Chapter 2
A New Set of Accurate Multi-level Methods Including
Parameterization for Heavy Elements
Abstract
We have developed a new series of multi-coefficient electronic structure methods
that including parameterization for heavy elements. The training set was taken from
our MLSE(Cn)-DFT method with 10 additional atomization energies of Br- and Icontaining molecules (Br2, I2, HI, IBr, HBr, ICl, NOBr CH3I, CH3Br, C2H5I ), and
ionization potentials and electron affinities of Br and I atoms. Several methods have
been developed, we called them MLSE(HAn) methods. The most important new
correction term was SCS-MP2(spin component scaled MP2) correction. The best
method MLSE(HA-1) gave an average mean unsigned error (MUE) 0.58 kcal/mol on
225 thermochemical kinetics data. It also gave average error less than 1.0 kcal/mol for
10 AEs of Br- and I-containing molecules. In comparison, the MLSE(C1)-DFT gave
an MUE of 1.84 kcal/mol on the 10 AEs. The new method MLSE(HA-1) cost
approximately 60% more computer time than our previous MLSE(C1)-DFT method.
The new set of methods is suitable for thermochemical kinetics study on systems
containing the first- and second-row elements as well as on systems containing heavy
halogens.
27
Introduction
Over past thirty years, the advancement of computer processing capabilities has
allowed for more accurate quantum chemical methods to be routinely applied to all
areas of chemical studies. The accuracy of quantum chemical calculation is very
important in comparing data with experiment and to make quantitative predictions .
To approach the high accuracy, one can use very high-level theories, such as
QCISD(T) and CCSD(T), with large basis sets, such as G3Large, aug-cc-pVTZ,
aug-cc-pVQZ, or even larger basis sets. However, the costs of these methods are
prohibitively high except for very small molecules. Finding high accuracy and
economic methods is one of the most important goals in the quantum chemical
calculation research. In the past two decades, this goal had been realized by the
so-called multi-level methods or multi-coefficient methods.1-18 Many of these
methods have achieved the traditional goal, that is achieving the so-called “chemical
accuracy” of ~1 kcal/mol. In the multi-level methods, additive corrections are applied
to a base energy to account for the incomplete treatment of the correlation energies
and the incompleteness of the basis-set sizes. One of the first multi-level methods was
published at 1985 by Truhlar et al.1 Various methods were published in the following
decades, such as Gaussian-n,6-8 CBS,9,10 and Wn methods16-18 and their various
variants. After, a series study by Truhlar et al.3,4,13,14 discovered that by using scaled
energy components in the multi-level methods, higher accuracy and sometimes higher
efficiencies can be achieved. In these so-called multi-coefficient (MC) methods, such
as G3S/3,4 G3SX,15 and those in MCCM/3 suite,4 all the scaling factors for various
energy components were optimized against experimental dates or high-level
theoretical energies. Our group also has developed a series multi-coefficient method,
which are called MLSEn5 and MLSEn+d19 in the last decade. However, different
28
with the Pople-type basis sets used in the MCG3 and G3S methods, our methods was
developed by using Dunning’s correlation consistent basis sets.20-22 Dunning’s
correlation consistent basis sets provide a well-defined hierarchy of basis sets and
they were designed for methods that consider high-level electron correlation and for
basis-set extrapolation. For neutral systems, our MLSE4+d method gives a mean
unsigned error (MUE) of 0.70 kcal/mol for a set of 109 AEs, 0.87 kcal/mol for a set of
38 hydrogen-transfer BHs, and 0.69 for a set of 22 non-hydrogen-transfer BHs. This
is compared with the MCG3/3 results of 1.04, 0.90 and 1.05 kcal/mol, respectively.
Recently, Truhlar et al. found that incorporating hybrid DFT energies into the
multi-coefficient methods would substantially increase the accuracy with only small
increases in the computational cost.23 Thus, we also improved our MLSEn+d
methods by combining hybrid DFT energies, and the resulting methods were called
MLSE-DFT.24 For example, the mean unsigned error (MUE) to a thermochemical
kinetics data set of 169 can be improved form 0.91 kcal/mol using MLSE1+d method
to 0.61 kcal/mol using MLSE-DFT(MLSE-TPSS1KCIS) method. However,
MLSE-DFT methods were also designed for neutral systems. Although MLSE-DFT
method can be applied for most chemical reactions that in the neutral conditions, in
order to be widely applicable, we also developed another set of MLSE(Cn)-DFT
methods25 for charged systems. Augmented with the MP2/aug-cc-pVTZ and
MP4SDQ/cc-pVTZ calculations, the MLSE(Cn)-DFT methods performed well both
on charged and neutral systems. The efficiency and accuracy of MLSE(Cn)-DFT
methods are better than the most existing multi-coefficient methods. The MUE on 211
thermochemical kinetics data was 0.56 kcal/mol using the best method
MLSE(C1)-M06-2X. A simplified method MLSE(C2)-M06-2X can achieve similar
accuracy (MUE =0.59 ) at 54% of the cost. The most economic method
29
MLSE(C3)-B3LYP gave MUE of 0.62 kcal/mol with 36% of cost only. This is
compared with the MUE of 0.73 kcal/mol by the MCG3-MPWB method4,23,24 at cost
similar to that of MLSE(C3)- B3LYP.
However, all the current multi-coefficient methods were designed and tested for
light atoms (from the first- to the third-row elements). There are many important
chemical reactions involving heavier atoms on the fourth and fifth rows of the
periodic table where the most important ones are the heavy halogens (bromine and
iodine). Some of the Pople-type basis sets, that were used in the MCG3 and related
methods do not contain basis functions for Br and I atoms. On the other hand, all the
halogens are now supported by Dunning’s correlation consistent basis sets, in
particular, the aug-cc-pVnZ-pp basis sets. Currently, to our knowledge, there are no
multi-level methods available to calculate accurate thermochemical kinetics data
involving these heavy halogens. Thus, as a further step to extend the applicability of
our methods, in the current study we developed a new set of multi-coefficient
methods, called MLSE(HA-n), which includes parameterization for the heavy
halogens (Br and I).
Recently, a new method called spin-component scaling MP2 (SCS-MP2) was
developed by Grimme et al,27-29 which used different scaling factors to the “same
spin” and “opposite spin” perturbational terms in the MP2 calculation. The “same
spin” energy term consist of long-range, nondynamical effects, and the opposite spin
consist of short-range, dynamical correlation. The SCS concept has recently been
applied in the double-hybrid DFT method DSD-PLYP by Martin et al.30 By using
SCS-MP2 correction term, this new functional achieve very significant improvement
on the accuracy over the previous B2GP-PLYP-D31 functional. For example, the
root-mean square deviation (RMSD) of the W4-08 database can be improved from
30
3.17 kcal/mol using B2GP-PLYP-D functional to 2.66 kcal/mol using the DSD-PLYP
functional.30 The SCS concept has also been extended to higher level ab initio method
such as SCS-MP332-34 and SCS-CCSD.35,36 Compared to 48 CCSD(T)/cc-pVQZ
reaction energies, the SCS-CCSD performed mean absolute deviation of 1.1 kcal/mol
batter than CCSD method of 1.9 kcal/mol.35 Thus, in our most recent development,
we also included the SCS-MP2 type corrections in our new methods. In the original
work of Grimme, the values for cs and co were fix to 6/5 and 1/3, respectively. In our
new methods, the second-order energies (E2) obtained using the cc-pV(D+d)Z,
/cc-pV(T+d)Z, aug-cc-pV(D+d)Z, and aug-cc-pV(T+d)Z basis sets were each
corrected by two scaling factors (for same-spin and opposite-spin components) which
were optimized against our training sets which consist
data.
31
experimental of high-level
Method
To simplify the notation, we first make the following appreviations for
Dunning’s correlation-consistent basis sets:
pdz
cc-pV(D+d)Z
apdz
aug-cc-pV(D+d)Z
ptz
cc-pV(D+d)Z
aptz
aug-cc-p(V+d)Z
The “+d” signifies that an additional set of d functions are added to the original
correlation-consistent basis sets for the second-row elements.37
We used the MLSE(C1)-M06-2X method we developed previously25 as the
starting point for our improved methods:
E(MLSE(C1)-M06-2X) = CWF { E(HF/pdz) +
C2
[E2/pdz] +
C34
[E(MP4SDQ/pdz) – E(MP2/pdz)] +
CQCID
[E(QCISD/pdz) － E(MP4SDQ/pdZ)] +
CQCI
[E(QCISD(T)/pdz) – E(MP4SDQ/pdz)] +
CB1E2
[MP2/ptz – MP2/pdz] +
CHF+
[E(HF/apdz) – E(HF/pdz]) +
CE2
[E2/apdz – E2/pdz] +
CB2E2
[MP2/aptz – MP2/apdz] +
CB1MP4 [E(MP4D/ptz) － E(MP4D/pdz)] } +
(1  CWF ) { E(DFT/pdz) +
CDFT+
[E(DFT/apdz – DFT/pdz] }
(1)
Where E(theory/basis set) denotes the single-point Born-Oppenheimer energy
32
calculated with the combination of the particular theory and basis set, and E2 set
denotes the second-order energy correction calculated at the MP2 theory using the
particular basis set. The DFTX denotes a particular hybrid DFT method with X% of
Hartree-Fock exchange energy. The MP4D theory was used instead of the MP4SDQ
in the CB1MP4 term because our tests showed that the MP4D theory provided better
overall performance at a slightly lower computational cost. The explicit treatment of
the spin-orbital coupling was found unnecessary for this method. For the DFTX terms,
several hybrid DFT methods, including MPW1B95,38,39 TPSS1KCIS,40,41
MPW1PW91,38 B1B95,39,42 M06-2X43 and B3LYP44,45 were tested in the method.
And the M06-2X functional performed best in this method.
The coefficients were determined by minimizing the MUEs with respect to a set
of 211 accurate thermochemical kinetics data (the “training set”) that listed in Table 1.
These data include 109 main-group atomization energies (AEs) from the
MGAE109/05 database,47 38 hydrogen-transfer barrier heights (HTBHs), 38
non-hydrogen-transfer barrier heights (NHTBHs) from the HTBH38/04,24 13
ionization potential (IPs) energies and 13 electron affinity (EAs) energies from the
IP13/3 and EA13/3 databases,47 respectively. The NHTBH38/04 database values
were determined using the W1 theory. All other database values were derived from
reliable experimental measurement.48 These databases were compiled by Truhlar and
coworkers and were previously used to determine the MCG3DFT and MLSEDFT
coefficients.23,24
In the current study, we prepared a new database for training set. The new
database consist of the 211 themochemical energies that used in the
MLSE(Cn)-DFT25 method and 14 new energies including bromine and iodine
elements. The new database called HA-225 database, totally 225 energies. We
33
searched the new data comprehensively in any themochemical database. The
experimental errors of new data we will use should less than 1 kJ/mol or 0.24
kcal/mol, which is necessary for used in our highly accurate methods. However,
molecules containing 4th and 5th rows elements are mostly difficult to vaporize, others
also come with large experimental errors. Therefore, the final data we used are 14
themochemical energies including bromine and iodine elements. These new data are
10 atomization energies of Br and I containing molecules, Br2, I2, HI, IBr, HBr, ICl,
NOBr CH3I, CH3Br, and C2H5I , ionization potentials and electron affinities of Br
and I. The dissociation energies of Br2 and I2, ionization potentials, and electron
affinities of Br and I at 0 K are taken directly from the NIST-JANAF Themochemical
Table.49 The energies of the other eight molecules were derived from the
0
experimental ∆H𝑓,298
values26 by removing the zero-point and thermal energies using
the same treatment in a previous study26. The new data are listed in Table 2.
In the MLSE(Cn)-DFT methods, the structures of training set were obtained
using QCISD/MG3S4,50 method. However, the MG3S basis set does not support for
Br and I atoms. The structures of Br2, I2, HI, IBr, HBr, ICl, NOBr CH3I, CH3Br, and
C2H5I in this study were obtained using QCISD/aug-cc-pVTZ method. According to
our experience, the accuracy did not significantly influence by the structures. A
pseudo-potential (described as -pp) was used to replace the core electrons of I atom,
including cc-pVDZ-pp, cc-pVTZ-pp, aug-cc-pVDZ-pp and aug-cc-pVTZ-pp.
We started by testing our previous MLSE(C1)-M06-2X method on the new
HA-225 database. The correction of spin-orbital coupling (ESO term) was not used in
the original MLSE(C1)-M06-2X method. However, the large spin-orbital coupling
effect on these heavy halogen atoms cannot be ignored. The ESO values are 3.5 and
34
7.2 kcal/mol for Br and I atoms, respectively. Thus, the ESO term was added to the
MLSE(C1)-M06-2X method and the coefficients have also been readjusted against the
same training set. The resulting method is called MLSE(C1)-M06-2X (Eso). Then, we
optimized coefficients of the MLSE(C1)-M06-2X (Eso) method by minimizing the
overall MUE with respect to HA-225 database and the resulting new method is called
MLSE(C1)-M06-2X-HA. However, both MLSE(C1)-M06-2X (Eso) and
MLSE(C1)-M06-2X-HA methods performed unsatisfactorily on the 10 heavy halogen
containing atomization energies. We also tested other DFT functionals in these
methods, but with no significantly improvement.
The SCS-MP2 was a new scaling method that was not used in our previous
methods. Here we improved the MLSE(C1)-M06-2X method by using different
scaling factors to the same spin and opposite spin perturbational terms in the MP2
calculation (cs and co, respectively) , and the resulting method is called the
MLSE(C1S)-M06-2X method. The MLSE(C1S)-M06-2X method was performed on
the original database (211 themochemical energies), for comparison with
MLSE(C1)-M06-2X method, and thus the ESO term was not used in this method.
To test the practical limit in accuracy our MLSE methods can achieve, we also
developed a new set of methods that include the accurate but also expensive
QCISD(T)/aptz energies. The QCISD(T)/aptz method is considered as a reliable
method to calculate relative energies in the most chemical reactions. Here we call the
new methods the “full” type methods because they include all the energies available
obtained after a QCISD(T) calculation using the pdz, apdz, ptz, and aptz basis sets.
Thus, these methods used all the energy components obtained from QCISD(T)/pdz,
QCISD(T)/ptz, QCISD(T)/apdz and QCISD(T)/aptz calculations. Each calculation
35
give 6 energy components that were HF, MP2, MP4D, MP4SDQ, QCISD and
QCISD(T) with a particular basis set. Two methods MLSE(full) and
MLSE(full)-M06-2X has been developed. The MLSE(full) method consisted of 24 ab
initio energies that consisted of HF, MP2, MP4D, MP4SDQ, QCISD and QCISD(T)
methods with pdz, apdz, ptz and aptz basis set. The MLSE(full)-M06-2X method was
base on MLSE(full) method with two additional DFT term, M06-2X/pdz and
M06-2X/apdz. The formula of MLSE(full) and MLSE(full)-M06-2X methods are:
E(MLSE(full) and MLSE(full)-M06-2X) =
CWF
{ E(HF/pdz) +
CHF+
[E(HF/apdz) – E(HF/pdz]) +
CB1HF
[E(HF/ptz) – E(HF/pdz]) +
CB2HF
[E(HF/aptz) – E(HF/ptz]) +
CE2
[E2/pdz] +
CE2
[E2/apdz] +
CB1E2
[E2/ptz] +
CB2E2
[E2/aptz] +
CMP4D
[E(MP4D/pdz) – E(MP2/pdz)] } +
CMP4D+ [E(MP4D/apdz) – E(MP4D/pdz)] } +
CB1MP4D [E(MP4D/ptz) – E(MP4D/pdz)] } +
CB2MP4D [E(MP4D/aptz) – E(MP4D/apdz)] } +
CMP4
[E(MP4SDQ/pdz) – E(MP4D/pdz)] } +
CMP4+
[E(MP4SDQ/apdz) – E(MP4SDQ/pdz)] } +
CB1MP4 [E(MP4MP4SDQ/ptz) – E(MP4SDQ/pdz)] } +
36
CB2MP4 [E(MP4MP4SDQ/aptz) – E(MP4SDQ/apdz)] } +
CQCID
[E(QCISD/pdz) – E(MP4SDQ/pdz)] +
CQCID+ [E(QCISD/apdz) – E(QCISD /pdz)] +
CB1QCID [E(QCISD/ptz) – E(QCISD /pdz)] +
CB2QCID [E(QCISD/aptz) – E(QCISD /apdz)] +
CQCI
[E(QCISD(T)/pdz) – E(QCISD /pdz)] +
CQCI+
[E(QCISD(T)/apdz) – E(QCISD(T) /pdz)] +
CB1QCI [E(QCISD(T)/ptz) – E(QCISD(T) /pdz)] +
CB2QCI [E(QCISD(T)/aptz) – E(QCISD(T) /apdz)] +
(1  CWF)
CDFT+
{ E(M06-2X/pdz) +
[E(M06-2X /apdz –M06-2X /pdz] }
(2)
In the MLSE(full) method, the CWF was fixed to 1, and the DFT terms make no
contribution. The coefficients of MLSE(full) and MLSE(full)-M06-2X method were
determined by minimizing the MUEs with respect to a set of 211 accurate
thermochemical kinetics data (Table 1), the same treatment in MLSE(C1)-M06-2X.
Furthermore, we modified these two methods by using SCS-MP2 method. The
resulting methods are called MLSE(fullS) and MLSE(fullS)-M06-2X. The formula of
MLSE(fullS) and MLSE(fullS)-M06-2X methods are:
E(MLSE(fullS) and MLSE(fullS)-M06-2X) =
CWF
{ E(HF/pdz) +
CHF+
[E(HF/apdz) – E(HF/pdz]) +
CB1HF
[E(HF/ptz) – E(HF/pdz]) +
37
CB2HF
[E(HF/aptz) – E(HF/ptz]) +
CE2S
[(E2aa+E2bb)/pdz] +
CE2O
[(E2ab)/pdz] +
CE2+S
[(E2aa+E2bb)/apdz] +
CE2+O
[(E2ab)/apdz] +
CB1E2S [(E2aa+E2bb)/ptz] +
CB1E2O [(E2ab)/ptz] +
CB2E2S [(E2aa+E2bb)/aptz] +
CB2E2O [(E2ab)/aptz] +
CMP4D
[E(MP4D/pdz) – E(MP2/pdz)] } +
CMP4D+ [E(MP4D/apdz) – E(MP4D/pdz)] } +
CB1MP4D [E(MP4D/ptz) – E(MP4D/pdz)] } +
CB2MP4D [E(MP4D/aptz) – E(MP4D/apdz)] } +
CMP4
[E(MP4SDQ/pdz) – E(MP4D/pdz)] } +
CMP4+
[E(MP4SDQ/apdz) – E(MP4SDQ/pdz)] } +
CB1MP4 [E(MP4MP4SDQ/ptz) – E(MP4SDQ/pdz)] } +
CB2MP4 [E(MP4MP4SDQ/aptz) – E(MP4SDQ/apdz)] } +
CQCID
[E(QCISD/pdz) – E(MP4SDQ/pdz)] +
CQCID+ [E(QCISD/apdz) – E(QCISD /pdz)] +
CB1QCID [E(QCISD/ptz) – E(QCISD /pdz)] +
CB2QCID [E(QCISD/aptz) – E(QCISD /apdz)] +
CQCI
[E(QCISD(T)/pdz) – E(QCISD /pdz)] +
CQCI+
[E(QCISD(T)/apdz) – E(QCISD(T) /pdz)] +
CB1QCI [E(QCISD(T)/ptz) – E(QCISD(T) /pdz)] +
38
CB2QCI [E(QCISD(T)/aptz) – E(QCISD(T) /apdz)] +
(1 - CWF)
CDFT+
{ E(M06-2X/pdz) +
[E(M06-2X /apdz –M06-2X /pdz] }
(3)
where the CWF was fixed to 1 in the MLSE(fullS) method. The E2aa/basis
denotes the alpha-alpha spin perturbational energy correction calculated at the MP2
theory using the particular basis set, and E2ab and E2bb energy terms signify the
alpha-beta spin and beta-beta spin respectively.
The coefficients of above four methods (MLSE(full), MLSE(full)-M06-2X,
MLSE(fullS) and MLSE(fullS)-M06-2X) were determined by minimizing the MUEs
with respect to a set of 211 accurate thermochemical kinetics data (Table 1). In order
to develop new methods for heavy halogens, the coefficients of these four methods
have also been adjusted for the HA-225 database. The simplest modification was the
addition of correction energy of spin-orbital coupling (ESO). The resulting methods
are called MLSE(full) (Eso), MLSE(full)-M06-2X (Eso), MLSE(fullS) (Eso) and
MLSE(fullS)-M06-2X (Eso). The coefficients of these methods were still readjusted
against 211 accurate thermochemical kinetics data (Table 1). Then, we optimized the
coefficients of these methods against to the HA-225 database. The resulting methods
are called MLSE(HA-full), MLSE(HA-full)-M06-2X, MLSE(HA-fullS) and
MLSE(HA-fullS)-M06-2X, with the names including “HA”. A similar method,
MLSE(HA-fullS)-MPW1, which uses the MPW1PW91 functional were also
presented for comparison. The formula for the MLSE(full) (Eso),
MLSE(full)-M06-2X (Eso), MLSE(HA-full) and MLSE(HA-full)-M06-2X methods
are the same as Eq.(2), and the formula for the MLSE(fullS) (Eso),
MLSE(fullS)-M06-2X (Eso), MLSE(HA-fullS) and MLSE(HA-fullS)-M06-2X
methods are the same as Eq.(3).
39
The computational cost of MLSE(HA-fullS)-MPW1 method or similar methods
that including QCISD(T)/aptz calculation are very large and can only be applied to
small systems. The economical, but reasonably accurate methods were needed for
larger systems. We tried various combinations of relatively economical calculations,
such as MP4/apdz, MP4/ptz and QCISD(T)/apdz. The first version we developed in
the current study, called the MLSE(HA-1), was developed by adding the MP4/apdz,
QCISD/apdz, QCISD(T)/apdz and SCS-MP2 correction terms to MLSE(C1)-M06-2X
method.25 MLSE(HA-1) method used all the energy components obtained from
QCISD(T)/pdz, QCISD(T)/apdz, MP4SDQ/ptz and MP2/aptz. The formula of
MLSE(HA-1) method is:
E(MLSE(HA-1)) = CWF { E(HF/pdz) +
CHF+
[E(HF/apdz) – E(HF/pdz]) +
CB1HF
[E(HF/ptz) – E(HF/pdz]) +
CB2HF
[E(HF/aptz) – E(HF/ptz]) +
CE2S
[(E2aa+E2bb)/pdz] +
CE2O
[(E2ab)/pdz] +
CE2+S
[(E2aa+E2bb)/apdz] +
CE2+O
[(E2ab)/apdz] +
CB1E2S [(E2aa+E2bb)/ptz] +
CB1E2O [(E2ab)/ptz] +
CB2E2S [(E2aa+E2bb)/aptz] +
CB2E2O [(E2ab)/aptz] +
CMP4D
[E(MP4D/pdz) – E(MP2/pdz)] } +
40
CMP4D+ [E(MP4D/apdz) – E(MP4D/pdz)] } +
CB1MP4D [E(MP4D/ptz) – E(MP4D/pdz)] } +
CMP4
[E(MP4SDQ/pdz) – E(MP4D/pdz)] } +
CMP4+
[E(MP4SDQ/apdz) – E(MP4SDQ/pdz)] } +
CB1MP4D [E(MP4MP4SDQ/ptz) – E(MP4SDQ/pdz)] } +
CQCID
[E(QCISD/pdz) – E(MP4SDQ/pdz)] +
CQCID+ [E(QCISD/apdz) – E(QCISD/pdz)] +
CQCI
[E(QCISD(T)/pdz) – E(QCISD/pdz)] +
CQCI+
[E(QCISD(T)/apdz) – E(QCISD(T)/pdz)] +
(1 - CWF) { E(DFT/pdz) +
CDFT+
[E(DFT/aptz – DFT/apdz] } + ESO
(4)
E2aa, E2ab and E2bb signify alpha- alpha spin, alpha-beta spin and beta-beta
spin perturbational energy correction terms calculated at the MP2 theory respectively.
The treatment of the spin-orbital coupling is explicit (ESO) for selected open-shell
species.26 For the DFT terms, several hybrid DFT functionals, including
MPW1PW91,38 M06-2X43 and B3LYP44,45 were tested in this method. The
MPW1PW91 functional with the apdz/aptz basis set combination was found to
provide the best results.
The computational cost of MLSE(HA-1) is significantly higher than that of
MLSE(C1)-M06-2X because of the expensive QCISD(T)/apdz and MPW1PW91/aptz
calculations. One way to lower the cost is to use the MP4SDQ/apdz calculation
instead of QCISD(T)/apdz calculation. On the other hand, getting accurate results for
the ten heavy halogen-containing atomization energies is still one of the most
41
important goals in the current study. It turns out that using the MP4SDQ/apdz
calculation instead of QCISD(T)/apdz calculation was difficult to achieve that goal.
Therefore, we gave a weight of three for the MUE of 10 heavy halogens atomization
energies in the training set during optimizing the coefficients for the second method.
The simplified MLSE(HA-2) method is very similar to MLSE(HA-1). We replaced
the QCISD(T)/apdz term with the MP4SDQ/apdz, and replaced the B3LYP term with
the pdz/apdz basis set combination. The formula of MLSE(HA-2) method is:
E(MLSE(HA-2)) = CWF { E(HF/pdz) +
CHF+
[E(HF/apdz) – E(HF/pdz]) +
CB1HF
[E(HF/ptz) – E(HF/pdz]) +
CB2HF
[E(HF/aptz) – E(HF/ptz]) +
CE2S
[(E2aa+E2bb)/pdz] +
CE2O
[(E2ab)/pdz] +
CE2+S
[(E2aa+E2bb)/apdz] +
CE2+O
[(E2ab)/apdz] +
CB1E2S [(E2aa+E2bb)/ptz] +
CB1E2O [(E2ab)/ptz] +
CB2E2S [(E2aa+E2bb)/aptz] +
CB2E2O [(E2ab)/aptz] +
CMP4D
[E(MP4D/pdz) – E(MP2/pdz)] } +
CMP4D+ [E(MP4D/apdz) – E(MP4D/pdz)] } +
CB1MP4D [E(MP4D/ptz) – E(MP4D/pdz)] } +
CMP4
[E(MP4SDQ/pdz) – E(MP4D/pdz)] } +
42
CMP4+
[E(MP4SDQ/apdz) – E(MP4SDQ/pdz)] } +
CB1MP4 [E(MP4MP4SDQ/ptz) – E(MP4SDQ/pdz)] } +
CQCID
[E(QCISD/pdz) – E(MP4SDQ/pdz)] +
CQCI
[E(QCISD(T)/pdz) – E(QCISD/pdz)] +
(1 - CWF)
CDFT+
{ E(DFT/pdz) +
[E(DFT/apdz – DFT/pdz] } + ESO
(5)
The computational cost of MLSE(HA-2) method was just slightly more than the
MLSE(C1)-M06-2X method. The coefficients in Eqs. (45) were determined by
minimizing the MUEs with respect to the HA-225 database that consisted of 225
accurate thermochemical kinetics data, which listed in Table 1 and 2. All the
calculations were performed using the Gaussian 09 program.51
43
Result and Discussion
(a) MLSE(C1S)-M06-2X method
In the MLSE(C1S)-M06-2X method, SCS-MP2 correction did not improve the
accuracy significantly as compared to the MLSE(C1)-M06-2X method. The overall
MUE of MLSE(C1S)-M06-2X method only ~0.01 kcal/mol lower than that of
MLSE(C1)-M06-2X method. The MUEs obtained by the MLSE(C1S)-M06-2X were
listed in Table 3. The results of MLSE(Cn)-DFT methods also listed for comparison.
(b) “full” type methods for 211 thermochemical energies
The MLSE(full) and MLSE(full)-M06-2X methods gave very low overall MUEs
of 0.55 and 0.50 kcal/mol on the training set (211 energies). These two methods
performed best especially for the EAs and NHTBHs, with errors below 0.4 kcal/mol.
Adding DFT terms to MLSE(full), gave small improvement of 0.05 kcal/mol. The
SCS-MP2 correction did not give significant improvement in MLSE(C1S)-M06-2X
method. However, SCS-MP2 played a more important role in the MLSE(fullS) and
MLSE(fullS)-M06-2X methods. Compared to the MLSE(full) method (overall MUE
= 0.55 kcal/mol), the MLSE(fullS) method predicted more accurate atomization
energies and significantly more accurate non-hydrogen-transfer barrier heights. The
overall MUE of MLSE(fullS) method is 0.50 kcal/mol and is ~0.05 kcal/mol less than
that of MLSE(full) method. The added DFT term in the MLSE(fullS)-M06-2X
method also gave a 0.04 kcal/mol improvement on the overall MUE. The overall
MUE of MLSE(fullS)-M06-2X method was 0.46 kcal/mol, that is also the best result
of these “full” type methods. From the results of these “full” methods, it is reasonable
to assume that the best accuracy that can be achieved using this kind methods is ~0.45
kcal/mol. The results of above four methods were also listed in the Table 3.
44
(c) The “full” type methods with ESO term for HA-225 database
For the HA-225 database, the overall MUE of 225 energies and 5 types of energy
components obtained by the various methods in the current study were listed in Table
4. That were the MUE of 119 AEs, 15 IPs, 15EAs, 38 HTBHs and 38 NHTBHs.
Compared to the original training set (211 energies), the added energies were 10
heavy halogen AEs (HHAEs), 2 heavy halogen IPs (HHIPs) and 2 heavy halogen EAs
(HHEAs), the MUEs of added energies (HHAE(10), HHIP(2) and HHEA(2)) were
also listed in this table. The MLSE(C1)-M06-2X method performed poorly on the
HHAE(10) with MUE of of 5.53 kcal/mol. Because the spin-orbital coupling
energiest of the open-shell bromine and iodine atoms were not considered in the
MLSE(C1)-M06-2X method. The MLSE(C1)-M06-2X (ESO) method performed
much better on the HHAE(10) with MUE of 1.84 kcal/mol. The overall MUE of
MLSE(C1)-M06-2X (ESO) method is 0.66 kcal/mol, also better than the
MLSE(C1)-M06-2X method (MUE = 0.80 kcal/mol). The four “full” type methods
MLSE(full), MLSE(full)-M06-2X, MLSE(fullS) and MLSE(fullS)-M06-2X also been
tested for HA-225 database. The error of these four methods for the HHAE(10) were
6.73-8.70 kcal/mol before corrected with spin-orbital coupling energies. After
corrected for the spin-orbital coupling energies, they gave MUE of 1.50-2.71 kcal/mol
on the HHAE(10). The MLSE(fullS)-M06-2X (ESO) method provided the lowest
overall MUE of 0.54 kcal/mol on the HA-225 database. The MLSE(full) (ESO),
MLSE(full)-M06-2X (ESO) and MLSE(fullS) (ESO) method gave overall MUEs of
0.56-0.60 kcal/mol on the HA-225 database.
(d) The re-optimized “full” type methods against HA-225 database
45
Although these methods provided excellent overall MUE on the HA-225
database, the treatment on the HHAE(10), HHIP(2) and HHEA(2) terms was still
unsatisfactory, and four methods gave MUEs of 1.50-2.71 kcal/mol on the HHAE(10).
One possible reason is that most of the molecules in the training set do not contain
heavy halogens. Thus, we re-optimized the coefficients in these methods against the
HA-225 database, as described in the previous section. The results were listed in
Table 5. Five methods MLSE(C1)-M06-2X-HA, MLSE(HA-full),
MLSE(HA-full)-M06-2X, MLSE(HA-fullS) and MLSE(HA-fullS)-M06-2X methods
are presented. The treatment of the spin-orbital coupling (ESO) in these methods is
explicit. The best MLSE(HA-fullS)-M06-2X method gave 0.51 kcal/mol on the
overall MUE(225). Compare to the MLSE(fullS)-M06-2X (Eso) method, the MUE of
10 HHAEs significantly decrease to 1.17 kcal/mol by the MLSE(HA-fullS)-M06-2X
method.
The overall MUE of MLSE(HA-fullS)-MPW1 method has no significant
improvement to the MLSE(HA-fullS)-M06-2X method. However, the MUE of 10
HHAEs decreased from 1.17 kcal/mol by the MLSE(HA-fullS)-M06-2X method to
the 0.95 kcal/mol by the MLSE(HA-fullS)-MPW1 method. We have also tested other
DFT functionals for HHAEs, but the MPW1PW91 performed best. Therefore,
MPW1PW91 functional was used in our new MLSE(HA-1) method.
(e) MLSE(HA-1) and MLSE(HA-2) methods
The MUEs obtained by the current MLSE(HA-1) and MLSE(HA-2) methods
were listed in Table 5. Among the choices of the density functionals, for MLSE(HA-1)
method, the methods using the MPW1PW91 performed best, while for MLSE(HA-2)
method, the method using the B3LYP functional performed best. The MLSE(HA-1)
46
method provided the lowest overall MUE of 0.58 kcal/mol on the training set.
Compared to the MLSE(C1)-M06-2X-HA method, the MLSE(HA-1) method
predicted more accurate HHAE and significantly more accurate HHIP and HHEA.
The MUE of MLSE(HA-1) method on the 119 AEs, 15 IPs, 15EAs, 38 HTBHs and
38 NHTBHs were 0.57, 0.71, 0.79, 0.46 and 0.65 kcal/mol, respectively. It is
remarkable that MLSE(HA-1) method impressively predicted the MUE of 0.87
kcal/mol on the HHAE, even better than the extremely time-consuming
MLSE(HA-fullS)-M06-2X and MLSE(HA-fullS)-MPW1 methods.
The MLSE(HA-2) method, which is a simplified version of the MLSE(HA-1)
method described in the previous section, also gave a satisfactory overall MUE of
0.64 kcal/mol. Compared to the MLSE(HA-1) method, the MLSE(HA-2) method
gave similar MUEs on barrier heights, but it gave higher MUE on atomization
energies. However, the errors in the IPs and EAs are ~0.1 kcal/mol lower than the
MLSE(HA-1) method. Based on the weighting treatment on 10 heavy halogen AEs in
optimizing coefficients, MLSE(HA-2) method also provided MUE of 0.98 kcal/mol
on the HHAE.
In order to further evaluate the new methods developed in the current study, we
tested MLSE(HA-1) method on the total noble-gas bond energy (TNGBE) database
our laboratory developed recently.52 The database consists of 31 energies
corresponding to the Born-Oppenheimer energy of the reaction XNgY  X + Ng + Y.
This is a very harsh test since the bonding in the noble gas molecules is very special,
and none of the energies is included in the training set used in the current study.
The
MLSE(HA-1) method gave an MUE of 2.71 kcal/mol on the TNGBE database. In
comparison, the MP2 and MPW1PW91 methods with aug-cc-pVTZ basis set gave
MUEs of 7.4 and 2.9 kcal/mol, respectively. Thus, the new MLSE(HA-1) method
47
developed in this study seem to be robust enough to treat noble gas with reasonable
accuracy.
(f) Coefficients
The optimized coefficients of the MLSE(HA-1) and MLSE(HA-2) method are
shown in Table 6. For the two methods, the coefficients of the E2S (same spin) are
obviously large than the E2O (opposite spin), the result is consistent with the previous
study by Grimme, where the values for cs and co were fix to 6/5 and 1/3, respectively.
The DFT energy only contributed to approximately 20% of the total energy in both
two methods, that is similar with MLSE(C1)-M06-2X method. The optimized
coefficients of other “full” type methods in Table 4 and Table 5 are also shown in
Table 7-9.
(g) Computational costs
Table 10 compares the computational cost and MUEs from various
multi-coefficient methods in this study and from recently developed other
multi-coefficient methods. The computational cost of MLSE(C1)-M06-2X is equal to
the MLSE(C1)-M06-2X-HA method. MLSE(HA-1) method take ~62% more
computer time than the MLSE(C1)-M06-2X-HA method. But the MUE of HHAE
obtained by MLSE(HA-1) method is remarkably better than the
MLSE(C1)-M06-2X-HA method . The MLSE(HA-1) method also provided slightly
better accuracy on the original 211 themochemical energies by ~0.03 kcal/mol than
the MLSE(C1)-M06-2X-HA method. However, the MLSE(HA-1) method achieved
similar accuracy on HHAE but required only 10% of the computational cost of the
MLSE(HA-fullS)-MPW1 and similar methods. The MLSE(HA-2) method performed
48
0.98 kcal/mol on the HHAE(10) and required only 4% more cost than the
MLSE(C1)-M06-2X-HA method. The MLSE(HA-2) method also gave a satisfactory
overall MUE of 0.64 kcal/mol for HA-225 database, better than the
MLSE(C1)-M06-2X-HA method. The MLSE(HA-2) method is comparable in cost to
the MLSE(C1)-M06-2X-HA method but achieves higher accuracy. Consequently, the
MLSE(HA-2) method is another economical alternative to the MLSE(HA-1) method,
especially for larger systems.
49
Concluding Remarks
In this chapter, we prepared a new database, called HA-225 database. They were
including 211 energies that were used in our previous MLSE(Cn)-DFT method and
with 10 additional atomization energies of Br- and I- containing molecules (Br2, I2,
HI, IBr, HBr, ICl, NOBr CH3I, CH3Br, C2H5I ), and ionization potentials and electron
affinities of Br and I atoms.
Two sets of methods have been developed. In order to test the practical limit in
accuracy our MLSE methods can achieve, we developed a new set of methods that
include the accurate but also expensive QCISD(T)/aptz energies, called the “full”
methods. From the results of these “full” methods, it is reasonable to assume that the
best accuracy that can be achieved using this kind methods is ~0.45 kcal/mol.
In addition, we developed two more economical methods, MLSE(HA-1) and
MLSE(HA-2). The MLSE(HA-1) method gave an average mean unsigned error
(MUE) 0.58 kcal/mol on HA-225 database. It also gave average error less than 1.0
kcal/mol for 10 AEs of Br- and I-containing molecules. In comparison, the
MLSE(C1)-DFT gave an MUE of 1.84 kcal/mol on the 10 AEs. The new method
MLSE(HA-1) cost approximately 60% more computer time than our previous
MLSE(C1)-DFT method. However, the second method, MLSE(HA-2) method,
performed 0.98 kcal/mol on the 10 AEs and required only 4% more cost than the
MLSE(C1)-M06-2X-HA method. The MLSE(HA-2) method also gave a satisfactory
overall MUE of 0.64 kcal/mol for HA-225 database. We expect that the new methods
can easily be applied to many types of interesting chemical systems containing heavy
halogens, and they will be invaluable for accurate study of thermochemistry and
kinetics.
50
Acknowledgements
This work is supported by the National Science Council of
Taiwan, grant number NSC 100-2113-M-194-007. We are grateful to the National
Center for High-Performance Computing (NCHC) for providing part of computation
resources.
51
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Table 1
211 thermochemical kinetics data for the training sets of MLSE(Cn)-DFT methods (kcal/mol)
a. AEs
CH
84.0
H2C2H
445.8
CH3Cl
394.6
CH3SH
473.8
HCOOH
501.0
C2H4O
650.7
OH
107.1
SH
87.0
F2
38.2
HOCl
164.4
NF3
204.5
CH3OCH3
798.1
NO
152.1
C2H5
603.8
N2
228.5
SO2
257.9
PF3
363.9
H3CCH2OH
810.4
CN
180.6
CH3CO
581.6
PH3
242.6
C2Cl4
466.3
BCl3
322.9
C3H4_a
704.8
ClO
64.5
(CH3)3C
1199.3
SC
171.3
C2F4
589.4
BF3
470.0
H3CCOOH
803.0
O2
120.0
(CH3)2CH
900.8
CH4
420.1
C4H4O
993.7
AlCl3
306.3
H3CCOCH3
978.0
SO
125.0
C2H5O
698.6
NH3
297.9
C4H6_d
987.2
SiCl4
384.9
H3CCHCH2
860.6
S2
101.7
FH
141.1
H2O
232.6
C4H6_e
1001.6
AlF3
426.5
C2H5OCH3
1095.1
NH
83.7
ClH
106.5
SH2
182.7
CCl4
312.7
SiF4
574.4
C4H10_h
1303.0
CH2 (T)
190.7
Cl2
58.0
CH2
181.4
CF3CN
639.9
C4H4S
962.7
C4H10_i
1301.3
CH3
307.5
CO2
389.1
SiH2
151.8
CF4
476.3
C4H5N
1071.6
C4H8_j
1149.0
56
(continue)
NH2
181.9
HCN
313.2
C2H2
405.4
CH3CN
615.8
C5H5N
1237.7
C4H8_k
1158.6
SiH2 (T)
131.1
ClF
61.4
C2H6
712.8
CH3NH2
582.6
C3H4_g
682.7
C5H8_l
1284.3
SiH3
227.4
SiH4
322.4
CO
259.3
CH3NO2
601.3
ClF3_2
125.3
C6H6
1367.6
PH2
153.2
H2O2
268.6
H2CO
373.7
CHCl3
343.2
C4H6_b
1012.4
H2CCO
532.3
HCO
278.4
P2
117.1
CH3OH
512.8
CHF3
457.5
C4H6_c
1004.1
C3H4_f
703.2
Si2_3
75.0
SiO
192.1
N2H4
438.6
H2
109.5
HCOCOH
633.4
C3H8
1006.9
C2H
265.1
C2H4
563.5
Si2H6
530.8
HCOOCH3
785.3
CH3CHO
677.0
C3H6
853.4
CH2OH
409.8
57
b. IPs
c. EAs
C
259.7
C
29.1
S
238.9
S
47.9
SH
238.9
SH
53.3
Cl
299.1
Cl
83.4
Cl2
265.3
Cl2
55.6
OH
299.1
OH
42.1
O
313.9
O
33.7
O2
278.9
O2
10.8
P
241.9
P
17.2
PH
234.1
PH
23.2
PH2
226.3
PH2
29.4
S2
216.0
S2
38.5
Si
187.9
Si
31.9
Br
272.4
Br
77.6
I
241.0
I
70.5
58
d. HTBHs
R1FBH
8.7
R10RBH
13.7
R1RBH
5.7
R11FBH
3.1
R2FBH
5.1
R11RBH
24.2
R2RBH
21.2
R12FBH
10.7
R3FBH
12.1
R12RBH
13.1
R3RBH
15.3
R13FBH
3.5
R4FBH
6.7
R13RBH
17.3
R4RBH
19.6
R14FBH
9.8
R5FBH
9.6
R14RBH
10.4
R5RBH
9.6
R15FBH
22.4
R6FBH
3.2
R15RBH
8.0
R6RBH
12.7
R16FBH
18.3
R7FBH
1.7
R16RBH
7.5
R7RBH
7.9
R17FBH
10.4
R8FBH
3.4
R17RBH
17.4
R8RBH
19.9
R18FBH
14.5
R9FBH
1.8
R18RBH
17.8
R9RBH
33.4
R19FBH
38.4
R10FBH
8.1
R19RBH
38.4
59
e. NHTBHs
H + N2O→OH + N2
18.1
HCN→HNC
48.2
REV. BH
83.2
REV. BH
33.1
H + FH→HF+H
42.2
F + CH3F→FCH3 + F
-0.34
REV. BH
42.2
REV. BH
-0.34
H + ClH→HCl +H
18.0
F...CH3F→FCH3 ... F
13.38
REV. BH
18.0
REV. BH
13.38
H + FCH3→HF+ CH3
30.4
Cl + CH3Cl→ClCH3 + Cl
3.1
REV. BH
57.0
REV. BH
3.1
H + F2→HF+F
2.3
Cl...CH3Cl→ClCH3...Cl
13.61
REV. BH
105.3
REV. BH
13.61
CH3 + FCl→CH3F+ Cl
7.4
F + CH3Cl→FCH3 + Cl
-12.54
REV. BH
60.2
REV. BH
20.11
H + N2→HN2
14.7
F...CH3Cl→FCH3...Cl
2.89
REV. BH
10.7
REV. BH
29.62
H + CO→HCO
3.2
OH + CH3F→HOCH3 + F
-2.78
REV. BH
22.7
REV. BH
17.33
H + C2H4→CH3CH2
1.7
OH...CH3F→HOCH3...F
10.96
REV. BH
41.8
REV. BH
47.2
CH3 + C2H4→CH3CH2CH2
6.9
REV. BH
33.0
60
Table 2
New data in the HA-225 database (kcal/mol)
AE
I2
35.87
HI
73.79
IBr
42.27
ICl
50.19
Br2
45.90
HBr
90.51
NOBr
181.64
CH3I
369.12
CH3Br
380.94
C2H5I
662.69
IP
I
241.01
Br
272.43
EA
I
70.54
Br
77.60
61
Table 3
Mean unsigned errors (kcal/mol) for the original 211 energies
IP
EA
MLSE(full)
MLSE(full)-M06-2X
MLSE(fullS)
MLSE(fullS)-M06-2X
0.62
0.62
0.65
0.57
0.57
0.52
0.55
0.54
0.75
0.59
0.77
0.55
0.63
0.48
0.34
0.37
0.36
0.29
0.47
0.5
0.46
0.45
0.46
0.48
0.43
0.43
0.39
0.37
0.31
0.30
0.56
0.55
0.55
0.50
0.50
0.46
MLSE-TSa
MCG3/3b
MCG3-MPWBb
G3SXb
0.62
1.04
0.75
0.85
0.95
0.67
1.07
0.92
0.86
1.06
0.55
0.84
0.54
0.67
0.69
1.00
0.84
0.60
0.61
0.98
0.73
0.80
MLSE(C1)-M06-2X
MLSE(C1S)-M06-2X
aFor
neutral systems only.
bObtained from ref. [4,24,53].
62
HTBH NHTBH
Overall
MUE
AE
Table 4
Mean unsigned errors (kcal/mol) of several methods for the HA-225 database
AE(119) IP(15) EA(15) HTBH(38) NHTBH(38) MUE(225) HHAE(10)
MLSE(C1)-M06-2X
1.03
0.55
0.97
0.47
0.44
0.80
5.53
MLSE(C1)-M06-2X (Eso)
0.74
0.68
0.82
0.52
0.49
0.66
1.84
HHIP(2)
HHEA(2)
0.61
1.22
3.07
2.21
MLSE(full)
MLSE(full) (Eso)
MLSE(full)-M06-2X
MLSE(full)-M06-2X (Eso)
MLSE(fullS)
MLSE(fullS) (Eso)
MLSE(fullS)-M06-2X
1.30
0.70
1.25
0.75
1.23
0.65
1.04
0.84
0.72
0.63
0.51
0.85
0.70
0.61
1.24
0.81
0.98
0.69
1.24
0.79
1.00
0.46
0.45
0.45
0.41
0.46
0.44
0.48
0.39
0.42
0.37
0.38
0.31
0.37
0.30
0.95
0.60
0.89
0.60
0.90
0.56
0.77
8.43
1.64
8.70
2.71
8.51
1.50
6.73
1.47
0.62
0.90
0.43
1.41
0.55
0.96
4.42
1.36
3.55
1.79
4.32
1.16
3.90
MLSE(fullS)-M06-2X (Eso)
0.65
0.52
0.68
0.44
0.32
0.54
1.85
0.37
1.59
63
Table 5
Mean unsigned errors (kcal/mol) of two new methods and several related methods for the HA-225 database
AE(119) IP(15)
EA(15) HTBH(38) NHTBH(38) MUE(225) HHAE(10)
MLSE(C1)-M06-2X-HA
0.72
0.68
0.86
0.54
0.49
0.66
1.66
HHIP(2)
HHEA(2)
2.30
1.44
1.64
1.16
1.53
1.06
1.07
1.04
MLSE(HA-full)
0.65
0.78
0.88
0.47
0.44
0.59
1.35
1.21
0.64
MLSE(HA-full)-M06-2X
MLSE(HA-fullS)
MLSE(HA-fullS)-M06-2X
MLSE(HA-fullS)-MPW1
MLSE(HA-1)
MLSE(HA-2)
0.61
0.65
0.58
0.57
0.57
0.65
0.59
0.71
0.51
0.70
0.71
0.61
0.75
0.79
0.66
0.77
0.79
0.68
0.45
0.44
0.47
0.40
0.46
0.49
0.44
0.36
0.33
0.41
0.65
0.69
0.55
0.56
0.51
0.51
0.58
0.64
1.17
1.38
1.17
0.95
0.87
0.98
0.47
0.54
0.35
0.48
0.49
0.48
64
Table 6
Optimized coefficients of the MLSE(HA-n) methods
MLSE(HA-1)
MLSE(HA-2)
CWF
CHF+
CB1HF
CB2HF
CE2S
CE2O
CE2+S
CE2+O
0.786928
1.459069
0.589013
-1.005416
1.957262
0.314312
-2.013948
0.422561
0.829308
1.069346
1.325074
-1.108107
1.951836
0.431388
-2.370384
0.201030
CB1E2S
CB1E2O
CB2E2S
-1.876080
0.659927
2.686884
-1.722345
0.385396
3.004871
CB2E2O
CMP4D
CMP4D+
CB1MP4D
CMP4
CMP4+
-0.276516
0.597872
-1.813549
2.685780
0.904651
0.095762
0.025068
0.685108
-0.661122
1.773237
0.705375
0.575352
CB1MP4
CQCID
CQCID+
CQCI
CQCI+
CDFT+
-1.619272
1.447285
-0.922835
0.667287
2.295483
3.333852
-0.616310
1.302820
0.898955
1.379385
65
Table 7
Optimized coefficients of the MLSE(full) and related methods
MLSE(full)
MLSE(full)
(Eso)
MLSE(full)
-M06-2X
MLSE(full)
-M06-2X
(Eso)
CWF
CHF+
CB1HF
CB2HF
CE2
CE2+
1
1.037829
0.493843
-1.126519
1.003176
0.000245
1
1.070880
0.520758
-0.823689
0.819161
-0.137225
0.870925
0.650580
0.675219
-1.140078
1.072170
0.126935
0.878519
0.541834
0.528563
-0.675190
0.991038
-0.147319
CB1E2
CB2E2
CMP4D
-0.844154
-0.844154
0.829441
-0.761518
-0.761518
1.049052
-0.875105
0.677745
0.899126
1.037470
-0.887498
0.892608
CMP4D+
CB1MP4D
CB2MP4D
CMP4
CMP4+
CB1MP4
-0.108677
0.571477
-0.168946
1.010427
-0.544262
0.475003
0.204636
0.341153
-0.160930
1.007510
-0.826130
0.076156
-0.109476
0.801761
-0.183723
1.039671
-0.245242
0.721351
0.320543
0.894578
-0.138093
0.973623
-0.639717
0.271239
CB2MP4
CQCID
CQCID+
CB1QCID
CB2QCID
CQCI
CQCI+
CB1QCI
CB2QCI
CDFT+
-2.391712
1.111177
3.031734
-0.068526
0.043619
1.171872
-2.883487
0.770581
2.740099
-2.704330
1.081755
2.809705
-0.265190
0.017438
1.197801
-2.691017
1.296029
3.012421
-2.154875
1.061348
2.828960
-0.355317
0.127422
1.352000
-2.919135
0.731192
2.313722
2.874405
-2.821769
1.110156
2.539067
-0.627058
0.108841
1.141174
-2.642808
1.018256
3.089372
2.602210
66
Table 8
Optimized coefficients of the MLSE(fullS) and related methods
MLSE(fullS)
MLSE(fullS)
(Eso)
MLSE(fullS)
-M06-2X
MLSE(fullS)
-M06-2X (Eso)
CWF
CHF+
CB1HF
CB2HF
CE2S
CE2O
CE2+S
1
0.883401
0.703184
-0.913475
1.382839
0.596671
-0.275642
1
0.612080
0.698557
-0.600850
1.951113
0.423185
-1.470842
0.883179
0.416649
0.674666
-0.733691
2.131216
0.440984
-1.306096
0.879725
0.418571
0.649687
-0.592647
2.041226
0.436953
-1.762987
CE2+O
CB1E2S
CB1E2O
0.032581
-1.173935
-0.305314
-0.066521
-2.123908
0.331396
0.268186
-2.924380
0.368213
0.262996
-2.580876
0.248282
CB2E2S
CB2E2O
CMP4D
CMP4D+
CB1MP4D
CB2MP4D
0.996106
0.645705
0.830052
0.048356
-0.161826
-0.285691
2.535541
0.286811
0.814974
0.150407
0.009809
-0.147972
2.962292
-0.075049
0.816386
0.323032
-0.302436
-0.239473
3.134308
0.072893
0.826959
0.371982
0.066923
-0.125712
CMP4
CMP4+
CB1MP4
CB2MP4
CQCID
CQCID+
CB1QCID
CB2QCID
CQCI
CQCI+
0.900704
-0.111807
-0.057520
-2.251636
1.039860
2.813779
-0.554685
0.052197
1.209001
-3.023357
0.879761
0.073514
-0.748149
-2.118263
1.066183
2.299143
-0.837736
-0.027041
1.104021
-2.488856
0.947535
0.124038
0.034873
-2.162555
1.052334
2.465793
-0.767259
0.014413
1.199952
-3.034517
0.927961
0.018081
-0.350364
-2.452403
1.085032
2.306737
-1.010055
0.065826
1.126697
-2.872572
CB1QCI
CB2QCI
CDFT+
2.200251
2.521614
2.618276
2.439316
2.413633
2.509004
2.220362
2.595462
2.767826
2.263888
67
Table 9
Optimized coefficients of the MLSE(HA-full), MLSE(HA-fullS) and related methods
MLSE
(HA-full)
MLSE
(HA-full)
-M06-2X
MLSE
(HA-fullS)
MLSE
(HA-fullS)
-M06-2X
MLSE
(HA-fullS)
-MPW1
1
1.266598
0.254205
-0.728768
0.898200
0.980842
0.486492
-0.711772
1
0.650849
0.710478
-0.619468
0.875028
0.565250
0.823825
-0.673838
0.847181
0.805937
0.567011
-0.712333
0.579623
0.584063
1.910358
1.870446
2.315592
0.390446
0.361555
0.176064
0.298863
0.258466
-1.384052
-1.648329
-2.132401
-0.005331
0.196881
0.675334
0.609823
0.582610
-2.076050
-2.413309
-2.192308
0.382770
0.385576
0.488093
-0.525417
-0.448739
2.436055
3.007931
2.750775
0.209358
0.066486
-0.256456
CMP4D
CMP4D+
CB1MP4D
0.909915
-0.166268
0.720742
0.946587
0.011374
0.960885
0.814464
0.104189
-0.014380
0.833752
0.139293
0.305951
0.740696
-0.390761
0.882816
CB2MP4D
CMP4
CMP4+
CB1MP4
CB2MP4
CQCID
CQCID+
CB1QCID
CB2QCID
CQCI
-0.187253
1.069257
-1.046107
-0.247101
-2.333597
1.021001
2.919915
-0.547524
0.013474
1.300890
-0.153381
1.156688
-0.863085
-0.305178
-2.448408
1.004261
2.701222
-0.743730
0.023410
1.305167
-0.150273
0.887121
0.085833
-0.828743
-2.080947
1.059439
2.280464
-0.769102
-0.036968
1.126684
-0.154377
0.991543
0.046032
-0.466325
-2.472926
1.050220
2.631438
-1.504552
0.019759
1.167994
-0.080397
0.740535
0.747883
-1.203428
-2.219091
1.177723
1.722150
-1.774898
-0.010091
0.810339
CQCI+
CB1QCI
CB2QCI
CDFT+
-2.641468
1.706159
2.902419
-2.632944
1.612230
2.897620
2.797512
-2.459164
2.659502
2.405594
-2.956299
2.843523
2.698097
2.323216
-2.187778
3.159057
2.522848
2.936087
CWF
CHF+
CB1HF
CB2HF
CE2S
CE2O
CE2+S
CE2+O
CB1E2S
CB1E2O
CB2E2S
CB2E2O
68
Table 10
Computational costa
costb (sec.) MUE(211) MUE(225) HHAE(10)
MLSE(C1)-M06-2X-HA
MCG3-MPWB
MCG3/3
MLSE(HA-fullS)
MLSE(HA-fullS)-M06-2X
MLSE(HA-fullS)-MPW1
MLSE(HA-1)
MLSE(HA-2)
1276
514
392
19271
19323
19599
2071
1324
aDetermined
0.59
0.73c
0.98c
0.51
0.47
0.49
0.56
0.62
0.66
N/A
N/A
0.59
0.51
0.51
0.58
0.64
1.66
N/A
N/A
1.38
1.17
0.95
0.87
0.98
using computers with an Intel i7 2600K CPU at 4.5GHz and a version of
Gaussian 03 program optimized for the hardware architecture.
b Total CPU time obtained from the calculation for C H N, C Cl , C H O, C H S,
5 5
2 4 4 4
4 4
C4H5N, CF3CN, and SiCl4.
cObtained from ref. [4,24].
69