Physics “Advanced Electronic Structure” Lecture 2. Density Functional Theory Contents: 1. Thomas-Fermi Theory. 2. Density Functional Theory. 3. Local Density Approximation. Supplement: Solving Kohn-Sham Equations

Thomas-Fermi Theory. Thomas and Fermi (1927) proposed Total Energy as Density Functional: Let’s evaluate kinetic energy for homogeneous electron gas: Hence we can write a simplest functional:

Thomas-Fermi Theory. To find the minimum we constrain the density Let’s evaluate the variational derivative Obtain equation for the density:

Thomas-Fermi Theory. Solution of Thomas Fermi equation produces desired density. The equation is remarkably simpler than full many-body Equation. Once density is found one can evaluate total energy. Corrections are easy to evaluate: First, kinetic energy corrections (Weiszsacker, 1935) Second, exchange energy via homogeneous electron Gas (Kohn-Sham, 1965)

Density Functional Theory, Hohenberg-Kohn, 1964 Many body wave function uniquely defines the density Hence, for a given external potetnial V ext (r) ground state total energy can be viewed as density functional E[n] Note that when density n is the true ground state density E[n] becomes true ground state total energy. Away from the minimum E[n] is not necessarily interpreted as the energy!

Density Functional Theory Density becomes a functional of the external potential V ext (r) Such densities are called V-representable. Various extensions have been proposed: Extensions to spin dependent systems (Barth, Hedin, 1972) Extension to relativistic systems (Vignale, Kohn, 1988) Extension to finite temperatures Time-Dependent DFT. (Runge, Gross, 1984)

Kohn-Sham Theory Key assumption: represent density by a set of independent particles moving in some effective field Leads to writing down kinetic energy for independent particles Kohn-Sham functional becomes

Kohn-Sham Equations Minimization subject to constrain Leads to Kohn-Sham equations

Meaning of Eigenvalues Unfortunately, DFT eigenvalues are not excitation energies since Therefore, Koopman’s theorem (valid in Hartree Fock) does not hold in DFT. However, it was proved by Janak (1977) that DFT eigenvalues are derivatives with respect to one-electron occupation numbers (Janak Theorem) Current puzzles: Fermi surface predictions, energy gaps, Metal-insulator transition (e.g. metallization pressure)

Approximations for Exchange Correlation Exchange Correlation Functional can be represented as where g xc (r,r’) describes exchange correlation hole. Approximations for g xc (r,r’) have led to Weighted Density Approximation (WDA, Gunnarsson, 1979) Implementation is not straightforward mainly because g xc (r,r’) is highly non spherical but doable (David Singh et.al) !

Local Density Approximation Exchange Correlation Functional can be represented as where  xc [n(r)] is the energy density in homogeneous electron gas. Exchange part is trivial: Correlations are more complicated but doable as well analytical forms: Barth, Hedin, 1972, Gunnarsson, Lundquvisit, 1974 QMC simulations by Ceperley, Alder (1980)and their parameterizations by Vosko, Wilk, Nussiar (1980).

Comparison with X-alpha method of Slater Slater proposed an approximation similar to LDA. Let’s estimate average Fock exchange potential for free-electrons: The constant C here is different from exchange potential found in LDA by a factor 3/2 because LDA comes from the energy while Slater found approximation for potential. Slater proposed X-Alpha method: If  =2/3 we come to Kohn-Sham exchange, if  =1, we come to Slater exchange.  can be treated as adjustable constant. Important: Kohn Sham theory is variational!

LDA and it accuracy 40 years of experience: what LDA can do and what it cannot do. Total energies, densities, P(V), elastic constants, phonons, response functions. Excitations, optical properties, transport – weakly correlated systems vs strongly correlated systems.

Generalized Gradient Approximations Perdew, Wang 1991, simplified version: Perdew, Wang, 1996 Corrections due to density gradients up to second order: GGA produces practically the same spectra as LDA. GGA gives slight improvement of the ground state properties such as lattice constants, bulk modulus, etc.

Solving Kohn-Sham Equations Self-Consistent Cycles with respect to Density Mixing Schemes: Linear Mixing: Pratt Scheme Non-linear Mixing: Broyden Scheme.