**1. Introduction**

Since pronounced by Dirac in 1929 that "The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws lead to equations much too complicated to be solved" [1]. So, a great effort has been devoted to develop the numerical methods to solve Schrödinger equation for atoms and molecules which help to reveal the physicochemical process and properties. The density functional theory has become an important tool for physicst, chemist, and material scientist. Over the past three decades, DFT has been developed successfully challenging traditional wavefunction-based methods for large scale quantum chemistry calculations. DFT has become an important method and suitable alternative to *ab initio* method as well as cheaper in terms of computational cost. The well-developed modern DFT is applicable to quantum as well as classical systems based on the theorems of Hohenberg

and Kohn [2]. The system is having N-particles allowing interactions with a given interparticle interaction, the total energy is completely derived by specification of external field φ(r) using Hamiltonian and ground-state wave function. Many electron wave functions j(r1, r2, rN) could be obtained by solving Schrödinger equation provided all the necessary information about the system. The density of single particle possibly acquired which on performing integration over any one of the single directions of coordinates of N-1 electrons,

$$\boldsymbol{\rho}\left(\mathbf{r}\right) = \mathbf{N}\left[\ldots\right] \boldsymbol{\psi}\left(\mathbf{r}\_1, \mathbf{r}\_2, \mathbf{r}\_3, \ldots, \mathbf{r}\_{\aleph}\right) \boldsymbol{\psi}\left(\mathbf{r}\_1, \mathbf{r}\_2, \mathbf{r}\_3, \ldots, \mathbf{r}\_{\aleph}\right) d\mathbf{r}\_2 \ldots \boldsymbol{\mathrm{d}r\_{\aleph}}\tag{1}$$

where,

$$\int \rho(\mathbf{r})d\mathbf{r} = \mathbf{N} \tag{2}$$

In other words, the functional of φ(r) gives the ground state energy of the system. During the development of DFT, Hohenberg and Kohn sham (HK) shown that the correspondence between external field φ(r) and the single-particle density ρ (r) and its consequence lead to the total ground state energy with the functional of ρ(r) using the following equation,

$$\mathbf{E}\left[\boldsymbol{\wp}\right] = \mathbf{E}\_{\rm o}\left[\boldsymbol{\wp}\right] + \int d\boldsymbol{r} \,\boldsymbol{\wp}(\mathbf{r})\boldsymbol{\wp}(\mathbf{r})\tag{3}$$

Moreover, Hohenberg and Kohn proved and called a second theorem which provides an energy variational principle. Further, they showed the trial density which satisfies ρ = (r N ) *dr* ∫

$$\mathbf{E}\left[\overline{\rho}\right] \ge \mathbf{E}\_{\mathfrak{g}} \tag{4}$$

Here, Eg is the ground state energy. In Eq. (4), the Left Hand Side (LHS) and Right Hand Side (RHS) attain equal when ρ (r) is the true ground state single particle density.

In case, E0() were known for an interacting electron of a given system, then the Eqs. (3) and (4) allows to calculate the ground state energy and density of electron of any multi-electron system in a given arbitrary external field. Though, HK approach produce the total energy calculation but it does not provide any prescription for its determination. So, it is ultimate goal of a researcher to establish and develop accurate approximate functionals. Of course, in later 1990s, a number of functionals was developed to produce the experimental observations. A familiar and few density functionals are hybrid density functional B3LYP [3–5] introduced by M.J.Frisch in 1994, further gradient-corrected correlation functional: Perdew-Burke-Ernzerhof (PBE) [6] was introduced by Ernzerhof, M. in 1996, the Global Hybrid Meta-GGAs Minnesota [7–11] functionals such as M05, M06-HF, M06, M06-2X, M08-HX, M08-SO, revM06, MN15-L introduced by Donald G. Truhlar during 2005 to 2016, dispersion corrected functionals DFT-D [12], DFT-D3 [13], then the First GGA functionals were introduced by Stefan Grimme in the period of 2006–2014, moreover the another dispersion corrected method ωB97XD [14] was introduced by M. Head-Gordon in 2008. The functionals played an important role

*Applications of Density Functional Theory on Heavy Metal Sensor and Hydrogen Evolution… DOI: http://dx.doi.org/10.5772/intechopen.99825*

to predict the properties of unknown materials and also explain the post-process analysis of experimental results. Selection of the method is the key in computational chemistry to achieve appropriate results.

In the past 30 years, the evolution and success of DFT for various chemical applications, emerged as the most popular electronic structure method for the chemists and plays a vital role in computational chemistry. To parameterize electronic structure theory methods and to give the guidance to chemist, nearly five different databases evolved so far *viz*, GMTKN, MGCDB84 [15], Minnesota2015B [16], DP284 [17], and W4–17 [18]. In addition to that the W4–17-RE [19], and MN-RE [20] are two newly developed databases for reaction energies. In 2018, Peverati P et al., comprised the above mentioned databases and published it in the name of ACCDB [21]-which includes data from 16 different research groups, for a total of 44,931 unique reference data points. Now a days, data points and databases plays an important role to proceed the research further to get clear picture on history. This kind of databases will be the standard reference for the future research. In computational chemistry, the various DFT methods have been implemented and studied in several systems including chemosensor [22, 23], hydrogen evolution reaction (HER) [24–26], oxygen reduction reaction (ORR) [27, 28], oxygen evolution reaction (OER) [29], molecular machines [30, 31], DNA mutation [32–34], selective etching [35, 36], atomic layer deposition (ALD) [37, 38] etc. Although a number of applications of DFT were reported, this book chapter precisely focused on heavy metal sensor and hydrogen evolution reaction.
