Toward Accurate Theoretical Thermochemistry of First Row Transition Metal Complexes

TLDR: The recently developed correlation consistent Composite Approach for transition metals (ccCA-TM) was utilized to compute the thermochemical properties for a collection of 225 inorganic molecules containing first row (3d) transition metals, ranging from the monohydrides to larger organometallics such as Sc(C(5)H(5))(3) and clusters such as (CrO(3))( 3).

Abstract: The recently developed correlation consistent Composite Approach for transition metals (ccCA-TM) was utilized to compute the thermochemical properties for a collection of 225 inorganic molecules containing first row (3d) transition metals, ranging from the monohydrides to larger organometallics such as Sc(C5H5)3 and clusters such as (CrO3)3. Ostentatiously large deviations of ccCA-TM predictions stem mainly from aging and unreliable experimental data. For a subset of 70 molecules with reported experimental uncertainties less than or equal to 2.0 kcal mol–1, regardless of the presence of moderate multireference character in some molecules, ccCA-TM achieves transition metal chemical accuracy of ±3.0 kcal mol–1 as defined in our earlier work [J. Phys. Chem. A 2007, 111, 11269–11277] by giving a mean absolute deviation of 2.90 kcal mol–1 and a root-mean-square deviation of 3.91 kcal mol–1. As subsets are constructed with decreasing upper limits of reported experimental uncertainties (5.0, 4.0, 3.0, 2.0, and 1...

Full text

Published: November 22, 2011
r
2011 American Chemical Society 870 dx.doi.org/10.1021/jp205710e
|
J. Phys. Chem. A 2012, 116, 870–885
ARTICLE
pubs.acs.org/JPCA
Toward Accurate Theoretical Thermochemistry of First Row Transition
Metal Complexes
Wanyi Jiang, Nathan J. DeYonker, John J. Determan, and Angela K. Wilson*
Center for Advanced Scientific Computing and Modeling (CASCaM), Department of Chemistry, University of North Texas,
Denton, Texas 76203-5070, United States
b
S Supporting Information
’ INTRODUCTION
Transition metal-containing species play a pivotal role in all
major areas of both chemical research and industry, including
catalysis, synthesis, combustion, and materials. Quantitative
energetic information of these compounds is useful in elucidating
the underlying mechanism of chemical processes involving transi-
tion metal-containing compounds, as well as providing guidance
about synthetic preferences. Unlike main group molecules,
accurate bond enthalpies and standard enthalpies of formation
(ΔH
f
) are still elusive for many fundamental transition metal-
containing molecules in the gas phase. Tabulated values of these
thermodynamic quantities
13
are not unequivocal even for some
classical metal compl exes such as Cr(CO)
x
and Fe(CO)
x
.
46
This is due to the age and scarce availability of experimental data,
as well as severe difficulties in assessing the quality of this data.
Consequently, computational methods are critical and some-
times provide the only means to obtain the energetic profiles of
transition metal species. The development of reliable and com-
putationally amenable methods to accurately predict the ener-
getic properties of transition metal species becomes an important
task for the computational chemistry community.
Many physi cal phenomena compound to make the investiga-
tion of the electronic structure of transition metal-containing
molecules an extremely difficult task. These effects are not limited
to competing low-lying excited states, the existence of spinorbit
coupling, strong relativistic effects, increased electron correlation
due to metalligand backbonding, and complexity of corecore
and corevalence electron interactions. General computational
approaches that can be used to efficiently and reliably compute
bond energies of transition metal molecules are not well estab-
lished, even though a great deal of progress has been made recently
(see a selected few examples in refs 714). Density functional
theory (DFT), more efficient as compared to wave function-based
electron correlation methods, has been extensively applied to a
great variety of transition metal-containing species with growing
success for a number of years (see recent reviews
15,16
and ref-
erences therein). However, DFT methods that are viable for
research on transition metal-containing species are often heavily
parametrized, for example, the M06 families.
1719
Pure func-
tionals such as PBE do not contain empirical parameters but fail
to predict the ground spin states of open shell transition metal-
containing systems such as iron complexes.
20
Despite their wide
applications, DFT methods may fail to provide quantitatively
useful information, for example, for the enthalpies of formation
calculated via atomization energies for a number of transition
metal complexes.
9
The problems could be alleviated by using
Received: June 17, 2011
Revised: November 16, 2011
ABSTRACT: The recently developed correlation consistent
Composite Approach for transition metals (ccCA-TM) was
utilized to compute the thermochemical properties for a collec-
tion of 225 inorganic molecules containing first row (3d) transi-
tion metals, ranging from the monohydrides to larger organo-
metallics such as Sc(C
5
H
5
)
3
and clusters such as (CrO
3
)
3
.
Ostentatiously large deviations of ccCA-TM predictions stem
mainly from aging and unreliable experimental data. For a subset
of 70 molecules with reported experimental uncertainties less
than or equal to 2.0 kcal mol
1
, regardless of the presence of moderate multireference character in some molecules, ccCA-TM achieves
transition metal chemical accuracy of (3.0 kcal mol
1
as defined in our earlier work [J. Phys. Chem. A 2007, 111,1126911277] by
giving a mean absolute deviation of 2.90 kcal mol
1
and a root-mean-square deviation of 3.91 kcal mol
1
. As subsets are constructed
with decreasing upperlimits of reported experimental uncertainties (5.0, 4.0,3.0, 2.0, and 1.0 kcal mol
1
), the ccCA-TMmean absolute
deviations were observed to monotonically drop off from 4.35 to 2.37 kcal mol
1
. In contrast, such a trend is missing for DFT methods
as exemplified by B3LYP and M06 with mean absolute deviations in the range 12.914.1 and 10.511.0 kcal mol
1
, respectively.
Salient multireference character, as demonstrated by the T
1
/D
1
diagnostics and the weights (C
0
2
) of leading electron configuration in
the complete active self-consistent field wave function, was found in a significant amount of molecules, which can still be accurately
described by the single reference ccCA-TM. The ccCA-TM algorithm has been demonstrated as an accurate, robust, and widely
applicable model chemistry for 3d transition metal-containing species with versatile bonding features.
This is an open access article published under an ACS AuthorChoice License, which permits
copying and redistribution of the article or any adaptations for non-commercial purposes.

871 dx.doi.org/10.1021/jp205710e |J. Phys. Chem. A 2012, 116, 870–885
The Journal of Physical Chemistry A
ARTICLE
reaction energies, which are not always available. Multireference
methods such as multireference configuration interaction
(MRCI) and complete active space perturbation theory
(CASPTn)
21,22
are free of spin contamination and can properly
address the multireference character that is prominent in many
transition metal compounds. These methods have been com-
monly used to obtain accurate energetic description of small
transition metal compounds. For relatively larger molecules, a
complete active space (CAS) that includes all relevant valence
electrons and valence molecular orbitals is comp utationally
intractable. The use of restricted active space is more computa-
tionally amenable, but is challenging to users
23
as there is no
generally applicable strategy of defining truncated active spaces
for a well-balanced multiconfiguration reference wave function.
While a rigorously size-consistent multireference analogue of
coupled cluster theory, especially coupled cluster including singles,
doubles, and perturbative treatment of triples [CCSD(T)], has
not yet been implemented or fully developed,
2426
the well-
developed and readily available MRCI (or MRACPF
27
and
MRAQCC
28
) and CASPT2 approaches are generally not suita-
ble for accurate predictions of dissociation energies and related
quantities such as enthalpies of formation due to the size-
consistency error
29,30
that increases with the system size. Because
of the above-mentioned difficulties with MR methods, single
reference wave function-based electronic structure theory, parti-
cularly CCSD(T), remains a potential method of choice for
accurate transition metal thermochemistry.
The scaling order of computational cost increases with the
electron excitation level of the employed theory; for example, the
cost of MP2 and CCSD(T) scales as N
5
and N
7
, respectively,
where N is related to t he size of the molecule. Composite
approaches can attain accuracy comparable to CCSD(T), cou-
pled cluster theory with excitation levels higher than triple, or full
CI with a very large basis set or extrapolated complete basis
set (CBS) limit, at only a small fraction of computational cost.
Our laboratory has created a composite method, the correlation
consistent Composite Approach (ccCA),
3133
that achieves
“chemical accuracy”, where energies and thermochemical prop-
erties of main group-containing species are reliably computed to
within 1.0 kcal mol
1
of experimental values on average, but the
cost is comparable to only a single point energy evaluation by
CCSD(T) with a triple-ζ quality basis set or MP2 with a quadruple-
ζ quality basis set. One distinguishable feature of ccCA is that it is an
MP2-based model chemistry, but no empirical corrections are in-
cluded in the energy computations. Consequently, the errors origi-
nate only intrinsically from the utilized theories, allowing unbiased
performance of ccCA for molecules outside the benchmark. Also, it
is expected that ccCA can serve as a pan-periodic model chemistry
that is, in principl e, applicable for all elements without regard to
the block in which they reside in the periodic table, that is, s-block
for alkali and alkaline metals,
33
p-block for group IIIAVIIIA
elements,
3134
and d-block for transition metals.
7,8
Because of the aforementioned issues, the development of an
accurate model chemistry for transition metal-containing species
has lagged behind their main group counterparts. This was
primarily due to significant advances in d-block basis sets
5,35
occurring over the last 5 years. Our research group
7,8
pioneered
the development of ccCA for computing energies and thermo-
chemical properties of a variety of 3d-containing molecules to be
within 3.0 kcal mol
1
of experimental values on average, using
the correlation consistent basis sets developed by Peterson and
coauthors.
35
In earlier work, we have coined the term “transition
metal chemical accuracy” to describe a mean absolute deviation
(MAD) of 3.0 kcal mol
1
or better for transition metal species.
7
This targeted accuracy is larger than for energetics of main group
species because greater uncertainties are common in the experi-
mental data for transition metal compo unds and greater errors
are expected with theory due to a number of factors including
increased valence electron space, stronger relativistic effects, and
increased complexity of metal ligand bonding. In a recent
study,
8
the earlier ccCA algorithm was modified to address more
effectively the corevalence correlation and scalar relativistic
effects. This modified ccCA algorithm for transition metal
chemistry is called ccCA-TM. Using ccCA-TM, a MAD of 2.85
kcal mol
1
was achieved for a set of 52 molecules, which
represents a variety of metalligand bonding and includes
species ranging from diatomics to transition metal complexes
with organic ligands. Unlike DFT methods,
36
ccCA-TM per-
forms consistently for all 10 3d metals without drastic variation.
Similar accuracy was also obtained for a subset of 20 molecules in
a recent study by Mayhall et al.
10
using the Gaussian 4(MP2)-tm
model chemistry. Additionally, Di xon and co-workers have
developed a normalized clustering energy approach
37,38
for small
clusters and have performed a series of theoretical studies of
bond energies and enthalpies of formation for oxides, clusters,
and other oxo-compounds of group IVB and VIB metals.
3740
Landis et al.
41
have also investigated bond enthalpies of a large set
of d-block transition metal compounds.
Despite the successes achieved recently, many problems in the
development of transition metal model chemistries remain to be
solved. Given the diversity of the 3d metal-containing com-
pounds, the sets are usually limited or heavily biased. For
example, the 20-molecule set employed by Mayhall et al.
10
does
not contain any molecules with more than four non-hydrogen
atoms, and the important class of organometallics is essen-
tially excluded. The systematic studies by Dixon and co-
workers
3740
are limited to oxo-compounds and clusters, and
most of these molecules do not have an experimentally deter-
mined enthalpy of formation. The relatively larger set of 58
molecules in the ccCA-TM study
8
is still not comprehensive in
that only four molecules are presented for Sc and V each, and not
all homologous compounds such as MX
n
(M = transition metal,
X = F, Cl, Br, O, and n = 1, 2, 3, etc.) for which experimental data
are known have been included. In contrast to most studies that
focus on specific types of transition metal compounds, some
DFT studies considered quite a few types of molecules such as
hydrides, halides, oxides, coordinate compounds, and metal
dimers. Furche and Perdew
12
investigated the thermochemical
properties of 74 species and systematically compared the per-
formance of DFT methods for homologous compounds. Riley
and Merz
36
studied the quality of a variety of DFT methods in the
prediction of ΔH
f
’s for 95 species. As compared to DFT
methods, the performance of parameter-free ccCA-TM is gen-
erally less dependent on transition metals or bonding in mol-
ecules. Nonetheless, a more extensive set is still helpful for
evaluating the overall quality of single reference ccCA-TM
against experimental data, and for statistical assessment of the
applicability, as well as limitations, of ccCA-TM for various
subcategories of transition metal molecules. To this end, 225
enthalpies of formation (referred to as the ccCA-TM/11all set)
are collected from available thermochemistry compendia and
journal literature, regardless of the magnitude of experimental
uncertainties. It is impossible to be exhaustive in considering all
molecules for which gaseous ΔH
f
values have been experimentally

872 dx.doi.org/10.1021/jp205710e |J. Phys. Chem. A 2012, 116, 870–885
The Journal of Physical Chemistry A
ARTICLE
reported. We have attempted to include (before removal of
outliers, vide infra) all existent and computationally tractable 3d
metal gas-phase enthalpies of formation from the current litera-
ture, JANAF tables, NIST webbook, and Yungman compendia.
This large collection of test molecules should provide a more
statistically meaningful analysis of ab initio methods than any sets
used in previous transition metal model chemistry studies. Given
the diversity of the molecules in the set, a number of ccCA-TM
predicted enthalpies of formation are found to significantly deviate
from reported experimental data. Possible causes for the outliers,
for example, the large uncertainty of experimental data and the
inappropriateness of single reference methods as indicated by the
T
1
/D
1
diagnostics
42,43
and the leading CI coefficient (C
0
2
)inCAS
CI wave functions, are discussed metal by metal. Unlikethe various
main group test sets, the only selection criterion for our overall
ccCA-TM/11all test set is tractability with the ccCA methodology.
As a result, the presence of outlier molecules will drastically change
the statistical results. For example, the metal dimer Fe
2
,whichis
notoriously difficult to study with single and multireference ab
initio methods, has a ccCA-TM deviation as large as 47 kcal mol
1
.
This statistically masks the possible advantages of applying ccCA-
TM toward determining the electronic structure of new iron-
containing molecules. To minimize masking effects on the statis-
tical results, the outliers are excluded from the set of 225 entries
to give a subset of 193 quantities (referred to as ccCA-TM/11),
still large enough for statistically significant analysis. The pre-
sence of large deviations but small coupled cluster diagnostics,
known sev ere multireference character, or ambiguous experi-
mental data (vide infra) is the criterion for removing species from
the ccCA-TM/11 subset.
The ccCA-TM/11 set of 193 molecules were then divided into
different subsets on the basis of ranges of experimental uncer-
tainties, bonding types, and size of molecules. Several extrapola-
tion schemes for complete basis set (CBS) limits were compared
for the ccCA-TM/11 set. The T
1
and D
1
coupled cluster
diagnostics,
42,43
along with C
0
2
from viable CASCI calculations,
are further discussed to a priori estimate the validity of single
reference computations for the overall set, as well as different
bonding types.
’ COMPUTATIONAL METHODS
The general ccCA-TM formulation has been described previ-
ously.
8
Here, it is briefly recapitulated with emphasis on the
possible variations of ccCA-TM. The equilibrium geometries are
optimized with B3LYP
4447
and the cc-pVTZ basis set. The total
ccCA-TM energy of a molecular species is calculated by
EðccCA TMÞ¼E
0
ðccCA TMÞþΔEðCCÞþΔEðCVÞ
þ ΔEðZPEÞþΔEðSOÞ ð1Þ
where E
0
(ccCA-TM) is the reference MP2 energy, ΔE(CC) is
the correction to the dynamic correlations not sufficiently
recovered by the MP2 method, ΔE(CV) is the correction for
the corecore and corevalence interactions, ΔE(ZPE) in-
cludes the zero point energy scaled by 0.989
48
and thermal
corrections to 298.15 K, and ΔE(SO) is the spinorbit coupling
correction, which is only calculated for smaller sized molecules
with the FOCI Stuttgart ECP
49
method. The scalar relativistic
correction is considered throughout this study by using the one-
electron DouglasKrollHess Hamiltonian
5052
and the DK
correlation consistent basis sets
35
in all single point energy
computations. The atomic energies are calculated in the same
fashion as molecules except that the spinorbit corrections are
taken from experimental data.
1,3
The ROHF - and UHF-ccCA-
TM atomic energies and the additive contributions of ccCA
components, as well as the experimental values of atomic
enthalpies of formation, are given in Tables S1 and S2.
The reference energy E
0
(ccCA-TM) is the extrapolate d
complete basis set (CBS) limit for MP2 with the aug-cc-pVnZ-
DK (n = 2 or D, 3 or T, and 4 or Q )
35
series of basis sets [aug-cc-
V(n+d)Z-DK for Si, P, S, and Cl].
53,54
A two-point extrapolation
of HF energies with TZ and QZ basis sets has proven to be very
effective and is adopted in the ccCA algorithm s.
55,56
EðnÞ¼EðCBSÞþA expð1:63nÞð2Þ
The dynamic correlation energy can be either extrapolated with
the mixed Gaussian formula by Peterson, Woon, and Dunning,
57
denoted as “P” in this study:
EðnÞ¼EðCBSÞþA exp½ðn  1Þ þ B exp½ðn  1Þ
2

ð3Þ
or the inverse cubic power of l
max
, the highest angular momen-
tum used in the basis set functions, by Schwartz,
58,59
by Halkier
et al.,
60
and by Helgaker et al.
61
denoted as “S3”:
EðnÞ¼EðCBSÞþAl
max
3
ð4Þ
or the inverse quartic power of l
max
+ 1/2 by Kutzelnigg et al.
62
and by Martin and co-worker,
63,64
denoted as “S4”:
EðnÞ¼EðCBSÞþAðl
max
þ 1=2Þ
4
ð5Þ
The value of l
max
is equal to the cardinal number n for main group
elements, denoted as “TQ”, and n + 1 for 3d metal elements,
denoted as “Q5” in this Article. Preliminary results show that
using Q5 for transition metal atoms and transition metal-contain-
ing molecules, while using TQ for main group atoms, results in a
bias of the main group atomic energies that are not consistent
with molecular energies. It is proposed that l
max
should be equal
to either n (TQ ) or n + 1 (Q5) in an individual ccCA-TM
calculation. A compromise was also made to replace l
max
by
(n+1/2), and two more extrapolations can be defined as
EðnÞ¼EðCBSÞþAðn þ 1=2Þ
3
ð6Þ
EðnÞ¼EðCBSÞþAðn þ 1Þ
4
ð7Þ
These two additional extrapolations are denoted as “S3h ” and
“S4h”, respectively. An averaged value of the P and S3 extrapola-
tions,
48
denoted as “PS3”, is also investigated. Different CBS
extrapolations of MP2 energies are summarized in Table 1.
The correlation correction ΔE(CC) is the energy difference
between CCSD(T) and MP2 with the same cc-pVTZ-DK
basis set.
ΔEðCCÞ¼E½CCSDðTÞ=cc pVTZ DK
 E½MP2=cc pVTZ DK ð8Þ
In the main group ccCA algorithm, MP2 is used for the
corecore and corevalence correlation corrections. However,
our earlier study revealed that MP2 is not sufficient for recovering
the corecore and corevalence correlation corrections for
larger transition metal compounds.
8
Significant improvements
in accuracy were found when CCSD(T) is applied in place of
MP2. The corecore and corevalence correlation corrections
were found less dependent on basis sets than the level of theory.
8

873 dx.doi.org/10.1021/jp205710e |J. Phys. Chem. A 2012, 116, 870–885
The Journal of Physical Chemistry A
ARTICLE
To reduce the computational cost, the aug-cc-pCVDZ-DK basis
set is used instead of aug-cc-pCVTZ-DK.
ΔEðCVÞ¼E½CCSDðT, FC1Þ=aug  ccpCVDZDK
 E½CCSDðTÞ=augccpCVDZDK ð9Þ
The notation “FC1” indicates tha t the inner shell closest to the
valence shell is included as active, which means, in addition to
valence electrons, 1s electrons are correlated for LiNe, 2s2p
electrons correlate d for NaAr, 3s3p electrons correlated for
KZn including 3d transition metals, and 3s3p3d electrons
correlated for GaKr. The aug-cc-pCVDZ-DK basis sets are
generated by adding the core/valence basis set functions to aug-
cc-pVDZ-DK without further optimization. All ccCA-TM var-
iants and their corresponding notations are detailed in Table 1.
The spin orbit coupling corrections
65
were calculated for
molecules when applicable and tractable. Spinorbit interac-
tions were approximately considered as the energy difference
between the lowest LS state and the state averaged, where the
energies of each LS state were obtained by diagonalizing the
effective spinorbit Hamiltonian on the basis of contracted
CASSCF wave functions.
In this study, all CASSCF computations are performed with
the cc-pVTZ-DK basis set, and T
1
/D
1
diagnostics and spin
contamination are extracted from CCSD/cc-pVTZ-DK calcula-
tions on the basis of HF or ROHF canonical orbitals. All
computations were performed using MOLPRO 2006.1
66
except
for UHF ccCA-TM energies, which were obtained in Gaussian
03.
67
In Gaussian 03 DKH calculations, the nuclei are simulated
as point charges so that the scalar relativistic energies for closed-
shell systems are equivalent to those computed in MOLPRO.
68
When the experimental enthalpies of formation from various
sources are in disagreement, the experimental value with the least
experimental uncertainty is usually selected for the molecules.
Values from recent literature are adopted when the experimental
uncertainties are comparable to earlier data. We note that
occasionally experimental data reported with a large uncer-
tainty are found in better agreement with theoretical prediction,
for example, the standard enthalpy of CrO
3
as discussed in ref 8.
No attempts were generally made to select the experimental data
in better agreement with theoretical predictions. Decisions made
on the adopted experimental value of specific cases (and the rare
exceptions) are explicitly detailed below, and a full listing of all
experimental values located in the literature is provided in Tables
S3 and S4 of the Supporting Information.
’ RESULTS AND DISCUSSION
A. Effects of CBS Extrapolation Schemes. The Peterson
extrapolation (eq 3) and its averaged PS3 extrapolation with S3
(eq 4) for correlation energies have been used previously in the
CCSD(T) CBS
69,70
and main group ccCA studies.
48
Although it
is inconclusive that on e extrapolation is generally superior to
others for the CCSD(T) CBS limits, it is beneficial to assess the
effectiveness of the possible schemes (Table 1). Consistent with
our earlier observations,
71
utilization of the ROHF reference
wave function for open-shell molecules gives substantially better
results than a UHF reference wave function by minimizing the
effect of spin contamination. Only ROHF-ccCA-TM results are
discussed below. For the ccCA-TM/11 set, the differences
among different extrapolations are less than 0.2 kcal mol
1
,
except for S3(Q5), which has a MAD of 4.72 kcal mol
1
, 0.37
kcal mol
1
greater than the ROHF-ccCA-TM-P MAD of 4.35
kcal mol
1
(Figure 1). For subsets of different experimental
uncertainty ranges, the S3(Q5) MAD is 0.4 1.1 kcal mol
1
larger than the best CBS extrapolation scheme. The S4(TQ ) and
PS3(TQ ) schemes remain close to ccCA-TM-P with a differ-
ence of less than 0.05 kcal mol
1
in MAD for all subsets as well as
the overall set, and thus can be considered equivalent alterna-
tives. As the size of set increases, the CBS extrapolations choice
has less impact on the accuracy on average. Different patterns are
found in mean signed deviation (MSD) of various extrapolation
schemes. While the P, S4(TQ ) variants have negative MSDs for
all subsets, S3(TQ ), PS3(TQ ), S4h, S4(Q5), and PS3(Q5)
show a change from positive MSD to negative MSD as the subset
size increases, and S3(Q5) and S3h have positive MSDs for all
subsets. Similar to the fin dings for CCSD(T) CBS energies,
69,70
historically preferred extrapolations do not show statistically
significant differences. In the following, only the ROHF -ccCA-
TM results with the mixed Gaussian/inverse exponential form
(“P”, eq 3) are discussed. “ROHF” is omitted unless specified
otherwise.
B. 3d Transition Metal-Containing Species and Metal
Dimers. The ccCA-TM algorithm has been applied to calculate
the standard ΔH
f
’s for the ccCA-TM/11all set of 225 transition
metal species, about 4 times larger than the set in our previous
study.
8
Although numerous theoretical studies have been per-
formed on some of the species in our test set, and the thermo-
dynamic properties and electronic structure of their ground state
have been well-established (for examples, see a comprehensive
review of the electronic structure of diatomic 3d block molecules
by Harrison
72
), there are also quite a few species for which little
information exists on the equilibrium geometry, spin multiplicity,
Table 1. Different Variants of the ccCA-TM Algorithm
a
cc-CA-TM
geometry optimization B3LYP/cc-pVTZ
ZPE experimental values for atoms
B3LYP/cc-pVTZ
harmonic frequencies scaled by 0.989
for molecules
HF/CBS HF/aug-cc-pVTZ-DK
HF/aug-cc-pVQZ-DK
E(n)=E(HF/CBS) + A exp(1.63n)
MP2/CBS MP2/aug-cc-pVDZ-DK
MP2/aug-cc-pVTZ-DK
MP2/aug-cc-pVQZ-DK
“P”:eq3
b
“S3”:eq4
b
“S4”:eq5
b
“PS3”: 1/2 (“P” + “S3”)
b
“S3h”:eq6
“S4h”:eq7
correlation corrections CCSD(T)/cc-pVTZ-DK - MP2/cc-pVTZ-DK
corevalence corrections CCSD(T,FC1)/aug-cc-pCVDZ
 CCSD(T)/aug-cc-pCVDZ
spinorbit corrections experimental values for atoms when applicable
FOCI Stuttgart ECP for molecules
when applicable
a
Both ROHF and UHF references can be considered for open-shell
molecules.
b
l
max
= n for “TQ” or (n + 1) for “Q5”.

874 dx.doi.org/10.1021/jp205710e |J. Phys. Chem. A 2012, 116, 870–885
The Journal of Physical Chemistry A
ARTICLE
and/or electronic configuration. This study may be useful in
addressing knowledge gaps in the electronic structure for many
small inorganic molecules, which can further guide experimental
and theoretical research. The complete listings of ccCA-TM
ground electronic states and experimental ground states (when
known) are also given in Tables S3 and S4 of the Supporting
Information. The deviations of the ccCA- TM predictions from
experimental ΔH
f
’s (calculated as experimental value minus
theoretical value) are collected in Table S5. The overall perfor-
mance of ROHF-ccCA-TM, by utilizing ROHF, RMP2,
73
and
R/UCCSD(T)
74
methods, is better than the UHF counterpart,
by utilizing UHF, UMP2, and UCCSD(T)
74
methods, for open-
shell mo lecular and atomic energies.
8
However, it has been found
that for most molecules, the ΔH
f
predictions by both methods
agree well with each other, even though both may be in close
agreement with or deviate significantly from experimental data
(Table S5). Single reference methods such as CCSD(T) and
MP2 suffer from the inaccurate or incorrect wave function
description of the ground electronic state of molecular systems
when the multireference character is prominent. As transition
metal compounds are more likely to involve complicated bond-
ing with quasidegenerate orbitals and to harbor a wealth of low-
lying electronic states, intrinsic errors rising from the multi-
reference character in the ground electronic state become more
prevalent than m olecules containing main group elements only.
To assess the reliability of the single reference coupled cluster
methods in this respect, the weights of leading configurations
(C
0
2
), T
1
/D
1
diagnostics,
42,43,75
and spin contamination
(ÆS
2
S
z
2
S
z
æ) are collected in Table 2 for challenging mol-
ecules. Given the complexity of the electronic structure of the
transition metal containing molecules, in the following paragraphs,
the diagnostics are analyzed and used in determining whether large
deviations (or outliers) of theoretical prediction with respect to
experimental data originate from inappropriateness of single
reference methods or large uncertainty of experiments. Even
though the diagnostics should be interpreted qualitatively, stringent
requirements have been recommended, especially for accurate
study of main group species. However, the size of test set will
be seriously restricted even if loose requirements of T
1
< 0.05,
D
1
< 0.10, and C
0
2
> 0.90 are reinforced in screening the
molecules. Thus, the diagnostics are rather used together with
other information in determining possible outliers. The cal-
culations of C
0
2
are restr ict ed due to the ex pone nti ally
increasing computational cost with increasing CASSCF active
space size, and, as such, only molecules of up to three atoms
and few tetratomics of high symmetry were considered. For
larger sized molecules, only the T
1
/ D
1
diagnostics are dis-
cussed. Another restriction of the C
0
2
diagnosticsisthat
excited configuratio n(s) including virtual orbitals may be
important for the electronic ground-state wave function, for
example, the 4p orbital of the transition metal atom. In the
following paragraphs, the species in the ccCA-TM/11all set
will be discussed metal by metal, followed by a separate
description of metal dimers.
1. Sc. In our ccCA-TM/11all set, 16 scandium-containing
compounds are included. The single reference ccCA-TM calcu-
lation for ScF
2
is expected to be reliable on the basis of the
diagnostics by the C
0
2
, T
1
, and D
1
values of 0.975, 0.020, and
0.046, respectively, and t
1
and t
2
amplitudes of less than 0.05. The
theoretical prediction (158.2 kcal mol
1
) is within the Yangman
value (157.4 ( 7.0 kcal mol
1
) but slightly deviates from the
experimental value (163.7 ( 5.3 kcal mol
1
) obtained by
Hildenbrand and Lau.
76
The ccCA-TM prediction (297.1 kcal
mol
1
) for ScF
3
agrees well with the value (300.4 ( 3.6 kcal
mol
1
) by Hildenbrand and Lau
76
but outside of the error bar
of ref 1 (302.9 ( 3.2 kcal mol
1
). In this study, the experi-
mental data from ref 76 are adopted for both ScF
2
and ScF
3
for
consistency.
The ΔH
f
’s by Gingerich for TiN and VN were found to be
810 kcal mol
1
lower than more recent experimental data.
1
This is somewhat consistent with our ccCA-TM prediction for
ScN, whi ch is 27.9 kcal mol
1
higher than the Gingerich
experimental ΔH
f
. Single reference results are considered
acceptable for ScN on the basis of the diagnostic requirements
for appropriate utility of single r eference methods for transition
metal-containing molecules. Experimentally, ScB
2
has been
mostly investigated in its solid form
77
and to our knowledge
has not been characterized in the gas phase. We observe that a
cyclic geometry with
2
A
1
symmetry is more stable than a linear
geometry of BSc B. However, neither geometry gives a ΔH
f
value as low as the experimental value.
1
Multireference char-
acter of ScB
2
, as evidenced by the diagnostics, may undermine
the reliability of MP2/CCSD-based methods. An e quilibrium
bent geometry was predicted for Sc
2
O, resulting in a ccCA-TM
ΔH
f
of 14.6 kcal mol
1
, which is within the experimental value
of 2.81 ( 18 kcal mol
1
by Kordis and Gingerich.
78
The
ccCA-TM ΔH
f
of ScBr
3
is 17.8 kcal mol
1
higher than the
experimental value.
79
However, the reliability of the ccCA-TM
value is supported by the small T
1
and D
1
diagnostics, and good
agreement with experime nt is found for the lighter analogues
ScF
3
and ScCl
3
, with deviations of 3.3 and +4.4 kcal mol
1
,
respectively. The large discre pancy between experiment and
theory raises que stion to the experimental ΔH
f
value for ScBr
3
.
Figure 1. The mean ccCA-TM deviations in kcal mol
1
from the
experimental enthalpies of formation for subsets with different ranges of
experimental uncertainties: (a) MADs; (b) MSDs. The numbers of
quantities are given in parentheses.