J. Nig. Soc. Phys. Sci. 3 (2021) 292–297

Journal of the
Nigerian Society

of Physical
Sciences

Application of computational chemistry in chemical reactivity: a
review

C. W. Chidieberea,∗, C. E. Durua, JP. C. Mbagwub

aDepartment of Chemistry, Imo State University, Surface Chemistry and Environmental Technology (SCENT) Research Unit, Department of Chemistry, Imo State
University, Owerri, Imo State, Nigeria

bDepartment of Physics, Faculty of Physical Sciences, Imo State University, Owerri, Nigeria.

Abstract

Molecular orbitals are vital to giving reasons several chemical reactions occur. Although, Fukui and coworkers were able to propose a
postulate which shows that highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) is incredibly important
in predicting chemical reactions. It should be kept in mind that this postulate could be a rigorous one therefore it requires an awfully serious
attention in order to be understood. However, there has been an excellent breakthrough since the introduction of computational chemistry which
is mostly used when a mathematical method is fully well built that it is automated for effectuation and intrinsically can predict chemical reactivity.
At the cause of this review, we’ve reported on how HOMO and LUMO molecular orbitals may be employed in predicting a chemical change by
the utilization of an automatic data processing (ADP) system through the utilization of quantum physics approximations.

DOI:10.46481/jnsps.2021.347

Keywords: HOMO-LUMO, Computational methods, and Chemical reactivity

Article History :
Received: 14 August 2021
Received in revised form: 30 September 2021
Accepted for publication: 01 October 2021
Published: 29 November 2021

c©2021 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: E. A. Emile

1. Introduction

Frontier molecular orbital theory was postulated in 1952
by Kenichi Fukui where he described highest molecular orbital
(HOMO) and lowest unoccupied molecular orbital (LUMO) in-
teraction from molecular orbital theory [1]. In 20th century,
electronic structure of a molecule was described using molec-
ular orbital theory from quantum physics [2], where electrons
are treated as moving under the control of the atomic nucleus
within the bulk molecule but aren’t treated as a personal at-
traction between atoms in line with molecular orbital theory.

∗Corresponding author tel. no: +234
Email address: oraclewizzy@gmail.com (C. W. Chidiebere )

The reason of the spacing of electrons was given by quantum
physics. However, quantum physics also explains the energetic
behaviour of electrons as molecular orbital which house two
or more atoms [3, 4]. There’s little question, that molecular
orbital theory has changed how chemical bonding is viewed
through the approximation of the states of bonded electrons
i.e., viewing the molecular orbital as linear combinations of
atomic orbital (LCAO) [5]. These approximations are made by
applying the density functional theory (DFT) model or Hartree-
Fork (HF) models to the time independent differential equation.
These approximations are the fundamental theories in compu-
tational chemistry [6]. They’ll be modeled and automatic in a
very automatic data processing system to help easy calculation.
However, computational chemistry may be utilized in different

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number of ways but the foremost important one is to model a
molecular system by synthesizing the molecule in a laboratory
[7]. Confine in mind that computational models don’t seem
to be always perfect but it can rule out about 90% of possible
molecules that don’t seem to be good for his or her intended use
[8], however, computational chemistry makes it easy to under-
standing a controversy fine.

In the study of chemical reactivity, both state and open sys-
tem (involving radicals, cations, and anions) calculations are
performed to establish a result [9]. These methods are fre-
quently unreliable, especially when diffuse functions must be
included in the basis set. As a result, it is useful to have a tech-
nique that can extract all of the information required directly
from the findings of the molecular system’s basis state calcu-
lation [10]. Thus, the knowledge of frontier molecular orbitals
has imparted on the understanding of chemical reactions and is
accustomed to distinguish between the parts of orbital that have
opposite signs.

2. Highest Occupied and Lowest Unoccupied Molecular Or-
bital (HOMO-LUMO)

Highest occupied molecular orbital (HOMO) is that outer-
most orbital where the very best energy orbital is contained,
the orbital acts as an electron donor while lowest unoccupied
molecular orbital (LUMO) is that orbital with the lowest un-
occupied energy level which acts as the electron receiver. Ac-
cording to the frontier molecular orbital theory, the formation
of a transition state is because of a synergy between the fron-
tier orbitals of the reactants [11]. The stability of a molecule
is determined by HOMO-LUMO gap which is known as the
dissimilarity in energy between the HOMO and LUMO of the
molecule [12]; thus, a large HOMO-LUMO gap implies high
stability for the molecule in chemical reactions while a tiny low
one implies low stability. Thus, the HOMO-LUMO gap may
be defined mathematically as [13]:

∆E = ELU MO − EHOMO (1)

where ∆E is the Energy gap, ELU MO is the Energy of LUMO
and EHOMO is the Energy of HOMO. Keep in mind that when
the HOMO-LUMO gap is large, the molecule is said to be hard
and a soft molecule has a small energy gap [14]. Thus, chemical
hardness is the opposition to change in electron cloud density
or electron distribution of a chemical system and chemical soft-
ness is the inverse of chemical hardness [15]. Chemical hard-
ness and chemical softness can be shown mathematically as

η =
I − EA

2
(2)

σ =
1
η

(3)

where η is known as chemical hardness, σ is known as chemi-
cal softness.

Ionization potential relates to energy of the HOMO while
the energy of the LUMO relates to the electron affinity thus the
HOMO and LUMO energies can be applied in the calculation
of the ionization potential (I) and electron affinity EA. From
Koopmans theorem [16]

I = −EH (4)

EA = −EL (5)

3. Density Functional Theory (DFT)

Density functional theory also referred chemical reactiv-
ity theory is a variational method that’s currently the foremost
promising approach which is employed to determining the elec-
tronic structure of matter [17]. The N-particle differential equa-
tion yields density functional theory, which is totally stated in
terms of the density distribution of the base state and the sin-
gle particle wave function. It’s been a helpful tool which is
employed in analysis, prediction and elucidation of the results
of chemical reactions. Following the work of Parr et al [18],
advantageous number of ideas has been derived from the reso-
lution of the density of molecular systems using DFT. its req-
uisite to own recourse on the Kohn-Sham theory by calculat-
ing the molecular density, the system’s energy and energies of
the orbital especially; those connected to the frontier orbitals.
The kohn-Sham method creates a precise connection between
the density and state energy of a non-interacting fermion sys-
tem and the real many bodies system given by the Schrodinger
wave equation [19].

−~
2m

∂2

∂x2
= i~

∂Ψ

∂t
(6)

The basis of DFT has confirmed the existence of a univer-
sal density functional, and the system’s properties are acquired
by completing calculations using the functionals [20]. Because
they’re approximate functionals rather than universal function-
als, some of them are good at predicting certain qualities while
others are good at predicting others. Density functionals that
are excellent at defining the features of each molecule system
with a specific functional group can occasionally be discovered.
For the study of molecular systems, it’s quite advantageous to
employ different density functionals for a separate functional
group that you merely want to require in [21]. DFT reduces
the time it takes to compute the bottom state parameters of sys-
tems of synergizing particles to the same precision as the results
of single-particle Hartree-type equations, which is why it’s so
beneficial for systems with many electrons. The goal of DFT
calculations is to show how the electron density is related to the
bottom state. By using the results of the Kohn-Sham equations,
the definition of electron density is given [22].

ρ(r) =
∑

i

Ψi(r)Ψ
∗
i (r) (7)

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Figure 1: Aromatic compounds with the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) were discovered using
DFT.

4. Hatree-Fock Equation

The Hartree-Fork Fork technique is an approximation ap-
proach used in computational chemistry to derive the wave func-
tion with the energy of a quantum body system in an immobile
mmobile condition [23]. Hartree-Fork theory is a fundamen-
tal notion in electronic structure theory; it is the cornerstone
of molecular orbital theory, which states that each electron’s
mobility may be described by a single-particle particle func-
tion orbital that is independent of the other electrons’immediate
motion [24].The Hartree-Fork resolves the set of spin orbitals
which reduce the energy and devote the best single determinant.
The equation is given as [25].

h(X1)χi(X1) + h
∑
j,i

[∫
d x2

∣∣∣x j(X−1)∣∣∣2 r−112
]
χi(X

−1)

−
∑
j,i

[
d x2χ

∗
j (X

2)χi(X
2)r−112 x j(X

1) = �iχi(X1)
]
,

(8)

where �i is the energy eigenvalue associated with orbital (χi).
Hence the Hartree-Fork equations can be resolved mathemat-
ically i.e., exact Hartree-Fork, or can be solved in the spaced
spanned by a set of fundamental functions (Hartree-Fork-Roothan
equations) where their solutions are dependent on the orbitals
[26].

Reducing Hartree-Fork equation to Hartree-Fork-Roothan
equation in matrix form, we have:∑

ν

FµνCνi = �i
∑
ν

S µνCνi (9)

5. Frontier molecular orbital calculation and application of
computer system in computational chemistry

Calculation of molecular orbital energy may be done us-
ing ab initio methods; the tactic applies electron and also the
nuclei synergy as its fundamental component. It attempts to
calculate the electronic differential equation by assigning the
locations of the nuclei and therefore the electrons surrounding
it. Hence, electronic energy of a molecule may be calculated by
treating the general electronic wave function as being a product
of molecular orbitals which is predicated on Slater determinant
of molecular orbitals so as to satisfy the Pauli exclusive princi-
ple [27]. So, we will write a molecular wave function as:

Ψ = φ1φ2φ3φ4 · · ·φn (10)

where each molecular orbital is written in form of a linear com-
bination of atomic orbitals that is,

Ψi =
∑

i

cikΦ (11)

The molecular orbitals will be calculated by using the Hartree-
Fock method [28]. The Hamiltonian operator has terms for the

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potential and mechanical energy of the electrons; it involves
the interactions between the nuclei and the electrons, and also
the electron-electron repulsion. Note the molecular orbitals are
conceived by making an initial supposition at the coefficients
[29]. Hence

H − ES = 0 (12)

H contains the Hamiltonian matrix elements and S is known as
the overlap matrix elements. Solving this determinant will give
the answer of molecular orbital energy E, which is substituted
into the overall equations to allow new values of the coefficients
(ci ). This whole process is repeated until the coefficients don’t
change from one cycle to the other. The orbitals can then be de-
scribed as self-consistent, and this process is thought as a self-
consistent field procedure [30] where each molecular orbital is
created from a mixture of atomic orbitals.

When one function is employed for every filled atomic or-
bital for instance, using, an orbital for an atom, this basis set
is termed a minimal basis [31]. Using more functions than
this may improve the calculation i.e., it’ll tend to extend the
pliability and quality of the general wave function. From the
variational principle we all know that a stronger wave function
will provides a lower energy [32]. The larger the idea set the
higher the calculation, if a large number of functions i.e., a re-
ally large basis set is employed to point out the atomic orbitals;
we approach the Hartree-Fock limiting energy. However, a very
cheap energy is achieved by the Hartree-Fock method [33].

Frontier molecular orbital theory is an exemplification of
molecular orbital theory which elucidates HOMO/LUMO syn-
ergies [34]. During 1952, Kenichi Fukui put forth a paper within
the Journal of Chemical Physics which he titled “A molecular
theory of reactivity in aromatic hydrocarbons”. His work exam-
ined the frontier orbitals and additionally the consequences of
the HOMO and also the LUMO on reaction mechanism specif-
ically which is the reason why it’s called frontier molecular or-
bital theory (FMOT) [35]. Fukui noticed that an honest ap-
proximation for reactivity can be established by observing the
HOMO-LUMO. This was founded on principal point obser-
vance of molecular orbital theory of two molecules which shows
that the occupied orbital of unlike molecules opposes one an-
other, thus, the negative charges of a molecule attract positive
charges of another molecule as such, the occupied orbital of an-
other molecule and the unoccupied orbital of the opposite syn-
ergize with one another per se causing attraction [36].

Though this theory could also be a rigorous narrative of
chemical phenomena, impediment within the mathematical as-
pect may well be too difficult that it’ll not be viable to resolve a
controversy accurately. Since the introduction of computational
chemistry, most rigorous theories can now be treated with little
difficulty. Computational chemistry is employed in a very num-
ber of various ways, one in all which is to model a molecular
system before synthesizing the molecule within the laboratory
[37]. However computational models don’t seem to be always
perfect but it can rule out about 90% of possible molecules that

don’t seem to be good for its intended use [38].

However, it is very helpful because to synthesizing only
one (1) compound could take months of labour and raw ma-
terials and furthermore generate waste matter. Computational
chemistry helps in comprehending an issue wholly. There are
some properties of a molecule that may be gotten computation-
ally more easily than by experimental means [39]. There also
are deep views into molecular bonding, which might be got-
ten from the inference of computations that can’t be obtained
from any scientific method. Thus, many experimental chemists
are now making use of computational modeling to induce addi-
tional understanding of the compounds being observed within
the laboratory [40].

Going back to history, Paul Cruzan, Mario Molina, and
Sherwood Rowland who are computational chemists were awarded
the laurels in chemistry in 1995 for forming mathematical mod-
els that applied the thermodynamic and chemical laws to eluci-
date how ozone were formed and decomposed within the atmo-
sphere [42]. Although, computational chemistry wasn’t usually
imagined as its own separate field of study until 1998, when
Walter Kohn and John Pople were awarded the Nobel prize in
Chemistry for their exhibition of density functional theory and
disclosure of computational methods using quantum chemistry
respectively [43].

Computational chemistry is not the same as engineering sci-
ence; however professionals from both fields frequently collab-
orate. Computer scientists spend the majority of their work as-
sembling and testing computer algorithms, data visualization
skills, and software and hardware solutions [44].Computational
chemists work alongside laboratory scientists and theoretical
scientists to put all of this modeling, simulation, data analysis,
and visualization expertise to good use in their study. Com-
putational chemists can employ supercomputers and comput-
ing clusters, which are high-performance computing systems,
to solve issues and run simulations that require a large quantity
of data [45]. Electronic structure approaches, molecular dy-
namics simulations, quantitative structure-activity connections,
and complete statistical analysis are among the tools used by
computational chemists.

To combine chemical theory and modeling with actual ob-
servations, computational chemists use statistics, mathemati-
cal algorithms, and massive databases. Some computational
chemists create physical step models and simulations, and then
utilize statistics and data analysis approaches to extract useful
data [46]. Progress in computer visualization allows computa-
tional chemists to highlight complex analyses in a more com-
prehensive format, which they may use to explain experiments
and current materials, as well as validate the results.

Computational chemistry advances our understanding of how
the world works, aids manufacturers in developing more ef-
ficient and productive operations, allows for the characteriza-
tion of new chemicals and materials, and aids other researchers
in extracting relevant knowledge from large amounts of data.

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C. W. Chidiebere et al. / J. Nig. Soc. Phys. Sci. 3 (2021) 292–297 296

Figure 2: Shows the aid of computer system in elucidation of chemical be-
haviour

Quantum physics and thermodynamics are used in computa-
tional chemistry to learn about the character and fundamental
properties of atoms, molecules, and chemical reactions [47].

Recent researches showed that quantum physics (QM) and
molecular mechanics are the foremost used models in compu-
tation. Most computation software’s utilized by researchers in-
clude:

• Integrated packages

• ab initio and DFT software

• Semi empirical software

• Graphics package

• Special purpose program

• Molecular mechanics/molecular dynamics/Monte Carlo
software

Software packages differ in cost, functionality, efficiency,
automation, robustness and ease to use. These concerns make
a difference in determining what computational projects is suc-
cessful and the way much work which will be involved [48].

6. Conclusion

The quest for understanding chemical reaction mechanism
has really caused many researchers to go into in-depth study
to source out path of reactions and the cause of the reactions.
HOMO-LUMO molecular orbitals have changed the perspec-
tive at which chemists view chemical reactions with the help of
quantum mechanics which has given the exact explanation on
the behaviour of electrons. Apart from this, scientists went fur-
ther to develop models from quantum mechanics approxima-
tions of molecular orbital theory. These approximations were
achieved by applying Hartree-Fork and density theory function-
als which are approximations to determine a many electrons
body system. A further study on the variational methods can
improve research work on computational chemistry thus giving
a more easy and reliable way for solving problems.

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