短距離と長距離を補正した密度汎関数法(LCgau－DFT)の開発
宋 鍾元, Mark A. Watson, 徳良 誠健, 佐藤 健, 平尾 公彦
東京大学大学院工学系研究科
sjoshua@qcl.t.u-tokyo.ac.jp

An improved long-range corrected hybrid exchange-correlation functional (LCgau-BOP) is presented with the inclusion of ashort-range two-parameter Gaussian attenuation. We show that LCgau-BOP provides a consistently more accurate description ofthermochemistry, chemical reactions and excitation energies than other existing Coulomb-attenuated functionals.

1Introduction

Recently, we developed the Long-range
Corrected Density Functional Theory (LC-DFT),
where the exchange functional is partitioned withrespect to the inter-electronic separation into
long-range and short-range parts using a standarderror function.1 The LC approach was shown toclearly solve many of the problems whichconventional DFT confronted: for example, theunderestimation of Rydberg excitation energies andcorresponding oscillator strengths,1 the poor
reproduction of charge-transfer excitations intime-dependent DFT calculations,1 and the poor
description of van der Waals interactions2 as well as
accurate reaction enthalpies and barrier heights.3
However, these LC-DFT functionals still haveroom for improvement. For example, it was foundthat LC-DFT could not produce atomizationenergies of the same quality as the B3LYP hybrid
functional in spite of its more sophisticatedinclusion of exact exchange, dependent on theinter-electronic distance.4
Recently, Savin et al. suggested the so-called
erfgau interaction which combines a Gaussiancorrection to the error function to describe thelong-range--short-range separation of the Coulomb
operator.5
In this paper we extend this work by
investigating the erfgau interaction in the context ofgeneralized-gradient-approximation(GGA)
calculations on molecules.

2Theory

In the LC scheme,1 the electron repulsion
operator, 1/r12, is divided into short- and long-range
parts using a standard error function:

1 1 − erf ( μ r12) erf ( μ r12)
=+r12
r12
r12

(1)

The first term of Eq. (1) is the short-range
interaction term, and the second term is thelong-range interaction term. The DFT exchange

functional is included through the first term, and thelong range orbital–orbital exchange interaction isdescribed using the HF exchange integral.
The erfgau interaction of Savin et al.5 modifies
the partitioning with a Gaussian correction, suchthat the long-range part takes the formerf ( μ r ) 2 μ − (1/ 3) μ 2 r 2−erπwith an obvious modification to thecomplementary short-range part.
However, we took the pragmatic approach ofintroducing 2 extra parameters. We replaced theexponent, 1/3, by a parameter, 1/a, and multiplied
the prefactor by a parameter, k, to give thelong-range formerf ( μ r )2μ − (1/ a ) μ 2 r 2−kerπwith the obvious modification to thecomplementary short-range part. We named thisshortand
long-range
corrected
methodLCgau-DFT.

3 Result and Discussion
We employed a 50%-50% summation ofatomization energy and barrier height errors andobtained an optimized parameter set with μ=0.42,a=0.011, and k=-18.0, which gave an RMS error of
5.3 kcal/mol, which is significantly better thanLC-BOP (10.3 kcal/mol) and even B3LYP (7.5kcal/mol) and gave slightly improved barrier heights(2.43 kcal/mol RMS error) compared to LC-BOP(2.53 kcal/mol). As a result, the errors ofLCgau-BOP for both atomization energies andbarrier heights were lower than those of B3LYP andLC-BOP. This shows that the inclusion of HF
exchange in the short-range part is of significant
importance in calculating both the atomizationenergies and the barrier heights.
Concerning the reaction enthalpies of Database/3,
LCgau-BOP also gives smaller errors (2.00kcal/mol) than LC-BOP (2.18 kcal/mol) and B3LYP

(3.03 kcal/mol).
In addition, LCgau-BOP gave the most accurate
reproduction of the optimized bond lengths andbond angles.
Finally, LCgau-BOP is also seen to improve
excitation energy calculations, giving generally
improved errors of the valence, Rydberg, and total
excitation energies compared to other LC
functionals (Table.2).
Table.1: Results from 148 atomization energies(kcal/mol), 329 bond lengths (Å), and 271 bond angles(degrees) in the G2 test set, plus 72 calculated barrierheights and 26 reaction enthalpies (kcal/mol) fromDatabase/3. Root mean square (RMS) and mean signederrors (MSE) are given for all calculations using the6-311++G(2d,2p) basis set and optimized geometries.
G2 setμkaB3LYP
LC-BOP

0.33
0.47
LCgau-BOP 0.42 -18.0 0.01

AtomEnergy

Bond Bond
Length Angle

RMS
7.5
21.0
10.1
5.3

RMS RMS
0.0244 1.5581
0.0235 1.5421
0.0244 1.5798
0.0233 1.5087

Database/3
Barrier Reaction
HeightEnthalpy
MSE RMS-3.91 4.74
-3.77 4.92
0.48 2.53-0.95 2.43

RMS
3.03
3.27
2.182.00

Table.2: Mean absolute Errors of Vertical excitationenergies of the molecule set calculated by TDDFT.
Error of SAC-CI is also shown for comparison. All
calculations used an augumented Sadlej pVTZ basis
set.
StateLC-BOP# of state LCgau-BOP
BOP B3LYP SAC-CI
μ =0.42 μ =0.33 μ =0.47
a =0.011
k =-18.0
Mean absolute deviation of exitation energries of N2 (eV)Valence(8)0.440.37 0.52 0.40 0.53
0.33
Rydberg(4)
0.19
0.79 0.17 2.37 1.14
0.25
Total
(12)
0.35
0.51 0.40 1.06 0.73
0.31
Mean absolute deviation of exitation energries of CO (eV)Valence(7)0.20
0.19 0.21 0.28 0.35
0.26
Rydberg(7)
0.07
0.64 0.23 2.06 1.15
0.27
Total
(15)
0.14
0.41 0.22 1.17 0.75
0.27
Mean absolute deviation of exitation energries of H2CO (eV)Valence(4)0.21
0.30 0.22 0.23 0.18
0.45
Rydberg(8)
0.15
0.47 0.29 1.66 0.83
0.13
Total
(12)
0.17
0.41 0.27 1.18 0.62
0.24
Mean absolute deviation of exitation energries of C2H4 (eV)Valence(2)0.410.31 0.47 0.47 0.45
0.09
Rydberg(12)0.16
0.15 0.32 1.41 0.96
0.17
Total
(14)
0.19
0.17 0.34 1.28 0.88
0.16
Mean absolute deviation of exitation energries of C6H6 (eV)Valence
Rydberg
Total

(6)
(10)
(16)

0.32
0.50
0.43

0.23
0.29
0.27

0.37
0.64
0.54

0.28
1.00
0.73

0.25
0.56
0.44

0.23
0.20
0.21

such functionals, including LC-BOP, CAM-BLYP,
and B3LYP, in terms of the reproduction ofatomization energies, barrier heights, reaction
enthalpies, geometrical properties, and excitationenergies (including charge-transfer excitations)over a wide range of molecular systems. The key
feature is the introduction of a two-parameter
Gaussian function to the Coulomb attenuation
which allows a more flexible description of exact
exchangeatshort-range
inter-electronic
separations. The new partitioning preserves 100%
exact exchange in the long range, which is known
to be important for the success of the LC-scheme,with an asymptotic attenuation described by astandard error function with parameter 0.42. The
LCgau partitioning was optimized for thereproduction of atomization energies and reaction
barrier heights over large fitting sets and theresults highlight the importance of including asubstantial portion of exact exchange in the shortrange.

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4 Conclusion
We have presented a new hybrid
exchange-correlation functional based on the LCscheme, named LCgau-BOP, which we havedemonstrated to be consistently superior to other「Long-range—short-range separation of theelectron-electron interaction in density functionaltheory」Phys. Rev. A 70, 062505 (2004).