Simplification of the CBS-QB3 Method
for Predicting Gas-Phase Deprotonation
Free Energies
RODRIGO CASASNOVAS, JUAN FRAU,
JOAQUÍN ORTEGA-CASTRO, ANTONI SALVÀ, JOSEFA DONOSO,
FRANCISCO MUÑOZ
Institut Universitari d’Investigació en Ciències de la Salut (IUNICS), Departament de Química,
Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain
Received 27 November 2008; accepted 4 February 2009
Published online 3 June 2009 in Wiley InterScience (www.interscience.wiley.com).
DOI 10.1002/qua.22170
ABSTRACT: Simplified versions of CBS-QB3 model chemistry were used to calculate
the free energies of 36 deprotonation reactions in the gas phase. The best such version,
S9, excluded coupled cluster calculation [CCSD(T)], and empirical (⌬Eemp) and spinorbit (⌬Eint) correction terms. The mean absolute deviation and root mean square thus
obtained (viz. 1.24 and 1.56 kcal/mol, respectively) were very-close to those provided
by the original CBS-QB3 method (1.19 and 1.52 kcal/mol, respectively). The highaccuracy of the proposed simplification and its computational expeditiousness make it
an excellent choice for energy calculations on gas-phase deprotonation reactions in
complex systems. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem 110: 323–330, 2010
Key words: deprotonation free energy; CBS-QB3; pKa
Introduction
T
hermodynamic parameters are highly useful
for the analysis and characterization of chemical reactions. This has fostered the development of
improved theoretical methods for their prediction.
Recently, the best pKa predictions were shown to be
Correspondence to: J. Frau; e-mail: juan.frau@uib.es
Contract grant sponsor: Spanish Government.
Contract grant number: CTQ2008-02207/BQU.
Contract grant sponsor: Balearic Government.
Contract grant number: PROGECIB-28A.
International Journal of Quantum Chemistry, Vol 110, 323–330 (2010)
© 2009 Wiley Periodicals, Inc.
provided by theoretical methods using thermodynamic cycles involving a combined gas-phase deprotonation reaction and solvation– desolvation reactions [1– 8].
Based on Scheme 1, the theoretical pKa for the
process can be calculated from the following equations:
0
⌬Gaq
pK a ⫽
2.303RT
(1)
0
0
⌬G aq
⫽ ⌬Ggas
⫹ ⌬⌬Gsolv
(2)
CASASNOVAS ET AL.
SCHEME 1. Example of a thermodynamic cycle combining a gas-phase deprotonation reaction with solvation of the species involved.
0
where ⌬Ggas
is the gas-phase deprotonation free
energy and ⌬⌬Gsolv is the solvation free energy
difference between the species involved.
The two main problems that affect the accuracy
of this procedure are that the solution phase calculations on ionic species are often not accurate and
the demanding time of model chemistry calculations to calculate gas-phase deprotonation free energies.
The solvation free energies of neutral species can
be experimentally determined with a precision of
0.1 kcal/mol but such a precision is only 2–5 kcal/
mol for charged species. Because theoretical solvation methods are parametrized with reference to
experimental data, the smallest possible errors are
comparable to the experimental error [9].
Shields and coworkers [1– 4, 10, 11] have used
high-level ab initio methods including G2, G3, CBSAPNO, CBS-QB3, W1, and CCSD(T) to obtain very
0
good estimates of ⌬Ggas
. However, these methods
are computationally expensive and completely unfeasible for calculations on relatively large molecules. This led the previous authors to use the density functional theory to reduce computational costs
at the expense of precision [12].
The goal of this work is to get an accurate prediction of the gas-phase deprotonation free energy
with a lower computational cost than traditional
model chemistry calculations. This achievement
will allow applying this methodology in complex
systems. We chose the CBS-QB3 method [13] as
reference and used various simplified versions of
this method to analyze how the elimination of some
steps of the original CBS-QB3 affects the accuracy
and the computational cost.
Theory
The CBS-QB3 method encompasses the so-called
complete basis set model chemistries originally de-
veloped by Peterson and coworkers. CBS models
involve low-level (SCF and ZPE) calculations on
large basis sets, mid-sized sets for second-order
corrections, and small sets for high-level corrections. They include extrapolation to the complete
basis sets to correct Møller–Plesset second-order
energies in addition to empirical and spin-orbit interaction corrections [13–20]. Specifically, the CBSQB3 method involves the following steps:
i. B3LYP/6-311G(2d,d,p) geometry optimization.
ii. B3LYP/6-311G(2d,d,p) frequencies with a
0.99 scale factor for the ZPE.
iii. UMP2/6-311⫹G(3d2f,2df,2p) energy and CBS
extrapolation.
iv. MP4(SDQ)/6-31⫹G(d(f),p) energy.
v. CCSD(T)/6-31⫹G† energy.
Finally, the total and free energy are calculated
from the following:
E CBS-QB3 ⫽ EMP2 ⫹ ⌬EMP4 ⫹ ⌬ECCSD(T) ⫹ ⌬EZPE ⫹ ⌬ECBS
⫹ ⌬Eemp ⫹ ⌬Eint (3)
G CBS-QB3 ⫽ EMP2 ⫹ ⌬EMP4 ⫹ ⌬ECCSD(T) ⫹ ⌬Ethermalcorrection
⫹ ⌬ECBS ⫹ ⌬Eemp ⫹ ⌬Eint (4)
where ⌬ECBS is the term correcting the basis set
truncation error in the second-order energies, and
the energy terms ⌬EMP4, ⌬ECCSD(T), ⌬Eemp, and
⌬Eint are calculated from the following respective
equations:
⌬E MP4 ⫽ EMP4(SDQ)/6-31⫹G(d(f),p) ⫺ EMP2/6-31⫹G(d(f),p) (5)
⌬E CCSD(T) ⫽ ECCSD(T)/6-31⫹G† ⫺ EMP4(SDQ)/6-31⫹G† (6)
冘 冘 C 兲 兩S兩
n␤ Nvirt⫹1
⌬E emp ⫽ ⫺ 0.00579 共
␮ii
2
2
ii
(7)
i⫽1 ␮⫽1
⌬E int ⫽ ⫺ 0.00954关具S2典 ⫺ Sz共Sz ⫺ 1兲兴
(8)
In model chemistries, the highest-order calculation and specific basis sets used not only determine
the precision of the calculated final energy but also
the computational cost, and hence the largest possible system size that can be addressed [14].
Obtaining theoretical pKa values precise to
within 0.05– 0.1 units requires a precision of ⬃10⫺1
324 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
DOI 10.1002/qua
VOL. 110, NO. 2
SIMPLIFICATION OF THE CBS-QB3 METHOD
kcal/mol in the corresponding aqueous phase
deprotonation free energies. Also, for the com0
bined error in ⌬⌬Gsolv and ⌬Ggas
to be acceptable,
the individual error in the later should be less than
10⫺1 kcal/mol. Taking into account the experimental gas-phase deprotonation free energy values (i.e.,
from 150 to 400 kcal/mol in the considered reactions) and the acceptable error considered, the calculated gas-phase deprotonation free energy
should involve at least the 99.9 (i.e., from 99.93 to
99.97%) of the exact energy.
In a sufficiently large basis set, the HF wave
function is able to account for the 99% approximately of the total energy, the remaining 1% is,
however, very important for describing chemical
phenomena as pKa values and can be reach through
one o more corrections with post HF methods. The
MP2 method is known to provide 80 –90% of the
correlation energy (i.e., 0.8 – 0.9% of the total energy); the MP3 method 90 –95% of the electronic
correlation energy; and full MP4 calculations
(SDTQ) as much as 95–98% [21]. Accordingly, MP2
is the minimum computational level required to
obtain precise enough free energies of deprotonation, but high-order computations may be required
in some cases.
Computational Methodology
The studied set comprised 64 molecules containing 1– 8 heavy atoms involved in a total of 36
deprotonation reactions. The studied species included inorganic and organic acids and bases encompassing 18 different functional groups. Some
were potentially capable of delocalizing charge by
aromatic or nonaromatic resonance.
Computations were done by using Gaussian 03
[22], which implements the CBS-QB3 method. All
structures were optimized; also, all exhibited no
imaginary frequencies and were thus true energy
minima.
Comparing the calculated deprotonation free energies with their experimental counterparts entailed considering the free energy of the proton in
the gas phase, Ggas(H⫹) (Scheme 1). Since the electronic energy of the proton is zero, H(H⫹) can be
obtained by combining its translational energy (E ⫽
3/2RT) with PV ⫽ RT. The resulting value at 298 K
is 1.48 kcal/mol. The entropy, S(H⫹), can be calculated from the Sackur-Tetrode equation for monoatomic species in the gas phase [23]; thus, TS ⫽
VOL. 110, NO. 2
DOI 10.1002/qua
⫺7.76 kcal/mol at 298 K and 1 atm. Therefore,
Ggas(H⫹) is ⫺6.28 kcal/mol under these conditions.
The calculated Ggas, ⌬EMP2, ⌬EMP4, ⌬ECCSD(T),
⌬ECCSD, ⌬ECBS, ⌬Eemp, and ⌬Eint values for acids
and conjugated bases, and Ggas(H⫹), allowed us to
obtain the free energies of deprotonation with each
proposed simplification and compare them with
their experimental counterparts. Analyzing the contribution of the different terms to the calculated
deprotonation free energy (GCBS-QB3), we notice
that the MP2 is the most important contribution
(0.3 ⫾ 0.1%) and this contribution should not be
neglected. The contribution of MP4(SDQ) and
CCSD(T) correction terms to GCBS-QB3 energy are
0.014 ⫾ 0.011% and 0.009 ⫾ 0.005%, respectively.
The second contribution to the calculated deprotonation free energy is the term correcting the basis
set truncation error in the second-order energies
(0.032 ⫾ 0.012%). Finally, empiric and spin-orbit
contributions are similar (0.016 ⫾ 0.007% and
0.011 ⫾ 0.004%, respectively).
The simplified versions of CBS-QB3 studied here
were as follows: S1 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬Eemp; S2 ⫽
⌬GCBS-QB3 ⫺ ⌬⌬Eint; S3 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬ECBS;
S4 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬Eemp ⫺ ⌬⌬Eint; S5 ⫽ ⌬GCBS-QB3
⫺ ⌬⌬Eemp ⫺ ⌬⌬Eint ⫺ ⌬⌬ECBS; S6 ⫽ ⌬GCBS-QB3 ⫺
⌬⌬ECCSD(T) ⫹ ⌬⌬ECCSD; S7 ⫽ ⌬GCBS-QB3 ⫺
⌬⌬ECCSD(T); S8 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬Eemp ⫺ ⌬⌬Eint ⫺
⌬⌬ECCSD(T) ⫹ ⌬⌬ECCSD; S9 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬Eemp
⫺ ⌬⌬Eint ⫺ ⌬⌬ECCSD(T); S10 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬Eemp
⫺ ⌬⌬Eint ⫺ ⌬⌬ECBS ⫺ ⌬⌬ECCSD(T) ⫹ ⌬⌬ECCSD;
S11 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬Eemp ⫺ ⌬⌬Eint ⫺ ⌬⌬ECBS ⫺
⌬⌬ECCSD(T); S12 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬ECCSD(T) ⫺
⌬⌬EMP4; S13 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬Eemp ⫺ ⌬⌬Eint ⫺
⌬⌬ECCSD(T) ⫺ ⌬⌬EMP4; S14 ⫽ ⌬GCBS-QB3 ⫺ ⌬⌬Eemp
⫺ ⌬⌬Eint ⫺ ⌬⌬ECBS ⫺ ⌬⌬ECCSD(T) ⫺ ⌬⌬EMP4; S15 ⫽
⌬GCBS-QB3 ⫺ ⌬⌬Eemp ⫺ ⌬⌬Eint ⫺ ⌬⌬ECBS ⫺
⌬⌬ECCSD(T) ⫺ ⌬⌬EMP4) ⫺ ⌬⌬EMP2. In the previous
expressions, ⌬⌬Ex denotes the difference between
the energy corrections for the conjugate base and
acid, x being the correction order [MP2, MP4(SDQ),
CCSD(T), etc.]. We used the difference between the
energy corrections ⌬⌬ECCSD(T) and ⌬⌬ECCSD to examine the contribution of triple perturbative excitations in coupled cluster calculations.
Results and Discussion
Table I shows the experimental free energies of
deprotonation for the studied reactions and the differences resulting from each simplification. As can
be seen, the errors obtained with simplifications
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 325
326 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
DOI 10.1002/qua
a
H2O
OH⫺
NH3
NH2⫺
F⫺
Cl⫺
Br⫺
HS⫺
CH3⫺
CH3OH
CH3O⫺
HO2⫺
CH3NH2
CH3NH⫺
CN⫺
HCC⫺
H2CCH⫺
H2NNH2
HCOO⫺
EtOH
EtO⫺
NO2⫺
DMSO
DMSO⫺
Cl3C⫺
C3H6O
C3H5O⫺
NO3⫺
CH3COO⫺
C5H5N
C6H5S⫺
C6H5NH2
C6H5NH⫺
C6H5O⫺
m-NH2C6H4O
p-NH2C6H4O
Base
157.7⫾0.7
383.7⫾0.2
195.7⫾2.0
396.9⫾0.4
365.5⫾0.2
328.1⫾0.2
318.4⫾0.2
344.9⫾1.2
408.6⫾0.9
173.2⫾2.0
375.0⫾1.8
368.6⫾0.6
206.6⫾2.0
395.7⫾0.7
343.7⫾0.3
370.0⫾1.8
401.0⫾0.5
196.6⫾2.0
338.3⫾1.5
178.0⫾2.0
371.3⫾1.1
333.7⫾0.3
204.0⫾2.0
366.8⫾2.0
349.7⫾2.0
186.9⫾2.0
362.2⫾2.0
317.8⫾0.2
341.4⫾2.0
214.7⫾2.0
333.8⫾2.0
203.3⫾2.0
359.1⫾2.0
342.9⫾1.3
344.3⫾2.0
345.6⫾2.0
⌬Gexpa
0.50
⫺1.71
⫺0.26
⫺0.49
⫺1.53
1.26
0.16
0.52
⫺2.23
1.35
⫺0.39
0.21
⫺0.79
⫺0.26
0.59
1.13
0.22
⫺4.68
1.75
0.80
⫺0.37
1.59
3.44
⫺2.19
⫺1.04
0.65
2.16
1.69
1.80
0.74
1.34
⫺0.51
⫺1.48
1.07
1.62
0.35
0.53
⫺1.58
⫺0.13
⫺0.15
⫺1.63
1.30
0.17
0.73
⫺1.77
1.40
⫺0.13
0.31
⫺0.64
0.16
0.75
1.49
0.68
⫺4.50
1.85
0.84
⫺0.05
1.60
3.54
⫺1.79
⫺0.44
0.76
2.49
1.58
1.94
0.94
1.67
⫺0.38
⫺1.06
1.35
1.90
0.64
0.89
⫺1.01
0.04
0.35
⫺1.17
1.93
0.55
1.36
⫺1.43
1.79
0.39
0.77
⫺0.46
0.60
0.98
1.64
0.88
⫺4.29
2.36
1.23
0.43
2.21
4.11
⫺1.54
⫺0.21
1.12
2.83
2.24
2.45
1.21
2.14
⫺0.28
⫺0.86
1.67
2.26
0.81
⫺0.88
⫺3.61
⫺0.98
⫺2.39
⫺3.19
⫺0.32
⫺0.06
⫺1.13
⫺3.72
⫺0.21
⫺2.36
⫺1.35
⫺1.61
⫺2.11
⫺0.38
0.39
⫺0.96
⫺5.59
⫺0.04
⫺0.74
⫺2.22
⫺0.52
1.62
⫺3.75
⫺2.43
⫺0.75
0.39
⫺0.40
0.01
⫺0.54
⫺0.13
⫺1.13
⫺2.50
⫺0.15
0.35
⫺0.36
0.92
⫺0.88
0.17
0.70
⫺1.27
1.97
0.56
1.56
⫺0.97
1.84
0.65
0.87
⫺0.32
1.02
1.14
2.00
1.34
⫺4.11
2.46
1.27
0.75
2.23
4.21
⫺1.14
0.39
1.24
3.16
2.14
2.59
1.41
2.47
⫺0.15
⫺0.43
1.96
2.55
1.10
⫺0.45
⫺2.78
⫺0.56
⫺1.21
⫺2.93
0.39
0.33
⫺0.09
⫺2.47
0.28
⫺1.32
⫺0.70
⫺1.14
⫺0.83
0.18
1.26
0.17
⫺5.02
0.67
⫺0.27
⫺1.09
0.12
2.39
⫺2.70
⫺1.00
⫺0.16
1.39
0.05
0.80
0.13
1.01
⫺0.78
⫺1.45
0.73
1.28
0.39
0.20
⫺3.15
⫺0.60
⫺2.03
⫺2.78
0.96
0.15
0.13
⫺3.56
0.77
⫺2.17
⫺1.47
⫺1.35
⫺2.09
⫺0.27
⫺0.36
⫺1.34
⫺5.34
0.20
0.25
⫺2.57
⫺0.18
1.72
⫺3.88
⫺3.61
⫺0.54
0.26
1.61
0.12
⫺0.43
⫺0.07
⫺1.28
⫺3.83
⫺1.22
⫺0.54
⫺1.90
0.34
⫺2.15
⫺0.48
⫺1.65
⫺1.51
0.99
0.29
0.22
⫺3.48
0.95
⫺1.39
⫺1.12
⫺1.22
⫺1.76
0.09
⫺0.04
⫺1.26
⫺5.25
0.87
0.42
⫺1.66
0.49
2.84
⫺3.69
⫺3.34
0.19
0.52
1.36
0.74
⫺0.31
0.46
⫺0.99
⫺2.99
0.15
0.71
⫺0.63
0.62
⫺2.32
⫺0.18
⫺0.84
⫺2.51
1.67
0.55
1.17
⫺2.31
1.26
⫺1.13
⫺0.81
⫺0.88
⫺0.81
0.29
0.51
⫺0.22
⫺4.77
0.91
0.72
⫺1.45
0.46
2.50
⫺2.84
⫺2.18
0.05
1.27
2.06
0.91
0.24
1.07
⫺0.92
⫺2.78
⫺0.33
0.39
⫺1.15
0.76
⫺1.32
⫺0.06
⫺0.47
⫺1.25
1.70
0.69
1.26
⫺2.22
1.44
⫺0.36
⫺0.46
⫺0.75
⫺0.48
0.64
0.83
⫺0.14
⫺4.68
1.58
0.89
⫺0.53
1.13
3.61
⫺2.64
⫺1.91
0.78
1.52
1.81
1.53
0.36
1.59
⫺0.64
⫺1.94
1.04
1.64
0.12
⫺0.75
⫺4.22
⫺0.90
⫺2.74
⫺4.17
0.09
0.32
⫺0.48
⫺3.80
⫺0.30
⫺3.10
⫺2.37
⫺1.70
⫺2.67
⫺0.68
⫺0.23
⫺1.39
⫺5.68
⫺0.87
⫺0.82
⫺3.29
⫺1.65
0.68
⫺4.40
⫺3.57
⫺1.35
⫺0.51
⫺0.04
⫺0.88
⫺1.04
⫺0.40
⫺1.55
⫺3.80
⫺1.55
⫺0.88
⫺1.87
⫺0.61
⫺3.22
⫺0.78
⫺2.37
⫺2.91
0.12
0.47
⫺0.39
⫺3.71
⫺0.12
⫺2.33
⫺2.03
⫺1.57
⫺2.34
⫺0.32
0.10
⫺1.31
⫺5.60
⫺0.20
⫺0.65
⫺2.38
⫺0.98
1.80
⫺4.20
⫺3.30
⫺0.62
⫺0.25
⫺0.28
⫺0.26
⫺0.92
0.13
⫺1.26
⫺2.96
⫺0.19
0.38
⫺0.59
1.89
1.55
1.07
2.06
1.25
3.77
3.17
3.24
⫺0.69
2.81
1.75
0.71
0.57
1.76
1.51
2.00
1.32
⫺3.48
4.36
2.37
1.62
5.46
6.25
0.65
⫺3.04
3.01
5.47
3.34
4.13
3.03
4.40
1.59
0.48
3.63
3.68
2.33
2.32
2.38
1.50
3.25
1.52
4.48
3.57
4.28
0.57
3.30
2.78
1.37
1.04
3.03
2.06
2.87
2.45
⫺2.92
5.07
2.84
2.74
6.10
7.02
1.70
⫺1.60
3.60
6.47
3.78
4.93
3.70
5.54
1.95
1.53
4.52
4.62
3.08
0.95
0.48
0.77
1.35
⫺0.14
2.90
3.35
2.63
⫺0.93
1.74
0.81
⫺0.19
0.22
1.18
1.09
2.13
1.27
⫺3.83
3.29
1.30
0.90
3.99
5.21
0.14
⫺2.99
2.20
4.70
1.69
3.13
2.42
4.07
1.32
0.51
3.29
3.35
2.36
⫺3.59
⫺10.14
⫺3.87
⫺9.88
⫺6.56
1.25
3.42
⫺1.81
⫺9.46
⫺4.57
⫺8.75
⫺5.61
⫺5.37
⫺10.14
1.15
⫺0.62
⫺5.56
⫺10.84
⫺4.56
⫺5.25
⫺10.32
⫺3.92
⫺5.36
⫺12.15
⫺9.72
⫺7.55
⫺7.15
2.18
⫺5.20
⫺5.59
⫺4.07
⫺4.42
⫺11.06
⫺6.49
⫺6.09
7.08
␦⌬GS0 ␦⌬GS1 ␦⌬GS2 ␦⌬GS3 ␦⌬GS4 ␦⌬GS5 ␦⌬GS6 ␦⌬GS7 ␦⌬GS8 ␦⌬GS9 ␦⌬GS10 ␦⌬GS11 ␦⌬GS12 ␦⌬GS13 ␦⌬GS14 ␦⌬GS15
Experimental values were taken from Ref. [24].
H3O⫹
H2O
NH4⫹
NH3
HF
HCl
HBr
H2S
CH4
CH3OH2⫹
CH3OH
H2O2
CH3NH3⫹
CH3NH2
HCN
HCCH
H2CCH2
H2NNH3⫹
HCOOH
EtOH2⫹
EtOH
HNO2
DMSOH⫹
DMSO
Cl3CH
C3H6OH⫹
C3H6O
HNO3
CH3COOH
C5H5NH⫹
C6H5SH
C6H5NH3⫹
C6H5NH2
C6H5OH
m-NH2C6H4OH
p-NH2C6H4OH
Acid
Experimental gas-phase free energies for the considered deprotonation reactions and tabulated errors from experiment (␦⌬GSX) in kcal/mol.
TABLE I _______________________________________________________________________________________________________________________________
CASASNOVAS ET AL.
VOL. 110, NO. 2
SIMPLIFICATION OF THE CBS-QB3 METHOD
FIGURE 1. Plot of the total computation time of the studied simplifications versus the number of basis functions of
the largest basis set of the model (SCF and MP2). In blue circles, S0 –S5; Purple triangles, S6, S8, and S10; Red
squares, S7, S9, and S11; Yellow rhombi, S12, S13, S14; Green triangles, S15. [Color figure can be viewed in the
online issue, which is available at www.interscience.wiley.com.]
S1–S11 were comparable to those made with the
original CBS-QB3 method (S0). With a few exceptions, the errors made with simplifications S1–S11
were less than the experimental error for 26 of the
36 reactions. Also, the errors for 17 of such reactions
were less than 1 kcal/mol and only those for methane, hydrazine, DMSOH⫹, DMSO, and acetone exceeded 2 kcal/mol.
Figure 1 shows the variation of the computation
time for each simplification with the number of
basis sets in the largest basis set used (SCF and
MP2). The overall times are exclusive of the optimization time as this depends strongly on the particular starting geometry. Also, the computation
time needed to extrapolate the results to the infinite
basis set (⌬ECBS), the empirical correction (⌬Eemp),
and the spin-orbit interaction correction (⌬Eint)
VOL. 110, NO. 2
DOI 10.1002/qua
were taken to be zero since they involved noniterative calculations.
Table II shows the analysis of errors of deprotonation free energies obtained with the original CBSQB3 method (S0) and 15 simplifications studied
(S1–S15). As can be seen, none of the simplifications
exceeded the precision of the original CBS-QB3
method in all statistics at once. In any case, those
excluding the higher-order terms (S1–S11) were
quite precise (mean absolute deviation (MAD) ⬍1.5
kcal/mol and root mean square (RMS) ⬍ 2.0 kcal/
mol). Also, omitting the empirical and spin-orbit
corrections (S1, S2, and S4) resulted in positive errors, whereas excluding CBS extrapolation (S3) led
to negative errors. Therefore, simultaneously excluding the previous three parameters (S5) caused
their associated errors to mutually cancel out. This
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 327
CASASNOVAS ET AL.
TABLE II _____________________________________
0
Analysis of errors in the predicted ⌬Ggas
(in kcal/
mol).
Simplification
S0
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
Mean
deviation
Mean
absolute
deviation
RMS
Standard
deviation
0.19
0.40
0.78
⫺1.22
0.98
⫺0.43
⫺1.12
⫺0.65
⫺0.33
0.14
⫺1.74
⫺1.27
2.19
2.98
1.57
⫺5.69
1.19
1.19
1.40
1.37
1.50
1.07
1.47
1.29
1.25
1.24
1.80
1.43
2.60
3.23
2.02
6.13
1.52
1.52
1.71
1.90
1.80
1.49
1.99
1.76
1.59
1.56
2.32
1.98
2.98
3.59
2.44
6.85
1.53
1.49
1.55
1.48
1.52
1.45
1.67
1.66
1.58
1.58
1.56
1.54
2.04
2.02
1.90
3.87
made simplification S5 less empirical and similarly
precise to the original CBS-QB3 method. However,
excluding these parameters resulted in no substantial reduction in computation time since the slowest
step of the process was the highest-level calculation
[CCSD(T)].
The next step involved excluding triple perturbative excitations in the coupled cluster calculation
(simplifications S6, S8, and S10). This reduced the
computation time by up to 50% for the larger molecules (Fig. 1). Exclusively, omitting triple excitations (S6) provided acceptable precision (RMS ⫽
1.99 kcal/mol). Additionally, omitting the empirical and spin-orbit corrections (S8) increased the
precision to MAD ⫽ 1.25 kcal/mol and RMS ⫽ 1.59
kcal/mol, both of which are similar to the values
provided by the original CBS-QB3 method. Finally,
omitting CBS extrapolation from S8 (S10) led to a
high mean error markedly departing from that for
the unaltered CBS-QB3 method.
Excluding the correction for the coupled cluster
calculation (S7) resulted in very good precision in
the calculated energies of deprotonation (MAD ⫽
1.29 kcal/mol and RMS ⫽ 1.76 kcal/mol). However, the precision was even better (MAD ⫽ 1.24
kcal/mol and RMS ⫽ 1.56 kcal/mol) if the empirical and spin-orbit corrections were additionally
omitted (S9). One advantage of S9 is that it is less
empirical than the parent method. As with S8, omitting CBS extrapolation from S9 detracted from precision (S11), which remained at acceptable levels,
however (MAD ⫽ 1.43 kcal/mol and RMS ⫽ 1.98
kcal/mol).
Excluding fourth-order corrections (⌬EMP4)
alone (S12–S14) or in combination with the secondorder correction (⌬EMP2, S15) in addition to the
highest-level terms [CCSD(T)] reduced the precision to unacceptably low-levels for pKa as per the
previously established requirements. Also, using
simplifications S12–S14 rather than S7, S9, or S11
only reduced the computation time by a small extent (Fig. 1).
0
Figure 2(A) shows the ⌬Ggas
values obtained
with simplification S9 (the best of the simplifications presented) against those provided by the CBSQB3 method. As can be seen, the protonation energies in the gas-phase predicted by S9 were quite
consistent with those of the original CBS-QB3
method throughout the studied range. Figure 2(B)
compares the energy values provided by S9 with
their experimental counterparts. As can be seen, the
fit was quite good: the slope was very close to 1 and
the profiles provided an accurate reproduction of
the general trend. Interestingly, omitting the coupled cluster calculation reduced the computation
time by up to 70% for the larger molecules examined (Fig. 1).
Based on the above-described results, omitting
empirical corrections resulted in substantially improved precision, probably because conjugate acids
and bases are similar species, so such corrections
are unnecessary to calculate relative energies. Also,
omitting spin-orbit interaction corrections had no
adverse affect on the precision of the calculated
deprotonation energies, even though the main correction was made via a UMP2 function—which
might thus contain some spin contamination.
Conclusions
Model chemistries methods allow highly precise
absolute energy values to be obtained from highlevel computations, but involve high-computational costs with large molecular systems. However, chemical problems involving the calculation
of relative energies can be solved by using lowerlevel theoretical methods since the highest-level energy corrections are smaller than the precision required to accurately quantify energy changes.
Moreover, electronic structure and geometric simi-
328 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
DOI 10.1002/qua
VOL. 110, NO. 2
SIMPLIFICATION OF THE CBS-QB3 METHOD
and performing MP2 and MP4(SDQ) calculations
on expanded sets.
ACKNOWLEDGMENTS
The authors wish to thank the management of
the Supercomputational Center of Catalunya
(CESCA) for access to its computer facilities. R.C. is
also grateful to the Spanish Government for award
of a research fellowship.
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FIGURE 2. Plot of the S9 ⌬G predicted energy versus
CBS-QB3 free energy (A) and experimental free energy
(B).
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