Density Functional Theory (DFT): Literature Review

meanness structural Theory (DFT) literature ReviewTheoretical Background and Literature Review2.1 compactness Functional TheoryThis section c overs basics some dumbness Functional Theory (DFT), which is the theoretical method behind our investigations. For those who atomic number 18 interested in a much more deep fellowship about the DFT we refer to textbooks such as 29 and 30.2.1.1 History of Density Functional TheoryTo get precise and accurate results from both theoretical and computational methods, the scale of physical phenomena must be easy defined. In natural philosophy and strong science the relevant scales of matter be time and size of it. In computational material science, for the multiscale understanding in both time and size scale the sm eitherest relevant scale of atomic interactions are best draw by ab initio techniques. These techniques are based on the determination of negatronic structure of the considered materials and an intelligent transfer of its characteristics to higher-order scales development multidisciplinary schemes. More specific al nonpareily, if the interaction of electrons is entirely described using public patterns such as the fundamental laws of quantum mechanics condensed in the Schrodinger comparison, these simulations are called firstprinciples, or ab initio methods. champion bottom of the inning likewise separate those methods as Hartree-Fock and post-HF techniques that mainly employments by quantum chemistry field and Density Functional Theory (DFT) which is typically utilise in of material science.Ab initio simulations are becoming remarkably popular in scientific look for fields. For example in DFT case, in a simple chase at Web Of Science31 or any other publication search tool, matchless kindle easily see that number of publications that include Density Functional Theory in their title or abstract is over 15000 in 2013. thitherfore, it fag end be concluded that, ab initio based seek alre ady an important third discipline that makes the connection between data-based blastes and theoretical knowledge.Figure 2.1 Usage trend of DFT over years inside ab initio simulations quantum mechanical equalitys for any schema that may be consistent or disordered are concluded. That actually gives matchless drawback which is, solving that cordial of equations is generally only possible for simple systems, beca uptake of the expensive electron-electron interaction term. So, in general, the ab initio simulations are restricted to 150-200 atoms calculations with to the highest degree powerful thinkr clusters. collectible to the that severe limitation, better techniques and methods are developed and implemented to bring the veritable materials into realm of ab initio simulations. The major development of ab initio methods with practical applications took identify when many electron interactions in a system was possible to be pretendd using a set of one electron equations (H artree-Fock method) or using stringency useable theoryIn 1927, doubting Thomas 32 and Fermi 33 disclose a statistical model to compute the elan vital of atoms by approximate the distri saveion of electrons in an atom. Their concept was quite similar to modern DFT but less rigorous because of the crucial manybody electronic interaction was not interpreted into account. The sentiment of the Thomas and Fermi was that, at the starting point for control that electrons do not interact with each other and using definitive call, therefore, one toilette describe the energising postcode as a functional of electron compactness of non-interacting electrons in a homogenised electron gas. 3 years later, in 1930, Dirac 34 succeeded to include the many-body exchange and coefficient of correlation terms of the electrons and actually he formulated the topical anesthetic tightness melodic theme (LDA), that is comfort used in our days. However, the Thomas-Fermi and Dirac model that ar e based on homogeneous electron gas do not cover the accuracy prerequisite in current applications.In same the years as Thomas and Fermi, Hartree 35 also introduce a procedure to purpose approximate wavefunctions and energies for atoms and that was called Hartree function. many years later, to deal with antisymmetry of the electron system, his students Fock 36 and Slater37, separately published self-consistent functions fetching into account Pauli exclusion principals and they expressed the multi-electron wavefunction in the form of single(a)-particle orbitals namely Slater-determinants. Since the calculations within the Hartree-Fock model are complicated it was not popular until 1950s.The fundamental concepts of immersion functional theory were proposed by Hohenberg and Kohn in their very substantially cognise paper in the year 1964 38. The main idea was trying to use the electron density instead of complex and complicated wavefunction. A wavefunction contains 3N variables, where N is the number of electrons and each electron has 3 spatial degrees of freedom. In short letter to that electron density contains only 3 variables. Therefore, the implementation of the electron density with 3 variables will be more easy to handle than 3N wavefunction variables. In their carry, Hohenberg and Kohn proved that all ground responsibility properties of a quantum system, in specific the ground allege match zip fastener, are unique functionals of the ground tell apart density. However, the Hohenberg-Kohn (HK) formulation is not useful foractual calculations of ground state properties with decorous accuracy.A major improvement was achieved one year later, in 1965. Kohn and mask 39 proposed a formulation by partially going back to a wavefunction description in terms of orbitals of independent quasi particles. The main idea was that the many-body problem can be mapped onto a system of non-interacting quasiparticles. This approach change the multi-electron pr oblem into a problem of non-interacting electrons in an effective potential. This potential includes the extraneous potential and the effects of the hundred interactions between the electrons, e.g., the exchange and correlation interactions. Since and then up to now the Kohn-Sham equations are used in practically all calculations based on DFT.2.1.2 Schrodingers equivalenceIn quantum mechanics, analogue to Newtons equations in classical mechanics, the Schrodinger equation is used. This is a partial differential equation and used to describe the physical quantities at the quantum level. The Schrodinger equation forms the basis of many ab initio approaches and its non-relativistic form is an eigenvalue equation of the formH(ri,Rj)= E(ri,Rj) (2.1)where (ri,Rj) is the wavefunction of the system depending on the electron coordinates ri,i =1N and the coordinates of all nuclei in the system Rj,j =1M. His the Hamiltonian of a system that contains M nuclei and N electrons. Therefore, the S chrodinger equation that involves both nuclei and electrons has to be understandd for the many-body eigenfunctions (r1,r2, , rN R1,R2, , RM ). The many-body Hamiltonian can be written in the formH= Te + Tn + Vnn + Ven + Vee (2.2)where all of parts are operators. Te and Tn are the kinetic energies of the electrons and nuclei, respectively. Ven, Vee and Vnn represent the attractive electrostatic interaction between the electron and the nuclei and the repulsive potential due to the electron-electron and nucleus-nucleus interactions.One can also salvage them pig explicitlyNf2Te = 2i(2.3)2mei=1M2Mnn=1f2Tn = 2n(2.4)11MZnZme2= (2.5)4 0 2 Rn Rm=1n,mn=mVnnVen = 11MNZne2(2.6)4 0 2 ri Rnn=1 i=1j=Me= (2.7)4 0 2 ri rji,j=1i211Veewhere me and Mn are the electron and nuclei masses, Zn is the nu assimilate number of the n-th atom, e is the electronic charge and f is the Planck constant. For simplicity one can also use atomic units. Then the Hamiltonian takes the formNMMZnZmin22 Rn Rmi=1 n=1 n,m=1n=m11H = 2 2+(2.8)j=MNMZn +ri Rnri rjn=1 i=1 i,j=1i2.1.3 Born-Oppenheimer ApproximationIt is clear that forces on both electrons or nuclei is in the same order of order of magnitude because of their electric charge. Therefore, the expected momen1tum changes due to that forces must be the same. However electrons are much smaller than nuclei (e.g. even for Hydorgen case nuclei nearly 1500 times larger than an electron) they must have higher velocity than nuclei. One can conclude that electrons will very rapidly adjust themselves to eye socket the ground state embodiment if the nuclei start moving. Born and Oppenheimer 40 published their imprint in 1927, they simply separated the nuclear exertion from electronic motion which is now known as the Born-Oppenheimer approximation. Therefore, while solving the Hamiltonian Equation in (2.8) one can simply assume nuclei as stationary and solve the electronic ground state at first then calculate the energy of the system in that co nfiguration and solve the nuclei motion. Then the dissolution of electronic and nuclear motion leads to an separation of the wavefunctions = of electrons and nuclei, respectively. Via the separation one can treat the nuclear motion externally by not including the Hamiltonian and the electronic Hamiltonian can be written asHe = Te + Ven + Vee (2.9)Solving the equation (2.9), one can get the wide energy of the ground state of the system, which can be defined asE0 = 0He0 + Vnn (2.10)where E0 is the ground state total energy of the system and 0 is the eigenfunction of the electronic ground state.2.1.4 Hohenberg-Kohn TheoremHowever, the Hamiltonian in Equation (2.9) is quite complicated to solve for existent systems due to the high number of electrons and especially the term Vee makes it impossible to solve the problem exactly. Therefore, instead of solving the many-body wavefunctions, Hohenberg-Kohn deal with that problem by step-down it to the electron density (r). This approach makes the fundamentals of DFT. According to Hohenberg and Kohn, the total energy of the system can be defined via the electron density as E = E(r) and it will be the minimum for the ground state electron distribution, namely 0(r). Therefore, the exact theory of many-body systems reduced to the electron density that can be defined as(r)= d3 r2d3 rN (r1, rN )2 (2.11)and has to attend the relation(r)d3 r = N (2.12)where N is the total number of electrons in the system. One can also summarize the HK theorem in the form of the ii main theorems,Theorem I The external potential vext(r), which is the potential energy generated by the nuclei, can be determine from the ground state electron density 0(r). Then Hamiltonian will be fully defined, also the wavefunction for the ground state will also be known.Theorem II E0,the ground state total energy of the system with a particular vext will be the orbiculate minimum when = 0.From the perspective of these two theorems one can make unnec essary down the total electronic energy asE= Te(r) + (r)vext(r) + EH (r) + Exc(r)d3 r (2.13)One can also add the kinetic energy of the electrons T e(r), the classical Coulomb interaction (or Hartree interaction) between electrons EH (r) and the remaining complex non-classical electron exchange correlations Exc(r) into an universal functional FHK (r)E= FHK + (r)vext(r)d3 r (2.14)The remaining will be to apply the variational principle to extract the ground state energyE(r)=0 = 0 (2.15)(r)2.2 Kohn-Sham EquationsHowever, the Equation (2.14) does not give an accurate solution. In that point, Kohn and Sham reformulated the current approach and introduced a new scheme by considering the orbitals by mapping the fully interacting electronic system onto a fictitious system of non-interacting quasi particles moving in an effective potential.The Kohn-Sham equations solution can be written asHKSi = ii (2.16) where the Hamiltonian isHKS = 1 2 + Veff (r) (2.17)2Therefore, the problem of finding the many-body Schrodinger equation is now replaced by solving single particle equations. Since the KS Hamiltonian is a functional of just one electron at the point r then the electron density can be defined harmonise to HK theoremocc.(r)= i(r)2 (2.18)i=1Besides, kinetic energy term and the classical Coulomb interaction energy of the electrons can be define asN1d3Te = ri(r)2 (2.19)2i=11 (r)(r )EH = d3rd3 r(2.20)2 r r Then the Hohenberg-Kohn ground state energy cn be written according to Kohn-Sham approachNExcEKS = i EH + Exc (2.21)(r)ii are the one electron energies and are coming from the results of KS equations results, however it has low physical meaning. The most significant term in the Equation (2.20) is the last term. which is the exchange correlation term that contains all the many-body interactions of exchange and interactions of the electrons. One can also write down it as in the form of Hohenberg-Kohn universal functional from the equationExc= FHK (Te+ EH ) (2.22)Th e total ground state energy can be obtained from EKS in Equation (2.21). Since it contains only the electronic energy, the total ground state energy of the system is calculated by adding the nuclei-nuclei repulsion termE0(R1, , RM )= i EH 0+ Exc0 vxc0dr + Vnn(R1, , RM )(2.23) where E0 is the total ground state energy for a given atomic configuration (R1, , R2). Therefore, the total energy depend on bonce positions that is actually depends on the volume and cell shape, so one can easily compute the ground state structure by minimizing the total energy. Also one can find the force acting on the particular atom, articulate atom A, by taking the derivative of the energy with respect to ionic position of AE0(R1, .., RM )FA(RA) = (2.24)RAwhich also shows the total energy dependence on atomic positions.2.3 Calculating the Exchange-Correlation EnergyThe derived and briey explained KS equations from the fundamentals of all modern DFT calculations today. The most important point in the so lution of KS equations are the exchange-correlation functional Exc which also determines the quality of the calculation. There are two well known approximation methods to get the exchange correlations local density approximation (LDA)39 and generalized side approximation (GGA)41, 42.2.3.1 Local Density ApproximationThe local density approximation starts with a very simple approximation that, for regions of material where the charge density is slowly varying, the exchange-correlation energy at that point can be considered as the same as for a local supply electron gas of the same charge density. In that case one can write the Exc asExc = (r) xc(r) (2.25)where xc(r) is the exchange correlation energy per electron in an homogenous electron gas of density (r). Even though the approximation is seemingly simple it is suprisingly accurate. However, it also has some drawbacks such as under-predict on of ground state energies and ionisation, while overpredicting binding energies as well as slightly favouring the high spin state structures and does not work fine for some systems where the charge density is rapidly changing.2.3.2 Generalized incline ApproximationKnowing the drawbacks of LDA the most logical step to go beyond LDA is not to limit oneself to the information about the charge densitiy (r) at a particular point r, but also adding the information about the gradient of the charge density (r) to be able to take into account the unhomogeneous density in the system. Then one can write the exchange correlation energy as Exc= f(, )dr (2.26)That way of description leads to an improvement over LDA, moreover in some systems LDA still works better. There also several(prenominal) different parameterizations of GGA while in LDA its only one. In GGA some of these parameterizations are semi-emprical, in that experimental data (e.g. atomization energies) is used in their derivation. Others are found entirely from first principles. A commonly used functional is the PW91 fu nctional, due to Perdew and Yang 43, 44 and most commonly used today is PBE 45, 46 by Perdew, murder and Ernzerhof.2.4 Ultra-Soft Pseudopotentials and the Projector-Augmented Wave MethodIn the previous section, the calculation of Exc is described. Nevertheless this is not the single sensitive point of DFT calculations. The other point is the treatment of the electron-nuclei interaction. There are several available methods that describes the electron-nuclei interaction, but the most effective