1
J. Chem. Phys., in press.
Final author version
Oct. 24, 2006
A New Local Density Functional for Main Group Thermochemistry,
Transition Metal Bonding, Thermochemical Kinetics,
and Noncovalent Interactions
Yan Zhao and Donald G. Truhlar
Department of Chemistry and Supercomputing Institute
University of Minnesota, 207 Pleasant Street S.E.
Minneapolis, MN 55455-0431
Abstract.
We present a new local density functional, called M06-L, for main-group and
transition element thermochemistry, thermochemical kinetics, and noncovalent
interactions. The functional is designed to capture the main dependence of the exchange-
correlation energy on local spin density, spin density gradient, and spin kinetic energy
density, and it is parametrized to satisfy the uniform-electron-gas limit and to have good
performance for both main-group chemistry and transition metal chemistry. The M06-L
functional and 14 other functionals have been comparatively assessed against 22
energetic databases. Among the tested functionals, which include the popular B3LYP,
BLYP, and BP86 functionals as well as our previous M05 functional, the M06-L
functional gives the best overall performance for a combination of main group
thermochemistry, thermochemical kinetics, and organometallic, inorganometallic,
biological, and noncovalent interactions. It is also does very well for predicting
geometries and vibrational frequencies. Because of the computational advantages of local
functionals, the present functional should be very useful for many applications in
chemistry, especially for simulations on moderate-sized and large systems and when long
time scales must be addressed.
2
I. INTRODUCTION
In the early days of density functional theory,
1
all density functionals were local.
Local functionals may depend on the local spin density,
2,3
its gradient
4-9
or Laplacian,
10
or even spin kinetic energy density approximated in terms of the kinetic energy of Kohn-
Sham spin orbitals.
11-13
Unfortunately functionals depending on the gradient of the spin
density were sometimes called nonlocal, but that semantic error appears to be
disappearing. More recently, nonlocal density functionals have been widely employed,
and in many cases they greatly improve the accuracy.
14-17
The present article returns to
the formulation of local density functionals.
It is important to clarify the significance of local density functionals. Some
researchers have voiced the opinion that nonlocal density functionals are to be tolerated
as an interim “fix” until the true density functional is derived/discovered/developed. This
is a misunderstanding. The theorem
18
that exact density functionals exist does not apply
to local density functionals; thus we must allow a density functional to be nonlocal if it is
to be exact.
19,20
Nonlocal density functionals include hybrid functionals that depend on
Hartree-Fock exchange.
14
Thus local functionals are important not because they
somehow represent a more theoretically justified solution. Rather they are important for
practical reasons because calculations on large complex systems may employ specialized
algorithms (including density fitting, also called resolution of the identity) that are tens or
hundreds of times faster if one employs local density functionals than if one employs
nonlocal ones.
21-30
For example, we found that a single-point energy calculation for a
C
104
H
30
N
4
fullerene-porphyrin complex
31
with a local functional employing density
fitting is 15 times faster (17 hours vs. 250 hours) than a hybrid density functional
3
calculation on the same system (using GAUSSIAN03
32
). Furthermore, calculations are
impractical for many solids when Hartree-Fock exchange is included,
33
unless some
special techniques
34-37
are employed.
A second reason to study local density functionals is their usefulness for modeling
the bonding of metallic elements. Two recent systematic studies of highly unsaturated
systems containing metal atoms, one for metal-metal bonds
38
and one for metal-ligand
bonds,
39
have shown that most functionals with more than about 10% Hartree-Fock
exchange fail badly for a large number of bonds involving transition metals, which is
understandable in that the density-based exchange functionals do a better job than
Hartree-Fock exchange in accounting for static correlation.
9,40-43
Even more recently we
have presented a new functional
44
called M05, that performs well for such cases even
with 28% Hartree-Fock exchange and also performs well for main group chemistry,
barrier heights, and non-covalent interactions. Nevertheless, we are still interested in
local density functionals because of their cost advantages discussed in the previous
paragraph.
In the present article, we present a local functional that has better general
performance than the most popular hybrid functional, B3LYP. The new functional is
called M06-L. We assess this new functional by applying it, along with eleven previous
local functionals (BP86,
5,7
BLYP,
7,8
BB95,
7,11
G96LYP,
8,45
PBE,
46
mPWPW,
16,47
VSXC,
12
HCTH,
48
OLYP,
41
τ-HCTH,
49
and TPSS
13
) and three previous nonlocal functionals
(B3LYP,
7,14,50
TPSSh,
51
and M05
44
), to the data in 22 diverse databases. In fact the
average performance of the M06-L functional over the 22 energetic databases considered
here is better than that of any local or hybrid functional that we know.
4
Most of the databases used in the present work have been introduced and
described in previous work. One new database introduced here is a database of five
metal-atom excitation energies containing two main group neutral metals, two neutral
transition metals, and one transition metal cation. The addition of this data set is
prompted by the increasing attention being paid by many workers to the relative energies
of spin states of transition metal systems because of the importance of spin states for
structures, properties, and chemical reactivities of organometallic complexes and for
functional nanotechnology.
52-62
Holthausen made a systematic study
63
of the ability of
density functionals to predict 3d–4s excitation energies in transition metal cations and
found that some functionals, even though performing well for main group atomization
energies, show large errors for transition metal atomic excitation energies. Because we
seek a functional that is accurate for both main-group and transition metal chemistry, our
small database has both types of atoms. We note that our transition metal excitation
energy comparisons take account of scalar relativistic effects,
64
whereas those of
Holthausen do not, even though such effects are sometimes large.
The paper is organized as follows. Section 2 presents our databases. Section 3
gives computational details. Section 4 discusses the theory and parametrization of the
new functionals. Section 5 presents results and discussion, including test for energies,
geometries, and frequencies not used in training, and Section 6 concludes the paper.
II. ENERGETIC DATABASES
Our general notation for databases is XN/V, where X is an acronym for the type
of data, N is the number of data, and V is a version number (sometimes omitted if there
5
has only been one version). The reason for version numbers is that historically we have
sometimes corrected (if possible) or eliminated data whose reliability has been disproved
or credibly challenged. All data in Subsections II.A through II.H are pure electronic
energies, i.e., zero-point energies and thermal vibrational-rotational energies have been
removed by methods discussed previously,
17,65-68
but nuclear repulsion is included. Since
the databases are based on experimental or accurate data, the values in our database
correspond to relativistic values. Thus our calculations must include relativistic
corrections as well, where these are not negligible; this is discussed in Section III.B.
II.A. MGAE109/05 test set
The MGAE109/05 test set
17
consists of 109 atomization energies (AEs) for main
group compounds. We always give the mean errors in atomization energies on a per bond
basis because that makes comparison between different test sets more portable. In the
past, as some workers increased the size of their test sets, they tended to add larger
molecules, and the resulting increase in mean errors due to increasing the average number
of bonds could not be distinguished from the increase in mean errors due to the added
diversity of the test molecules. To make it possible for readers to compare other workers’
values to our mean errors on a per bond basis, we always compute the mean errors in
atomization energies by computing the mean error per molecule and then dividing by the
average number of bonds per molecule in the test set. The latter value is 4.71 for
MGAE109/05.
II.B. Ionization potential, electron affinity, and proton affinity test sets
The zero-point-exclusive ionization potential (IP) and electron affinity (EA) test
sets are called IP13/3 and EA13/3, respectively, and they have been explained and
