0.0016 Å.
Additional improvement in the level of agreement with
experiment may be possible by addressing the difference between CCSD共T兲 and FCI. Conceptually, this might be accomplished in the coupled cluster realm by performing
CCSD, CCSDT, and CCSDTQ calculations and extrapolating to the CC⬁limit. However, the n 10 scaling of CCSDTQ,
where n is the number of basis functions, would currently
preclude its use for all but the smallest systems. We have
previously described a CI procedure for estimating FCI wave
functions in cases where explicit FCI is intractable, but software limitations effectively restrict its use to di- and
triatomics.17,75
An empirically motivated approach based on the use of a
continued fraction 共cf兲 approximant
E CCcf⫽
␦1
,
␦2 /␦1
1⫺
1⫺ ␦ 3 / ␦ 2
共4兲
where ␦ 1 ⫽ESCF , ␦ 2 ⫽ECCSD⫺ESCF , and ␦ 3 ⫽ECCSD(T)
⫺ECCSD , has been proposed for extrapolating CCSD共T兲 energies to the full CI limit. In his paper introducing the
CCSD共T兲-cf approximation, Goodson76 grouped his results
into two categories, characterized by whether the energies
from perturbation theory converge monotonically 共class A兲
or not 共class B兲. Whether a chemical system is identified as
class A or class B is determined by computing the location of
the dominant singularity, z d , in the complex plane of the
perturbation parameter from a fourth-order perturbation
theory calculation. Class A molecules have values of z d in
the positive half plane, whereas class B systems have z d in
the negative half plane. For 20 class A chemical systems, the
CCSD共T兲-cf energies were always closer to the FCI energy
than CCSD共T兲, although sometimes the differences were
small. For example, for AlH ( 1 ⌺ ⫹ ) with a cc-pVDZ basis,
the differences with respect to FCI were 0.39 关CCSD共T兲兴 vs
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3516
Feller et al.
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
TABLE II. Comparison of selected theoretical CCSD共T兲 bond lengths with experiment.
Molecule
F2 ( 1 ⌺ ⫹
g )
Cl2 ( 1 ⌺ ⫹
g )
Br2 ( 1 ⌺ ⫹
g )
I2 ( 1 ⌺ ⫹
g )
HF ( 1 ⌺ ⫹ )
HCl ( 1 ⌺ ⫹ )
HBr ( 1 ⌺ ⫹ )
HI ( 1 ⌺ ⫹ )
ClF ( 1 ⌺ ⫹ )
BrF ( 1 ⌺ ⫹ )
IF ( 1 ⌺ ⫹ )
BrCl ( 1 ⌺ ⫹ )
ICl ( 1 ⌺ ⫹ )
CH3 F ( 1 A 1 )
CH3 Cl ( 1 A 1 )
CH3 Br ( 1 A 1 )
CH3 I ( 1 A 1 )
C2 F4 ( 1 A ⬘g )
Basis set/method
aug-cc-pV6Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV(6⫹d)Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV5Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pRV5Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV6Z
CBS(FC)⫹CV⫹FCIc
aug-cc-pV(6⫹d)Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV5Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pRV5Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV(6⫹d)Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV5Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pRV5Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV(5⫹d)Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV(5⫹d)Z
CBS(FC)⫹CV
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV5Z
CBS共FC兲
aug-cc-pV(5⫹d)Z
CBS共FC兲
aug-cc-pV5Z
CBS共FC兲
aug-cc-pV5Z
aug-cc-pV5Z
CBS共FC兲
C2 Cl4 ( 1 A ⬘g )
aug-cc-pV5Z
CBS共FC兲
r e 共Å兲
Expt. r e 共Å兲a
1.4103
1.4085
1.4112
1.9907
1.9849
1.9857
2.2945
2.2809
2.2816
2.6728
2.6465
2.6473
0.9173
0.9167
1.2761
1.2763
1.2765
1.4212
1.4155
1.4157
1.6134
1.5995
1.5998
1.6288
1.6254
1.6267
1.7598
1.7538
1.7548
1.9110
1.8980
1.8991
2.1435
2.1334
2.1343
2.3235
2.3065
2.3075
1.3848d
1.3847
1.7817e
1.7808
1.9429f
1.9425
2.1418g
1.3345h
1.3134d
1.3344h
1.3132d
1.3441h
1.7115e
1.3440h
1.7109e
1.4119 HH
1.9879 HH
2.2810 HH
2.6663 HH
0.9168 HH
1.2746 HH
1.4144 HH
1.6092 HH
1.6283 HH
1.7589 HH
1.9098 HH
2.1361 HH
2.3209 HH
1.391 JANAF
1.7810 Kuchitsu
1.9340 Kuchitsu
2.1358 Kuchitsu
1.313 JANAF
1.313 JANAF
1.327 JANAF
1.724 JANAF
a
HH⫽Huber and Herzberg, Ref. 61. JANAF⫽M. W. Chase Jr., Ref. 73. Kuchitsu⫽Kuchitsu, Ref. 74.
Frozen core complete basis set estimate based on an exponential fit. Core/valence effects were obtained with
aug-cc-pCVQZ basis set. Estimated FCI based on the continued fraction approximation applied to the largest
basis set CCSD共T兲 results.
c
Correction for full CI effects based on explicit FCI/VTZ calculations involving 3.8⫻109 determinants.
d
r(CF).
e
r(CCl).
f
r(CBr).
g
r(CI).
h
r(CC).
b
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J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
Thermochemical calculations
3517
TABLE III. Comparison of FCI and CCSD共T兲-cf estimates of FCI total energies 共a.u.兲. Errors 共⌬兲 with respect
to FCI are given in mE h . Values in parentheses are estimated FCI energies taken from R. J. Cave et al., Ref.
75, D. Feller and J. A. Sordo, Ref. 77, and D. Feller 共unpublished兲.
System
Basis set
E(FCI)
E(CCSD(T))
⌬ CCSD(T)
E(CCSD(T)cf)
⌬ CCSD(T)cf
H2 O 1 A 1
VDZ
VDZ/VTZ
VDZ
VTZ
VDZ
VTZ
VDZ
VTZ
VQZ
VDZ
VTZ
VQZ
VTZ
VTZ
VTZ
VTZ
VQZ
⫺76.241 802
⫺76.317 404
⫺75.729 821
共⫺75.785 72兲
⫺92.494 413
共⫺92.568 89兲
⫺109.278 340
共⫺109.373 42兲
共⫺109.405 01兲
共⫺113.056 31兲
共⫺113.156 09兲
共⫺113.188 64兲
⫺39.078 343
⫺39.062 394
⫺99.620 536
⫺128.802 533
⫺38.419 484
⫺76.241 164
⫺76.317 002
⫺75.727 818
⫺75.782 634
⫺92.490 858
⫺92.566 198
⫺109.276 478
⫺109.373 266
⫺109.403 405
⫺113.054 814
⫺113.155 578
⫺113.187 860
⫺39.077 586
⫺39.061 384
⫺99.620 216
⫺128.802 453
⫺38.418 640
0.64
0.40
2.00
3.09
3.56
2.69
1.86
0.15
1.61
1.50
0.51
0.78
0.76
1.01
0.32
0.08
0.84
⫺76.241 816
⫺76.318 137
⫺75.732 119
⫺75.788 258
⫺92.492 645
⫺92.568 979
⫺109.277 990
⫺109.375 763
⫺109.406 205
⫺113.056 066
⫺113.157 704
⫺113.190 262
⫺39.078 221
⫺39.062 270
⫺99.620 763
⫺128.803 094
⫺38.419 262
⫺0.01
⫺0.73
⫺2.30
⫺2.54
1.77
⫺0.09
0.35
⫺2.34
⫺1.20
0.24
⫺1.61
⫺1.62
0.12
0.12
⫺0.23
⫺0.56
0.22
C2 1 ⌺ ⫹
g
CN 2 ⌺
N2 1 ⌺ ⫹
g
CO 1 ⌺ ⫹
CH2 3 B 1
CH2 1 A 1
F 2P
Ne 1 S
CH 2 ⌸
0.35 关CCSD共T兲-cf兴 mE h and, with the aug-cc-pVQZ basis
set, the differences are 0.43 关CCSD共T兲兴 versus 0.36
关CCSD共T兲-cf兴 mE h . For class B systems the level of agreement between CCSD共T兲-cf and FCI was worse. In six
out of 19 cases the CCSD共T兲-cf energy was further from
FCI than CCSD共T兲. For example, for CN⫹ the E(approx.)
⫺E(FCI) differences were ⫺0.23 关CCSD共T兲兴 versus ⫺5.41
关CCSD共T兲-cf兴 mE h with the cc-pVDZ basis set.
In Table III we compare the CCSD共T兲-cf approximation
with 17 explicit FCI and estimated FCI energies.17,77 Without
attempting to classify the molecules as A or B, the results are
mixed. In ten of the cases, CCSD共T兲-cf energies are in closer
agreement with the FCI energies than CCSD共T兲.
Very recent work by Goodson and Zheng78 suggests that
CCSD共T兲-cf provides an improved approximation to FCI,
compared to CCSD共T兲, not only near r c but also as a function of bond distance. This conclusion was based on an
analysis of the energies of HF, BH, and CH3 as a function of
bond distance. In the present work we have investigated the
effect of using CCSD共T兲-cf to estimate bond distances at the
FCI limit for a subset of the molecules. In lieu of a larger
body of FCI benchmarks, we can use agreement with experiment as the measure of accuracy for CCSD共T兲-cf. As expected based on other FCI results, the estimated FCI correction based on the CCSD共T兲-cf technique typically produces a
slight lengthening of the diatomic bond distances 共see Table
II兲. For HF, the FCI correction was based on an explicit
FCI/VTZ calculation involving 3.8⫻109 determinants. In
general, this bond lengthening is insufficient to compensate
for the shrinkage due to CBS extrapolation and CV effects.
Thus, the overall effect of the CBS⫹CV⫹est. FCI set of
corrections is to reduce the frozen core bond lengths. The
mean absolute deviation for the bond distances is 0.004 Å
and the signed average error is ⫺0.003 Å, nearly identical to
the raw frozen core results, although the latter overestimated
the experimental values and the CBS⫹CV⫹est. FCI results
underestimate experiment.
Comparing CCSD共T兲-cf atomization energies with ex-
periment is complicated by the individual uncertainties arising from the CBS extrapolation and each of the smaller energetic corrections. As discussed below, CCSD共T兲-cf
improved agreement with experiment for some molecules
but, in a substantial number of cases, it worsened agreement.
To what extent this reflects a failure of the CCSD共T兲-cf approximation or other errors in our theoretical approach 共or in
the experimental values兲 is unclear. Thus, at this point there
is insufficient evidence to indicate whether properties based
on energy differences, such as those associated with thermochemical properties, would be generally improved by application of the CCSD共T兲-cf approximation.
In Table IV, we compare the theoretical harmonic frequencies for the 13 diatomic molecules examined in this
study with the corresponding experimental values.61 Also
shown are CBS⫹CV⫹est. FCI values, based on an exponential fit of the three best ␻ e values, plus core–valence and
estimated FCI corrections using the CCSD共T兲-cf approximation. For those molecules not displaying monotonic convergence of the frequencies to the CBS limit 共HBr and HI兲, and
which therefore precluded CBS extrapolation, the largest basis set value of ␻ e was used. Core/valence corrections were
obtained from CCSD共T兲/aug-cc-pwCVQZ calculations. The
mean absolute deviation is 12.3 cm⫺1 and the maximum error is 45.7 cm⫺1 共HI兲. In general, the CBS and CV corrections both tend to increase ␻ e and the FCI correction tends to
decrease it, as expected from the preceding discussion of
trends in bond lengths. In order to investigate the magnitudes
of the errors, we also present the percent errors in Table IV.
In general, the calculated values are greater than the experimental ones. The errors for the molecules HF, HCl, Cl2 , F2 ,
and ClF are essentially 1% or less. The molecules containing
Br have a slightly larger error, on the order of up to 1.5%
except for the CBS(FC)⫹CV⫹est. FCI calculation for
BrCl, where the error is 1.9%. The molecules containing the
I atom have the largest errors, with values up to 5%. The
CBS(FC)⫹CV⫹est. FCI results for molecules containing I
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3518
Feller et al.
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
TABLE IV. Comparison of theoretical CCSD共T兲 frequencies with experiment.
Molecule
Basis set/Method
␻ e 共cm⫺1兲
Expt. ␻ e 共cm⫺1兲a
Error 共%兲
F2 ( 1 ⌺ ⫹
g )
aug-cc-pV6Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV(6⫹d)Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV5Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pRV5Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV6Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV(5⫹d)Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV5Z
aV5Z⫹CV⫹est. FCIb
aug-cc-pRV5Z
aV5Z⫹CV⫹est. FCIb
aug-cc-pV(6⫹d)Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV5Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pRV5Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV(5⫹d)Z
CBS(FC)⫹CV⫹est. FCIb
aug-cc-pV5Z
CBS(FC)⫹CV⫹est. FCIb
927.0
911.1
562.7
563.7
327.4
330.3
220.8
225.2
4141.9
4151.8
2995.3
2997.5
2655.0
2666.0
2317.3
2354.7
789.9
784.4
678.7
678.8
627.3
634.7
447.1
451.9
393.4
391.6
916.64
⫹1.1
⫺0.6
⫹0.5
⫹0.7
⫹0.6
⫹1.5
⫹2.9
⫹5.0
⫹0.1
⫹0.3
⫹0.1
⫹0.2
⫹0.2
⫹0.6
⫹0.4
⫹2.0
⫹0.5
⫺0.2
⫹1.2
⫹1.2
⫹2.8
⫹4.0
⫹0.8
⫹1.9
⫹2.4
⫹1.9
Cl2 ( 1 ⌺ ⫹
g )
Br2 ( 1 ⌺ ⫹
g )
I2 ( 1 ⌺ ⫹
g )
HF ( 1 ⌺ ⫹ )
HCl ( 1 ⌺ ⫹ )
HBr ( 1 ⌺ ⫹ )
HI ( 1 ⌺ ⫹ )
ClF ( 1 ⌺ ⫹ )
BrF ( 1 ⌺ ⫹ )
IF ( 1 ⌺ ⫹ )
BrCl ( 1 ⌺ ⫹ )
ICl ( 1 ⌺ ⫹ )
559.72
325.32
214.50
4138.32
2990.95
2648.98
2309.01
786.15
670.75
610.24
443.35
384.29
a
Huber and Herzberg, Ref. 61.
Frozen core complete basis set estimate with aug-cc-pCVQZ correction for core/valence effects. The estimated
FCI correction is based on the CCSD共T兲-cf approximation.
b
generally are worse than the ones obtained at the CCSD共T兲
valence only level with the largest basis set.
Comparison of the current relativistic corrections 共molecular scalar, molecular second order, and atomic SO兲 to
relativistic corrections reported for previously published
four-component Dirac Coulomb CCSD共T兲 calculations79– 81
shows agreement to within 0.2 kcal/mol. The errors increase
slightly as one progresses down the halogen column of the
periodic table. For example, the total DKH/VQZ relativistic
correction for Br2 computed in this work was ⫺7.1 kcal/mol,
whereas the four-component CCSD共T兲/VTZ value of Visscher and Dyall79 is somewhat smaller, at ⫺6.9 kcal/mol.
However, a new CCSD共T兲 four-component calculation 共described below兲 with the larger VQZ basis set yields a relativistic correction of ⫺7.0 kcal/mol, just 0.1 kcal/mol smaller
in magnitude than the DKH/VQZ value. The relativistic correction for I2 is ⫺12.5 共DKH/VTZ, this work兲 vs ⫺12.7
共4-component, CCSD共T兲/VTZ兲 kcal/mol, where the nonrelativistic result is taken from the VTZ D e of Visscher and
Dyall.
The fully relativistic 共four-component兲 and nonrelativistic CCSD共T兲/VQZ calculations on Br2 were done using the
latest parallel version of the MOLFDIR program package.82,83
A Br2 bond distance of 2.2986 Å was used and the dissociation limit was calculated for the 2 P 3/2 ground state of the Br
atom. A Gaussian distribution with an exponent of
241 301 637.1909 was used to represent the spatial extent of
the nucleus in both the relativistic and nonrelativistic calculation. The speed of light 共in atomic units兲 was taken to be
137.035 989 5. Both the nonrelativistic and relativistic contracted VQZ bromine basis set for use in MOLFDIR were generated by recontracting the exponents of the nonrelativistic
cc-pVQZ basis set42 with the basis set extension feature of
the atomic GRASP code.84 The set of exponents for the s- and
p functions is flexible and tight enough to provide a good
description of the relativistically contracted s- and p orbitals
as well as for the p-orbital spin–orbit splitting; hence, no
tight functions needed to be added to the basis set. In the
CCSD共T兲 calculations, the 4s and 4p electrons were correlated and the full virtual space was included.
In Tables V and VI, the calculated atomization energies
at 0 K, ⌺D 0 , and the heats of formation at 298 K,
⌬H f (298), are compared with the available experimentally
derived values.61,73,85– 88 Also shown in Table V are the variations of the frozen core, estimated CBS values of ⌺D e as a
function of the underlying basis sets used in the 1/ᐉ max , Eq.
共3a兲, extrapolation. This information sheds light on the relative stability of the extrapolation, allowing one to crudely
gauge the degree of convergence in CBS ⌺D e . In many
cases, the variation is no more than a few tenths of a kcal/
mol. The largest variations occur for CF4 and C2 F4 , where
the aVTQ and aVQ5 CBS estimates differ by 1.3 and 1.8
kcal/mol, respectively. This behavior in CF4 has previously
been reported.23 To put these numbers in perspective, the
average change in the raw ⌺D e values that occurs when
expanding the basis set from aVQZ to aV5Z is ⬃2 kcal/mol
for the molecules in this study. For example, the change for
Cl2 is 1.9 kcal/mol and for C2 F4 is 2.2 kcal/mol. Although
Downloaded 11 Feb 2003 to 134.121.44.19. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
Thermochemical calculations
3519
TABLE V. CCSD共T兲 and experimental atomization energies. Results are given in kcal/mol. The atomic asymptotes were calculated with the RCCSD共T兲
method. Experimental values: HH⫽Huber and Herzberg, Ref. 61; JANAF⫽Chase, Ref. 73; GGBVMKY⫽Glushko et al., Ref. 85; Ferguson⫽Ferguson
et al., Ref. 86; Cox⫽Cox and Pilcher, Ref. 87; DeMore⫽DeMore et al., Ref. 88.
Est. CBS ⌺D e Basis Setsa
Corrections
Theoretical
⌺D 0 共0 K兲g
Molecule
aVTQ
aVQ5
aV56
aV67
⌬EZPEb
⌬ECVc
⌬ESRd
⌬ESOe
F2 ( 1 ⌺ ⫹
g )
Cl2 ( 1 ⌺ ⫹
g )
Br2 ( 1 ⌺ ⫹
g )
I2 ( 1 ⌺ ⫹
g )
HF ( 1 ⌺ ⫹ )
HCl ( 1 ⌺ ⫹ )
HBr ( 1 ⌺ ⫹ )
HI ( 1 ⌺ ⫹ )
ClF ( 1 ⌺ ⫹ )
38.6
59.4
52.4
46.8
142.0
107.4
93.7
79.6
62.8
38.6
60.0
52.6
47.0
141.6
107.7
93.7
79.5
62.8
38.6
59.8
38.6
⫺1.3
⫺0.8
⫺0.5
⫺0.3
⫺5.9
⫺4.3
⫺3.8
⫺3.3
⫺1.1
⫺0.1
0.2
0.4
2.0
0.2
0.7
0.6
1.7
0.1
0.0
⫺0.2
⫺0.5
0.0
⫺0.2
⫺0.3
⫺0.6
0.0
⫺0.2
⫺0.8
⫺1.7
⫺7.0
⫺14.4
⫺0.4
⫺0.8
⫺3.5
⫺7.2
⫺1.2
BrF ( 1 ⌺ ⫹ )
IF ( 1 ⌺ ⫹ )
64.2
70.4
64.0
70.2
⫺1.0
⫺0.9
⫺0.2
0.2
⫺0.6
⫺0.1
⫺3.9
⫺7.6
0.3
1.7
58.6
63.5
BrCl ( 1 ⌺ ⫹ )
ICl ( 1 ⌺ ⫹ )
CH3 F ( 1 A 1 )
56.2
56.0
422.4
56.6
56.3
422.0
⫺0.6
⫺0.6
⫺24.2
0.4
0.8
1.1
⫺0.4
⫺0.1
⫺0.5
⫺4.4
⫺8.0
⫺0.5
0.2
1.3
51.8
49.7
397.9
CH3 Cl ( 1 A 1 )
395.3
395.3
⫺23.3
1.2
⫺0.5
⫺0.9
372.2
CH3 Br ( 1 A 1 )
CH3 I ( 1 A 1 )
CF4 ( 1 A 1 )
CCl4 ( 1 A 1 )
CH2 F ( 2 A ⬘ )
CH2 Cl ( 2 A ⬘ )
CH2 F2 ( 1 A 1 )
CH2 Cl2 ( 1 A 1 )
C2 F4 ( 1 A g⬘ )
C2 Cl4 ( 1 A g⬘ )
383.9
372.3
479.2
315.1
313.5
288.1
437.7
371.1
589.0
471.0
383.8
⫺22.8
⫺22.4
⫺10.9
⫺6.0
⫺15.6
⫺14.2
⫺20.4
⫺18.4
⫺13.4
⫺9.6
1.6
3.7
1.0
1.4
1.0
1.1
1.0
1.2
2.5
2.7
⫺0.8
⫺0.9
⫺0.9
⫺0.6
⫺0.4
⫺0.5
⫺0.6
⫺0.6
⫺1.1
⫺1.1
⫺3.6
⫺7.3
⫺1.6
⫺3.4
⫺0.5
⫺0.9
⫺0.9
⫺1.8
⫺1.7
⫺3.5
358.2
345.4
465.5
307.2
297.6
273.6
416.1
351.3
573.5
459.9
477.9
315.8
313.1
288.1
437.0
371.3
587.2
471.4
141.6
107.5
62.9
141.6
⌬ESOf
0.4
2.0
0.1
0.5
36.4
57.3
45.4
36.3
135.3
102.8
86.5
71.2
60.5
Experimental
⌺D 0 共0 K兲
36.9⫾0.1 JANAF
57.18⫾0.01 JANAF
45.44⫾0.03 JANAF
35.57⫾0.02 JANAF
135.2⫾0.2 JANAF
102.23⫾0.05 JANAF
86.64⫾0.04 JANAF
70.42⫾0.05 JANAF
60.4⫾0.07 HH
59.1⫾0.1 JANAF
58.8⫾0.4 JANAF
65.3⫾0.9 JANAF
66.4 HH
51.5⫾0.3 JANAF
49.64⫾0.03 JANAF
397.4⫾7.9 JANAF
397⫾1 DeMore
371.6⫾0.5 JANAF
371.0 GGBVMKY
357.6⫾0.2 Ferguson
344.8⫾0.3 Cox
465.5⫾0.3 JANAF
307.3⫾0.5 JANAF
298.9⫾2.0 DeMore
272.2⫾1.0 DeMore
416.1⫾0.4 JANAF
351.6⫾0.3 JANAF
570.4⫾0.7 JANAF
457.1⫾0.7 JANAF
Extrapolated by using E(ᐉ max)⫽ECBS⫹b/(ᐉ max⫹0.5) 4 . The basis set designations refer to the two basis sets that were used in performing the extrapolation.
For example, aVTQ means that the aug-cc-pVTZ and aug-cc-pVQZ basis set molecular and atomic energies were extrapolated to obtain a CBS value of
⌺D e .
b
The zero-point energies are taken from the following sources: 共1兲 for diatomics, the anharmonic ZPEs were taken from Huber and Herberg, Ref. 61, and
computed as 0.5␻ e ⫺0.25␻ e xe ; 共2兲 for polyatomics, the anharmonic ZPEs were taken as the average of the zero-point energies based on the experimental
fundamental and the CCSD共T兲 harmonics. If a complete set of experimental frequencies was not available, as was the case for the halomethyl radicals (CH2 F
and CH2 Cl), the zero-point energy is based simply on the harmonic CCSD共T兲 frequencies.
c
Core/valence corrections were obtained with the cc-pCVQZ 共C and F兲 and cc-pwCVQZ 共for Cl, Br, and I兲 basis sets at the optimized CCSD共T兲/aug-cc-pVTZ
geometries.
d
For those molecules that do not contain iodine, the scalar relativistic correction is based on DK-CCSD共T兲共FC兲/cc-pVQZ calculations evaluated at the
CCSD共T兲共FC兲/aug-cc-pVQZ geometry. For the molecules containing iodine, the scalar relativistic correction is based on a CISD共FC兲/cc-pRVQZ-cc-pVQZ
MVD calculation and is expressed relative to the CISD result without the MVD correction, i.e., including the existing relativistic effects resulting from the
use of a relativistic effective core potential.
e
Correction due to the incorrect treatment of the atomic asymptotes as an average of spin multiplets. Values are based on Moore’s Tables, Ref. 65.
f
Second-order molecular spin–orbit effects obtained with an aRVTZ basis set and a relativistic ECP.
g
The theoretical value of ⌺D 0 共0 K兲 was computed with the best available CBS estimate.
a
the change in CBS ⌺D e for C2 F4 is in the direction of improving agreement with experiment, it is unlikely that a CBS
extrapolation based on an aug-cc-pV6Z energy would reconcile the large remaining difference between theory and experiment. With our present hardware and software, it is not
possible to answer this question definitively, since a
CCSD共T兲/aug-cc-pV6Z calculation with 1134 basis functions
is prohibitively expensive.
Among the energetic corrections to ⌺D 0 listed in Table
V, the atomic spin–orbit corrections dominate for the diatomics, whereas zero-point vibrational energies are clearly
the largest correction for the polyatomic species. The importance of the latter can be seen in molecules like CH3 F
(⌬E ZPE⫽⫺24.2 kcal/mol) and CH3 Cl, (⌬E ZPE⫽⫺23.3
kcal/mol), where this correction is more than an order of
magnitude larger than the other corrections. Uncertainty in
⌬E ZPE can easily be the largest source of error in the theoretical atomization energy, thus underlining the general need
for the development of new methods that will allow us to
accurately determine ZPEs for polyatomic species.
The inclusion of the CCSD共T兲-cf estimate for the FCI
correction to CCSD共T兲 uniformly increased the atomization
energies, improving agreement with experiment for some
and worsening it for others. Among the diatomics, the magnitude of this correction ranges from as little as 0.003 kcal/
mol 共HCl兲 to as much as 0.80 kcal/mol (F2 ).
Downloaded 11 Feb 2003 to 134.121.44.19. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
3520
TABLE VI. CCSD共T兲 and experimental heats of formation at 298 K 共kcal/
mol兲.
Theory
Experimenta
F2 ( 1 ⌺ ⫹
g )
Cl2 ( 1 ⌺ ⫹
g )
Br2 ( 1 ⌺ ⫹
g )
I2 ( 1 ⌺ ⫹
g )
HF ( 1 ⌺ ⫹ )
HCl ( 1 ⌺ ⫹ )
HBr ( 1 ⌺ ⫹ )
HI ( 1 ⌺ ⫹ )
ClF ( 1 ⌺ ⫹ )
⫺0.5
⫺0.1
7.4
14.2
⫺65.2
⫺22.6
⫺8.6
5.5
⫺13.4
BrF ( 1 ⌺ ⫹ )
IF ( 1 ⌺ ⫹ )
⫺13.8
⫺20.9
BrCl ( 1 ⌺ ⫹ )
ICl ( 1 ⌺ ⫹ )
CH3 F ( 1 A 1 )
3.2
4.1
⫺56.5
CH3 Cl ( 1 A 1 )
⫺20.6
CH2 Br ( 1 A 1 )
CH3 I ( 1 A 1 )
CF4 ( 1 A 1 )
CCl4 ( 1 A 1 )
CH2 F ( 2 A ⬘ )
CH2 Cl ( 2 A ⬘ )
CH2 F2 ( 1 A 1 )
CH2 Cl2 ( 1 A 1 )
C2 F4 ( 1 A ⬘g )
C2 Cl4 ( 1 A g⬘ )
⫺8.8
2.8
⫺223.0
⫺22.3
⫺6.7
27.6
⫺107.7
⫺22.4
⫺160.5
⫺5.7
0.0
0.0
7.39⫾0.03
14.92⫾0.02
⫺65.1⫾0.2
⫺22.06⫾0.05
⫺8.71⫾0.04
6.30⫾0.05
⫺13.3⫾0.1
⫺12.0⫾0.1
⫺14.0⫾0.4
⫺22.7⫾0.9
⫺24.1
3.5⫾0.3
4.18⫾0.03
⫺56⫾7
⫺56⫾1
⫺20.0⫾0.5
⫺19.4
⫺8.2⫾0.2
3.4⫾0.3
⫺223.0⫾0.3
⫺22.4⫾0.5
⫺8.0⫾2.0
29.0⫾1.0
⫺107.7⫾0.4
⫺22.8⫾0.3
⫺157.4⫾0.7
⫺3.0⫾0.7
Molecule
a
Feller et al.
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
Experimental references are given in Table V.
The mean absolute deviation with respect to experiment
for the atomization energies shown in Table V is 0.8 kcal/
mol, roughly equivalent to the value obtained for the much
larger collection of molecules taken from the G2/97 benchmark set.15 Errors exceeding 2 kcal/mol were found in only
two molecules (C2 F4 and C2 Cl4 ), of which one (C2 F4 ) has
an experimental value that has previously been questioned.
In contrast to the present results, Lazarou et al.2 reported
CCSD共T兲 mean absolute deviations for heats of formation in
the range of 4 to 10 kcal/mol, depending on the choice of
basis set. In subsequent work, Lazarou et al.3 used the infinite basis set extrapolation technique of Truhlar and
co-workers,57,89 in conjunction with double and triple zeta
basis sets and an empirical correction to account for errors in
the extrapolation, to obtain a root-mean-square deviation of
1.4 kcal/mol for 57 molecules.
Although, in general, the errors for the dissociation energies of the 13 diatomic molecules are small, for IF ( 1 ⌺ ⫹ )
the error is ⫺1.8 kcal/mol. On the basis of their study of
halogenated compounds, Lazarou et al.2 have previously
questioned the accuracy of the experimental value for IF. We
raise the same question and propose that the heat of formation of IF be revised to a less negative value of ⫺20.9
⫾1.0 kcal/mol. We note that IF is difficult to work with experimentally and that the IF heat of formation is based on a
spectral observation of the dissociation limit.72 There are
questions about the value originally assigned to the dissocia-
tion energy.72 This suggests that the experimental value may
have larger error bars than are currently employed.
We and others have suggested that there is an error in the
experimental heat of formation of C2 F4 based on high level
computational results.23,90,91 In all cases, the heat of formation is predicted to be more negative than the currently accepted experimental value. The present theoretical value of
⫺160.5 kcal/mol for ⌬H 298
f (C2 F4 ) is consistent with the previously calculated values of ⫺160.6 kcal/mol from Dixon
et al.23 and ⫺160.5 kcal/mol from Bauschlicher and Ricca.91
Again, we would recommend using the more negative calculated value for ⌬H f (C2 F4 ).
For C2 Cl4 , which has a similar valence electronic structure to that of C2 F4 , the calculated heat of formation is too
negative as compared to the experimental result. This was
also the case for C2 F4 . Based on the C2 F4 result, it may also
be appropriate to revise the heat of formation of C2 Cl4 to be
more negative in agreement with the theoretical value. In
addition, we note that if there is a problem in the calculation
of the heats of formation of C2 F4 and C2 Cl4 , it must be in
the interaction of the halogen atoms with the double bond,
since we can calculate the heat of formation of C2 H4 to
within a few tenths of a kcal/mol.21
The current calculated value for the heat of formation of
CH3 F is consistent with previous isodesmic reaction
calculations92 on fluorinated methanes, as well as with the
NASA value of ⫺56⫾1 kcal/mol. 88 We suggest that the
NASA error bars for ⌬H 298
f (CH3 F) of ⫾1.0 kcal/mol are
realistic and should be used in the future.
The heats of formation of the radicals CH2 F 共⫺6.7 kcal/
mol兲 and CH2 Cl 共27.6 kcal/mol兲 are consistent with previously calculated values of ⫺6.8⫾1.3 kcal/mol and 29.4
⫾0.8 kcal/mol, respectively, based on MP2 calculations and
isodesmic reactions.93 The previous calculated value for
CH2 Cl is closer to the experimental value (29.0
⫾1.0 kcal/mol) than to the current calculated value. The current calculations are reliable enough that we suggest revising
the heats of formation at 298 K of CH2 F and CH2 Cl to
⫺6.7⫾1.0 and 27.6⫾1.0 kcal/mol, respectively.
The predicted value for ⌬H 298
f (CH3 I) is 0.6 kcal/mol
below the experimental value, which is similar to the error
for HI, where the predicted value is 0.8 kcal/mol below the
experiment. This is consistent with the similarity in bonding
for HI and CH3 I.
IV. CONCLUSION
A composite CCSD共T兲-based approach was used to
compute geometries, normal mode frequencies, atomization
energies, and heats of formation for a collection of 25 small
halogenated compounds in an effort to determine if this class
of molecules presented systematic problems to our theoretical approach. Basically, this approach involves extrapolating
the results of large basis set, frozen core CCSD共T兲 calculations to the complete basis set limit, followed by inclusion of
a number of smaller energetic corrections. For the molecules
examined in the present study, the mean absolute deviation
with respect to experiment for heats of formation was 0.8
kcal/mol. This value is very similar to the mean absolute
Downloaded 11 Feb 2003 to 134.121.44.19. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp
J. Chem. Phys., Vol. 118, No. 8, 22 February 2003
deviation of ⬃1 kcal/mol accuracy found for a large number
of molecules with reliable experimental heats of formation.
Thus, in combination with the results of previous studies, the
present results on halogenated compounds provide further
evidence that our approach is generally capable of predicting
thermochemical properties to chemical accuracy for a wide
range of molecular systems. Errors with respect to experiment greater than 2 kcal/mol were found for only two molecules (C2 F4 , and C2 Cl4 ). We suggest that for these molecules the experimental values be revised to be more in line
with the calculated values. For the other molecules, the largest errors in the calculated heats of formation are found in
molecules containing the I atom.
ACKNOWLEDGMENTS
Dr. Y. Lazarou is thanked for early access to his infinite
basis set work. Professor D. Z. Goodson is thanked for early
access to his work on the reliability of the CCSD共T兲-cf approximation as a function of bond length. Professor H. Stoll
is thanked for helpful discussions on the mixed RECP/allelectron basis set problem with ⌬E SR . Dr. S. Hirata is
thanked for a critical reading of this manuscript prior to publication. This research was supported, in part, by the U.S.
Department of Energy, Office of Basis Energy Research,
Chemical Sciences, under Contract No. DE-AC06-76RLO
1830. This research was performed, in part, using the Molecular Science Computing Facility 共MSCF兲 in the William
R. Wiley Environmental Molecular Sciences Laboratory at
the Pacific Northwest National Laboratory. The MSCF is a
national user facility funded by the Office of Biological and
Environmental Research in the U.S. Department of Energy.
The Pacific Northwest National Laboratory is a multiprogram national laboratory operated by Battelle Memorial Institute.
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