Applications of Density Functional Theory

# Introduction:

All materials system we study essentially consist of electrons and nuclear charge. And it is due to electron and its interaction with other electrons which results in various mechanical, electronic, magnetic etc. properties. In order to define electron and their interaction Schrodinger equation is best tool. If Schrodinger equation of many electron problem can be solved accurately and efficiently then almost any property of the materials can be determined determined accurately. But unfortunately there is neither a accurately nor efficient method to solve these problems.

As we saw in the class there are various methods developed to solve Schrodinger equation. Relatively simple system as Hydrogen atom and H2+ could be solved analytically. To solve relatively complex system methods as Nearly free electron method and Tight binding method have been developed. These methods are not accurate as we have to take a lot approximation to simplify the problem.

First successful method to solve this problem was Wigner-Seitz method. Using this they solved lattice parameter 4.2 Å (compared to experimental 4.23 Å), binding energy 25.6 kCal/mole (compared to 26.9 kCal/mole mesaured experimentally) and bulk modulus of 1.6x10-11 N/m2(compared to 10-11 N/m2 calculated experimentally).

After this grand success many methods have been developed to compute various properties.

1. Quantum Chemistry (Hartree-Fock)
2. Quantum Monte Carlo
3. Perturbation theory
4. Density Functional Theory (DFT).

Here I would discuss DFT most successful of them all. By successful I mean it is best combination of accuracy and efficiency.

First of all I would discuss formalism of DFT, then extensively discuss various applications of DFT and for sake of completeness some limitations.

# Formalism:

In many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed generating a static external potential V in which the electrons are moving. A stationary electronic state is then described by a wavefunction $\Psi(\overrightarrow{r_{1}},...,\overrightarrow{r_{n}})$ satisfying the many-electron Schrödinger equation

where $\hat H$ is the electronic molecular Hamiltonian, $\ N$ is the number of electrons, $\hat T$ is the $\ N$ -electron kinetic energy, $\hat V$ is the $\ N$ -electron potential energy from the external field, and $\hat U$ is the electron-electron interaction energy for the $\ N$ -electron system. Due to various interactions parameters involved many electron problem is very difficult to solve.

This problem is reduced to single electron problem which is relatively easy to solve by using Kohn-Sham equation, expressed as:

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which yields the orbitals $\,\!\phi_i$ that reproduce the density $n(\vec r)$ of the original many-body system and $V_{s}(\overrightarrow{r})$ is given by:

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where the second term denotes the so-called Hartree term describing the electron-electron Coulomb repulsion, while the last term $\,\!V_{\rm XC}$ is called the exchange-correlation potential. Here, $\,\!V_{\rm XC}$ includes all the many-particle interactions. Since the Hartree term and $\,\!V_{\rm XC}$ depend on $n(\vec r )$, which depends on the $\,\!\phi_i$, which in turn depend on $\,\!V_s$, the problem of solving the Kohn-Sham equation has to be done in a self-consistent (i.e., iterative) way.

# Applications:

Listed below are various applications of DFT. It finds application in determining various properties which can be found using various experimental setup. For comparison I talk about experimental setup by which given property is determined. I would discuss how to compute phase transformation in detail.

### Phase Transformations:

We can predict various phase transformations that takes place under mechanical strain. Crystal structure changes even when dimension of the system is altered for example ZnO in Wurtzitic in bulk but it is Graphitic in nanowire this can be predicted by DFT; calculating cohesive energy per atom of the system we can predict the phase transformations.

Phase transformation are determined using X-ray and neutron diffraction.

#### Phase transformation in ZnS under hydrostatic pressure:

Here I illustrate how phase transformation can be estimated using DFT in ZnS. ZnS exist in the B3 (zinc blende) at equilibrium and B1 (rocksalt) crystal structures at high pressure. Transformation takes place when pressure of both crystal structure is same. Now to calculated pressure at what pressure Zinc Blend structure of ZnS changes to rocksalt we use Birch–Murnaghan equation of state that relates pressure to volume and energy.

(1)
\begin{align} P = - \left( \frac{\partial E}{\partial V} \right) \end{align}

Where P is pressure, E is energy of the system (in our case energy per unit cell) and V volume of system (in our case volume of unit cell).

To calculate pressure at various volume of unit cell, calculate energy per unit volume at various volume of unit cell of both Zinc Blend and Rocksalt crystal structure. Energy per unit cell can be easily calculated using DFT to very high degree of accuracy by choosing suitable input parameters. Figure below shows energy per unit cell vs volume of unit cell of two crystal structure. 

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Slope of the curve give pressure at various volume of unit cell. Common tangent to two curves give the pressure of phase transformation since at that point pressure is same for two crystal structure. Thus calcualte value of phase transformation is 16.1 Gpa. Expremental value of pressure of phase transformation is 15-18 Gpa. 

### Structural properties:

It can predict the best possible crystal structure with out any input of experimental data what so ever. Geometries of various organics molecules can be determined. Defect structure as vacancy, grain boundary etc can be estimated. Surface structures and adsorption can be predicted for variety of systems.

Experimentally structural properties are determined using X-ray and neutron diffraction. Various types of microscopy can also determine structural properties

### Mechanical properties:

We can predict various mechanical properties of given material.

1. Elastic Modulus:
2. Compressibility:
3. Thermal expansion coefficients:

Experimentally mechanical properties can be determined using tensile test, hardness test (Vicker and Rockwell). Thermal expansion can be determined using X-ray diffraction.

### Thermodynamic properties:

Formation energy of the materials can be estimated.

### Phase Diagram:

By calculating which is phase is stable at various composition and various temperatures we can plot phase diagram.

Phase diagram can be made by finding crystal structure at various composition and temperature. Experimentally various alloy composition have to be made and crystal structure should be found.

### Transport properties:

Diffusion rate of vacancy, interstitial etc. can be quantified by calculating migration barrier to diffusion of these species.

Vacancy, interstitial have to be measured with various composition to get diffusion rate.

### Electronic and electrical properties:

Band structure can be calculated of almost any given system. Dipole moment of molecule, interface can be calculated. Conductivity of materials can be found. Ionization energies and electron affinity can be estimated. Dharma describe the method to calculate electron affinity in his term paper. Band offset can be calculated at hetero-junctions.

Band strucure can be plotted using technique as Auger effect. Conductivity can be measured Four-probe method. Ionization energy and affinity is measured using Low Energy Electron Ionization and Chemical Ionization Mass Spectrometry.

### Optical properties:

Though band gap is underestimated by DFT. But change in band gap (trend in band gap) with strain can be very well approximated. Hence optical spectra can be estimated. Luminescence and fluorescence of materials can be found using DFT.

Band gap, luminescence and fluorescence can be measured using UV-Visible spectroscopy.

### Magnetic properties:

Magnetic properties as frequency-dependent polarizabilities, frequency-dependent optical rotation parameters and magnetic susceptibilities. 

### Chemical properties:

Rate of reaction can be determined if we know what is formation energy. Chemi-soprtion of various gases at various surface can be established.

Rate of reaction and surface adsorption properties can be determined by measuring compounds one is looking for.

# Deffeciency:

DFT can not be used to find every single property. Here are some area in which DFT can not predict properties.

### Band Gap:

Absolute value of band gap can not be predicted.

### High temperature properties:

It is not possible to predict various high temperature properties close to melting point of the material. But a very recently published paper have improved the formalism which can even account for high temperature properties . Even after this improvement it would not be possible to predict various properties close to and above melting point.

### Superconductivity:

Superconductivity is attributed to Cooper pairs, pairs of electrons interacting through the exchange of phonon. Since it is long distance interaction which is not taken into account in DFT hence DFT can not accurately predict superconductivity.

Bibliography
1. Emilio Artacho, FUNDAMENTALS: The quantum-mechanical many-electron problem and Density Functional Theory, PPT.
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4. B. Grabowski, Ab initio up to the melting point: Anharmonicity and vacancies in aluminum, PHYSICAL REVIEW B 79, 134106 (2009)
6. J. E. Jaffe, R. Pandey and M. J. Seel, Ab initio high pressure structural and electronic properties of ZnS, PHYSICAL REVIEW B 47, 6299 (1993)
7. G. A. Samara and H. G. Drickamer, J. Physical Chemistry Solids, 23, 475, (1962).