Fundamentals of DFT R. Wentzcovitch U of Minnesota VLab Tutorial Hohemberg-Kohn and Kohn-Sham theorems Self-consistency cycle Extensions of DFT

BO approximation - Basic equations for interacting electrons and nuclei Ions (R I ) + electrons (r i ) This is the quantity calculated by total energy codes.

Pseudopotentials Nucleus Core electrons Valence electrons V(r) Radial distance (a.u.) rR l (r) s orbital of Si Real atom Pseudoatom r Ion potential Pseudopotential 1/2 Bond length

BO approximation Born-Oppenheimer approximation (1927) Ions (R I ) + electrons (r i ) Molecular dynamicsLattice dynamics forces stresses phonons

Electronic Density Functional Theory (DFT) (T = 0 K) Hohemberg and Kohn (1964). Exact theory of many-body systems. DFT1 Theorem I: For any system of interacting particles in an external potential V ext (r), the potential V ext (r) is determined uniquely, except for a constant, by the ground state electronic density n 0 (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 V ext (r). For any particular V ext (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 n 0 (r).

Proof of theorem I Assume V ext (1) (r) and V ext (2) (r) differ by more than a constant and produce the same n(r). V ext (1) (r) and V ext (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)

Proof of theorem II Each V ext (r) has its Ψ(R) and n(r). Therefore the energy E el (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)

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

Kohn-Sham equations : (one electron equation) With ε i s as Lagrange multipliers associated with the orthonormalization constraint and and df t2 Minimizing E[n] expressed in terms of the non-interacting system w.r.t. Ψs, while constraining Ψs to be orthogonal:

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

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

Self consistency cycle until

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.

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)

Wentzcovitch, Martins, Allen, PRB 1991 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].

Dissociation phase boundary

Umemoto, Wentzcovitch, Allen Science, 2006

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…