J. Nig. Soc. Phys. Sci. 5 (2023) 1094

Journal of the
Nigerian Society

of Physical
Sciences

Effects of Hybrid Exchange Correlation Functional (Vwdf3) on
the Structural, Elastic, and Electronic Properties of Transition

Metal Dichalogenides

S. A. Yamusaa, A. Shaarib, I. Isahc, U. B. Ibrahimd, S. I. Kunyac, S. Abdulkarimd, Y. S. Itase,∗, I.
M. Alsalamhf

a Department of Physics, Federal College of Education Zaria, P.M.B 1041, Zaria, Kaduna State, Nigeria
b Department of Physics, Faculty of Science, Universiti Teknologi Malaysia

c Department of Science Laboratory Technology, Jigawa State Polytechnic, Dutse, Jigwa State Nigeria
d Faculty of Science, Physics Department Kano University of Science and Technology, Wudil, Kano, Nigeria

e Department of Physics, Bauchi State University, Gadau, P.M.B. 65 Bauchi, Nigeria
f Physics Department, Faculty of Science, University of Hail, Saudi Arabia

Abstract

In this research, the effects of Van der Waals forces on the structural, elastic, electronic, and optical properties of bulk transition metals dichalco-
genides (TMDs) were studied using a novel exchange-correlation functional, vdW-DF3. This new functional tries to correct the hidden Van der
Waals problems which are not reported by the previous exchange functionals. Molybdenum dichalcogenide, MoX2 (X = S, Se, Te) was chosen
as a representative transition metal dichalcogenide to compare the performance of the newly designed functional with the other two popular
exchange-correlation functional; PBE and rVV10. From the results so far obtained, the analysis of the structural properties generally revealed
better performance by vdW-DF3 via the provision of information on lattice parameters very closer to the experimental value. For example, the
lattice constant obtained by vdW-DF3 was 3.161 Å which is very close to 3.163 Å and 3.160 Å experimental and theoretical values respectively.
Calculations of the electronic properties revealed good performance by vdW-DF3 functional. Furthermore, new electronic features were revealed
for MoX2 (x = S, Se, Te). In terms of optical properties, PBE functional demonstrates lower absorption than vdW-DF3, as such it can be reported
that vdW-DF3 improves photon absorption by TMDs. However, our results also revealed that vdW-DF3 performed well for MoS2 than for MoSe2
and MoTe2 because of the lower density observed for the S atom in MoS2.

DOI:10.46481/jnsps.2022.1094

Keywords: Van der Waals, PBE, Hexagonal, vdW-DF3, Dichalcogenides.

Article History :
Received: 29 September 2022
Received in revised form: 21 November 2022
Accepted for publication: 02 December 2022
Published: 14 January 2023

c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: Taoreed Owolabi

∗Corresponding author tel. no: +2348069316888
Email address: yitas@basug.edu.ng ( Y. S. Itas )

1. Introduction

The study of transition metal dichalcogenides has grown
in popularity as a condensed matter physics research area and
as a potential resource for a range of applications that could

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S. A. Yamusa et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1094 2

affect scientific and high-tech advancement. DFT explanations
on how electrons in the many-body systems interact with
each other depending on the so-called approximations of
the exchange-correlation (XC) functional [1]. Moreover,
much of the successes in DFT came from the fact that these
functions often produce accurate results. However, there are
some situations where failures are reported by many of these
functional [2]. Therefore, there is a need to understand the
accurate XC functional to adequately describe the behavior
of the many-body electronic system. One good example of
failure by XC is the inability to fully describe a long-range
electron interaction also called dispersion forces. Van der
Waals problem by local density approximation (LDA) and
Perdew-Burke-Ernzerh (PBE) exchange functional remains a
challenge that needs urgent improvement [3] The problem of
lack of dispersion forces otherwise referred to as van der Waals
(vdW) forces, is one of the most disturbing problems in DFT.

Therefore it becomes one of the most traded topics in
condensed matter physics and material science. It can be un-
derstood by the fact that over 800 dispersion-based DFT studies
were reported in 2011 compared to fewer than 80 in the whole
of the 1990s [4]. Recent studies have revealed that LDA
exhibits overestimations of the lattice constants for a and
c of 0.4 % and 5.2 %, and GGA underestimates c by 4 %
while overestimating a by 0.58 %, respectively. This problem
persisted in TMDs, especially in terms of their mechanical,
electronic, and optical characteristics [5]. In this research, a
full demonstration of the effects of Van der Waals forces on the
mechanical, electronic, and optical properties of TMDs was
carried out with MoX2 (X = S, Se, and Te).

Transition metal dichalcogenides are types of materials that
can be metallic or semiconducting [6]. The semiconducting
TMDs represent the layered materials [7], they can also be
classified as direct or indirect energy band gap materials. For
example, MoS2, MoSe2, WS2, and WSe2 are direct band gap
semiconductors.

In this study MoS2, MoSe2 and MoTe2 were considered.
Molybdenum dichalcogenides MoX2 (X = S, Se, and Te)
in the most stable phase 2H-MX2 belong to the space group
P63/mmc (194), it has a hexagonal crystal structure with Wyck-
off positions of 2c and 4f with z coordination value of 0.3779.
Literature studies revealed that the study of the effect of Van
der Waals forces on this material using the new exchange func-
tional vdW-DF3 has not been conducted. Although MoS2 has
been reported as good material for hydrogen storage [8], there
are need to explore the potentials of other di-chalcogenides
such as MoSe2 and MoTe2. Based on the obtained results,
vdW-DF3 may demonstrate promising results on the electronic
and optical properties of TMDs. Table 1 demonstrates the
results on the effect of the pre-existing exchange functional
with the experimental values. Reports from Table 1, revealed
that there are still more problems to solve regarding the effect
of Van der Waals, for example, the obtained value for the
lattice parameter with PBE is 3.680 Å, that of rVV10 is 2.186

Å which is far away from the experimental value of 3.163 Å
obtained.

Therefore there is a need to use better exchange functional
such as vdW-DF3 for better improvement in the next gener-
ations’ sustainability science and technology to pave way for
optoelectronic applications.

2. Research Method

The three materials were first optimized using three selected
exchange-correlation functional (PBE, rVV10, and vdW-DF3)
by setting the Brillouin zone sample 12 × 12 × 3 Monkhorst
Pack k-mesh and 800 eV plane wave cut-off energy. The op-
timization was performed till the total energy and force con-
verged to 10−3 eV, this is true for all three materials (MoS2,
MoSe2, and MoTe2). The calculations were performed via ab
initio density functional theory (DFT) using plane-wave basis
as implemented in the Quantum-ESPRESSO package, this in-
cludes the optimization, and determination of equilibrium lat-
tice parameters, and electronic band gaps. An auxiliary pack-
age to the quantum ESPRESSO thermo_pw [1] was also used
in the calculation of the elastic constants and optical proper-
ties. To make an accurate comparison, Kohn-Sham equations
were applied by implementing the DFT ab initio quantum com-
puting framework within the Perdiew-Burke-Emzahope (PBE)
exchange functional [9], rVV10 and our novel VdW-DF3 func-
tional. Calculations were performed using a non-spin polarized
DFT to save computational costs. To ensure accurate results in
this study, TMDs were appropriately relaxed to appropriate ge-
ometries. For all three systems, the length and the height were
chosen as 12.03 Å each. The chiral/translation vectors were
constructed such that the maximum force, stress, and displace-
ments were set at 0.05 eV/Å each.

3. Results and Discussion

3.1. Structural and Elastic Properties

The equilibrium lattice parameters for the three systems
were determined by fitting energy volume in the standard equa-
tion of the state. This can also be obtained by polynomial fit
to the energy-volume data [10]. The lattice parameter can be
determined from equilibrium volume as:

a0 =
( V

K

)1/3
(1)

where K is the ratio c/a for the materials MoX2 (X = S, Se, and
Te). The crytallogrphic structure of MoS2, MoSe2 and MoTe2
are presented in Figure 1.

The lattice parameters were calculated such that the three
systems can be viewed as having a hexagonal P6_3/mmc sym-
metry with a lattice constant of 3.66 Å. The Mo-S and S-S bond
lengths are 2.415 Å and 3.131 Å respectively which agrees with
the available literature [11]. In the case of MoSe2, the bond
lengths of Mo-Se and Se-Se atoms were 2.424 Å and 3.113 Å,
respectively. To obtain significant results from our calculations,

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S. A. Yamusa et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1094 3

Figure 1: Crystal structure of 2H-MX2: (a) unit cell, (b) Top view

lattice parameters of MoTe2 were also studied, these were en-
sured to be 2.423 Å and 3.116 Å respectively. To further under-
stand the vdW-DF3 effects, we calculated the formation energy
based on the lattice parameters earlier reported [12, 13]. The
results are presented in Table 2. The result predicted the out-
put of the effects of Van der Waals under PBE, rVV10, and
the novel vdW-DF3 exchange functional. Table 1 presented the
result of the calculated equilibrium lattice parameter of Molyb-
denum Chalcogenide MoX2 for the three exchange-correlation
functional (PBE, rVV10, and vdW-DF3) compared to the avail-
able experimental and theoretical results.

Table 1: The calculated lattice parameters with the three functional are com-
pared with the available experimental and theoretical results

PBE
(Å)

rVV10
(Å)

VdW-
DF3
(Å)

Theor.
results
[15]

Expt. Ref.

a(Å) 3.680 2.186 3.161 3.163 3.160 [14]
MoS2 c(Å) 13.37 13.394 12.296 12.442 12.290 [16]

a(Å) 2.314 3.523 3.293 3.295 3.288 [17]
MoSe2 c(Å) 13.001 13.036 12.918 13.088 12.920 [18]

a(Å) 3.874 3.489 3.551 3.617 3.520 [19]
MoTe2 c(Å) 13.906 13.965 13.817 14.261 13.970 [11]

It can be seen that the vdW-DF3 successfully describe
accurately the lattice parameter of the three systems with only
1 % error. This shows that the functional performed excellently
in the determination of the lattice parameters of a bulk MoX2.

Molybdenum chalcogenide MoX2 (X = S, Se, Te) is a
hexagonal crystal with 2H-MoX2 as the most stable phase.
For this type of crystal, there are only five independent elastic
constants. The five elastic constants were used to check the
stability of the optimized structure using the Born stability
criteria [20] and to determine the Mechanical properties of the
materials for the three correlation functional. The calculated
properties were compared with the available literature both ex-
perimentally and theoretically [21, 22, 23], which is presented
in Table 2. The Born stability criteria were checked using
equations (2) - (4 [24].

C11 > |C12| (2)

2C213 < C33 (C11 + C12) (3)

C44, C66 > 0, (4)

where C66 = (C11 − C12) /2 and Ci j are the five independent
elastic constants for the hexagonal materials.

As presented in Table 2, VdW-DF3 revealed high mechan-
ical stability for all systems. Therefore it can be reported that
VdW-DF3 XC functional significantly improves correction to
Van der Waals problem in TMDs.

3.2. Electronic Properties
The electronic band structures of the Molybdenum chalco-

genides MoX2 (X = S, Se, Te) were calculated along the high
symmetry point of the Brillouin zone by following the k- paths
Γ − M − K − Γ for all the three systems. Results of the three
XC functional (PBE, rVV10 and vdW-DF3) were obtained as
presented in Figure 2. The valence band maximum (VBM) and
conduction band minimum (CBM) located at Γ and between
Γ − K, respectively, were used to determine the band gaps as
shown in Table 3.

To further explain the efficiency of VdW-DF3 XC func-
tional, the electronic band structure and density of states were
calculated for MoS2, MoSe2 and MoTe2 systems, the results
presented in Figure 2 show that MoS2 demonstrated band gap
of 0.79 eV, MoSe 2 revealed 0.88 eV and MoTe2 was found
to be 0.67 eV. This results showed significant improvement in
narrowing the band gap of TMDs by vdW-DF3 XC functional
which brought them to new applications for optoelectronics
[25]. Therefore vdW-DF3 XC functional make significant con-
tribution towards turning TMDs from wide gap to narrow gap
semiconductors. In terms the partial density of states (PDOS),
calculations were performed to determine contributions by
different s, p, d, f orbitals, the results are illustrated in Figure 2
(b, d and f).

To further elaborate on the nature of the band gap of the
three systems, the total density of State (TDOS) and partial
density of state for the MoS2, MoSe2, and MoTe2, are illus-
trated in Figure 2. In terms of MoS2 (Figure 2(b)), the lower
valance bands at -6.95 to 0.28 eV are composed mainly of Mo-
4d states and S-3p states, zero states were seen from 0.28 to
0.58 eV. The conduction bands are mainly due to Mo-4d and
S-3p states located at 0.68 to 3.92 eV, there are lower contribu-
tions above 4.002 eV up to the conduction bands. For MoSe2
(Figure 2(d)), the valance bands are composed of Mo-4d and
Se-4p states located at -5.99 to 0.24 eV, the width 0.47 eV to
0.0.36 eV is the Fermi level of zero states and above 0.35 eV
is mainly composed of Mo-4d and Se-4p states for conduction
bands. For MoTe2, valance band contribution starts from the
energy range of -5.69 to 0 eV mainly by Mo-4d Te-5p states,
Fermi level was show from 0 eV to 0.2 eV, and the conduction
bands contribution is the energy range above 0.74 eV mainly
by Mo-4d and Te-5p. it can beseen that vdW-DF3 was able to

3



S. A. Yamusa et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1094 4

Table 2: Elastic constants in Gpa, Bulk Modulus B in Gpa, Young modulus E in Gpa, and Shear Modulus G in Gpa For three functional for MoX2, (X = S, Se, and
Te)

Material Funct. C11 C12 C13 C33 C44 B G E B/G σ
PBE 214.39 51.92 13.43 55.27 17.73 57.860 97.784 40.13 1.442 0.218

MoS2 rVV10 218.01 53.89 18.86 69.27 4.48 65.303 73.342 27.933 2.338 0.313
vdW-DF3 94.21 76.71 25.85 305.07 5.00 79.171 50.102 17.964 4.407 0.395
PBE 78.34 64.68 46.85 403.90 2.36 83.32 83.698 47.07 1.046 5.001

MoSe2 rVV10 88.32 68.71 53.27 387.93 3.53 88.38 89.207 50.233 1.118 4.994
vdW-DF3 87.94 69.88 47.15 459.24 4.31 91.39 91.845 58.691 1.495 4.361
PBE 116.84 30.71 36.83 111.78 25.16 61.577 85.590 33.741 1.825 0.268

MoTe2 rVV10 119.70 33.14 20.33 69.05 26.98 48.538 82.769 34.039 1.426 0.216
vdW-DF3 122.44 31.08 13.33 54.48 27.91 42.193 82.021 34.873 1.210 0.176

Figure 2: Electronic band structure and density of state of MoS2 (a) and (b),
MoSe2 (c) and (d), and (e) and (f) for MoTe2 using three the fuctionals

Table 3: The energy gap of MoS2, MoSe2, and MoTe2 with the three exchange-
correlation functionals

System PBE
(eV)

rVV10
(eV)

vdW-DF3
(eV)

Ref.

MoS2 0.84 0.85 0.79 This work
MoSe2 0.84 0.75 0.88 This work
MoTe2 0.73 0.67 0.67 This work

figure out all orbitals contributions as against PBE and rVV10
XC functionals.

3.3. Optical Properties
The optical properties of Mos2, MoSe2, and MoTe2 bulk

crystals with polarization along x-direction (in-plane) are cal-
culated using independent particle approximation by solving
time-dependent density-functional theory (TDDFT) and linear
response technique [3], using the Sternheimer approach within
Thermo_ pw code [1], a proprietary branch of the quantum
ESPRESSO project [4]. The calculated real and imaginary parts
of frequency-dependent microscopic dielectric function in the
energy range of 0 to 21 eV were plotted. The imaginary parts
of the dielectric function of MoS2 (Figure 4), MoSe2 (Figure 5),
and MoTe2 (Figure 6) were obtained from interband transition
for the parallel and perpendicular direction of the electric field
as computed from equation (3) [5], however, the real part of
the frequency-dependent dielectric function was obtained from
Kramers-kroning relation as shown in equation (6) [2]:

�2(ω) =
2πe2

Ω�0
Σκ,ν,c

∥∥∥∥λ̄. 〈ψck|u.r|ψνk〉∥∥∥∥2 δ (Eck − Eνk − E) . (5)
where λ̄ is the polarization vector of light and the integral

is over the Brillouin zone, u, ω, e, ψck, ψ
ν
k are the polarization

vector of the incident electric field, frequency of light, the elec-
tronic charge, and conduction and valance band wave function
at k, respectively.

�1(ω) = 1 +
2
π

P
∫
∞

0

ω′�2(ω′)
ω′2 −ω2

dω′, (6)

where P denotes the integral’s principle value. The computed
real and imaginary dielectric functions of MoS2, MoSe2, and

4



S. A. Yamusa et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1094 5

Figure 3: Real and imaginary dielectric functions for MoS2 with respect to the (a) PBE (b) rVV10 and (c) vdW-DF3 functionals

Figure 4: Real and imaginary dielectric functions for MoSe2 with respect to the (d) PBE (e) rVV10 and (f) vdW-DF3 functionals

MoTe2 within the three functionals are plotted in Figures
3, 4 and 5, respectively. The result shows that the interband
transition due to Mo-4d and S-3p, Se-4p, and Te-5p states move

to lower energies from MoS2, MoSe2, and MoTe2, respectively.
Both MoS2, MoSe2 and MoTe2 materials show anisotropy [8]
in the energy range from 0 to 7.5 eV and isotropy at higher

5



S. A. Yamusa et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1094 6

Figure 5: Real and imaginary dielectric functions for MoTe2 with respect to the (a) PBE (b) rVV10 and (c) vdW-DF3 functionals

energy.

To further confirm the band gap in the three systems, a
bound state can be seen at 2.0 eV, 1.1 eV and 1.2 eV for
PBE, rVV10 and vdW-DF3 functional respectively, the results
obtained by vdW-DF3 was found to be in good agreement
with previous theoretical results [15]. Similar results were also
obtained for MoSe2, MoTe2, these were presented in Figures 4
and 5, respectively.

To describe optical absorption, the imaginary dielectrics for
all systems were studied. Favourable results were obtained by
vdW-DF3 functional, for example higher optical absorptions
were observed for MoS2 (Figure 3c) at 22.5 cm−1 which cor-
responds to 2.80 eV, this is the absorption in the visible range,
other functionals only demonstrated absorption in the infra-red
range, which significantly underestimates the absorption char-
acteristcs of TMDs.

4. Conclusion

To conclude this work, results so far obtained brought out
new hidden properties of TMDs which failed to be reported by
PBE and rvv10 functionals. Calculation of the elastic properties
revealed that TMDs are more stable with vdW-DFT3, higher
moduli of elasticity such as Young’s Moduli, shear moduli and
bulk moduli were significantly improved with well agreement
with theoretical results. From the results of electronic proper-
ties, it revealed that bulk TMDs can be turned from being a wide
gap semi conductors to being a narrow gap semiconductors, it
also shows that d’orbital majorly contributed to narrowing the

band gap in all the sysyems. Higher optical absorption are also
repoted by vdW-DF3, this brought the systems under study as
potential candidates for optoelectronics [25].

Acknowledgement

The authors acknowledge Dr. Abdu Barde College of Vo-
cational and Technical study, Department of Science and Lab-
oratory Technology, Dutse, Jigwa State, and the University of
Technology Malaysia for financial support, facilities, and ser-
vices of high-performance computing on this research work.

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