The predictive performances for maximum absorption wavelength of the PBE, BP86, PBE0, B3LYP, M06, M06
2X, CAMB3LYP, LCwPBE, APDF, wB97XD, and PW6B9D3 functionals have been benchmarked through
comparison of maximum absorption wavelength values between calculation and experiment of 21 coumarin derivatives.
For the results obtained from direct calculation by these functionals, the predictive performance decreases gradually in
the following order: B3LYP > APDF > M06 > PW6B9D3 > PBE0 > BP86 > PBE > M062X > CAMB3LYP >
wB97XD > LCwPBE. B3LYP functional gives the best predictive performance, with the smallest value of the mean
absolute error (MAE = 15 nm) and the root mean square deviation (RMSD=19 nm). When using the results obtained
through correction based on the linear correlation, the predictive performance decreases gradually in the following
order: M062X > PBE0, M06, PW6B9D3 > B3LYP, APDF > CAMB3LYP, LCwPBE, wB97XD > PBE and BP86.
M062X functional gives the best predictive performance, with the smallest values of MAEfix (7 nm) and RMSDfix
(9nm). The correction is very necessary because the values of the corrected maximum absorption wavelengths are
closer to the experimental maximum absorption wavelengths. The values of MAEfix and RMSDfix are much smaller than
those of MAE and RMSD.
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Cite this paper: Vietnam J. Chem., 2021, 59(2), 203210 Article
DOI: 10.1002/vjch.202000200
203 Wiley Online Library © 2021 Vietnam Academy of Science and Technology, Hanoi & WileyVCH GmbH
TDDFT benchmark for UVVis spectra of coumarin derivatives
Mai Van Bay
1,2
, Nguyen Khoa Hien
3
, Phan Thi Diem Tran
3
, Nguyen Tran Kim Tuyen
4
,
Doan Thi Yen Oanh
5
, Pham Cam Nam
6*
, Duong Tuan Quang
1*
1
University of Education, Hue University, 34 Le Loi, Hue City, Thua Thien Hue 49000, Viet Nam
2
The University of Danang University of Science and Education, 41 Le Duan, Hai Chau, Da Nang City
50000, Viet Nam
3
Mientrung Institute for Scientific Research, Vietnam Academy of Science and Technology, 312 Huynh Thuc
Khang, Phu Hoa, Hue City, Thua Thien Hue 49000, Viet Nam
4
Kontum Community College, 704 Phan Dinh Phung, Kon Tum City, Kon Tum 60000, Viet Nam
5
Publishing House for Science and Technology, Vietnam Academy of Science and Technology, 18 Hoang
Quoc Viet, Cau Giay, Hanoi 10000, Viet Nam
6
The University of Danang  University of Science and Technology, 41 Le Duan, Hai Chau, Da Nang City
50000, Viet Nam
Submitted December 9, 2020; Accepted December 30, 2020
Abstract
The predictive performances for maximum absorption wavelength of the PBE, BP86, PBE0, B3LYP, M06, M06
2X, CAMB3LYP, LCwPBE, APDF, wB97XD, and PW6B9D3 functionals have been benchmarked through
comparison of maximum absorption wavelength values between calculation and experiment of 21 coumarin derivatives.
For the results obtained from direct calculation by these functionals, the predictive performance decreases gradually in
the following order: B3LYP > APDF > M06 > PW6B9D3 > PBE0 > BP86 > PBE > M062X > CAMB3LYP >
wB97XD > LCwPBE. B3LYP functional gives the best predictive performance, with the smallest value of the mean
absolute error (MAE = 15 nm) and the root mean square deviation (RMSD=19 nm). When using the results obtained
through correction based on the linear correlation, the predictive performance decreases gradually in the following
order: M062X > PBE0, M06, PW6B9D3 > B3LYP, APDF > CAMB3LYP, LCwPBE, wB97XD > PBE and BP86.
M062X functional gives the best predictive performance, with the smallest values of MAE
fix
(7 nm) and RMSD
fix
(9nm). The correction is very necessary because the values of the corrected maximum absorption wavelengths are
closer to the experimental maximum absorption wavelengths. The values of MAE
fix
and RMSD
fix
are much smaller than
those of MAE and RMSD.
Keywords. DFT, TDDFT, UVVis, coumarin, benchmark.
1. INTRODUCTION
Coumarins are from the benzoryrone family, an
important class of phytochemicals. Over 1300
different coumarin derivatives have been identified,
including natural and synthetic derivatives.
[1]
Coumarin derivatives are widely used in medicine,
food, and industry.
[2]
In particular, many
applications are based on the outstanding optical
properties of coumarin derivatives such as dyes,
color indicators, sunscreens, organic lightemitting
diodes, fluorescent markers, and fluorescence
sensors
[36]
Therefore, the prediction of the optical
properties of coumarin derivatives such as UVVis
and fluorescence spectra is necessary in order to
provide a scientific basis for the design and
development of new materials from
thesederivatives.
[3,7,8]
Based on methods of electronic structure theory
for studying the electron excited states, theoretical
calculations allow predicting quite accurately optical
properties of molecules. Some recent publications
show that the UVVis spectra of some large organic
molecules are predictable with high accuracy and
reasonable computing costs by using the time
dependent density functional theory (TDDFT)
method. Therefore, TDDFT is the current popular
method for calculation of the electron excited states
Vietnam Journal of Chemistry Duong Tuan Quang et al.
© 2021 Vietnam Academy of Science and Technology, Hanoi & WileyVCH GmbH www.vjc.wileyvch.de 204
of large molecules
[913]
So far, many DFT methods
(pure, hybrid, longrangecorrected, dispersion
corrected, etc.) have been developed.
[14]
However,
accurate prediction of UVVis spectra using DFT
methods depends on the structure of molecules.
Therefore, it is necessary to choose suitable DFT
functional for each group of compounds to get the
accurate calculation results.
[15,16]
Denis Jacquemin et
al. used 29 DFT functionals to predict the UVVis
spectra of about 500 compounds including bio
organic molecules and dyes. The obtained results
were compared with the previous theoretical
calculations or experimental measurements. The
results show that the predictive performance of the
DFT functionals strongly depends on the molecular
structure of organic compounds. For example, PBE0
and LCωPBE(20) give very good results in
predicting the UVVis spectra of many neutral
organic dyes but are completely inconsistent with
cyaninelike derivatives.
[17]
Azzam CharafEddin et
al used the six hybrid functionals, including B3LYP,
PBE0, M06, M062X, CAMB3LYP and LCPBE
for calculation of the UVVis spectra of 20
conjugated organic compounds in solvent.
[18]
The
results show that B3LYP functional gives the
smallest mean signed error (MSE) and mean
unsigned error (MUE) for the position of absorption
peaks, but is not the most efficient in terms of root
mean square deviation (RMSD) or largest standard
deviation (SD). Meanwhile, M062X functional
provides the most appropriate results with the
experiment for the band shapes corresponding to the
absorption spectra of conjugated compounds.
[18]
So
far, some studies, but not many, have been
successful in using TDDFT method to calculate the
UVVis spectra of coumarin derivatives.
[19,20]
In this work, the predictive performances of the
PBE, BP86, PBE0, B3LYP, M06, M062X, CAM
B3LYP, LCwPBE; APDF, wB97XD, and
PW6B9D3 functionals for maximum absorption
wavelength of 21 coumarin derivatives (figure 1) are
benchmarked through comparison of maximum
absorption wavelength values between calculation
and experiment.
Figure 1: Coumarin derivatives for this investigation
Vietnam Journal of Chemistry TDDFT benchmark for UVVis spectra of
© 2021 Vietnam Academy of Science and Technology, Hanoi & WileyVCH GmbH www.vjc.wileyvch.de 205
2. COMPUTATIONAL METHODS
In this study, the optimized geometry of all
molecules in the ground state is carried out by using
density functional theory (DFT) at the PBE0/6
311++G(d,p) level of theory.
[15]
Here, it should be
emphasized that the PBE0 density functional has
been evaluated to accurately predict the geometrical
structure of most of the organic molecules in the
ground states.
[21]
The UVVis spectra of coumarin
derivatives are carried out by using timedependent
density functional theory (TDDFT)
[9,10]
at the X/6
311++G(d,p) level of theory, where X: PBE, BP86,
PBE0, B3LYP, M06, M062X, CAMB3LYP, LC
wPBE; APDF, wB97XD and PW6B9D3. The
polarizable continuum model (PCM) is applied for
all DFT and TDDFT calculations, with the solvent
used in the calculations being the same as that used
in the experiment.
[22]
All calculations are performed
using Gaussian 16 Revision A.03 program.
[23]
The predictive performance of DFT functionals
for maximum absorption wavelength of coumarin
derivatives is benchmarked through comparison
between the calculated maximum absorption
wavelength (
) and experimental maximum
absorption wavelength (
). In this case, both the
mean absolute error (MAE) (or mean unsigned error,
MUE), and root mean square deviation (RMSD) (or
root mean square error, RMSE) are frequently used
to evaluate the performance of these models.
[24]
The
mean signed error (MSE) (or average signed
difference, MSD) is also used for the purpose of
examining the direction of the errors.
[14,25]
In
addition, the linear correlation relationship between
the experimental and the calculated values of the
maximum absorption wavelength is also used to
evaluate the method's predicted performance.
[26]
Then, the corrected calculated maximum absorption
wavelength
is also determined based on the
linear correlation equations between
and
. The predictive performance of DFT
functionals is once again evaluated through
comparison between values of
and
.
[14,25]
3. RESULTS AND DISCUSSION
3.1. TDDFT benchmark based on comparison
between the
and
The results of calculation of maximum absorption
wavelength of 21 coumarin derivatives using 11
DFT functionals, difference between the calculated
maximum absorption wavelength and the
experimental maximum absorption wavelength
(Δλmax), and statistical analysis data of MAE,
RMSM, and MSE are presented in table 1 and
figure 2.
Figure 2: Graph of MSE and MAE values by DFT functionals based on comparison between the
and
of coumarin derivatives
29 29
22
12
20
42 45
62
19
49
21
30 29
23
15
21
43 45
62
20
49
22
70
50
30
10
10
30
50
70
E
rr
o
r
re
la
ti
v
e
to
e
x
p
er
im
en
t
(n
m
)
MSE MAE
Vietnam Journal of Chemistry Duong Tuan Quang et al.
© 2021 Vietnam Academy of Science and Technology, Hanoi & WileyVCH GmbH www.vjc.wileyvch.de 206
Table 1: Results of comparison between the
and
of coumarin derivatives
Compound
Δλmax =

(nm)
[Ref] PBE BP86 PBE0 B3LYP M06 M062X
CAM
B3LYP
LC
wPBE
APDF wB97XD
PW6B9
D3
M1 6 6 26 19 20 39 36 48 24 38 25 326 [27]
M2 30 30 1 7 6 11 9 19 3 10 2 280 [27]
M3 22 21 24 15 21 41 41 56 21 43 22 351 [27]
M4 25 25 28 17 25 52 50 70 24 54 27 384 [27]
M5 46 46 8 3 4 28 26 42 5 29 7 360 [27]
M6 23 22 15 8 11 31 29 42 13 31 14 324 [27]
M7 17 16 20 13 16 35 33 45 18 35 19 355 [27]
M8 50 49 13 2 13 39 39 59 9 43 13 370 [27]
M9 50 49 15 4 16 42 43 63 11 47 15 376 [27]
M10 46 45 16 5 16 42 43 64 14 47 16 400 [27]
M11 39 38 22 11 22 46 47 64 18 50 21 373 [27]
M12 16 17 38 27 37 63 66 88 35 70 38 459 [27]
M13 25 25 37 25 37 59 61 79 33 65 36 436 [27]
M14 36 37 23 11 21 50 52 75 19 57 23 436 [27]
M15 32 32 28 17 26 52 55 75 25 59 28 408 [27]
M16 4 4 46 36 45 64 66 81 43 69 44 420 [27]
M17 63 63 11 3 11 43 44 65 7 49 11 376 [27]
M18 34 34 3 10 11 3 57 67 5 59 3 278 [27]
M19 53 52 3 6 0 27 25 41 0 26 3 340 [27]
M20 5 6 50 41 48 66 68 83 47 71 48 445 [27]
M21 0 0 46 37 44 62 64 81 43 68 44 436 [27]
MSE 29 29 22 12 20 42 45 62 19 49 21
MAE 30 29 23 15 21 43 45 62 20 49 22
RMSD 34 34 27 19 25 46 48 65 24 51 26
The calculation results show that the DFT
functionals used to predict the maximum absorption
wavelength of coumarin derivatives can be divided
into three groups with different performance. The
best performing group includes 5 functionals, sorted
in descending order B3LYP (MAE = 15, RMSD =
19) > APDF (MAE = 20, RMSD = 24) > M06
(MAE = 21, RMSD = 25) > PW6B9D3 (MAE = 22,
RMSD = 26) > PBE0 (MAE = 23, RMSD = 27).
The average performing group includes 2
functionals, sorted in descending order BP86 (MAE
= 29, RSMSD = 34) > PBE (MAE = 30, RMSD =
34). The worst performing group includes 4
functionals, sorted in descending order M062X
(MAE = 43, RMSD = 46) > CAMB3LYP (MAE =
45, RMSD = 48) > wB97XD (MAE = 49, RMSD =
51) > LCwPBE (MAE = 62, RMSD = 65).
In this study, B3LYP and PBE0 belong to the
group of functionals that give the best predictive
performance of the maximum absorption
wavelength, consistent with previous
publications.
[14,25]
B3LYP functional gives better
performance than PBE0 functional for coumarin
derivatives, whereas in some previous studies, PBE0
functional gave better performance than B3LYP
functional for other derivatives.
[14,25]
This indicates
that it is necessary to choose right functionals for
each compound. The studied results also show that
there is no difference between MAE and RMSD
when using them for these model evaluation studies.
When considering the direction of the errors,
PBE and BP86 functionals give values of
systematically higher than the values of
.
Meanwhile, the remaining functionals give values of
systematically smaller than the values of
. The results indicate that the values of λmax
systematically obey the order of PBE > BP86 >
B3LYP > APDF > M06 > PW6B9D3 > PBE0 >
M062X > CAMB3LYP > wB97XD > LCwPBE.
3.2. TDDFT benchmark based on comparison
between the (
) and (
)
The results of the linear correlation analysis between
experimental maximum absorption wavelength and
calculated maximum absorption wavelength are
shown in figure 3. The results show that all the DFT
functionals used above yield a very good linear
correlation between
and
. The evidence is
Vietnam Journal of Chemistry TDDFT benchmark for UVVis spectra of
© 2021 Vietnam Academy of Science and Technology, Hanoi & WileyVCH GmbH www.vjc.wileyvch.de 207
that the linear correlation coefficient (R) ranges from
0.932 to 0.982, sorted in descending order
corresponding to the functionals as follows: M062X
(R = 0.982) > PW6B9D3 (R = 0.979) > M06 (R =
0.977) > PBE0 (R = 0.976) > APDF (R = 0.975) >
B3LYP (R = 0.971) > CAMB3LYP (R = 0.967) >
LCwPBE (R = 0.9654) > wB97XD (R = 0.9649) >
BP86 (R = 0.933) > PBE (R = 0.932). Thus, if it is
based on MAE and RMSE (obtained from
comparison between
and
), M062X
belongs to the group of functionals with the worst
predictive performance (with the largest error), but
from linear correlation analysis, M062X belongs to
the group of functionals with the best predictive
performance (with the largest linear correlation
coefficient). When considering the direction of the
errors, the PBE and BP86 functionals result in all the
points on the graph as being below a line with the
equation y = x. Meanwhile, the remaining
functionals result in all the points on the graph as
being above a line with the equation y = x. This
shows that whole recommended DFT functionals
give the calculated maximum absorption wavelength
systematically larger, or smaller than the
experimental maximum absorption wavelength.
On the basis of the linear relationship between
experimental maximum absorption wavelengths and
calculated maximum absorption wavelengths (figure
3), the calculated maximum absorption wavelengths
can be determined according to the equations in
table 2. The calculated differences between the
corrected calculated maximum absorption
wavelengths and experimental maximum absorption
wavelengths ∆
and statistical analysis data of
MAE, RMSM, and MSE are presented in table 3 and
figure 4.
Figure 3: Diagrams of linear correlation between the
and
The calculation results show that the MAE
fix
values decrease from 33.8 % (B3LYP) to 84.6 %
(LCwPBE) compared to those without correction,
the RMSD
fix
values decrease from 37.1 % (B3LYP)
to 79.8 % (M062X) compared to those without
correction. The values of MAE
fix
and RMSD
fix
obey
Vietnam Journal of Chemistry Duong Tuan Quang et al.
© 2021 Vietnam Academy of Science and Technology, Hanoi & WileyVCH GmbH www.vjc.wileyvch.de 208
a M062X (MAE
fix
= 7, RMSD
fix
= 9) < PBE0, M06,
PW6B9D3 (MAE
fix
= 9, RMSD
fix
= 11) < B3LYP,
APDF (MAE
fix
= 10, RMSD
fix
= 12) < CAM
B3LYP, LCwPBE, wB97XD (MAE
fix
= 10,
RMSD
fix
= 13) < PBE, BP86 (MAE
fix
= 15, RMSD
fix
= 19) order. These results indicate that M062X
functional gives the best predictive performance for
maximum absorption wavelength of coumarin
derivatives. These results show the importance of
correcting the calculated results for the maximum
absorption wavelength of coumarin derivatives by
TDDFT, in order to obtain the values of corrected
maximum absorption wavelength that are closer to
experimental maximum absorption wavelengths.
Table 2: Equations for calculation of
according to
of DFT functionals
Functionals Equations
PBE
 17.802
BP86
 16.424
PBE0
 66.822
B3LYP
 59.939
M06
 89.388
M062X
 93.783
CAMB3LYP
 24.851
LCwPBE
 31.871
APDF
 65.196
wB97XD
 28.301
PW6B9D3
 69.671
Figure 4: Graph of MAE values with/without linear fixing
4. CONCLUSION
The predictive performances of the PBE, BP86,
PBE0, B3LYP, M06, M062X, CAMB3LYP, LC
wPBE; APDF, wB97XD, and PW6B9D3
functionals for maximum absorption wavelength
have been benchmarked through comparison of
maximum absorption wavelength values between
calculation and experiment of 21 coumarin
derivatives. The results show that all the DFT
functionals yield a very good linear correlation
between
and
. For the results obtained
from direct calculation by functionals, B3LYP
functional gives the best predictive performance of
the maximum absorption wavelengths, with the
smallest values of MAE and RMSD. However,
when using the results obtained through correction
based on the linear correlation between the
experimental maximum absorption wavelengths and
the calculated maximum absorption wavelengths,
the M062X functional gives the best predictive
performance, with the smallest values of MAE
fix
and
30 29
23
15
21
43 45
62
20
49
22
15 15
9 10 9 7 10 10 9 10 9
0
10
20
30
40
50
60
70
E
rr
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re
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x
p
er
im
en
t
(n
m
)
MAE without fixing MAE with linear fixing
Vietnam Journal of Chemistry TDDFT benchmark for UVVis spectra of
© 2021 Vietnam Academy of Science and Technology, Hanoi & WileyVCH GmbH www.vjc.wileyvch.de 209
RMSD
fix
. This correction is very necessary because
the values of the corrected maximum absorption
wavelengths are closer to the experimental
maximum absorption wavelengths. The values of
MAEfix and RMSDfix are much smaller than MAE
and RMSD.
Table 3: Results of comparison between the
and
of coumarin derivatives
Compounds
=

(nm)
P
B
E
B
P
8
6
P
B
E
0
B
3
L
Y
P
M
0
6
M
0
6
2
X
C
A
M
B
3
L
Y
P
L
C
w
P
B
E
A
P
D
F
w
B
9
7
X
D
P
W
6
B
9
D
3
M1 21 21 18 18 16 16 0 3 18 1 18
M2 4 4 4 4 4 4 23 27 4 25 4
M3 6 7 9 8 10 9 0 0 9 1 8
M4 4 4 6 4 5 11 4 8 5 5 5
M5 17 17 13 15 15 13 20 21 13 20 14
M6 4 5 5 5 5 6 9 10 5 9 4
M7 11 12 3 5 2 1 10 16 4 11 3
M8 20 20 9 11 7 2 6 2 11 5 9
M9 20 20 9 10 4 0 3 2 10 2 8
M10 16 15 13 13 12 9 8 4 12 7 12
M11 10 9 1 1 4 7 3 4 0 3 1
M12 15 14 1 1 2 4 8 9 1 7 0
M13 6 5 4 4 5 0 7 4 3 7 3
M14 5 6 14 13 16 13 4 1 14 3 13
M15 2 2 0 0 1 1 5 8 0 6 1
M16 26 25 19 20 20 13 16 12 20 15 18
M17 33 33 14 18 11 2 1 4 15 1 13
M18 7 8 6 7 10 23 35 36 6 36 5
M19 24 24 14 14 14 6 17 16 15 19 13
M20 35 36 18 21 16 6 13 7 19 12 16
M21 30 30 15 18 14 4 10 7 16 10 14
MSE
fix
0 0 0 0 0 0 0 0 0 0 0
MAE
fix
15 15 9 10 9 7 10 10 10 10 9
RMSD
fix
19 19 11 12 11 9 13 13 12 13 11
Acknowledgment. This research was funded by the
Vietnam National Foundation for Science and
Technology Development (NAFOSTED) under grant
number 104.062017.51 (Duong Tuan Quang).
REFERENCES