Fundamentals of DFT R. Wentzcovitch U of Minnesota VLab Tutorial

1. Fundamentals of DFTR. WentzcovitchU of MinnesotaVLab Tutorial • Hohemberg-Kohn and Kohn-Sham theorems • Self-consistency cycle • Extensions of DFT

2. BO approximation - Basic equations for interacting electrons and nuclei Ions (RI ) + electrons (ri ) This is the quantity calculated by total energy codes.

3. Pseudopotentials 1.0 3s orbital of Si rRl (r) 0.5 Pseudoatom Real atom 0.0 -0.5 0 1 2 3 4 5 Radial distance (a.u.) V(r) 1/2 Bond length r Nucleus Core electrons Valence electrons Pseudopotential Ion potential

4. BO approximation • Born-Oppenheimer approximation (1927) Ions (RI ) + electrons (ri ) phonons forces stresses Molecular dynamics Lattice dynamics

5. Electronic Density Functional Theory (DFT) (T = 0 K) • Hohemberg and Kohn (1964). Exact theory of many-body systems. Theorem I: For any system of interacting particles in an external potential Vext(r), the potential Vext(r) is determined uniquely, except for a constant, by the ground state electronic density n0(r). Theorem II: A universal functional for the energy E[n] in terms of the density n(r) can be defined, valid for any external potential Vext(r). For any particular Vext(r), the exact ground state energy is the global minimum value of this functional, and the density n(r), that minimizes the functional is the ground state density n0(r). DFT1

6. Proof of theorem I Assume Vext(1)(r) and Vext(2)(r) differ by more than a constant and produce the same n(r). Vext(1)(r) and Vext(2)(r) produce H(1) and H(2) , which have different ground state wavefunctions, Ψ(1) and Ψ(2) which are hypothesized to have the same charge density n(r). It follows that Then and Adding both which is an absurd! Hohemberg and Kohn, Phys. Rev. 136, B864 (1964)

7. Proof of theorem II Each Vext(r) has its Ψ(R) and n(r). Therefore the energy Eel(r) can be viewed as a functional of the density. Consider and a different n(2)(r) corresponding to a different It follows that Hohemberg and Kohn, Phys. Rev. 136, B864 (1964)

8. The Kohn-Sham Ansatz Kohn and Sham, Phys. Rev. 140, A1133 (1965) Hohemberg-Kohn functional: How to find n? Replacing one problem with another…(auxiliary and tractable non-interacting system) • Kohn and Sham(1965)

9. dft2 • Kohn-Sham equations: (one electron equation) Minimizing E[n] expressed in terms of the non-interacting system w.r.t. Ψs, while constraining Ψs to be orthogonal: With εis as Lagrange multipliers associated with the orthonormalization constraint and and

10. • Exchange correlation energy and potential: By separating out the independent particle kinetic energy and the long range Hartree term, the remaining exchange correlation functional Exc[n] can reasonably be approximated as a local or nearly local functional of the density. and with • Local density approximation (LDA) uses εxc[n] calculagted exactly for the homogeneous electron system Quantum Monte Carlo byCeperley and Alder, 1980 • Generalized gradient approximation (GGA) includes density gradients in εxc[n,n’]

11. Meaning of the eigenvalues and eigenfunctions: • Eigenvalues and eigenfunctions have only mathematical meaning in the KS approach. However, they are useful quantities and often have good correspondence to experimental excitation energies and real charge densities. There is, however, one important formal identity • These eigenvalues and eigenfunctions are used for more accurate calculations of total energies and excitation energy. • The Hohemberg-Kohn-Sham functional concerns only ground state properties. • The Kohn-Sham equations must be solved self-consistently

12. Self consistency cycle until

13. Extensions of the HKS functional • Spin density functional theory The HK theorem can be generalized to several types of particles. The most important example is given by spin polarized systems.

14. Finite T and ensemble density functional theory The HK theorem has been generalized to finite temperatures. This is the Mermin functional. This is an even stronger generalization of density functional. D. Mermim, Phys. Rev. 137, A1441 (1965)

15. Use of the Mermin functional is recommended in the study of metals. Even at 300 K, states above the Fermi level are partially occupied. It helps tremendously one to achieve self-consistency. (It stops electrons from “jumping” from occupied to empty states in one step of the cycle to the next.) This was a simulation of liquid metallic Li at P=0 GPa. The quantity that is conserved when the energy levels are occupied according to the Fermi-Dirac distribution is the Mermin free energy, F[n,T]. Wentzcovitch, Martins, Allen, PRB 1991

16. Dissociation phase boundary

17. Umemoto, Wentzcovitch, Allen Science, 2006

18. Few references: • -Theory of the Inhomogeneous electron gas, ed. by • S. Lundquist and N. March, Plenum (1983). • Density-Functional Theory of Atoms and Molecules, • R. Parr and W. Yang, International Series of Monographs • on Chemistry, Oxford Press (1989). • - A Chemist’s Guide to Density Fucntional Theory, • W. Koch, M. C. Holthause, Wiley-VCH (2002). • Much more ahead…