Regular Article PHYSICAL
CHEMISTRY
RESEARCH
Published by the
Iranian Chemical Society
www.physchemres.org
info@physchemres.org
Phys. Chem. Res., Vol. 1, No. 2, 104-110, December 2013.
Does One-Third Scheme of PBE0 Functional Dominate Over PBE0 for Electronic
Properties of Transition Metal Compounds?
M. Alipour*
Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71454, Iran
(Received 27 April 2013, Accepted 27 May 2013)
The one-third paradigm of PBE0 density functional, PBE0-1/3, has shown to be a successful method for various properties of
molecules containing main group elements. In this paper, the applicability of PBE0-1/3 is put into broader perspective for transition metals
chemistry. As a comparative study, the performance of PBE0 and PBE0-1/3 has been assessed for geometries and vibrational frequencies
of some transition metal hydrides and molecules containing transition metals, and static dipole polarizabilities and dipole moments for
transition metal halides. The numerical data show that although PBE0-1/3 performs better than the parent PBE0 for response properties of
small molecules, it does not approach the quality of PBE0 for structural parameters. Overall, the results of this investigation suggest that
there is no real incentive to use PBE0-1/3 in place of PBE0 for calculations involving transition metals.
Keywords: DFT, PBE0, PBE0-1/3, Transition-metal, Polarizability
1
INTRODUCTION E xc U xc , d (1)
0
where
The computational simplicity and reasonable accuracy
of Kohn-Sham density functional theory (DFT) [1-3] has led 1 n r n r (2)
U xc , Vˆee dr dr
to its current status as the most frequently applied method 2 r r
for a wide range of systems and properties. The story behind
the success of DFT is the search for the exchange- and is the wave function that minimizes
correlation (XC) functional. Therefore, finding new accurate Tˆ Vˆee under the constraint of producing the given
XC functionals is of paramount importance in DFT. density n r ,
Currently, computational chemists have a wide variety of Min Tˆ Vˆee (3)
n r
density functionals at their disposal, ranging from local
density approximation (LDA) to double-hybrid (DH)
A first-principle rationale of the HF and KS mixing has
density functionals. In the present work, the class of density been given by Perdew, Ernzerhof, and Burke [5]. Later on,
functionals under study is the hybrid functionals. Adamo and Barone [6] and, in parallel, Ernzerhof and
In 1993, Becke [4] presented the combination of the Scuseria [7] defined the PBE0 model as a parameter-free
generalized gradient approximation (GGA) and the Hartree- hybrid functional,
Fock (HF) exchange in a single XC scheme. This idea was
further developed to create a lineage of functionals, hybrid E xc E xcPBE a 0 E xHF E xPBE (4)
functionals, which are rooted in the adiabatic connection
formula, in which a0 1 / 4 . This model is one of the widely used
functionals and gives good performances for a wide variety
*Corresponding author. E-mail: malipour@shirazu.ac.ir of systems [8-12].
Alipour/Phys. Chem. Res., Vol. 1, No. 2, 104-110, December 2013.
In general, a0 = n-1 is considered as the lowest order of the COMPUTATIONAL DETAILS
perturbation theory providing a realistic description of the
systems under investigation. However, since for atomization The systems and molecular properties investigated in
energies of the molecules of the G1 dataset the fourth-order our study included the geometries and vibrational
Møller-Plesset perturbation theory is adequate, n = 4 was frequencies for 3d transition metal hydrides (following
considered in PBE0. On the other hand, Cortona [13] has previous suggestion these molecules provide a sensitive test
shown that other values of the theoretical mixing coefficient for XC functionals [15]) and several molecules containing
a0 can be chosen which actually have the same theoretical transition metals, static dipole polarizabilities for some of
argument. On this basis, in a most recent communication, 4d and 5d transition metal halides, and dipole moments for
Guido et al. [14] proposed the recipe of one-third (a0 = 1/3) 4d transition metal monofluorides and monochlorides for
as a mixing coefficient for the PBE0 functional and which experimental data were available [16-19]. Several
presented the PBE0-1/3 model. Using various benchmarks basis sets were used in our calculations. Geometries and
including atomization energies, weak interactions, vibrational frequencies were computed using Pople’s 6-
hydrogen-bond length optimizations, dissociation energies, 311++G(3df,2pd) basis set [20,21]. In the case of dipole
and vertical excitation energies they have claimed that polarizabilities and dipole moments calculations, Dunning’s
PBE0-1/3 generally performs better than the parent PBE0. aug-cc-pVTZ [22-24] and SDD [25-28] basis sets were used
In light of these findings, we became interested in studying for main group elements and transition metals, respectively.
the performance of PBE0-1/3 for transition metal We employed the Gaussian09 suite of codes [29] for all the
compounds. Accordingly, the main concern of the present runs.
work is a comparative investigation on the performance of
PBE0 and PBE0-1/3 models for some properties of RESULTS AND DISCUSSION
molecules containing transition metals.
The rest of this paper is organized as follows. In the The numerical results for the mentioned properties are
following section, we expose the details of calculations. reported in Tables 1-4. Table 5 summarizes the statistical
Then, a section is provided in which the results and measures for the performance of PBE0 and PBE0-1/3
discussion of the general trends of the benchmark functionals. Figure 1 is the graphical representation of mean
calculations are covered. Finally, we conclude the paper unsigned relative error (MUE) and root mean square relative
highlighting the main inferences in the last section. error (RMSE) computed on different test sets. First, a glance
Table 1. Bond Lengths (r/Å) and Vibrational Frequencies (ω/cm-1) for the First-row Transition Metal
Hydrides (MH) Computed with PBE0 and PBE0-1/3 Models Compared with Experiment
PBE0 PBE0-1/3 Experimenta
MH r ω r ω r ω
ScH 1.748 1631 1.745 1638 1.775 1596
CrH 1.663 1618 1.667 1605 1.662 1615
MnH 1.727 1538 1.731 1528 1.740 1547
FeH 1.554 1752 1.561 1729 1.609 1821
CoH 1.540 1758 1.550 1727 1.513 1927
NiH 1.521 1785 1.532 1748 1.454 2001
CuH 1.485 1855 1.490 1833 1.463 1942
a
Refs. [30-37].
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Does One-Third Scheme of PBE0 Functional Dominate Over PBE0/Phys. Chem. Res., Vol. 1, No. 2, 104-110, December 2013.
Table 2. Bond Lengths (r/Å) for Some First-row Transition Metal Compounds Computed with PBE0 and
PBE0-1/3 Models Compared with Experiment
Molecule (point group) PBE0 PBE0-1/3 Experimenta
CrO (C∞v)
r (C-O) 1.607 1.613 1.615
TiCl4 (T d)
r (Ti-Cl) 2.162 2.157 2.170
CuCN (C∞v)
r (Cu-C) 1.843 1.852 1.832
r (C-N) 1.156 1.152 1.157
FeCO (C∞v)
r (Fe-C) 1.718 1.725 1.727
r (C-O) 1.163 1.160 1.159
VOF3 (C3v)
r (V-O) 1.541 1.531 1.569
r (V-F) 1.715 1.710 1.729
Ni(CO)4 (Td)
r (Ni-C) 1.822 1.822 1.838
r (C-O) 1.132 1.128 1.141
a
Refs. [38-43].
Table 3. Computed Values of the Static Dipole Polarizabilities (au) for Second- and Third-row Transition
Metal Halides Computed with PBE0 and PBE0-1/3 Models Compared with CCSD(T) Results
Molecule PBE0 PBE0-1/3 CCSD(T)
YF 127.55 128.08 126.27
NbF 73.99 74.91 85.03
MoF 55.71 55.77 64.67
TcF 54.64 54.30 54.97
RuF 50.52 50.12 43.60
RhF 27.99 26.56 26.18
PdF 27.21 25.73 24.69
AgF 24.07 22.69 22.54
CdF 39.56 39.34 41.68
YCl 140.05 140.26 137.56
NbCl 83.39 83.63 95.77
MoCl 67.57 66.56 75.39
TcCl 70.14 69.51 70.11
RuCl 69.41 62.06 52.32
RhCl 50.98 49.32 48.71
PdCl 45.02 43.53 42.67
AgCl 41.82 40.45 40.68
CdCl 55.97 55.63 57.76
AuF 28.93 28.04 30.68
HfF 91.35 91.79 82.58
HgF 35.25 35.15 35.82
OsF 44.62 36.95 41.09
ReF 46.77 46.20 47.04
WF 52.85 52.84 59.32
106
Alipour/Phys. Chem. Res., Vol. 1, No. 2, 104-110, December 2013.
Table 4. Computed Values of the Dipole Moments (D) for Some Second-row Transition Metal
Halides Computed with PBE0 and PBE0-1/3 Models Compared with Experiment
Molecule PBE0 PBE0-1/3 Experimenta
YF 1.975 1.981 1.828
AgF 5.984 6.182 6.22
YCl 2.593 2.638 2.587
AgCl 5.798 6.021 6.22
a
Refs. [16-19].
Table 5. Comparison of the Performance of PBE0 and PBE0-1/3 Functionals for the Studied Properties in
This Investigation. The Statistical Descriptors are Mean Unsigned Relative Error (MUE) and
Root Mean Square Relative Error (RMSE).
PBE0 PBE0-1/3
Test sets MUE RMSE MUE RMSE
BLTMHa 0.020 0.024 0.021 0.027
b
BLTM 0.007 0.008 0.008 0.010
VFTMHc 0.044 0.058 0.055 0.069
DP4d-5dTMXd 0.077 0.104 0.064 0.084
e
DM4dTMX 0.047 0.056 0.035 0.046
a
Bond length of 3d transition metal hydrides. bBond length of transition metal molecules. cVibrational
frequency of 3d transition metal hydrides. dDipole polarizability of 4d and 5d transition metal halides.
e
Dipole moment of 4d transition metal halides.
to our statistical analysis (Table 5 and Fig. 1) is sufficient to in PBE0-1/3 does not affect the geometries in this case.
conclude that the two functionals PBE0 and PBE0-1/3 have Usually, including a larger percentage of HF exchange
almost the same accuracy in all cases. However, let us deteriorates the structures, if this effect is not compensated
compare the performance of these functionals in details. by correlation [14,44]. Interestingly, both functionals
The results of Tables 1 and 2 show that both PBE0 and outperform parameter-free double-hybrid PBE0-DH model
PBE0-1/3 yield bond lengths which are in agreement with (MUE = 0.038 and RMSE = 0.05) for geometries of metal
the experimental values. Moreover, observe, from Table 1, hydrides. Note that the PBE0-DH includes 50% HF
that there are no clear geometrical trends in metal hydrides. exchange and 12.5% MP2 contribution to the correlation
As shown in Figs. 1a and 1b, for geometries of metal energy. Since hybrid density functional calculations are
hydrides and molecules containing transition metals, the much less expensive than double-hybrid ones, it is
MUEs of PBE0 and PBE0-1/3 are very close and their encouraging that some of the hybrid functionals perform
differences are not significant. These results reveal that better than their DH counterparts. In the case of vibrational
increasing the HF exchange from 25% in PBE0 to 33.33% frequencies of metal hydrides, Fig. 1c, the MUE of PBE0
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Does One-Third Scheme of PBE0 Functional Dominate Over PBE0/Phys. Chem. Res., Vol. 1, No. 2, 104-110, December 2013.
BLTMH VFTMH
0.03
MUE RMSE 0.08
MUE RMSE
0.02 0.06
Relative Error
Relative Error
0.04
0.01
0.02
0.00
PBE0-1/3 PBE0 0.00
PBE0-1/3 PBE0
PBE-based hybrid functionals
PBE-based hybrid functionals
(a)
(c)
BLTM
DP4d-5dTMX
0.012 0.12
MUE RMSE
MUE RMSE
R elative Error
0.008 0.08
Relative Error
0.004 0.04
0.00
0.000
PBE0-1/3 P BE0
PBE0-1/3 PBE0
PBE-base d hybrid functionals
PBE-based hybrid functionals
(b) (d)
DM4dTMX
0.06
MUE RMSE
0.04
Relative Error
0.02
0.00
PBE0-1/3 PBE0
PBE-based hybrid functionals
(e)
Fig. 1. Pictorial representation of mean unsigned relative error (MUE) and root mean square relative error (RMSE) for various
test sets. (a) Bond length of 3d transition metal hydrides (BLTMH), (b) Bond length of transition metal molecules
(BLTM), (c) Vibrational frequency of 3d transition metal hydrides (VFTMH), (d) Dipole polarizability of 4d and 5d
transition metal halides (DP4d-5dTMX), and (e) Dipole moment of 4d transition metal halides (DM4dTMX).
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Alipour/Phys. Chem. Res., Vol. 1, No. 2, 104-110, December 2013.
(0.044) is smaller than those of PBE0-1/3 (0.055). However, We found that although PBE0-1/3 works better than PBE0
the PBE0-1/3 MUE is close to those obtained from PBE0- for dipole polarizabilities and dipole moments of small
DH (0.052). molecules, it provides results for geometries and vibrational
Next, the performance of PBE0-1/3 is evaluated for frequencies of about the same quality as PBE0. On the
response properties of several transition metal systems. The whole, our results suggest that there is no real incentive to
calculation of how a small perturbation can affect the use PBE0-1/3 in place of PBE0 for calculations involving
ground state energy allows one to calculate some important transition metals, at least for properties studied here. Lastly,
properties from linear response theory with the use of DFT it remains challenging to develop a generally density
and compare the applicability of functionals in this respect. functional resolving all the qualitative failures of previous
Considering the corresponding results for static dipole approximations at a reasonable computational cost.
polarizabilities and dipole moments of 4d and 5d transition
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