Density Functional Theory Yasir Ali
Density Functional Theory Yasir Ali
Quantum Mechanics: SWE •
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Waves and particles loose their identity in quantum Waves mechanics and can not be localized without loosing their notion of o f energy en ergy.. There is no locality of particle-wav part icle-wave e but they are every where with different probabilities. The probabilities are find out from wave functions () corresponding to particle. Wave Wa ve function is found from basic principle pr inciple of QM, i.e. SWE
Hamiltonian: A quantum mechanical Operator •
For Free and non interacting
particles. Only K.E. •
But in case of few electrons having some interactions.
Hamiltonian: Cont.… •
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But there is also some effect of nucleus on electrons.
But there are many atoms, so there is some interaction between different nuclei as well.
Complexity: How to solve this SWE •
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In a single crystal, there are many electrons. This large number of electrons Hamiltonian more complex.
cause
So SWE can not be solved in this form. Some approximations and special methods are needed.
Hamiltonian Under Born-Oppenheimer Approximation •
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Due to heavy nuclear masses, the nuclei move much slower than the electrons. So we can consider the electrons as moving in the fixed nuclei.
Still much of complexity. 10 ZnO atoms, 2+6=8*10=80, valence electrons=80, SWEs in 3D. 80*3=240 spatial coordinates.
Hartree Method •
N-electrons
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Assume they have ‘no effect’ on each other
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Hamiltonian ∑ = contains kinetic and potential terms.
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So, we can solve for single electron:
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The resulting solution are called spin-orbitals.
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For each electron there are multiple Eigen functions/orbitals (). Where is vector of coordinates of electron . is spin orbital.
The Hartree Product
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is orbital/state to position 1 for electron 1.
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is orbital/state to position N for electron N.
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j=1 means lowest energy, j=2 means next highest energy orbital.
Problem with Hartree Product •
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Hartree method is inconsistent with Pauli Exclusion principle. Electrons are Fermions. Their wave function must change sign if two electrons change places with each other.
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This is known as the anti-symmetry principle.
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Exchanging two electrons does not change the sign of the Hartree product.
Slater Determinant •
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This inconsistency is removed by Slater Determinant.
1 , 2
(1) (1)
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(2) (2)
2 1 − (1) (2)
It changes sign if two electrons are exchanged
Hartree-Fock Method •
There is some interaction between electrons.
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Let’s consider
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N-electrons with N-fixed nuclei.
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The Schrodinger wave equation is
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Where Hartree potential is
Hartree-Fock Method •
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In this method electron-electron interaction is considered in Hartree potential. In HF, the overall wave function is found by Slater determinant. This method is mainly based on wavefunction. Single electron wave function should be defined.
Basis for single particle wave function •
In practice, for individual electron is written in terms of a user-defined basis set
() ∶ 1 , 2, ⋯ ,
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The coefficients actually shape the singleelectron solutions. Larger basis means more accuracy.
Hartree Fock Cycle •
Make an initial estimate of the spin orbitals
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∑= , ()
by specifying the expansion coefficients, . , From the current estimate of the spin orbitals, define the electron density, (′). Using the (′) from step 2, solve the singleelectron equations for the spin orbitals. Compare the orbital of step 3 with step 1, if they are same then this is solution to problem. If they are not same, then try another basis set.
The Hartree- Fock limit •
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The solution (and corresponding energy eigenvalues) for infinitely large value of is the HF-limit. Practically, suitable extrapolation scheme needs to be employed for reaching such a limit. However, even in HF-limit, the results can be (and are) away from actual solution! It neglects of exchange and correlation term.
Exchange and Correlation •
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In mathematics, correlation means that two distribution functions & are not independent of each other. In atom, in actual, coulomb interaction of electron with electronic structure. HF uses interaction of an electron with an average electron density instead of electronic structure. This reduces energy of atom from actual energy.
Exchange and Correlation •
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Exchange is due to indistinguishable nature of electrons. Exchange increases expectation value of distance for fermions and decreases that for bosons. For electrons, exchange interaction is due to Pauli principle. Both exchange and correlation make a hole in electronic structure in which no electron exists.
Beyond Hartree-Fock Method •
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Exchange interactions can be included in HF. Schemes like Configuration interaction (CI) or coupled cluster (CC) do this job. But such refinements progressively require more computer time.
From Wave Function to DFT •
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Density functional theory attempts to address both the inaccuracy of HF and the high computational demands of post-HF methods by replacing the many-body electronic wave function with the electronic density as the basic quantity. It replaces problem of 3N variables with 3 variables.
Hohenberg and Kohn Theorems •
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DFT was given a formal footing by the two theorems introduced by Hohenberg and Kohn. The ground-state energy from Schrödinger’s equation is a unique functional of the electron density . The first H –K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only 3 spatial coordinates.
Hohenberg and Kohn Theorems •
Extending it further, we may say that ( ) determines all other properties of the system.
• Note that it will reduce complexity of problem. In place of solving for 240 variables, solve it for () only (3D Problem now). • This is the basis of Density Functional Theory
The Second Hohenberg –Kohn theorem •
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First H-K Theorem gives idea of a functional but does not tell how to find it. Second HK theorem gives the idea. The electron density that minimizes the energy of the overall functional is the true electron density corresponding to the full solution of the Schrödinger equation. This H –K theorem defines an energy functional for the system and proves that the correct ground state electron density minimizes this energy functional.
The Second Hohenberg –Kohn theorem •
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It is a prescription for finding ()… Vary the electron density to search for energy minimum. The corresponding density is solution!
Energy Functional •
Energy functional can be defined as
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Where is energy that can be calculated.
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Electron KE, electron-nuclei interaction, electronelectron interaction, nuclei interaction.
Energy Functional •
[{}], is the exchange –correlation functional.
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It includes all the quantum mechanical effects that are not included in the “known” terms.
Solving SWE, Kohn-Sham Equations •
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Still, there is difficulty, how to find electron density which can minimize energy functional. This difficulty was solved by Kohn and Sham. They showed that finding the right electron density can be expressed in a way that involves solving a set of equations. In this set, each equation only involves a single electron.
Kohn-Sham Equations •
() electron interaction with nuclei.
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Hartree potential
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It describes interaction of electron with electron density. Exchange-correlation potential.
Cyclic process in solving KS equations
Algorithm 1. Define an initial, trial electron density, n(r). 2. Solve the Kohn –Sham equations defined using the trial electron density to find the single-particle wave functions, (). 3. Calculate the electron density defined by the Kohn – Sham single particle wave functions from step 2, () ∑∗ ()(). Compare the calculated electron density, nKS(r), with the electron density used in solving the Kohn –Sham equations, n(r). If it is not same with that, then update the trial density. •
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KS-with other approximations •
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In actual Kohn-Sham equations are implemented to calculate properties of materials. Different approximations are used to approximate exchange-correlation and system’s potentials. LDA, GGA are common approximations used. MBJ is an improved form of potential. A new method, TB-MBJLDA potential gives band gap values that are much correct than LDA potential.
Softwares for DFT •
VASP (HF, DFT)
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Quantum EXPRESSO
(HF, DFT) •
Gaussian (HF, DFT,
Molecular Mechanics) •
Wien2k (HF, DFT)
Some Comparison
Comparison
Applications and Advantages •
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Can be used to predict molecular and structure . It is being used in band structure study and Xray spectrum study. It predicts electronic properties of atoms or molecules from the densities of their electron clouds, rather than from the solution of the Schrödinger equation for each individual electron's motion.
Some limitations •
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DFT is not an exact method, it approximates band gap etc., because exact form of functional is not known. For complex structure and many atoms calculations, powerful computer is needed.
Conclusion •
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DFT is giving correct results but still inaccurate. Solutions to SWE are still not exact but approximate. For exact solution if SWE and hence for accurate results, exact for of exchange and correlation should be known.
References 1.
Density Functional Theory, A Practical Introduction by D. S. Sholl & J. A. Steckel
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Density Functional Theory- An Introduction by N. Argaman & G. Makov.
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An Introduction of Density Functional Theory & Its Application by Di Zhou.
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The Abc Of Dft by K. Burke.
