TD-DFT benchmark for UV-Vis spectra of coumarin derivatives

Cite this paper: Vietnam J. Chem., 2021, 59(2), 203-210 Article 
DOI: 10.1002/vjch.202000200 
203 Wiley Online Library © 2021 Vietnam Academy of Science and Technology, Hanoi & Wiley-VCH GmbH 
TD-DFT benchmark for UV-Vis 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, CAM-B3LYP, LC-wPBE, 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 > M06-2X > CAM-B3LYP > 
wB97XD > LC-wPBE. 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: M06-2X > PBE0, M06, PW6B9D3 > B3LYP, APDF > CAM-B3LYP, LC-wPBE, wB97XD > PBE and BP86. 
M06-2X 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, TD-DFT, UV-Vis, 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 light-emitting 
diodes, fluorescent markers, and fluorescence 
sensors
[3-6]
 Therefore, the prediction of the optical 
properties of coumarin derivatives such as UV-Vis 
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 UV-Vis spectra of some large organic 
molecules are predictable with high accuracy and 
reasonable computing costs by using the time-
dependent density functional theory (TD-DFT) 
method. Therefore, TD-DFT 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 & Wiley-VCH GmbH www.vjc.wiley-vch.de 204 
of large molecules
[9-13]
 So far, many DFT methods 
(pure, hybrid, long-range-corrected, dispersion 
corrected, etc.) have been developed.
[14]
 However, 
accurate prediction of UV-Vis 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 UV-Vis 
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 LC-ωPBE(20) give very good results in 
predicting the UV-Vis spectra of many neutral 
organic dyes but are completely inconsistent with 
cyanine-like derivatives.
[17] 
Azzam Charaf-Eddin et 
al used the six hybrid functionals, including B3LYP, 
PBE0, M06, M06-2X, CAM-B3LYP and LC-PBE 
for calculation of the UV-Vis 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, M06-2X 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 TD-DFT method to calculate the 
UV-Vis spectra of coumarin derivatives.
[19,20]
In ... ssary 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 > 
M06-2X > CAM-B3LYP > wB97XD > LC-wPBE. 
3.2. TD-DFT 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 TD-DFT benchmark for UV-Vis spectra of 
 © 2021 Vietnam Academy of Science and Technology, Hanoi & Wiley-VCH GmbH www.vjc.wiley-vch.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: M06-2X 
(R = 0.982) > PW6B9D3 (R = 0.979) > M06 (R = 
0.977) > PBE0 (R = 0.976) > APDF (R = 0.975) > 
B3LYP (R = 0.971) > CAM-B3LYP (R = 0.967) > 
LC-wPBE (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 
), M06-2X 
belongs to the group of functionals with the worst 
predictive performance (with the largest error), but 
from linear correlation analysis, M06-2X 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 % 
(LC-wPBE) compared to those without correction, 
the RMSD
fix
 values decrease from 37.1 % (B3LYP) 
to 79.8 % (M06-2X) 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 & Wiley-VCH GmbH www.vjc.wiley-vch.de 208 
a M06-2X (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, LC-wPBE, wB97XD (MAE
fix 
= 10, 
RMSD
fix 
= 13) < PBE, BP86 (MAE
fix 
= 15, RMSD
fix 
= 19) order. These results indicate that M06-2X 
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 
TD-DFT, 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 
M06-2X 
 - 93.783 
CAM-B3LYP 
 - 24.851 
LC-wPBE 
 - 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, M06-2X, CAM-B3LYP, 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 M06-2X 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
o
r 
re
la
ti
v
e 
to
 e
x
p
er
im
en
t 
(n
m
) 
MAE without fixing MAE with linear fixing
Vietnam Journal of Chemistry TD-DFT benchmark for UV-Vis spectra of 
 © 2021 Vietnam Academy of Science and Technology, Hanoi & Wiley-VCH GmbH www.vjc.wiley-vch.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 104.06-2017.51 (Duong Tuan Quang). 
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Corresponding authors: Duong Tuan Quang 
University of Education, Hue University 
34 Le Loi, Hue City, Thua Thien Hue 49000, Viet Nam 
E-mail: duongtuanquang@gmail.com. 
Pham Cam Nam 
The University of Danang - University of Science and Education 
41 Le Duan, Hai Chau, Da Nang City 50000, Viet Nam 
E-mail: pcnam@dut.udn.vn.