Excerpt

## Table of contents

List of figures

1 Introduction

1.1 Introduction to pervoskite structure

1.1.1 Introduction to CaTiO

1.1.2 Structure

1.1.3 Formation of band gap

1.2 Scope of present study

1.3 How we approach?

2 Theoritical Background

2.1 General Consideration

2.2 Many-Body Hamiltonian and Born-Oppenheimer Approximation

2.3 Hartree-Fock Method of Self Consistent Fields

2.4 Density Functional Theory

2.4.1 General Consideration

2.4.2 Thomas-Fermi Model

2.4.3 Hohenberg-Kohn Theorem

2.4.4 The Kohn-Sham Formalism

2.5 The Exchange -Correlation Functional

2.5.1 Local Density Approximation

2.5.2 Generalized Gradient Approximation

2.5.3 Solution of the Kohn-Sham equations: Self-consistency iteration procedure

2.6 LDA+U Method

3 Computational Details

3.1 General Consideration

3.1.1 PWscf

3.1.2 Post Processing

4 Results and Discussion

4.1 Convergence Tests

4.1.1 Kinetic energy cut-off(ecutwfc)

4.1.2 K-points grid

4.1.3 Lattice parameter

4.1.4 Band structure

4.2 LDA Method

4.2.1 Calculation of band structure of perovskite CaTiO3 by LDA method

4.2.2 Effect of strain on atomic displacement by LDA method

4.2.3 Effect of strain on indirect band gap by LDA method

4.2.4 Effect of strain on direct band gap by LDA method

4.2.5 Partial density of states of perovskite structure of CaTiO3 by LDA method

4.3 LDA+U Method

4.3.1 Determination of Hubbard potential

4.3.2 Calculation of band structure of perovskite CaTiO3 by LDA+U method

4.3.3 Effect of strain on indirect band gap by LDA+U method

4.3.4 Effect of strain on direct band gap by LDA+U method

4.3.5 Partial density of states of perovskite structure of CaTiO3 by LDA+U method

4.3.6 Comparision of band gap between LDA and LDA+U method

5 Conclusions and Concluding Remarks

References

## Acknowledgements

At first I want to offer my sincere gratitude to my supervisior Dr. Rajendra Prasad Adhikari for aspiring continuously and supporting guidance throughout my work. His extreme motivation and continuous good support made me to realize my potential in this research work and made complicated problems to solve step by step.

I would like to extend my deep respect to Professor Homnath Paudel, Prof. Dr. Bhadra Pokherel for their kind cooperation and support. Also, I would like to acknowledge my friends Mr. Aanand Sapkota, Mr. Amrit Kumar Aryal, Mr. Pramod K.C., Mr. Sudip Basnet and Mr. Bhojraj Bhandari for their kind support. I also want to give special thanks to my dear wife Mrs. Tara Poudel and dear aunt Mrs. Samjhana Adhikari for their kind help during my study.

This work would not have been possible without love, consistence help and patience of my family. So, I want to give a huge respect to my parents: Mr. Rishi Ram Sharma and Mrs. Sita Adhikari. Finally, I would also like to give a lot of thanks to my supervisor Dr. Rajendra Prasad Adhikari for computational support.

## Abstract

CaTiO3 is generally used for semiconductor, laser, microwave, biochemical applications and photovoltanics. We have carried out the first-principles calculation to determine the optimized geometry as well as electric properties of perovskite CaTiO3. The first-principles calculation was perforperformed using Density Functional Theory (DFT) under Local Density Approximation (LDA) and LDA with Hubbard potential (LDA+U), employed with Quantum ESPRESSO code.

During present calculation, the kinetic energy cut-off energy is found to be 60 Ry, k-point grid is found to be 8 x 8 x 8 and lattice parameter is found to be 7.2 Bohr, which fairly agrees with experimental as well as previous calculated data. At first, we studied the CaTiO3 to investigate the effects of inplane strain from -5% to +5% of its standard (optimized) lattice constant by LDA method. The properties studied are band gap and polarization of the crystal. We obtained the crystal as indirect bandgap insulator in every case which is also verified experimentally. The obtained band gap by LDA method was 1.98 eV, which is about 42% less than the experimental band gap. It is mainly due to the fact that LDA underestimate the band gap. The strain characteristics are very important to determine the deformation of a physical part. From the calculation of effect of strain on the bandgap, it is found that the bandgap increases with the contraction and decreases with the expansion of the crystal from the equilibrium value. Next, we studied the atomic displacements of inplane and out of plane Oxygen atoms and central Titanium atoms. Their displacements behave differently during contraction and expansion. Even during contraction and expansion, their displacements are very different. After that, we calculated Partial Density Of States (PDOS) and observed that band edges of the valence band arise from O 2p states, whereas the conduction band arises from Ti 3d orbitals, which is also verified by other works.

Finally, we calculated electronic band structure with and without strain by using LDA+U method and compared with our LDA and other results. The obtained band gap by using LDA+U method was 3.34 eV, which is in close to the experimental band gap. Next, we observed that PDOS calculation by LDA+U method is not different than our LDA results but the band gap is wider than that of LDA.

## List of figures

1.1 Unit cell of simple cubic CaTiO3. This unit cell is generated from VESTA(version:3.3.9)

1.2 Mineral composed of Calcium Titanate. Photo credit:Valery Voennyy

4.1 The plot of total energy versus kinetic energy cut-off value for perovskite CaTiO

4.2 Plot of total energy versus k-points grid for the simple cubic perovskite CaTiO

4.3 Plot of total energy versus lattice parameter

4.4 The first brillouin zone of simple cubic CaTiO3. M-r-X-R-r reperesents irreducible Brillouin zone. This figure is taken from the Ref.[1]

4.5 Band structure of perovskite CaTiO

4.6 Effect of strain on atomic displacement of inplane Oxygen atoms

4.7 Effect of strain on atomic displacement of out of plane Oxygen atoms

4.8 Effect of strain on atomic displacement of Titanium atoms

4.9 Effect of strain on indirect band gap of CaTiO3 by LDA method

4.10 Effect of strain on direct bandgap of CaTiO3 by LDA method

4.11 Partial density of states for the perovskite CaTiO3 by LDA method

4.12 Plot of graph between Hubbard potential and its corresponding bandgap

4.13 Band structure of perovskite CaTiO3 by LDA+U method

4.14 Effect of strain on indirect bandgap of CaTiO3 by LDA+U method

4.15 Effect of strain on direct bandgap of CaTiO3 by LDA+U method

4.16 Partial density of states for the perovskite CaTiO3 by LDA+U method

4.17 Comparision of band structure of perovskite CaTiO3 by LDA and LDA+U method

## Chapter Introduction

### 1.1 Introduction to pervoskite structure

A family of oxides having the general formula ABO3[2]with A = a metallic cation, B = a transition metal ion and O = Oxygen[3], are called perovskites. It is named after the Russian mineralogist, L. A. Perovski (1792-1856)[4]. Here, in ideal form, the crystal structure of simple cubic perovskite compounds (ABO3) can be explained as consisting of corner sharing BO6 octahedra with ’A’ cation that occupies 12-fold coordination site formed in the middle of the cube of octahedra[5]. Many of the perovskites are cubic, but they can exhibit one or more structural phase transitions at low temperature. They can exhibit variety of solid state phenomenon like metals, semiconductors, insulators and even superconductors. Some have localized electrons, whereas some of them have delocalized energy band states and the remaining others show transitions between these two types of behaviour[3]. The atomic arrangement of perovskite structure was first found for the mineral perovskite, Calcium titanate (CaTiO3)[6]. Some examples of the compounds containing perovskite structure are: BaSiO3, BaTiO3, PbTiO3, BaGeO3 etc[7]. The atomic position in cubic perovskites is given below[8]:

illustration not visible in this excerpt

Figure(1.1) shows the structure of a simple cubic perovskite structure of CaTiO3, which also represents the perovskite structure with a chemical formula ABO3, where the smallest spheres are Oxygen atoms, the middle spheres are Ti atoms and the biggest spheres are the Ca atoms.

#### 1.1.1 Introduction to CaTiO3

Calcium Titanate (CaTiO3) is a ceramic material[9]having simple cubic crystal with per- voskite structure. It is also known as Calcium Titanium Oxide[10]. It was first material discovered in Russia in ABO3 composition[11]. Its molar mass is 135.943 g/mol with density 3.98 gmcm—[3]. Its melting point is 1975°C and boiling point is 3000°C. It is a diamagnetic colourless solid but appears coloured due to the presence of impurities[4].

Calcium titanate (CaTiO3) powders can be prepared by reacting Calcium carbonate (CaCO3) and Titanium dioxide (TiO2) under controlled conditions by taking the molar ratio 1:1 and 3:2. The obtained mixture are calcinated at 1150°C in successive thermal cycles. Then, CaTiO3 is finally prepared at 1960° C[9].

This material can be used for semiconductor, laser, microwave, biochemical applications and photovoltanics[10].

#### 1.1.2 Structure

illustration not visible in this excerpt

Fig. 1.1 : Unit cell of simple cubic CaTiO3. This unit cell is generated from VESTA(version:3.3.9).

The structure of CaTiO3 which we are going to discuss, has a simple cubic structure with lattice constant of a = 3.81A and the space group 01 — Pm3m with a = b = c and a = 90°,

p = 90° and Y = 90°. The unit cell has the volume of 55.3063 A[3]. The Ca atoms are located at the centre of the cube whereas Ti atoms are located at the corners and the Oxygen is placed at the centeres of the twelve cube edges. The TiO6 are perfect with the angle of 90° with six equal Ti-O bonds at 1.905A. Each Ca atom is surrounded by 12 equidistant Oxygen atoms at 2.694A. The structure can be distorted to complex structure on the basis of temperature, pressure and chemical composition[10].

illustration not visible in this excerpt

Fig. 1.2: Mineral composed of Calcium Titanate. Photo credit:Valery Voennyy

#### 1.1.3 Formation of band gap

The band gap is formed by Oxygen and Titanium orbitals. The band edges of the valence band arise from O 2p states, whereas the conduction band arises from Ti 3d orbitals[12, 13], which is also verified from our calculation. The band gap is established by the Ti-O transistion and is tuned by the surrounding grid of Ca atoms[14].

For bulk crystrals, transmission measurements showed a band gap of about 3.4eV, whereas for powdered form of CaTiO3 it was measured between 3.86eV and 3.97 eV[10].

Density functional theory (DFT) calculations with various computing methods revealed band gaps near 2eV for the Ti-O transitions, which is a typical underestimation for the used methods. Calculations with adapting semi-empirical fitting factors showed a band gap between 3.8-4.2eV depending on the selected permittivity[10].

### 1.2 Scope of present study

Nowadays, the use of electronics as well as the use of alternative energy source like solar energy is increasing day by day. The demand of raw materials to prepare semiconductors and photovoltanic cells are being increased in the same order. Calcium titanate (CaTiO3), when doped with trivalent or pentavalent impurities, changes it from an insulator to a semiconductor or even show metallic behaviour and can be used as a transparent semiconductor[15]. Similarly, for the communication purposes like radar, global positioning systems, the materials with the high permittivity is required that can operate at microwave frequencies. For these purposes, CaTiO3-based ceramics can be used[15]. Also, they can be used for biomechanical applications due to their high elastic limit and low Young’s modulus[16], as well as they can be used for laser application due to high chemical stability at room temperature[17].

Nowadays, more developed countries are producing electricity from the nuclear reactor and there is the chance for the leakage of radioactive particles which cause a serious problem on the health of organisms and the environment. To control this harmful effect of radioactive materials, CaTiO3 can be used to immobilize fission products and other radioactive waste materials[18].

### 1.3 How we approach?

We performed this research work by using Quantum espresso (QE) suite. We have studied sructural and electronic properties of simple cubic Calcium titanate (CaTiO3) perovskite by using the Local Density Approximation (LDA) and LDA with Hubbard potential(U) (LDA+U). We first optimized the lattice parameter, band structure, and partial density of states (PDOS). Next, we applied strains in the range of ±5% to study the variation of direct and indirect band gaps. In the second phase, we estimated the value of Hubbard potential reference to experimental bandgaps. This way we calculated the new band gaps as a function of strains.

The structure of CaTiO3 can be simple cubic, orthorhombic or tetragonal[12, 18]. In this study, we choose simple cubic because of its simplicity of structure and computational cost.

## Chapter 2

## Theoritical Background

### 2.1 General Consideration

Solids comprises of dynamics of mutually interacting electrons and nuclei. By determining the eigenfunction, many properties of the interacting system can be obtained, which is the fundamental problem in many body system. In condensed matter physics, the solution to any problem lies in the exact determination of the electronic structure of atoms, molecules and solids as a whole. The starting point to investigate properties of materials is to solve many-body Schrondinger wave equation (SWE). The main problem associated to find the solution of SWE is the number of particles involved (about 10[23]atoms/cm[3]) and various interactions and coupling between them.

### 2.2 Many-Body Hamiltonian and Born-Oppenheimer Approximation

If a system contains N mutually interacting electrons and N’ nuclei interacting, then the total Hamiltonian[19] of that system is given by,

illustration not visible in this excerpt

where the indices i, j refer to electrons; indices l, J refer to nucleus; me represents the mass of electron and Ml represents the mass of nucleus; whereas ri and Rl represents the position of ith electron and Ith nucleus respectively; e is the charge of electron; Z/ is the atomic number of Ith atom and Z/e is the charge of Ith nucleus.

In Equation(2.1), the first term is the kinetic energy contribution to the Hamiltonian due to electrons and the second term is the kinetic energy contribution to the Hamiltonian due to nuclei; third term is the potential energy due to interaction between electrons and nuclei; whereas the fourth term is the electron-electron repulsive potential energy and the last term is repulsive potential energy between the nuclei. The term 2 appears in the last two terms to avoid the double counting of the inter-electronic and inter-nuclei interactions. The terms containing i=j in fourth term and /=J in the last term are excluded to avoid the self interaction terms. It is a difficult task to obtain the exact solutions of the Schrodinger equation of N-electron system using the Hamiltonian (H) given by Equation(2.1) as the equation is quite complex. Hence, certain approximations should be made to reduce it.

The many-body SWE related with the Hamiltonian(2.1) can be written as,

illustration not visible in this excerpt

where xi represents position co-ordinate of N electrons and R/ represents the position co-ordinate of N’ nuclei. It is very difficult to obtain the exact solution of SWE as the many body Hamiltonian in Equation(2.1) contains a large number of variables associated with large number of particles along with mutual interaction terms between electron-electron, electron- nucleus and nucleus-nucleus. So, to simplify the many body hamiltonian wavefunction, certain approximations are made to obtain at least numerical solution of SWE and to analyze the properties of the the system. The first and simplest approximation of the many body Hamiltonian is the elimination of dependency of the nuclear and electronic dynamics by breaking it into two sub-systems; one for electrons and other for the nuclei, which is called Born-Oppenheimer approximation.

Even for a simple nuclei, we know that M ≥ 10[3]. The electronic motion is so rapid that they instanstly adjust almost any change in the ionic (lattice) geometry i.e. the massive nuclei can be considered at rest with respect to the motion of electrons. And we consider the motion of electrons in fixed field of nuclei, where the electronic wavefunction depends only on the position of nuclei and not on their momenta. Thus, one can solve the SWE for electrons moving in the stationary potential generated by the nuclei frozen in a single arrangement, instead of solving SWE for a collection of mobile electrons and nuclei[20]. Different conformations can be taken and the SWE can be solved for each conformation. Now, the total wavefunction can be written as,

### 2.2 Many-Body Hamiltonian and Born-Oppenheimer Approximation

illustration not visible in this excerpt

Here, We(x; R) is the electronic wave function and WN(R) is the nuclear wavefunction. In this case, the electronic wavefunction is not independent of nuclear co-ordinates but it depends parametrically on the nuclear co-ordinates, which is called zeroth order Born-Oppenheimer approximation. As the KE is inversely proportional to the mass of nuclei (second term in the Hamiltonian 2.1) can be neglected. Also, under the frozen nuclear approximation, the repulsive potential energy between the nuclei (last term in the Hamiltonian 2.1) is merely a constant additive term and can be dropped out as it changes only the phase of the eigenfunction but not its behavior. This is the main approximation of Born-Oppenheimer method which is called adiabatic approximation. Now, the SWE with simplified Hamiltonian becomes,

illustration not visible in this excerpt

We can simply write He as H, Te as T , We as W and in atomic units we write, h = e = me =1 for our convinence. Using these atomic units, the energy can be expressed in hartree (lhartree = 27.2138 eV) and length in Bohr (1 Bohr = 0.529177A) Then,

illustration not visible in this excerpt

Here, the first two terms in the Hamiltonian(2.6) are considered as universal terms and the Hamiltonian is completely governed by the third term, such that Hamiltonian depends on the nuclear charge number Z, number of electrons N and parametrically on the nuclear co-ordinate RI . We have to add the nuclear energy part to this Hamiltonian to obtain the correct expression for the total energy. Now, under Born-Oppenheimer approximation, the total energy can be written as,

illustration not visible in this excerpt

and the nuclei which depends upon nuclear co-ordinate as,

illustration not visible in this excerpt

### 2.3 Hartree-Fock Method of Self Consistent Fields

Hartree formulated an approximate method to obtain the solution of SWE for many electron system called Hartree method of self consistent fields. In this method, each electron is assumed to move independently in the effective field generated by all the nuclei and remaining N-1 electrons such that the motion of each electron is governed by a one particle SWE. The Schrodinger equation for N-electron system under this approximation is given by;

illustration not visible in this excerpt

where, H is the Hamiltonian given by(2.6) and E is the total energy of the system within this approximation. The total wavefunction of N electron system is taken as the simple product of N-one electron wavefunction in Hartree self-consistent field method. i.e.;

illustration not visible in this excerpt

Now, the total energy of the system becomes,

illustration not visible in this excerpt

Here, the electronic wavefunction y (r) can be obtained by minimizing the Hartree energy under the normalization condition,

illustration not visible in this excerpt

From variational principle,

illustration not visible in this excerpt

Here, the third term inside the curly bracket is known as Hartree field. i.e.,

illustration not visible in this excerpt

The set of N-coupled integro-differential equations given by Equation(2.14) are known as Hartree equations. Hartree used the method of successive approximation to solve Equa- tion(2.14). He used hydrogen-like wavefunction to calculate VSC in the zeroth order approximation. Then using the calculated VSC , Equation(2.14) is again solved to get new set of wavefunctions. These wavefunctions are used to calculate new VSC and this process continues until we obtain VSC of desired accuracy. This method of solving Schrodinger wave equation is known as Self-Consistent Field (SCF) method and the finally obtained VSC is called self-consistent Hartree field[21].

The total electronic wavefunction is written as the simple product of individual electronic wavefunctions in Hartree method of self-consistent field, which violates the antisymmetric requirement of fermionic (electronic) wavefunction. It fails to explain the instantaneous interaction between the electrons due to their motion. Hartree method of self-consistent fields fails to meet the requirement of Pauli’s exclusion principle for fermions[21]. V. Fock used the correct antisymmetric wavefunction meeting the requirement of Pauli’s exclusion principle in the form of Slatter determinant;

illustration not visible in this excerpt

where -→Nr is a normalization constant. With the composite spin-orbital functions and combined integration over space coordinates and summation over spin coordinates defined by,

illustration not visible in this excerpt

It is clear that the Equation(2.16) shows the anti-symmetric properties of the fermions i.e; electrons. By using this determinant form of wave function, we get a set of N-coupled integrodifferential equation, which is known as Hartree-Fock equations[22], given by;

illustration not visible in this excerpt

where, Pij is permutation operator that exchanges the indices such that

illustration not visible in this excerpt

In the LHS of Equation(2.18), the last term is known as Exchange potential VExc which is a non-local potential related to the interaction between all electrons in the system. So, it is difficult to obtain the exact value of VExc.

The Hartee-Fock wave function includes the correlation effect that arises from the electrons of same spin, but the motion of electrons with opposite spin remains uncorrelated; which overestimates the electron-electron repulsion. It also satisfies the anti-symmetric nature of fermions i.e; electrons. There are two types of electron-electron repulsion: the first one is the classical Coulomb repulsion that arises from electric charge and the second one is Fermi repulsion due to Pauli’s exclusion principle. So, it can be stated that HF method cannot explain the correlation effect due to the dynamic nature of electrons. The repulsion energy between two electrons is calculated between an electron and the average electron density for the other electrons in HF method. This discrepancy arises due to the fact that the electron will push away the other electrons as it moves around by diminishing the repulsion energy. So, the correct solution of Schrodinger equation is not obtained from HF method. The difference between the exact solution of Schrodinger equation and Hartree-Fock limit energy is known as Correlation energy. So, to explain the dynamical behaviour of electrons (electronic correlation), HF approximation must be improved. The methods like DFT, MP2 etc are used to account correlation of electrons.

### 2.4 Density Functional Theory

#### 2.4.1 General Consideration

One of the most remarkable and fundamental approach to explain the the electron correlation is Density Functional Theory(DFT). It is a quantum mechanical theory that replace the complicated N-electron wave function (xi,x2,,xN) and the associated Schrodinger equation by electron density n0(r) to investigate the electronic structure of systems in particular atoms, molecules and condensed phases. According to the DFT, any property of a system of many interacting particles can be viewed as a functional of the ground state density n0(r); i.e; in principle, one scalar function of position n0(r), determines all the information in the many-body wavefunctions for the ground state and all excited states. In this theory, the emphasis shifts from the ground-state wavefunction to the manageable ground-state one body electron density n0(r). DFT can predict various molecular properties like molecular structures, vibrational frequencies, atomization energies, ionization energies, electric and magnetic properties, reaction paths, etc.[23].

The first attempts to use the electron density instead of wave function was Thomas and Fermi methods in 1920s[24, 25] which was aimed to obtain information about the atomic and molecular systems. But, the accurate and complete theory was illustrated by Hohenberg and Kohn-Shan in 1960s[26, 27].

#### 2.4.2 Thomas-Fermi Model

Thomas and Fermi proposed the density functional theory for the first time in 1927. According to Thomas-Fermi model, the kinetic energy of the electron is treated as non-interacting electron in homogeneous gas with density equal to the local density at a given point. It states that ”Electrons are distributed uniformly in six-dimensional phase space for the motion of an electron at the rate of two for each h[3]of volume”, and there is an effective potential field such that it is itself determined by the nuclear charge and the distribution of electrons.

The Thomas-Fermi kinetic energy functional in terms of atomic units is given by;

illustration not visible in this excerpt

Dirac extended this theory in 1930 to encompass the local approximation for the exchange among electrons. According to Dirac, the exchange energy due to local density distribution of electrons can be written as[24];

illustration not visible in this excerpt

Now, the total energy functional for electrons will be,

illustration not visible in this excerpt

Eventhough Thomas-Fermi model comprises some limitations for its practical use, it is a step forward towards the density functional theory. Density functional theory can be understood more clearly by incorporating the idea of Density operator formalism. But, we directly discuss about Hohenberg-Kohn Theorem in this chapter.

#### 2.4.3 Hohenberg-Kohn Theorem

In 1964, P. Hohenberg and W. Kohn published a paper by introducing the exactness and viability of n(r) instead of much more complex y[23]. They proved two mathematical theorems that comprises DFT. They showed that a special role can be assigned to the density of particles in the ground state of quantum many-body system: the density can be considered as a “basic variable”, i.e. that all properties of the system can be considered to be unique functionals of the ground state density. Since, exchange and correlation and the major contribution from those to the kinetic energy are still unknown, HK-DFT did not achieve a high accuracy in calculations. But in 1965, Kohn and Sham found a way of approaching this problem to good accuracy and thus created DFT in the form that we know it today. As the reasoning leading to the HK theorem is quite instructive, it is worthwhile to study this prototype of an existence theorem in some detail.

The First Hohenberg-Kohn Theorem

According to the first Hohenberg-Kohn Theorem, "The ground state density n(r) of a bound system of interacting electrons in some external potential v(r) determines this potential uniquely"[26]. This means that, as v(r) fixes the Hamiltonian H, n(r) also uniquely determines all other properties of the system. After 40 years, it has finally been rigorously proved that it is indeed physically justified to use the electron density n(r) as a basic variable. The original proof of first Hohenberg-Kohn Theorem is very simple and is done by a reductio ad absurdum.

Let the ground-state density for a system of N electrons be n(r) in an external potential v(r), where the ground-state wave-function is y and the energy E. Then the energy is given by.

illustration not visible in this excerpt

Suppose that another potential v' (r) = v(r) + constant with ground state y' results in the same density n(r). Here y' cannot be equal to y because they are the solutions of different Schrodinger equations. Then, E’ is given by,

illustration not visible in this excerpt

From the variational principle,

illustration not visible in this excerpt

If we swap the prime and unprimed quantities, we find in the same way that;

illustration not visible in this excerpt

Now, adding Equation(2.25) and (2.26) together implies;

illustration not visible in this excerpt

This is a contradictory equation which establishes the fact that there cannot be two different Vext that yield the same ground state density. This means that, there is only one v(r) that produces the ground-state density, and conversely that the ground-state density n(r) uniquely determines the external potential v(r).

**[...]**

- Quote paper
- Upendra Adhikari (Author), 2017, The Study of Effects of Strain on Band Gap of Perovskite Crystal using LDA and LDA+U Method, Munich, GRIN Verlag, https://www.grin.com/document/419048

Comments