Introduction to density functional theory
- 1. Introduction to Density Functional Theory SARTHAK HAJIRNIS M.SC-II 1
- 2. Many Particle Problem Now, our Hamiltonian operator is given by: 𝐻 = 𝑇 + 𝑉 Where, 𝑇 is the Kinetic energy operator and 𝑉 is given by, 𝑉 = 𝑞 𝑖 𝑞 𝑗 𝑟 𝑖−𝑟 𝑗 𝐻ψ 𝑟𝑖, 𝑅𝐼 = 𝐸ψ 𝑟𝑖, 𝑅𝐼 e e - - + + 2
- 3. Born-Oppenheimer Approximation According to this approximation, the nucleus is large and slow as compared to electrons which are small and fast. Thus we can separate out our general wavefunction into a products of electron wavefunction and nuclei wavefunction, ψ 𝑟𝑖, 𝑅𝐼 = ψ 𝑒 𝑟𝑖 ∗ ψ 𝑁 𝑅𝐼 So now we first solve for ground state of electrons by considering fixed nuclei centres. 𝐻ψ 𝑟1, 𝑟2, … , 𝑟 𝑁 = 𝐸ψ 𝑟1, 𝑟2, … , 𝑟 𝑁 Where, 𝐻 = −ℏ2 2𝑚 𝑒 𝑖 𝑁 𝑒 ∇𝑖 2 + 𝑖 𝑁 𝑒 𝑉𝑒𝑥𝑡 𝑟𝑖 + 𝑖=1 𝑁 𝑒 𝑗>1 𝑈 𝑟𝑖, 𝑟𝑗 e + e e e _ _ _ _ 3
- 4. From wavefunctions to electron densities We now define the electron density in a region as, 𝑛 𝑟 = ψ∗ 𝑟1, 𝑟2, … , 𝑟 𝑁 ψ 𝑟1, 𝑟2, … , 𝑟 𝑁 Where n(r) is the electron density. Since you just need 3 coordinates to define density of a charge configuration, clearly our problem now reduces from 3N dimensions to 3 dimensions. The jth electron is treated as a point charge in the field of all other electrons. This reduces our many electron problem to single electron problem. 4
- 5. From wavefunctions to electron densities Again we can make simplifications, ψ 𝑟1, 𝑟2, … , 𝑟 𝑁 = ψ 𝑟1 ∗ ψ 𝑟2 ∗ ψ 𝑟3 ∗ ⋯ ∗ ψ 𝑟 𝑁 This is nothing but the Hartree Product. So now we can define electron density in terms of single electron wavefunctions: 𝑛 𝑟 = 2 𝑖 ψ∗ 𝑟 ψ 𝑟 5
- 6. Hohenberg and Kohn Theorem 1: The ground state energy E is a unique functional of the electron density. 𝐸 = 𝐸 𝑛 𝑟 The external potential corresponds to a unique ground state electron density. - A given ground state electron density corresponds to a unique external potential. - In particular, there is a one to one correspondence between the external potential and the ground state electron density. 6 𝑉𝑒𝑥𝑡 𝑟 ψ 𝐺 𝑟1, 𝑟2, … n𝐺 𝑟
- 7. Hohenberg and Kohn Theorem 2: The electron density that minimizes the energy of the overall functional is the true ground state electron density. 𝐸 𝑛 𝑟 > 𝐸0 𝑛0 𝑟 7 𝐸 𝑛 𝑟 𝑛 𝑟𝑛0 𝑟 𝐸0
- 8. The Energy Functional 𝐸 {ψ𝑖} = 𝐸 𝑘𝑛𝑜𝑤𝑛 {ψ𝑖} + 𝐸 𝑋𝐶 {ψ𝑖} Where 𝐸 𝑘𝑛𝑜𝑤𝑛 {ψ𝑖} = −ℏ 𝑚 𝑒 𝑖 ψ𝑖 ∗ ∇2ψ𝑖 𝑑3 𝑟 + 𝑉 𝑟 𝑛 𝑟 𝑑3 𝑟 + 𝑒2 2 𝑛 𝑟 𝑛 𝑟′ 𝑟−𝑟′ 𝑑3 𝑟 𝑑3 𝑟′ + 𝐸𝑖𝑜𝑛 And 𝐸 𝑋𝐶 {ψ𝑖} : Exchange-Correlational Functional which includes all the quantum-mechanical terms and this is what needs to be approximated 8 e + e e e + + _ _ _ _
- 9. Kohn-Sham Approach Solve a set of single electron wavefunctions that only depend on 3 spatial variables, ψ 𝑒(𝑟) −ℏ2 2𝑚 𝑒 ∇2 + V r + 𝑉𝐻 𝑟 + 𝑉𝑋𝐶 𝑟 ψ𝑖 𝑟 = ϵ𝑖ψ𝑖 𝑟 In terms of Energy Functional, we can write the Kohn-Sham equation as: 9 Exchange Correlation potentiale + e - - n(r) 𝐸 𝑛 = 𝑇𝑠 𝑛 + 𝑉 𝑛 + 𝑊𝐻 𝑛 + 𝐸 𝑋𝐶 𝑛
- 10. Self Consistency Scheme Step 1: Guess initial electron density n(r) Step 2: Solve Kohn-Sham equation with n(r) and obtain ψ𝑖 𝑟 Step 3: Calculate electron density based on single electron wavefunction Step 4: Compare; if n1(r) == n2(r) then stop else substitute n1(r) = n2(r) and continue with step 2 again 10 −ℏ2 2𝑚 𝑒 ∇2 + V r + 𝑉𝐻 𝑟 + 𝑉𝑋𝐶 𝑟 ψ𝑖 𝑟 = ϵ𝑖ψ𝑖 𝑟 𝑛 𝑟 = 2 𝑖 ψ∗ 𝑟 ψ 𝑟 n1(r) n2(r)
- 11. Kohn-Sham Approach with LDA The exchange-correlation functional is clearly the key to success of DFT. One of the great appealing aspects of DFT is that even relatively simple approximations to VXC can give quite accurate results. The local density approximation (LDA) is by far the simplest and known to be the most widely used functional. 𝐸 𝑋𝐶 𝐿𝐷𝐴 = n 𝑟 ϵ 𝑋𝐶 𝑢𝑛𝑖𝑓 n 𝑑𝑟 Where ϵ 𝑋𝐶 𝑢𝑛𝑖𝑓 n is the exchange correlation energy per particle of infinite uniform electron gas with density n. Thus, in LDA, the exchange correlation energy per particle of an inhomogeneous at spatial point r of density n(r) is approximated as the exchange-correlation energy per particle of the uniform electron gas of the same density. 11
- 12. Kohn-Sham Approach with LDA We can write, 𝜖 𝑋𝐶 𝑢𝑛𝑖𝑓 𝑛 = ϵ 𝑋 𝑢𝑛𝑖𝑓 𝑛 + ϵ 𝐶 𝑢𝑛𝑖𝑓 𝑛 ϵ 𝐶 𝑢𝑛𝑖𝑓 𝑛 cannot be calculated analytically. This quantity has been obtained numerically using Quantum Monte-Carlo calculations and fitted to a parameterized function of n. 12 − 3 4 3 π 1 3 𝑛1 3 LDA Exchange functional
- 13. Ionic Ground State Forces on atoms can be easily calculated once the electronic ground state is obtained. By moving along the ionic forces (steepest descent), the ionic ground state can be calculated. We can then displace ion from ionic ground state and calculate the forces on all other ions. 13 𝐹𝑙 = − ⅆ𝐸 ⅆ𝑟𝑙 = − 𝜓𝑖 𝜕 𝐻 𝜕𝑟𝑙 𝜓𝑖
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- 15. References •Fundamentals and applications of density functional theory, https://www.youtube.com/watch?v=SXvhDLCycxc&t=1166s •Local-density approximation (LDA), https://www.youtube.com/watch?v=GApI1I9AQMA&t=242s •Gritsenko, O. V., P. R. T. Schipper, and E. J. Baerends. "Exchange and correlation energy in density functional theory: Comparison of accurate density functional theory quantities with traditional Hartree–Fock based ones and generalized gradient approximations for the molecules Li 2, N 2, F 2." The Journal of chemical physics 107, no. 13 (1997): 5007-5015. •Wikipedia. •An Introduction to Density Functional Theory, N. M. Harrison, https://www.ch.ic.ac.uk/harrison/Teaching/DFT_NATO.pdf 15
- 16. 16 Thank You